Arithmetic
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- Integers - Addition and Subtraction - Part 1
- Integers - Addition and Subtraction - Part 2
- Multiplication and Division
- Operations with Zero
- Exponents and Roots - Part 1
- Exponents and Roots - Part 2
- PEDMAS - Part 1
- PEDMAS - Part 2
- Exercise 1
- Exercise 2
- Exercise 3
- Exercise 4
- Exercise 5
- Exercise 6
- Exercise 7 Parts a-c
- Exercise 7 Parts d-g

Fractions
0/22 completed

- Fractions
- Decimals - Part 1
- Decimals - Part 2
- Percent
- Fractions: Addition and Subtraction - Part 1
- Fractions: Addition and Subtraction - Part 2
- Fractions: Addition and Subtraction - Part 3
- Exercise 1
- Exercise 2
- Exercise 3
- Exercise 4
- Exercise 5
- Exercise 6
- Exercise 7
- Exercise 8
- Exercise 9
- Exercise 10
- Exercise 11
- Exercise 12
- Exercise 13
- Exercise 14
- Exercise 15

Algebraic Expressions
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Special Binomial Products
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Rational Expressions
0/4 completed

Complex Numbers
0/11 completed

{"Free":0,"Sample":1,"Paid":2}

[{"Name":"Arithmetic","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Integers - Addition and Subtraction - Part 1","Duration":"9m 19s","ChapterTopicVideoID":4869,"CourseChapterTopicPlaylistID":3855,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/4869.jpeg","UploadDate":"2020-04-03T14:10:16.5170000","DurationForVideoObject":"PT9M19S","Description":null,"MetaTitle":"Integers - Addition and Subtraction - Part a: Video + Workbook | Proprep","MetaDescription":"Intro to Algebra - Arithmetic. Watch the video made by an expert in the field. Download the workbook and maximize your learning.","Canonical":"https://www.proprep.uk/general-modules/all/intermediate-and-advanced-algebra/intro-to-algebra/arithmetic/vid4855","VideoComments":[{"CommentDate":"10:37 03/08/2022","Text":"Wow! that\u0027s really helpful\r\nThanks!","CommentStatus":1,"UserID":0,"ChapterTopicPlaylistVideoID":0,"CourseChapterTopicPlaylistID":0,"FirstName":"Mattan","LastName":"Kedem","IP":null,"ID":0}],"Subtitles":[{"Start":"00:00.020 ","End":"00:06.330","Text":"In this clip, we\u0027ll be talking about signed numbers and how to add, and subtract them."},{"Start":"00:06.330 ","End":"00:08.490","Text":"This really belongs to arithmetic,"},{"Start":"00:08.490 ","End":"00:12.000","Text":"but we need to review some arithmetic before we get into algebra and some of"},{"Start":"00:12.000 ","End":"00:15.930","Text":"the principles here in arithmetic will apply to algebra also."},{"Start":"00:15.930 ","End":"00:19.680","Text":"Let\u0027s start with a total review of the number system."},{"Start":"00:19.680 ","End":"00:22.320","Text":"We started out with numbers like,"},{"Start":"00:22.320 ","End":"00:25.100","Text":"historically with, 1, 2,"},{"Start":"00:25.100 ","End":"00:27.870","Text":"3, 4, 5, etc."},{"Start":"00:27.870 ","End":"00:32.380","Text":"These are sometimes called counting numbers,"},{"Start":"00:38.180 ","End":"00:41.145","Text":"and they go on forever."},{"Start":"00:41.145 ","End":"00:45.539","Text":"Then the 0 was introduced,"},{"Start":"00:45.539 ","End":"00:48.095","Text":"and that was a big step in history."},{"Start":"00:48.095 ","End":"00:51.740","Text":"Though it seems obvious to us 0 is a number many years,"},{"Start":"00:51.740 ","End":"00:56.300","Text":"and if you throw in 0 to the set of numbers that we have,"},{"Start":"00:56.300 ","End":"00:58.163","Text":"0, 1, 2, 3,"},{"Start":"00:58.163 ","End":"01:01.250","Text":"4 etc., we now have what"},{"Start":"01:01.250 ","End":"01:09.990","Text":"are called whole numbers."},{"Start":"01:13.240 ","End":"01:17.030","Text":"That expanded our number system some,"},{"Start":"01:17.030 ","End":"01:21.810","Text":"but it still wasn\u0027t enough because there are also negatives,"},{"Start":"01:21.810 ","End":"01:23.750","Text":"which you learned as,"},{"Start":"01:23.750 ","End":"01:28.445","Text":"for example, negative in money can mean a debt,"},{"Start":"01:28.445 ","End":"01:31.310","Text":"or negative temperature is below 0,"},{"Start":"01:31.310 ","End":"01:36.230","Text":"in the elevator negative floor might be a packing level,"},{"Start":"01:36.230 ","End":"01:39.630","Text":"so we got the negatives also."},{"Start":"01:42.940 ","End":"01:48.935","Text":"These are written with a minus sign, which is this in front."},{"Start":"01:48.935 ","End":"01:51.590","Text":"We have minus 1,"},{"Start":"01:51.590 ","End":"01:53.060","Text":"minus 2, minus 3,"},{"Start":"01:53.060 ","End":"01:54.980","Text":"minus 4, and so on."},{"Start":"01:54.980 ","End":"01:58.595","Text":"If we take the whole numbers together with the negative numbers,"},{"Start":"01:58.595 ","End":"02:03.510","Text":"then we get something called the integers."},{"Start":"02:04.920 ","End":"02:08.500","Text":"The integers, it\u0027s like whole numbers,"},{"Start":"02:08.500 ","End":"02:10.285","Text":"but minus is allowed also."},{"Start":"02:10.285 ","End":"02:11.860","Text":"We go in both directions."},{"Start":"02:11.860 ","End":"02:13.918","Text":"From 0 we can go 1, 2,"},{"Start":"02:13.918 ","End":"02:16.930","Text":"3 and so on and we can also go minus 1,"},{"Start":"02:16.930 ","End":"02:20.040","Text":"minus 2, minus 3 and so on."},{"Start":"02:20.040 ","End":"02:21.910","Text":"This is the integers."},{"Start":"02:21.910 ","End":"02:26.485","Text":"Just like to mention that if you hear the term natural numbers,"},{"Start":"02:26.485 ","End":"02:31.900","Text":"I won\u0027t be using it because there\u0027s no general agreement."},{"Start":"02:31.900 ","End":"02:34.150","Text":"Some people, when they say natural numbers,"},{"Start":"02:34.150 ","End":"02:37.510","Text":"they mean from 1, and onwards,"},{"Start":"02:37.510 ","End":"02:41.080","Text":"and some people include the 0 with natural numbers,"},{"Start":"02:41.080 ","End":"02:45.330","Text":"so because there\u0027s some controversy or different usages, I won\u0027t be using it."},{"Start":"02:45.330 ","End":"02:48.785","Text":"Counting numbers, whole numbers, integers."},{"Start":"02:48.785 ","End":"02:55.220","Text":"Of course, the integers wasn\u0027t everything because you must have fractions"},{"Start":"02:55.220 ","End":"03:05.010","Text":"and usually they teach it with some model like pizza and so on."},{"Start":"03:05.010 ","End":"03:08.974","Text":"But, I\u0027ll just mention because you\u0027ve seen them before."},{"Start":"03:08.974 ","End":"03:14.270","Text":"We have stuff like 2/3, 3/8."},{"Start":"03:14.270 ","End":"03:17.360","Text":"If you have a pizza and it\u0027s divided into 8 slices,"},{"Start":"03:17.360 ","End":"03:18.500","Text":"and you eat 3 of them,"},{"Start":"03:18.500 ","End":"03:20.855","Text":"then you\u0027ve had 3/8 of the pizza."},{"Start":"03:20.855 ","End":"03:24.380","Text":"Then there\u0027s numbers which are written as partly whole,"},{"Start":"03:24.380 ","End":"03:26.470","Text":"like 5 and 1/2."},{"Start":"03:26.470 ","End":"03:27.800","Text":"Ask a kid how old he is,"},{"Start":"03:27.800 ","End":"03:29.840","Text":"he won\u0027t say 5 or 6 if 5 and 1/2."},{"Start":"03:29.840 ","End":"03:32.045","Text":"Some of them will even say, \u0027\u0027Well,"},{"Start":"03:32.045 ","End":"03:36.020","Text":"I\u0027m 4 and 3/4,\u0027\u0027 and so on."},{"Start":"03:36.020 ","End":"03:38.449","Text":"Of course, fractions can also be negative."},{"Start":"03:38.449 ","End":"03:48.880","Text":"We could have minus 7/9 and we can have minus 8, and 1/2, and so on."},{"Start":"03:48.880 ","End":"03:51.749","Text":"What about the signed numbers?"},{"Start":"03:51.749 ","End":"03:53.910","Text":"Well, I\u0027ll explain."},{"Start":"03:53.910 ","End":"03:56.893","Text":"You notice some numbers don\u0027t have anything in front of them,"},{"Start":"03:56.893 ","End":"03:58.040","Text":"and some have a minus."},{"Start":"03:58.040 ","End":"04:00.995","Text":"Well, there\u0027s another way of writing the numbers."},{"Start":"04:00.995 ","End":"04:09.125","Text":"For example, I can write plus 3 when I want to emphasize that it\u0027s positive."},{"Start":"04:09.125 ","End":"04:11.405","Text":"This is the same as 3."},{"Start":"04:11.405 ","End":"04:15.875","Text":"This is to contrast it with minus 3."},{"Start":"04:15.875 ","End":"04:21.736","Text":"These are the same if I write a plus,"},{"Start":"04:21.736 ","End":"04:23.285","Text":"or I don\u0027t write a plus."},{"Start":"04:23.285 ","End":"04:28.160","Text":"Usually we don\u0027t write the plus unless is some particular emphasis that we want to"},{"Start":"04:28.160 ","End":"04:33.200","Text":"put on the fact that it\u0027s plus. That\u0027s where the sign is."},{"Start":"04:33.200 ","End":"04:35.546","Text":"The sign is the plus, or the minus."},{"Start":"04:35.546 ","End":"04:41.370","Text":"All these numbers, pluses and minuses are what we call signed numbers."},{"Start":"04:42.050 ","End":"04:44.400","Text":"One way of representing them,"},{"Start":"04:44.400 ","End":"04:45.560","Text":"and we\u0027re going to do this a lot,"},{"Start":"04:45.560 ","End":"04:47.090","Text":"is something called the number line,"},{"Start":"04:47.090 ","End":"04:48.410","Text":"which I\u0027m going to show next."},{"Start":"04:48.410 ","End":"04:52.085","Text":"I threw these over here just to make room."},{"Start":"04:52.085 ","End":"04:54.575","Text":"This will be the number line,"},{"Start":"04:54.575 ","End":"04:56.540","Text":"and I\u0027ll show you how we put numbers on it."},{"Start":"04:56.540 ","End":"05:03.725","Text":"There\u0027s some spot that we mark as 0 and then we count."},{"Start":"05:03.725 ","End":"05:06.595","Text":"The squares are very small, so I\u0027ll take 2 squares to a unit."},{"Start":"05:06.595 ","End":"05:08.800","Text":"This will be 1,"},{"Start":"05:08.800 ","End":"05:14.250","Text":"this will be 2, 3, 4."},{"Start":"05:14.250 ","End":"05:16.415","Text":"These marks are sometimes called ticks."},{"Start":"05:16.415 ","End":"05:19.920","Text":"You\u0027ll learn about them when you learn graphs,"},{"Start":"05:19.930 ","End":"05:23.090","Text":"6, 7, and so on."},{"Start":"05:23.090 ","End":"05:25.445","Text":"I usually put an arrow here."},{"Start":"05:25.445 ","End":"05:29.360","Text":"This also indicates this is the direction that the numbers are increasing."},{"Start":"05:29.360 ","End":"05:30.890","Text":"On the other side, of course,"},{"Start":"05:30.890 ","End":"05:34.690","Text":"we have minus 1, minus 2,"},{"Start":"05:34.690 ","End":"05:38.310","Text":"minus 3, minus 4,"},{"Start":"05:38.310 ","End":"05:41.830","Text":"minus 5 and so on."},{"Start":"05:41.840 ","End":"05:45.435","Text":"Of course, each one of them I could write,"},{"Start":"05:45.435 ","End":"05:47.690","Text":"like the 3 I could write as plus 3,"},{"Start":"05:47.690 ","End":"05:53.390","Text":"but it\u0027s customary not to write the plus if it\u0027s understood."},{"Start":"05:53.390 ","End":"06:00.875","Text":"Now, the idea is that there is a definite direction and we always use it to the right,"},{"Start":"06:00.875 ","End":"06:03.095","Text":"increasing to the right."},{"Start":"06:03.095 ","End":"06:05.510","Text":"Now we can compare 2 numbers,"},{"Start":"06:05.510 ","End":"06:07.940","Text":"and say which one is bigger than another."},{"Start":"06:07.940 ","End":"06:10.830","Text":"Bigger means further to the right."},{"Start":"06:10.830 ","End":"06:12.590","Text":"Certain things are obvious,"},{"Start":"06:12.590 ","End":"06:15.590","Text":"like we would always say that 5 is bigger than"},{"Start":"06:15.590 ","End":"06:19.025","Text":"3 because 5 is to the right of 3, and so on."},{"Start":"06:19.025 ","End":"06:24.460","Text":"We would say that 2 is bigger than 0."},{"Start":"06:24.460 ","End":"06:26.870","Text":"But, what about if I asked you about,"},{"Start":"06:26.870 ","End":"06:30.265","Text":"let\u0027s say minus 5 and minus 4."},{"Start":"06:30.265 ","End":"06:34.280","Text":"Well, minus 5 is actually less than minus 4,"},{"Start":"06:34.280 ","End":"06:39.260","Text":"even though your intuition tells you that 5 is bigger than 4 because it\u0027s to the left."},{"Start":"06:39.260 ","End":"06:45.265","Text":"For example, minus 3 is less than 0."},{"Start":"06:45.265 ","End":"06:49.628","Text":"Certainly every negative number is less than any positive number,"},{"Start":"06:49.628 ","End":"06:54.510","Text":"so minus 1 is going to be less than 2,"},{"Start":"06:54.510 ","End":"06:56.385","Text":"for example, and so on."},{"Start":"06:56.385 ","End":"06:58.460","Text":"We have a comparison of numbers."},{"Start":"06:58.460 ","End":"07:02.855","Text":"Of course, each direction we have the opposite direction."},{"Start":"07:02.855 ","End":"07:06.605","Text":"For example, instead of saying minus 1 is less than 2,"},{"Start":"07:06.605 ","End":"07:13.020","Text":"I can rephrase that by saying that 2 is bigger than minus 1."},{"Start":"07:13.190 ","End":"07:15.510","Text":"This means less than,"},{"Start":"07:15.510 ","End":"07:18.580","Text":"and this means greater than depending which way you look at it."},{"Start":"07:18.580 ","End":"07:19.780","Text":"For example, this one,"},{"Start":"07:19.780 ","End":"07:24.355","Text":"I could rewrite as 3 is less than 5."},{"Start":"07:24.355 ","End":"07:29.750","Text":"But in each case, the bigger one is the one that\u0027s further to the right."},{"Start":"07:30.210 ","End":"07:34.930","Text":"Now the fractions are also going to be put in on the number line."},{"Start":"07:34.930 ","End":"07:37.420","Text":"We\u0027ll talk in more detail about fractions later,"},{"Start":"07:37.420 ","End":"07:39.730","Text":"but I just want to give you the general idea."},{"Start":"07:39.730 ","End":"07:42.400","Text":"The easy ones are the ones with the 1/2,"},{"Start":"07:42.400 ","End":"07:43.690","Text":"like 5, and 1/2,"},{"Start":"07:43.690 ","End":"07:44.890","Text":"I count 5,"},{"Start":"07:44.890 ","End":"07:47.725","Text":"and then halfway between 5 and 6."},{"Start":"07:47.725 ","End":"07:51.744","Text":"This one might be 5 and 1/2."},{"Start":"07:51.744 ","End":"07:52.980","Text":"3/8, well,"},{"Start":"07:52.980 ","End":"07:55.065","Text":"it\u0027s hard to say exactly,"},{"Start":"07:55.065 ","End":"07:56.580","Text":"but 3/8 of a pizza,"},{"Start":"07:56.580 ","End":"07:58.372","Text":"is certainly less than 1/2 a pizza."},{"Start":"07:58.372 ","End":"08:00.335","Text":"Let\u0027s choose some point here."},{"Start":"08:00.335 ","End":"08:05.735","Text":"What I would really do would be divide this up into 8 equal bits and count 3 bits."},{"Start":"08:05.735 ","End":"08:09.875","Text":"I\u0027m sorry, strike the last bit, take 2."},{"Start":"08:09.875 ","End":"08:13.040","Text":"Let\u0027s say 2/3."},{"Start":"08:13.040 ","End":"08:15.240","Text":"2/3 is somewhere between 0 and 1."},{"Start":"08:15.240 ","End":"08:20.630","Text":"2/3 of a pizza means I divide my pizza up into 3 equal pieces."},{"Start":"08:20.630 ","End":"08:23.582","Text":"You got to imagine I\u0027ve divided this up into 1,"},{"Start":"08:23.582 ","End":"08:25.385","Text":"2, 3 equal pieces."},{"Start":"08:25.385 ","End":"08:32.215","Text":"Then this point here would be the point 2/3, and so on."},{"Start":"08:32.215 ","End":"08:34.640","Text":"If I had a negative number,"},{"Start":"08:34.640 ","End":"08:36.200","Text":"well, not 8 and 1/2,"},{"Start":"08:36.200 ","End":"08:41.030","Text":"let\u0027s change that to minus 2 and 1/2, that will fit in better."},{"Start":"08:41.030 ","End":"08:45.515","Text":"So minus 2 and 1/2 would be here."},{"Start":"08:45.515 ","End":"08:51.260","Text":"I should really put these ticks on every whole number."},{"Start":"08:51.260 ","End":"08:53.675","Text":"Well, that\u0027ll be more important when we do graphs."},{"Start":"08:53.675 ","End":"08:56.660","Text":"We see that all these numbers,"},{"Start":"08:56.660 ","End":"09:00.395","Text":"positive and negative, all fit on the number line."},{"Start":"09:00.395 ","End":"09:02.570","Text":"If I take this thing here,"},{"Start":"09:02.570 ","End":"09:05.120","Text":"everything to the right of 0 is a plus,"},{"Start":"09:05.120 ","End":"09:07.175","Text":"everything to the left of 0 is a minus,"},{"Start":"09:07.175 ","End":"09:08.870","Text":"but as we said, if it\u0027s a plus,"},{"Start":"09:08.870 ","End":"09:10.690","Text":"we don\u0027t have to write the plus."},{"Start":"09:10.690 ","End":"09:12.435","Text":"I could write plus 3,"},{"Start":"09:12.435 ","End":"09:14.770","Text":"but we leave the plus out."},{"Start":"09:14.770 ","End":"09:16.310","Text":"That\u0027s for the numbers."},{"Start":"09:16.310 ","End":"09:20.100","Text":"Now let\u0027s get onto the addition, and subtraction part."}],"ID":4855},{"Watched":false,"Name":"Integers - Addition and Subtraction - Part 2","Duration":"11m 31s","ChapterTopicVideoID":4870,"CourseChapterTopicPlaylistID":3855,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.325","Text":"I\u0027d like to talk about addition before I talk about subtraction,"},{"Start":"00:05.325 ","End":"00:07.200","Text":"addition is what everyone knows."},{"Start":"00:07.200 ","End":"00:10.590","Text":"If I say what is 5 plus 3?"},{"Start":"00:10.590 ","End":"00:12.960","Text":"If I say 3 plus 5,"},{"Start":"00:12.960 ","End":"00:14.220","Text":"you\u0027ll know it\u0027s the same thing,"},{"Start":"00:14.220 ","End":"00:16.350","Text":"and you\u0027ll know that the answer is 8."},{"Start":"00:16.350 ","End":"00:19.410","Text":"But, I want to introduce sign numbers into this."},{"Start":"00:19.410 ","End":"00:21.720","Text":"I want to take each of the combinations of 5 and"},{"Start":"00:21.720 ","End":"00:24.870","Text":"3 once with a plus and once to the minus,"},{"Start":"00:24.870 ","End":"00:27.460","Text":"with all the brackets and with the plus sign,"},{"Start":"00:27.460 ","End":"00:30.225","Text":"so bear with me though this is a bit artificial."},{"Start":"00:30.225 ","End":"00:34.530","Text":"If you think about it, I have 4 combinations to try."},{"Start":"00:34.530 ","End":"00:40.870","Text":"I would like to know what is positive 5 plus positive 3 or plus 3."},{"Start":"00:40.870 ","End":"00:42.545","Text":"What does that equal?"},{"Start":"00:42.545 ","End":"00:46.490","Text":"I\u0027d like to know what negative 5,"},{"Start":"00:46.490 ","End":"00:49.925","Text":"and positive 3, what does that equal?"},{"Start":"00:49.925 ","End":"00:59.550","Text":"I\u0027d like to know what is positive 5 and negative 3 what does that equal?"},{"Start":"00:59.550 ","End":"01:02.825","Text":"I\u0027d like to know what negative 5,"},{"Start":"01:02.825 ","End":"01:07.390","Text":"and negative 3 equals?"},{"Start":"01:09.140 ","End":"01:15.750","Text":"Let\u0027s see now, in the first case this is exactly 5 plus 3,"},{"Start":"01:15.750 ","End":"01:20.455","Text":"just written with brackets on pluses, so this is 8."},{"Start":"01:20.455 ","End":"01:24.260","Text":"But, what I\u0027d like to show you is some pattern."},{"Start":"01:24.260 ","End":"01:27.455","Text":"Why the first 1 in some ways is like the second 1."},{"Start":"01:27.455 ","End":"01:29.375","Text":"I\u0027m going to think about."},{"Start":"01:29.375 ","End":"01:30.845","Text":"Plus, and minus."},{"Start":"01:30.845 ","End":"01:36.860","Text":"I\u0027ll use the financial model that plus is a credit money you have,"},{"Start":"01:36.860 ","End":"01:39.950","Text":"and minus will be a debit, or a debt."},{"Start":"01:39.950 ","End":"01:41.720","Text":"If I have $5,"},{"Start":"01:41.720 ","End":"01:43.265","Text":"and I have $3,"},{"Start":"01:43.265 ","End":"01:46.565","Text":"or 2 credits of 5 and 3 altogether,"},{"Start":"01:46.565 ","End":"01:52.670","Text":"the credit strengthen each other and I have an even bigger credit of $8."},{"Start":"01:52.670 ","End":"01:56.540","Text":"The thing is that the last case is similar to this in the sense"},{"Start":"01:56.540 ","End":"02:00.325","Text":"that both sides are equal here it\u0027s 2 pluses or 2 credits."},{"Start":"02:00.325 ","End":"02:02.585","Text":"Here is 2 minuses are 2 debits."},{"Start":"02:02.585 ","End":"02:05.570","Text":"In this case, they also strengthen each other a debt of 5,"},{"Start":"02:05.570 ","End":"02:08.890","Text":"and a debt of 3 is altogether a debt of 8."},{"Start":"02:08.890 ","End":"02:11.330","Text":"Essentially what I\u0027m doing is adding, first of all,"},{"Start":"02:11.330 ","End":"02:13.870","Text":"I ignore the fact that its debts and I add 5,"},{"Start":"02:13.870 ","End":"02:16.670","Text":"and 3 is 8, and then I noticed, oh,"},{"Start":"02:16.670 ","End":"02:17.960","Text":"it\u0027s not accredited to debit,"},{"Start":"02:17.960 ","End":"02:19.940","Text":"so I put a minus in front."},{"Start":"02:19.940 ","End":"02:22.970","Text":"What I\u0027m saying is, when the 2 signs are the same,"},{"Start":"02:22.970 ","End":"02:26.300","Text":"both positive or both negative in each case I"},{"Start":"02:26.300 ","End":"02:29.810","Text":"do an addition first of the numbers without the sign 5 and 3,"},{"Start":"02:29.810 ","End":"02:32.915","Text":"5 and 3, and I get the 8, and I get the 8."},{"Start":"02:32.915 ","End":"02:37.430","Text":"But then the sign is determined by if they were all pluses, it\u0027s a plus."},{"Start":"02:37.430 ","End":"02:39.290","Text":"If they were all minuses, it\u0027s a minus."},{"Start":"02:39.290 ","End":"02:42.739","Text":"But, there\u0027s a different rule that we\u0027ll cover these 2 middle cases."},{"Start":"02:42.739 ","End":"02:45.290","Text":"In this case I have a debt of 5,"},{"Start":"02:45.290 ","End":"02:47.850","Text":"and someone gives me 3."},{"Start":"02:48.230 ","End":"02:52.110","Text":"You can see 2 in the hole or 2 in debt."},{"Start":"02:52.110 ","End":"02:54.740","Text":"What we do is we say in each of these cases,"},{"Start":"02:54.740 ","End":"02:56.285","Text":"and you\u0027ll see that it\u0027s here too."},{"Start":"02:56.285 ","End":"02:58.640","Text":"We take the difference between the 2 numbers."},{"Start":"02:58.640 ","End":"03:00.544","Text":"The larger less, the smaller,"},{"Start":"03:00.544 ","End":"03:05.460","Text":"5 less 3, 5 subtract 3."},{"Start":"03:05.480 ","End":"03:07.850","Text":"The difference between 5 and 3,"},{"Start":"03:07.850 ","End":"03:10.355","Text":"whatever way you want to say it is 2."},{"Start":"03:10.355 ","End":"03:14.315","Text":"Here also, the difference between 5 and 3 is 2."},{"Start":"03:14.315 ","End":"03:17.209","Text":"But, when they\u0027re pulling in opposite directions,"},{"Start":"03:17.209 ","End":"03:20.345","Text":"once a debit to once a credit or the other way round,"},{"Start":"03:20.345 ","End":"03:22.610","Text":"then I take the sign from which is the bigger one."},{"Start":"03:22.610 ","End":"03:25.190","Text":"If I have more debt than credit,"},{"Start":"03:25.190 ","End":"03:26.780","Text":"I\u0027ve got a debit."},{"Start":"03:26.780 ","End":"03:28.580","Text":"If I have more credit, and debit,"},{"Start":"03:28.580 ","End":"03:29.945","Text":"then I\u0027ve got a credit."},{"Start":"03:29.945 ","End":"03:37.130","Text":"It\u0027s plus, and at the end we would say that the ones with the plus,"},{"Start":"03:37.130 ","End":"03:40.370","Text":"we just rewrite them without, we don\u0027t have to,"},{"Start":"03:40.370 ","End":"03:44.150","Text":"but the final answer is not customary to leave the plus another."},{"Start":"03:44.150 ","End":"03:46.310","Text":"You could if want to emphasize it."},{"Start":"03:46.310 ","End":"03:48.470","Text":"The hurray, I\u0027ve got $2,"},{"Start":"03:48.470 ","End":"03:50.940","Text":"and I\u0027m not in debt or something."},{"Start":"03:51.520 ","End":"03:56.240","Text":"Now that\u0027s addition with slightly artificial examples,"},{"Start":"03:56.240 ","End":"03:58.190","Text":"we\u0027ll get to more natural examples in a moment."},{"Start":"03:58.190 ","End":"04:04.410","Text":"I just want to say a few words about subtraction."},{"Start":"04:04.410 ","End":"04:05.920","Text":"What we\u0027re going to do,"},{"Start":"04:05.920 ","End":"04:11.345","Text":"what we do in arithmetic is a way of converting a subtraction into an addition."},{"Start":"04:11.345 ","End":"04:13.715","Text":"I\u0027ll just do 2 of the basic cases."},{"Start":"04:13.715 ","End":"04:20.425","Text":"Suppose I have 5 subtract 3."},{"Start":"04:20.425 ","End":"04:26.390","Text":"The way to convert a subtraction into addition is we say that this is equal to 5."},{"Start":"04:26.390 ","End":"04:30.230","Text":"Instead of taking away $3,"},{"Start":"04:30.230 ","End":"04:35.645","Text":"you give me, what a gift, a debt of $3."},{"Start":"04:35.645 ","End":"04:39.530","Text":"That\u0027s how we convert a subtraction into an addition."},{"Start":"04:39.530 ","End":"04:41.360","Text":"In this case, of course,"},{"Start":"04:41.360 ","End":"04:46.400","Text":"we would reduce it to the case here and the answer would be 2,"},{"Start":"04:46.400 ","End":"04:48.110","Text":"but that\u0027s not the point of the answer."},{"Start":"04:48.110 ","End":"04:50.630","Text":"The point is to show how to convert subtraction to addition,"},{"Start":"04:50.630 ","End":"04:52.400","Text":"so I can always do addition."},{"Start":"04:52.400 ","End":"04:57.650","Text":"The case that we haven\u0027t covered is what happens when you subtract a negative number,"},{"Start":"04:57.650 ","End":"05:00.545","Text":"and this is where people sometimes get confused."},{"Start":"05:00.545 ","End":"05:03.380","Text":"I have 5, and I take away something."},{"Start":"05:03.380 ","End":"05:05.890","Text":"I say the answer is going to be 8."},{"Start":"05:05.890 ","End":"05:10.865","Text":"The reason is that subtracting a negative is like removing a debt."},{"Start":"05:10.865 ","End":"05:13.110","Text":"If you can think of it as someone takes on your debt,"},{"Start":"05:13.110 ","End":"05:14.470","Text":"or removes that debt,"},{"Start":"05:14.470 ","End":"05:16.550","Text":"or the bank loses your debt of,"},{"Start":"05:16.550 ","End":"05:19.040","Text":"takes away that debt of $3."},{"Start":"05:19.040 ","End":"05:21.155","Text":"It\u0027s like giving you $3."},{"Start":"05:21.155 ","End":"05:23.465","Text":"It\u0027s 5 plus 3."},{"Start":"05:23.465 ","End":"05:26.975","Text":"If you see a minus followed by a minus,"},{"Start":"05:26.975 ","End":"05:30.545","Text":"then you make that into a plus."},{"Start":"05:30.545 ","End":"05:33.829","Text":"I\u0027m just going to emphasize that we have a minus, a minus."},{"Start":"05:33.829 ","End":"05:35.960","Text":"We\u0027re subtracting a minus, it\u0027s an addition."},{"Start":"05:35.960 ","End":"05:38.593","Text":"I\u0027ve got all the cases I need here,"},{"Start":"05:38.593 ","End":"05:43.825","Text":"and let\u0027s do some typical examples the way you might run across them."},{"Start":"05:43.825 ","End":"05:47.970","Text":"I\u0027m going to go with 7, and 4."},{"Start":"05:47.970 ","End":"05:50.810","Text":"I\u0027m going to think of all the combinations that you might get."},{"Start":"05:50.810 ","End":"05:55.245","Text":"You might get 7 plus 4 if you\u0027re lucky."},{"Start":"05:55.245 ","End":"05:57.015","Text":"I might ask you,"},{"Start":"05:57.015 ","End":"06:01.110","Text":"what is minus 7 plus 4 equal to?"},{"Start":"06:01.110 ","End":"06:04.855","Text":"I could ask what is 7 minus 4 equal to?"},{"Start":"06:04.855 ","End":"06:06.980","Text":"We\u0027ll answer, do all the answers later."},{"Start":"06:06.980 ","End":"06:09.380","Text":"I\u0027ll write the questions as I can think of them."},{"Start":"06:09.380 ","End":"06:10.970","Text":"What are the combination could we have,"},{"Start":"06:10.970 ","End":"06:13.625","Text":"minus 7 minus 4?"},{"Start":"06:13.625 ","End":"06:16.325","Text":"Now, what else can I think of?"},{"Start":"06:16.325 ","End":"06:22.845","Text":"How about 7 minus, minus 4?"},{"Start":"06:22.845 ","End":"06:25.480","Text":"What is that equal to?"},{"Start":"06:28.610 ","End":"06:32.550","Text":"I know, what about 4 minus 7?"},{"Start":"06:32.550 ","End":"06:35.420","Text":"What does that equal to? Let\u0027s see."},{"Start":"06:35.420 ","End":"06:42.780","Text":"If you can do those, you can do pretty much anything with just 2 terms,"},{"Start":"06:42.780 ","End":"06:45.260","Text":"and let\u0027s get to the first one."},{"Start":"06:45.260 ","End":"06:47.960","Text":"The first one, I know you know how to do it,"},{"Start":"06:47.960 ","End":"06:49.250","Text":"and you\u0027re going to shout out 11."},{"Start":"06:49.250 ","End":"06:52.490","Text":"Yeah, but I just wanted to show systematically how we can do it later on."},{"Start":"06:52.490 ","End":"06:54.680","Text":"We can dispense with the system once you use to it."},{"Start":"06:54.680 ","End":"06:58.000","Text":"But if you\u0027re not sure, the system will always work."},{"Start":"06:58.000 ","End":"07:00.210","Text":"We see that this is a 7,"},{"Start":"07:00.210 ","End":"07:05.380","Text":"is a positive 7 and we\u0027re adding a positive 4."},{"Start":"07:05.500 ","End":"07:09.260","Text":"I\u0027ll just go through like this and then we\u0027ll do all the answers at the end."},{"Start":"07:09.260 ","End":"07:13.990","Text":"Here I have a negative 7 and a positive 4,"},{"Start":"07:13.990 ","End":"07:16.670","Text":"here I have a positive 7."},{"Start":"07:16.670 ","End":"07:18.950","Text":"Remember, taking away a number,"},{"Start":"07:18.950 ","End":"07:23.250","Text":"subtracting a number is like adding, it\u0027s negative."},{"Start":"07:23.530 ","End":"07:27.630","Text":"Here I have a negative 7,"},{"Start":"07:27.630 ","End":"07:29.175","Text":"and I\u0027m subtracting 4."},{"Start":"07:29.175 ","End":"07:34.480","Text":"Subtracting 4 means that I\u0027m adding negative 4."},{"Start":"07:39.650 ","End":"07:45.150","Text":"Let\u0027s see. Yeah, 7 minus, minus 4."},{"Start":"07:45.150 ","End":"07:50.130","Text":"What we said this was one of those cases of removing a debt is like giving you money."},{"Start":"07:51.320 ","End":"07:55.310","Text":"We write this immediately as 7 plus 4."},{"Start":"07:55.310 ","End":"07:59.585","Text":"The minus, minus gets converted and here, 4 minus 7."},{"Start":"07:59.585 ","End":"08:02.970","Text":"We have irregular 4,"},{"Start":"08:03.310 ","End":"08:09.110","Text":"and where adding a negative 7."},{"Start":"08:09.110 ","End":"08:13.040","Text":"To be consistent, I should have written the pluses here,"},{"Start":"08:13.040 ","End":"08:17.315","Text":"but you know, I sometimes forget to do that and so you\u0027ll just forgive me for that."},{"Start":"08:17.315 ","End":"08:22.580","Text":"You also will stop not writing the pluses where we don\u0027t need to equals."},{"Start":"08:22.580 ","End":"08:24.305","Text":"I do want to write though."},{"Start":"08:24.305 ","End":"08:29.075","Text":"Now let\u0027s go with each one and see what we get."},{"Start":"08:29.075 ","End":"08:32.855","Text":"Here we have always have additions."},{"Start":"08:32.855 ","End":"08:37.430","Text":"Here, we see the first thing is that they have the same sign,"},{"Start":"08:37.430 ","End":"08:39.920","Text":"a positive and a positive credit and credit,"},{"Start":"08:39.920 ","End":"08:41.690","Text":"we add them when we take the sign,"},{"Start":"08:41.690 ","End":"08:44.465","Text":"let me just write the final answer, so that will be 11."},{"Start":"08:44.465 ","End":"08:47.085","Text":"Here we have opposite signs,"},{"Start":"08:47.085 ","End":"08:48.390","Text":"negative and a positive."},{"Start":"08:48.390 ","End":"08:52.600","Text":"We take the difference, so difference between 7 and 4 is 3."},{"Start":"08:52.600 ","End":"08:57.250","Text":"But the minus beats the plus so it\u0027s minus 3."},{"Start":"08:57.250 ","End":"08:59.110","Text":"Here again, opposite signs,"},{"Start":"08:59.110 ","End":"09:01.180","Text":"a plus and a minus we\u0027re adding."},{"Start":"09:01.180 ","End":"09:03.820","Text":"We take the difference which is 3,"},{"Start":"09:03.820 ","End":"09:05.560","Text":"and this time the positive wins,"},{"Start":"09:05.560 ","End":"09:07.525","Text":"so it\u0027s plus 3."},{"Start":"09:07.525 ","End":"09:10.150","Text":"Here we have a negative and a negative,"},{"Start":"09:10.150 ","End":"09:12.290","Text":"the same sign 2 debts."},{"Start":"09:12.290 ","End":"09:16.600","Text":"They strengthen each other so 7 plus 4 is 11."},{"Start":"09:16.600 ","End":"09:18.775","Text":"But they were debts after all,"},{"Start":"09:18.775 ","End":"09:21.650","Text":"not credit so minus."},{"Start":"09:21.950 ","End":"09:25.470","Text":"7 plus a debt removal of 4,"},{"Start":"09:25.470 ","End":"09:30.840","Text":"which is worth $4, is 11 and 4 minus 7."},{"Start":"09:30.840 ","End":"09:32.700","Text":"So add 4 to minus 7,"},{"Start":"09:32.700 ","End":"09:34.725","Text":"it\u0027s like plus 4."},{"Start":"09:34.725 ","End":"09:38.860","Text":"I could write this as plus 4 plus negative 7."},{"Start":"09:38.860 ","End":"09:41.480","Text":"Opposite signs, they cancel."},{"Start":"09:41.480 ","End":"09:47.315","Text":"The difference between 4 and 7 is 3 just the same as the difference between 7 and 4, 3."},{"Start":"09:47.315 ","End":"09:49.775","Text":"Just right is a difference."},{"Start":"09:49.775 ","End":"09:52.085","Text":"But then I take care of the sign and say,"},{"Start":"09:52.085 ","End":"09:54.140","Text":"my negative be my positive,"},{"Start":"09:54.140 ","End":"09:55.550","Text":"so it\u0027s minus 3."},{"Start":"09:55.550 ","End":"09:57.710","Text":"Now you know what 4 minus 7 is,"},{"Start":"09:57.710 ","End":"10:00.680","Text":"but at first grade they tell you that this is not possible to do,"},{"Start":"10:00.680 ","End":"10:01.910","Text":"but now we know it is,"},{"Start":"10:01.910 ","End":"10:03.920","Text":"the answer is just negative."},{"Start":"10:03.920 ","End":"10:07.325","Text":"You can\u0027t take a bigger number from a smaller number."},{"Start":"10:07.325 ","End":"10:10.625","Text":"This is all the cases with 2 numbers."},{"Start":"10:10.625 ","End":"10:14.735","Text":"Let\u0027s just extend it now to 3 or more."},{"Start":"10:14.735 ","End":"10:18.890","Text":"I have time for 1 more example when we have more than 2 things,"},{"Start":"10:18.890 ","End":"10:21.200","Text":"let\u0027s do it down here."},{"Start":"10:21.200 ","End":"10:31.150","Text":"Suppose I have minus 5 minus 13 plus 9."},{"Start":"10:31.340 ","End":"10:35.060","Text":"One way you can do it always is just by"},{"Start":"10:35.060 ","End":"10:37.850","Text":"going from left to right on each time taking a pair,"},{"Start":"10:37.850 ","End":"10:40.575","Text":"adding one more thing."},{"Start":"10:40.575 ","End":"10:44.000","Text":"If I do minus 5 minus 13,"},{"Start":"10:44.000 ","End":"10:45.980","Text":"that\u0027s similar to this one."},{"Start":"10:45.980 ","End":"10:47.953","Text":"I do an addition 5,"},{"Start":"10:47.953 ","End":"10:51.725","Text":"and 13 is 18 so it\u0027s minus 18."},{"Start":"10:51.725 ","End":"10:55.730","Text":"Now I have minus 18 plus 9,"},{"Start":"10:55.730 ","End":"11:00.195","Text":"and now I do the second last part here,"},{"Start":"11:00.195 ","End":"11:02.080","Text":"minus 18 plus 9,"},{"Start":"11:02.080 ","End":"11:05.340","Text":"I have an example that it looks like this."},{"Start":"11:05.950 ","End":"11:09.785","Text":"The signs oppose each other,"},{"Start":"11:09.785 ","End":"11:11.600","Text":"but the 18 wins."},{"Start":"11:11.600 ","End":"11:14.705","Text":"The difference between 18 and 9 is also 9."},{"Start":"11:14.705 ","End":"11:18.560","Text":"So the answer is minus 9."},{"Start":"11:18.560 ","End":"11:20.780","Text":"This is not the only technique,"},{"Start":"11:20.780 ","End":"11:23.240","Text":"but it\u0027s very practical."},{"Start":"11:23.240 ","End":"11:32.670","Text":"You\u0027ll see it in the exercises in the solve problems after this clip. We\u0027re done here."}],"ID":4856},{"Watched":false,"Name":"Multiplication and Division","Duration":"11m 39s","ChapterTopicVideoID":4864,"CourseChapterTopicPlaylistID":3855,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.590","Text":"Now that you\u0027ve learned how to add and subtract sign numbers,"},{"Start":"00:04.590 ","End":"00:07.275","Text":"let\u0027s go on to multiplication and division,"},{"Start":"00:07.275 ","End":"00:09.615","Text":"and we\u0027ll start with multiplication."},{"Start":"00:09.615 ","End":"00:15.790","Text":"I just copy paste to the word multiplication and that\u0027s what we\u0027ll start with."},{"Start":"00:17.170 ","End":"00:20.610","Text":"I\u0027ll tell you in words what the rule is."},{"Start":"00:20.610 ","End":"00:25.910","Text":"That if you multiply like signs plus times plus or minus times minus, you get a plus,"},{"Start":"00:25.910 ","End":"00:28.200","Text":"and if you multiply opposite signs,"},{"Start":"00:28.200 ","End":"00:31.485","Text":"a plus times a minus or vice versa, you get a minus."},{"Start":"00:31.485 ","End":"00:33.450","Text":"I don\u0027t want to write down that rule."},{"Start":"00:33.450 ","End":"00:34.860","Text":"You can write it down if you want."},{"Start":"00:34.860 ","End":"00:36.255","Text":"Plus times plus is plus,"},{"Start":"00:36.255 ","End":"00:38.025","Text":"minus times minus is plus,"},{"Start":"00:38.025 ","End":"00:39.810","Text":"and plus times minus is minus."},{"Start":"00:39.810 ","End":"00:42.375","Text":"I\u0027ll just do it through examples."},{"Start":"00:42.375 ","End":"00:44.495","Text":"I\u0027m assuming you know,"},{"Start":"00:44.495 ","End":"00:49.190","Text":"without a calculator that if I ask you what is 2x5,"},{"Start":"00:49.190 ","End":"00:51.275","Text":"you can tell me that that\u0027s 10,"},{"Start":"00:51.275 ","End":"00:55.415","Text":"but now we want to introduce the concept of sign numbers into this,"},{"Start":"00:55.415 ","End":"00:58.175","Text":"so really I have four possible combinations."},{"Start":"00:58.175 ","End":"01:02.371","Text":"I have (+2)x(+5),"},{"Start":"01:02.371 ","End":"01:09.335","Text":"I notice that we write multiplication with a dot most commonly in Algebra though,"},{"Start":"01:09.335 ","End":"01:14.580","Text":"in school you probably learned it as 2x5=10,"},{"Start":"01:14.580 ","End":"01:17.450","Text":"but we\u0027re not going to use the cross symbol."},{"Start":"01:17.450 ","End":"01:19.980","Text":"We\u0027re going to use the dot symbol."},{"Start":"01:21.440 ","End":"01:24.545","Text":"Let\u0027s see what this is and then I\u0027m going to have,"},{"Start":"01:24.545 ","End":"01:28.190","Text":"the other possibility is (-2)x(-5)."},{"Start":"01:28.190 ","End":"01:29.735","Text":"What is that equal?"},{"Start":"01:29.735 ","End":"01:36.250","Text":"The third possibility is (+2)x(-5)."},{"Start":"01:36.250 ","End":"01:40.040","Text":"Well really there\u0027s four possibilities because I could also have"},{"Start":"01:40.040 ","End":"01:45.380","Text":"(-2)x(+5) although I could also reverse the order,"},{"Start":"01:45.380 ","End":"01:46.520","Text":"instead of minus times plus,"},{"Start":"01:46.520 ","End":"01:48.110","Text":"I can say plus times minus."},{"Start":"01:48.110 ","End":"01:52.105","Text":"Let\u0027s just keep it to all four possible combinations."},{"Start":"01:52.105 ","End":"01:54.740","Text":"Now, like I said, like signs,"},{"Start":"01:54.740 ","End":"01:56.690","Text":"if you multiply, you get a plus."},{"Start":"01:56.690 ","End":"01:58.070","Text":"These two alike,"},{"Start":"01:58.070 ","End":"01:59.455","Text":"so I\u0027m going to get a plus."},{"Start":"01:59.455 ","End":"02:02.590","Text":"These two alike, a minus times a minus is also a plus,"},{"Start":"02:02.590 ","End":"02:05.885","Text":"and remember that minus times minus is plus."},{"Start":"02:05.885 ","End":"02:09.200","Text":"A plus times minus is going to be a minus and a minus times"},{"Start":"02:09.200 ","End":"02:12.465","Text":"a plus is going to be a minus,"},{"Start":"02:12.465 ","End":"02:15.950","Text":"and the actual value without the sign"},{"Start":"02:15.950 ","End":"02:21.040","Text":"called the absolute value is going to be 10 in each case."},{"Start":"02:21.040 ","End":"02:22.820","Text":"It\u0027s always going to be 10."},{"Start":"02:22.820 ","End":"02:24.815","Text":"It\u0027s just a matter of getting the sign right,"},{"Start":"02:24.815 ","End":"02:28.070","Text":"so if it\u0027s like signs,"},{"Start":"02:28.070 ","End":"02:32.790","Text":"it\u0027s plus, and if it\u0027s opposite signs, then it\u0027s minus."},{"Start":"02:32.830 ","End":"02:37.540","Text":"Now I want to show you what happens if we multiply more than two."},{"Start":"02:37.540 ","End":"02:41.990","Text":"I\u0027m assuming that if I asked you, like in school,"},{"Start":"02:41.990 ","End":"02:48.735","Text":"what is 2x3x4,"},{"Start":"02:48.735 ","End":"02:55.550","Text":"or I might say 2x3x4,"},{"Start":"02:55.550 ","End":"03:03.365","Text":"that you can do this in your heads and say 2x3 is 6 and then times 4 is 24."},{"Start":"03:03.365 ","End":"03:05.780","Text":"Again, I want to introduce sign numbers."},{"Start":"03:05.780 ","End":"03:07.460","Text":"I won\u0027t do all the combinations,"},{"Start":"03:07.460 ","End":"03:16.260","Text":"but let\u0027s do (-2)x(-3)x(-4)."},{"Start":"03:16.260 ","End":"03:22.410","Text":"Let\u0027s see what that equals and what is (-2)x3,"},{"Start":"03:22.410 ","End":"03:26.070","Text":"which I\u0027ll just write as 3, times (-4)."},{"Start":"03:26.070 ","End":"03:28.330","Text":"What does that equal?"},{"Start":"03:28.340 ","End":"03:32.500","Text":"Well, we can do them two at a time and say,"},{"Start":"03:32.500 ","End":"03:33.880","Text":"in this first example,"},{"Start":"03:33.880 ","End":"03:40.025","Text":"minus times minus is plus and then plus times minus is minus,"},{"Start":"03:40.025 ","End":"03:44.320","Text":"so the answer here would be minus and as for the rest of it,"},{"Start":"03:44.320 ","End":"03:49.525","Text":"you just ignore the signs and multiply 2x3x4 is 24."},{"Start":"03:49.525 ","End":"03:54.640","Text":"Whereas here, minus times plus is minus."},{"Start":"03:54.640 ","End":"03:57.340","Text":"I\u0027m looking at the table here all the time."},{"Start":"03:57.340 ","End":"03:59.470","Text":"Minus times plus is minus,"},{"Start":"03:59.470 ","End":"04:03.210","Text":"and then I\u0027ve got a minus times a minus and then I look here,"},{"Start":"04:03.210 ","End":"04:06.765","Text":"it\u0027s a plus, so the answer is going to be plus 24."},{"Start":"04:06.765 ","End":"04:08.670","Text":"I\u0027ll write it as plus 24."},{"Start":"04:08.670 ","End":"04:13.945","Text":"At the end, just write it as 24."},{"Start":"04:13.945 ","End":"04:16.580","Text":"There is another way of looking at this."},{"Start":"04:16.580 ","End":"04:19.445","Text":"Every time I have a pair of minuses,"},{"Start":"04:19.445 ","End":"04:22.280","Text":"they cancel each other out and become a plus,"},{"Start":"04:22.280 ","End":"04:25.040","Text":"so when we have three or more,"},{"Start":"04:25.040 ","End":"04:28.355","Text":"the trick is just to count the number of minuses."},{"Start":"04:28.355 ","End":"04:31.280","Text":"Here I have 1,2,3 minuses,"},{"Start":"04:31.280 ","End":"04:33.410","Text":"3 is an odd number."},{"Start":"04:33.410 ","End":"04:37.330","Text":"If it\u0027s an odd number of minuses, you get a minus,"},{"Start":"04:37.330 ","End":"04:40.310","Text":"and if it\u0027s an even number of minuses as it is here,"},{"Start":"04:40.310 ","End":"04:42.019","Text":"we have 2 minuses,"},{"Start":"04:42.019 ","End":"04:44.040","Text":"and 2 is an even number, you get a plus."},{"Start":"04:44.040 ","End":"04:46.310","Text":"So odd number of minuses is minus,"},{"Start":"04:46.310 ","End":"04:50.165","Text":"even number of minuses in a product gives you a plus."},{"Start":"04:50.165 ","End":"04:52.850","Text":"Now let\u0027s say the reason is because every minus times minus"},{"Start":"04:52.850 ","End":"04:55.645","Text":"is a plus and if there\u0027s one leftover,"},{"Start":"04:55.645 ","End":"04:59.750","Text":"then it\u0027s a minus and if there isn\u0027t one leftover, then it\u0027s a plus."},{"Start":"04:59.930 ","End":"05:04.460","Text":"That\u0027s basically what I have to say right now for multiplication."},{"Start":"05:04.460 ","End":"05:06.755","Text":"Let\u0027s go on to division."},{"Start":"05:06.755 ","End":"05:11.555","Text":"We now come to division of sign numbers,"},{"Start":"05:11.555 ","End":"05:16.525","Text":"but I would like to say something about the notation."},{"Start":"05:16.525 ","End":"05:19.130","Text":"If I want to divide 8 by 4,"},{"Start":"05:19.130 ","End":"05:21.615","Text":"there are many ways to write this."},{"Start":"05:21.615 ","End":"05:29.860","Text":"One is to write 8/4 with a slash and the answer is 2."},{"Start":"05:29.860 ","End":"05:35.780","Text":"In the United States in elementary schools,"},{"Start":"05:35.780 ","End":"05:42.200","Text":"you would write it as this symbol with a line,"},{"Start":"05:42.200 ","End":"05:43.520","Text":"with a dot above and below,"},{"Start":"05:43.520 ","End":"05:45.425","Text":"it has a name it\u0027s called the obelisk,"},{"Start":"05:45.425 ","End":"05:49.050","Text":"but doesn\u0027t matter, divided by 4."},{"Start":"05:49.180 ","End":"05:55.980","Text":"You could also just write it as 8/4 this way."},{"Start":"05:55.980 ","End":"06:03.950","Text":"You can also use the ratio symbol, the colon, 8:4."},{"Start":"06:03.950 ","End":"06:09.740","Text":"Now, because I want this course to be international,"},{"Start":"06:09.740 ","End":"06:13.040","Text":"and since not every country uses this symbol,"},{"Start":"06:13.040 ","End":"06:15.620","Text":"even though in United States and in England,"},{"Start":"06:15.620 ","End":"06:17.330","Text":"this is frequently used,"},{"Start":"06:17.330 ","End":"06:19.580","Text":"I\u0027m not going to use this symbol,"},{"Start":"06:19.580 ","End":"06:25.470","Text":"and I\u0027m going to use generally any one of the other three,"},{"Start":"06:25.470 ","End":"06:27.870","Text":"but typically one of the last two,"},{"Start":"06:27.870 ","End":"06:33.180","Text":"I\u0027ll either write 8/4 or I\u0027ll write 8 then a colon sign and 4,"},{"Start":"06:33.180 ","End":"06:37.490","Text":"and any event, the answer to all four of these, this, this, this,"},{"Start":"06:37.490 ","End":"06:41.750","Text":"or this is 2, but as I said,"},{"Start":"06:41.750 ","End":"06:45.200","Text":"I will generally use one of the last two,"},{"Start":"06:45.200 ","End":"06:50.690","Text":"either the colon or the dividing line."},{"Start":"06:50.690 ","End":"06:53.090","Text":"While I\u0027m on the subject of notation,"},{"Start":"06:53.090 ","End":"06:54.905","Text":"let\u0027s go back to the multiplication."},{"Start":"06:54.905 ","End":"06:59.660","Text":"I mentioned that multiplication is indicated either by"},{"Start":"06:59.660 ","End":"07:05.720","Text":"a dot or in the elementary schools by the multiplication sign."},{"Start":"07:05.720 ","End":"07:08.330","Text":"There is actually a third way."},{"Start":"07:08.330 ","End":"07:13.550","Text":"In algebra, you often indicate multiplication by no symbol at all,"},{"Start":"07:13.550 ","End":"07:16.340","Text":"but just two things being next to each other."},{"Start":"07:16.340 ","End":"07:24.470","Text":"For example, this, I could write an algebra as just (-2)x(-3)x(-4)."},{"Start":"07:24.470 ","End":"07:28.509","Text":"This is more commonly done when using letters in algebra."},{"Start":"07:28.509 ","End":"07:34.310","Text":"The only thing is you mustn\u0027t get confused in situations where there\u0027s no brackets,"},{"Start":"07:34.310 ","End":"07:35.645","Text":"least on one of them."},{"Start":"07:35.645 ","End":"07:37.310","Text":"You can\u0027t do it."},{"Start":"07:37.310 ","End":"07:44.300","Text":"You can\u0027t say 2x3x4 like this because then it would be 234,"},{"Start":"07:44.300 ","End":"07:47.363","Text":"so that\u0027s not, but you could say"},{"Start":"07:47.363 ","End":"07:55.190","Text":"(2)(3)(4) and you would be understood that this would be a multiplication."},{"Start":"07:55.190 ","End":"08:00.380","Text":"Multiplication is either dot or cross or"},{"Start":"08:00.380 ","End":"08:06.110","Text":"just nothing as long as there\u0027s no possibility of confusion as here."},{"Start":"08:06.110 ","End":"08:08.930","Text":"And then division, one number over the other,"},{"Start":"08:08.930 ","End":"08:11.525","Text":"or the colon for divided."},{"Start":"08:11.525 ","End":"08:15.080","Text":"Now the rule for division is the same as for multiplication."},{"Start":"08:15.080 ","End":"08:19.415","Text":"If you multiply or divide likes,"},{"Start":"08:19.415 ","End":"08:25.410","Text":"like two pluses or two minuses,"},{"Start":"08:25.410 ","End":"08:27.690","Text":"it\u0027s going to be plus, and if it\u0027s mixed,"},{"Start":"08:27.690 ","End":"08:28.920","Text":"then it\u0027s going to be minus."},{"Start":"08:28.920 ","End":"08:31.030","Text":"I\u0027ll give some examples now."},{"Start":"08:31.030 ","End":"08:34.980","Text":"Let me just get myself some more space here."},{"Start":"08:35.020 ","End":"08:37.850","Text":"I\u0027m going to give four examples."},{"Start":"08:37.850 ","End":"08:41.260","Text":"The first one, we\u0027ll just repeat this one here,"},{"Start":"08:41.260 ","End":"08:48.300","Text":"8:4=2, and I\u0027m going to use the colon,"},{"Start":"08:48.300 ","End":"08:51.495","Text":"the 2 dots form of the dividing."},{"Start":"08:51.495 ","End":"09:00.690","Text":"The next one will be -50:-10,"},{"Start":"09:00.690 ","End":"09:03.350","Text":"and like I said, with same as the multiplication,"},{"Start":"09:03.350 ","End":"09:04.950","Text":"if the signs are alike,"},{"Start":"09:04.950 ","End":"09:08.945","Text":"minus and minus, then the answer will be plus."},{"Start":"09:08.945 ","End":"09:13.085","Text":"I\u0027ll do 50/10 as if it was plus and it\u0027s 5."},{"Start":"09:13.085 ","End":"09:16.580","Text":"Notice also that I\u0027ve contradicted"},{"Start":"09:16.580 ","End":"09:20.225","Text":"myself and I said we don\u0027t usually put two signs together."},{"Start":"09:20.225 ","End":"09:22.820","Text":"I have a division and a minus next to each other."},{"Start":"09:22.820 ","End":"09:28.350","Text":"With division, it\u0027s often acceptable and it saves you writing brackets,"},{"Start":"09:28.350 ","End":"09:29.510","Text":"but if in doubt,"},{"Start":"09:29.510 ","End":"09:32.945","Text":"you could always put brackets around the 10,"},{"Start":"09:32.945 ","End":"09:35.675","Text":"but it\u0027s optional in this case."},{"Start":"09:35.675 ","End":"09:37.880","Text":"There\u0027s no hard and fast rules."},{"Start":"09:37.880 ","End":"09:40.665","Text":"If you think that might be a misunderstanding,"},{"Start":"09:40.665 ","End":"09:42.345","Text":"always put brackets in."},{"Start":"09:42.345 ","End":"09:47.690","Text":"The next one will be -15:3,"},{"Start":"09:47.690 ","End":"09:48.830","Text":"here there\u0027s no confusion,"},{"Start":"09:48.830 ","End":"09:50.329","Text":"no need for brackets."},{"Start":"09:50.329 ","End":"09:52.640","Text":"This is a minus and this is a plus."},{"Start":"09:52.640 ","End":"09:55.685","Text":"I\u0027ve stopped writing the plus all the time now."},{"Start":"09:55.685 ","End":"09:57.320","Text":"We just know that it\u0027s a plus,"},{"Start":"09:57.320 ","End":"09:59.270","Text":"a minus and plus they\u0027re opposite signs."},{"Start":"09:59.270 ","End":"10:04.340","Text":"Opposite signs means minus and then you do the calculation without the sign,"},{"Start":"10:04.340 ","End":"10:06.560","Text":"15:3 is 5,"},{"Start":"10:06.560 ","End":"10:08.975","Text":"so this gives me minus 5."},{"Start":"10:08.975 ","End":"10:16.405","Text":"The last example would be 6:(-2)."},{"Start":"10:16.405 ","End":"10:18.830","Text":"Once again, this is a plus, this is a minus,"},{"Start":"10:18.830 ","End":"10:24.480","Text":"opposite signs is minus and 6 divided by 2 is 3."},{"Start":"10:25.550 ","End":"10:28.499","Text":"That\u0027s it for division."},{"Start":"10:28.499 ","End":"10:32.945","Text":"I\u0027ll give one example of a mixture of multiplication and division."},{"Start":"10:32.945 ","End":"10:33.560","Text":"How"},{"Start":"10:33.560 ","End":"10:44.500","Text":"about (-4)x2,"},{"Start":"10:44.500 ","End":"10:54.190","Text":"let\u0027s say minus 6 and all this divided by 12."},{"Start":"10:54.490 ","End":"10:58.115","Text":"I have multiplications."},{"Start":"10:58.115 ","End":"11:00.200","Text":"Here is a multiplication,"},{"Start":"11:00.200 ","End":"11:01.400","Text":"here is a multiplication,"},{"Start":"11:01.400 ","End":"11:03.320","Text":"and here is a division."},{"Start":"11:03.320 ","End":"11:09.140","Text":"What I do is the usual rule with the sign is I count the number of minuses,"},{"Start":"11:09.140 ","End":"11:11.180","Text":"I have two minuses,"},{"Start":"11:11.180 ","End":"11:13.205","Text":"that\u0027s an even number of minuses,"},{"Start":"11:13.205 ","End":"11:16.140","Text":"so the answer is going to be plus."},{"Start":"11:16.480 ","End":"11:20.420","Text":"Then I do the multiplication and division from left to right."},{"Start":"11:20.420 ","End":"11:22.250","Text":"4x2 is 8,"},{"Start":"11:22.250 ","End":"11:26.135","Text":"times 6 is 48 divided by 12 is 4."},{"Start":"11:26.135 ","End":"11:28.950","Text":"It\u0027s going to be a plus 4."},{"Start":"11:29.050 ","End":"11:32.885","Text":"That\u0027s enough about multiplication and division."},{"Start":"11:32.885 ","End":"11:39.240","Text":"Let\u0027s go on to the next subject, which is zero."}],"ID":4857},{"Watched":false,"Name":"Operations with Zero","Duration":"7m 2s","ChapterTopicVideoID":4865,"CourseChapterTopicPlaylistID":3855,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.430","Text":"Up to now we\u0027ve talked about signed numbers, positive and negative."},{"Start":"00:05.430 ","End":"00:09.330","Text":"But there\u0027s actually another number called 0,"},{"Start":"00:09.330 ","End":"00:11.189","Text":"which is a signed number,"},{"Start":"00:11.189 ","End":"00:13.230","Text":"though it\u0027s neither positive nor negative."},{"Start":"00:13.230 ","End":"00:16.380","Text":"It\u0027s just included with the package on the number line,"},{"Start":"00:16.380 ","End":"00:18.120","Text":"we have 0 positives and negatives,"},{"Start":"00:18.120 ","End":"00:19.620","Text":"and altogether signed numbers."},{"Start":"00:19.620 ","End":"00:22.455","Text":"I\u0027m not going to argue whether 0 has a sign or not,"},{"Start":"00:22.455 ","End":"00:24.030","Text":"the 0 is 0."},{"Start":"00:24.030 ","End":"00:26.760","Text":"Now, 0 can be added,"},{"Start":"00:26.760 ","End":"00:30.510","Text":"we can say 0 plus 5,"},{"Start":"00:30.510 ","End":"00:33.090","Text":"and that would just be 5."},{"Start":"00:33.090 ","End":"00:35.205","Text":"If you add 0 to anything,"},{"Start":"00:35.205 ","End":"00:38.290","Text":"it\u0027s just that number."},{"Start":"00:38.860 ","End":"00:43.760","Text":"We can say subtraction with 0."},{"Start":"00:43.760 ","End":"00:49.210","Text":"Now, notice that if I say 7 minus 0,"},{"Start":"00:49.210 ","End":"00:55.110","Text":"it\u0027s 7, but 0 minus"},{"Start":"00:55.110 ","End":"01:01.680","Text":"7 is not 7, it\u0027s minus 7."},{"Start":"01:01.680 ","End":"01:03.390","Text":"I take 7 away from nothing,"},{"Start":"01:03.390 ","End":"01:06.465","Text":"come 7 in the whole minus 7."},{"Start":"01:06.465 ","End":"01:08.895","Text":"That\u0027s addition and subtraction."},{"Start":"01:08.895 ","End":"01:10.845","Text":"Now multiplication."},{"Start":"01:10.845 ","End":"01:15.650","Text":"7 times 0 is 0,"},{"Start":"01:15.650 ","End":"01:19.350","Text":"anything times 0 is 0."},{"Start":"01:22.030 ","End":"01:25.535","Text":"I\u0027m leaving a blank here for the moment."},{"Start":"01:25.535 ","End":"01:33.600","Text":"0 divided by 7 is 0."},{"Start":"01:33.600 ","End":"01:36.090","Text":"But there\u0027s something missing here,"},{"Start":"01:36.090 ","End":"01:37.860","Text":"one of the combinations."},{"Start":"01:37.860 ","End":"01:40.009","Text":"I don\u0027t know why I\u0027m using 7 all the time,"},{"Start":"01:40.009 ","End":"01:41.615","Text":"I\u0027ll go to another number."},{"Start":"01:41.615 ","End":"01:43.220","Text":"What would be, say,"},{"Start":"01:43.220 ","End":"01:47.445","Text":"6 divided by 0?"},{"Start":"01:47.445 ","End":"01:52.295","Text":"Now, that\u0027s the big question and I\u0027m going to talk a bit more about this."},{"Start":"01:52.295 ","End":"01:58.175","Text":"But before that, I\u0027d like to just get a little bit into algebra already,"},{"Start":"01:58.175 ","End":"02:00.469","Text":"so far we\u0027ve just been doing arithmetic."},{"Start":"02:00.469 ","End":"02:03.230","Text":"What I want to say is I can generalize this,"},{"Start":"02:03.230 ","End":"02:05.995","Text":"and it\u0027s not just 0 plus 5 is 5,"},{"Start":"02:05.995 ","End":"02:08.325","Text":"0 plus 9 is 9,"},{"Start":"02:08.325 ","End":"02:11.555","Text":"0 plus negative 100 is negative 100,"},{"Start":"02:11.555 ","End":"02:13.250","Text":"and so in general,"},{"Start":"02:13.250 ","End":"02:18.900","Text":"0 plus a is equal"},{"Start":"02:18.900 ","End":"02:25.785","Text":"to a. I want to prefer to write it the other way round, somehow looks nicer."},{"Start":"02:25.785 ","End":"02:34.500","Text":"The second one could be stated in general that a minus 0 equals a,"},{"Start":"02:34.800 ","End":"02:43.170","Text":"but 0 minus a is not a, it\u0027s minus a."},{"Start":"02:43.170 ","End":"02:51.980","Text":"This one I could write in general that a times 0 is 0."},{"Start":"02:51.980 ","End":"02:54.830","Text":"Here again, I\u0027ll leave the question mark."},{"Start":"02:54.830 ","End":"03:00.605","Text":"A divided by 0 equals what,"},{"Start":"03:00.605 ","End":"03:08.055","Text":"and 0 divided by a is equal to 0."},{"Start":"03:08.055 ","End":"03:11.010","Text":"Let\u0027s get to this mystery,"},{"Start":"03:11.010 ","End":"03:14.670","Text":"in fact, I\u0027ll go back to the numerical example."},{"Start":"03:14.860 ","End":"03:17.435","Text":"What\u0027s the mystery here?"},{"Start":"03:17.435 ","End":"03:19.740","Text":"Let\u0027s take an example."},{"Start":"03:24.710 ","End":"03:28.620","Text":"8 divided by 2 is equal to 4."},{"Start":"03:28.620 ","End":"03:36.420","Text":"Because 8 over 2 asks what number times 2 would equal 8 and the answer would be 4."},{"Start":"03:36.420 ","End":"03:43.445","Text":"But what I\u0027m saying, is that every division calculation is really a multiplication."},{"Start":"03:43.445 ","End":"03:47.785","Text":"If I said that minus 12"},{"Start":"03:47.785 ","End":"03:54.030","Text":"divided by 4 is equal to minus 3,"},{"Start":"03:54.030 ","End":"04:05.710","Text":"it is so because minus 3 times 4 is equal to minus 12."},{"Start":"04:05.710 ","End":"04:09.050","Text":"I don\u0027t think I need the brackets here actually."},{"Start":"04:09.050 ","End":"04:12.260","Text":"Now, if you understand the logic in these 2 exercises,"},{"Start":"04:12.260 ","End":"04:14.735","Text":"then I can apply it over here."},{"Start":"04:14.735 ","End":"04:21.425","Text":"I can ask, 6 over 0 is something,"},{"Start":"04:21.425 ","End":"04:23.240","Text":"let\u0027s put a question mark,"},{"Start":"04:23.240 ","End":"04:29.005","Text":"has to mean that just like here we said 2 times 4, and here,"},{"Start":"04:29.005 ","End":"04:30.525","Text":"it doesn\u0027t matter which order,"},{"Start":"04:30.525 ","End":"04:36.900","Text":"I have to have 0 times something is got to equal 6."},{"Start":"04:36.900 ","End":"04:42.240","Text":"What number could I possibly put here to make this work?"},{"Start":"04:42.240 ","End":"04:50.910","Text":"I can\u0027t because we\u0027ve already got this rule that a times 0 or I could write also,"},{"Start":"04:50.910 ","End":"04:54.560","Text":"where 0 times a is also equal to 0."},{"Start":"04:54.560 ","End":"04:56.240","Text":"Whatever I put here,"},{"Start":"04:56.240 ","End":"04:58.415","Text":"0 times it is not going to be 6,"},{"Start":"04:58.415 ","End":"05:00.080","Text":"so there\u0027s no answer."},{"Start":"05:00.080 ","End":"05:05.685","Text":"That\u0027s why we say that this is undefined,"},{"Start":"05:05.685 ","End":"05:11.520","Text":"this is not defined or undefined or meaningless."},{"Start":"05:14.920 ","End":"05:18.200","Text":"There is no answer to 6 divided by 0,"},{"Start":"05:18.200 ","End":"05:19.580","Text":"we just have to accept it,"},{"Start":"05:19.580 ","End":"05:21.680","Text":"so we\u0027ll say in general,"},{"Start":"05:21.680 ","End":"05:25.185","Text":"that a divided by 0,"},{"Start":"05:25.185 ","End":"05:28.845","Text":"they call it undefined rather than meaningless."},{"Start":"05:28.845 ","End":"05:30.275","Text":"Even you get the idea,"},{"Start":"05:30.275 ","End":"05:33.740","Text":"there is no answer in plain words."},{"Start":"05:33.740 ","End":"05:36.259","Text":"I\u0027m basically done with this clip,"},{"Start":"05:36.259 ","End":"05:39.980","Text":"but I\u0027d like to give you a bit of extra about division by 0,"},{"Start":"05:39.980 ","End":"05:42.539","Text":"so you\u0027re welcome to stay."},{"Start":"05:42.870 ","End":"05:46.900","Text":"I want to give another example of why it\u0027s undefined,"},{"Start":"05:46.900 ","End":"05:50.785","Text":"and it\u0027s even worse possibly than the 6 divided by 0."},{"Start":"05:50.785 ","End":"05:52.195","Text":"What if I asked you,"},{"Start":"05:52.195 ","End":"05:56.945","Text":"what is 0 divided by 0 equal to?"},{"Start":"05:56.945 ","End":"05:59.680","Text":"Well, following the same example,"},{"Start":"05:59.680 ","End":"06:01.870","Text":"this times this has to equal this,"},{"Start":"06:01.870 ","End":"06:06.130","Text":"so I\u0027d say 0 times what,"},{"Start":"06:06.130 ","End":"06:08.870","Text":"is equal to 0?"},{"Start":"06:09.570 ","End":"06:12.085","Text":"In contrast to this,"},{"Start":"06:12.085 ","End":"06:14.905","Text":"where there was no solution for question mark,"},{"Start":"06:14.905 ","End":"06:18.730","Text":"here every number you put in is going to make it right."},{"Start":"06:18.730 ","End":"06:21.265","Text":"0 times 3 is 0,"},{"Start":"06:21.265 ","End":"06:23.470","Text":"0 times 9 is 0,"},{"Start":"06:23.470 ","End":"06:25.270","Text":"0 times 1,000,000 is 0,"},{"Start":"06:25.270 ","End":"06:27.835","Text":"0 times minus 17 is 0,"},{"Start":"06:27.835 ","End":"06:29.660","Text":"so 0 over 0 is everything."},{"Start":"06:29.660 ","End":"06:32.975","Text":"It could be 17 or 100 or minus 1,000,000 or whatever,"},{"Start":"06:32.975 ","End":"06:36.065","Text":"so that\u0027s even worse predicament."},{"Start":"06:36.065 ","End":"06:41.105","Text":"This has no solution and every number is a solution,"},{"Start":"06:41.105 ","End":"06:43.370","Text":"so that\u0027s confusing also."},{"Start":"06:43.370 ","End":"06:48.025","Text":"Once again, it shows that something divided by 0 is undefined,"},{"Start":"06:48.025 ","End":"06:50.987","Text":"whether a is 0 or whether it\u0027s non-zero,"},{"Start":"06:50.987 ","End":"06:53.060","Text":"we have a different problem in each case,"},{"Start":"06:53.060 ","End":"06:57.035","Text":"but we still can\u0027t define anything divided by 0,"},{"Start":"06:57.035 ","End":"07:00.390","Text":"so that\u0027s the end of the bonus bit."}],"ID":4858},{"Watched":false,"Name":"Exponents and Roots - Part 1","Duration":"11m 36s","ChapterTopicVideoID":4863,"CourseChapterTopicPlaylistID":3855,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.729","Text":"In this clip, we\u0027ll learn another couple of arithmetic operations, exponents,"},{"Start":"00:05.729 ","End":"00:09.270","Text":"and roots, and this added to the list of addition,"},{"Start":"00:09.270 ","End":"00:12.090","Text":"subtraction, multiplication, and division."},{"Start":"00:12.090 ","End":"00:15.255","Text":"Let\u0027s get started straightaway with an example."},{"Start":"00:15.255 ","End":"00:17.865","Text":"I\u0027m going to write something,"},{"Start":"00:17.865 ","End":"00:24.165","Text":"a 2 here, and a smaller 6 in this position."},{"Start":"00:24.165 ","End":"00:27.450","Text":"We say that this is 2^6,"},{"Start":"00:27.450 ","End":"00:29.580","Text":"and I\u0027m writing it down how we say it, 2 to the 6th,"},{"Start":"00:29.580 ","End":"00:39.105","Text":"or 2 to the power of 6."},{"Start":"00:39.105 ","End":"00:42.045","Text":"That\u0027s a couple of ways of saying it,"},{"Start":"00:42.045 ","End":"00:44.415","Text":"and now what does it mean?"},{"Start":"00:44.415 ","End":"00:47.818","Text":"Well, you look at the exponent,"},{"Start":"00:47.818 ","End":"00:50.000","Text":"this part is called the exponent,"},{"Start":"00:50.000 ","End":"00:51.680","Text":"and this part is called the base."},{"Start":"00:51.680 ","End":"00:53.900","Text":"We look at the exponent which is 6,"},{"Start":"00:53.900 ","End":"00:56.840","Text":"and when we see 6,"},{"Start":"00:56.840 ","End":"01:01.340","Text":"and this has to be a positive whole number,"},{"Start":"01:01.340 ","End":"01:06.780","Text":"then we count 6 times the base,"},{"Start":"01:06.780 ","End":"01:09.455","Text":"and we put a multiplication in between."},{"Start":"01:09.455 ","End":"01:12.665","Text":"I\u0027ll just write it, and you\u0027ll see immediately what I mean."},{"Start":"01:12.665 ","End":"01:14.630","Text":"I\u0027m counting 1,"},{"Start":"01:14.630 ","End":"01:16.730","Text":"2, 3, 4, 5, 6."},{"Start":"01:16.730 ","End":"01:20.867","Text":"So I\u0027ll just say that there are 6 of them,"},{"Start":"01:20.867 ","End":"01:22.155","Text":"and all of them are 2,"},{"Start":"01:22.155 ","End":"01:26.040","Text":"so it\u0027s 2 times 2*2*2*2*2."},{"Start":"01:26.040 ","End":"01:30.770","Text":"If you compute it it comes out to be 64,"},{"Start":"01:30.770 ","End":"01:34.265","Text":"so 2^6 is 64."},{"Start":"01:34.265 ","End":"01:38.995","Text":"This whole thing is called an exponent or just the"},{"Start":"01:38.995 ","End":"01:46.190","Text":"top part, 2^6."},{"Start":"01:46.190 ","End":"01:52.495","Text":"Another example, 2 to the power of 4 or 2^4."},{"Start":"01:52.495 ","End":"01:55.114","Text":"Let\u0027s see what this equals."},{"Start":"01:55.114 ","End":"01:57.278","Text":"We look at the 4,"},{"Start":"01:57.278 ","End":"02:00.015","Text":"and we count off 4 factors of 2;"},{"Start":"02:00.015 ","End":"02:02.190","Text":"2, 2,"},{"Start":"02:02.190 ","End":"02:05.965","Text":"2, 2 with multiplication in between."},{"Start":"02:05.965 ","End":"02:08.485","Text":"Remember we write multiplication as a dot,"},{"Start":"02:08.485 ","End":"02:09.678","Text":"not as a cross."},{"Start":"02:09.678 ","End":"02:11.500","Text":"So I\u0027m counting off 1, 2, 3,"},{"Start":"02:11.500 ","End":"02:15.334","Text":"4 of them and the answer is 16."},{"Start":"02:15.334 ","End":"02:19.090","Text":"So 2^4 is 16, 2 to the power of 4 is 16."},{"Start":"02:19.090 ","End":"02:23.245","Text":"Some thing. Now here\u0027s another example."},{"Start":"02:23.245 ","End":"02:25.330","Text":"I don\u0027t want you to think the base is always 2,"},{"Start":"02:25.330 ","End":"02:27.250","Text":"let us take a base of 7."},{"Start":"02:27.250 ","End":"02:30.670","Text":"This time we\u0027ll let 2 be the exponent."},{"Start":"02:30.670 ","End":"02:33.970","Text":"First of all I\u0027d like to say how I pronounce it."},{"Start":"02:33.970 ","End":"02:35.920","Text":"I was a bit lazy."},{"Start":"02:35.920 ","End":"02:37.060","Text":"I did a copy paste,"},{"Start":"02:37.060 ","End":"02:40.407","Text":"so let\u0027s just fix it. This not, this not, this not, this not."},{"Start":"02:40.407 ","End":"02:44.710","Text":"Here we need a 7,"},{"Start":"02:44.710 ","End":"02:46.090","Text":"here we need a 7,"},{"Start":"02:46.090 ","End":"02:48.630","Text":"and here we need a 2."},{"Start":"02:48.630 ","End":"02:50.325","Text":"We don\u0027t say the twoth,"},{"Start":"02:50.325 ","End":"02:52.650","Text":"we say the second,"},{"Start":"02:52.650 ","End":"02:55.650","Text":"7^2 or the 7 to the power of 2."},{"Start":"02:55.650 ","End":"02:57.940","Text":"But, there\u0027s a third way for 2 and 3,"},{"Start":"02:57.940 ","End":"02:59.155","Text":"there are special words."},{"Start":"02:59.155 ","End":"03:02.450","Text":"We also say 7^2,"},{"Start":"03:03.440 ","End":"03:06.640","Text":"2 and 3 get a special name."},{"Start":"03:06.640 ","End":"03:16.555","Text":"Anyway, the answer is 7*7 because it\u0027s I like accounted of here 4 factors."},{"Start":"03:16.555 ","End":"03:19.620","Text":"Here I count off 2 factors,"},{"Start":"03:19.620 ","End":"03:21.075","Text":"each of them is 7,"},{"Start":"03:21.075 ","End":"03:23.760","Text":"and I get 49."},{"Start":"03:23.760 ","End":"03:27.505","Text":"An example for the cube."},{"Start":"03:27.505 ","End":"03:31.030","Text":"I\u0027m going to leave Some extra space so I can copy paste again."},{"Start":"03:31.030 ","End":"03:36.930","Text":"Let\u0027s take this time 5 to the power of 3. Here we are."},{"Start":"03:36.930 ","End":"03:43.522","Text":"I need to change these to 5s and the 2 has to be changed,"},{"Start":"03:43.522 ","End":"03:46.725","Text":"and this also has to be changed."},{"Start":"03:46.725 ","End":"03:51.525","Text":"So we have 5, 5, 5."},{"Start":"03:51.525 ","End":"03:55.940","Text":"It\u0027s 5^3, or could have written it out in words,"},{"Start":"03:55.940 ","End":"03:58.270","Text":"5 to the power of 3,"},{"Start":"03:58.270 ","End":"04:00.360","Text":"and in the case of 3,"},{"Start":"04:00.360 ","End":"04:02.935","Text":"we also say cubed."},{"Start":"04:02.935 ","End":"04:10.050","Text":"Later on you\u0027ll see with the x\u0027s that when we write x to the power of 2,"},{"Start":"04:10.050 ","End":"04:12.105","Text":"we usually say x^2."},{"Start":"04:12.105 ","End":"04:14.630","Text":"When we write x to the power of 3,"},{"Start":"04:14.630 ","End":"04:16.820","Text":"we usually say x^3."},{"Start":"04:16.820 ","End":"04:19.330","Text":"Cubed and square,"},{"Start":"04:19.330 ","End":"04:23.265","Text":"they are old-fashioned, but they are still very much in use."},{"Start":"04:23.265 ","End":"04:25.880","Text":"I use these words squared and cubed,"},{"Start":"04:25.880 ","End":"04:27.470","Text":"so just bear in mind that means 2 and 3."},{"Start":"04:27.470 ","End":"04:29.195","Text":"I see people get confused."},{"Start":"04:29.195 ","End":"04:31.550","Text":"Anyway, there we are."},{"Start":"04:31.550 ","End":"04:34.520","Text":"I have to compute actually what it is."},{"Start":"04:34.520 ","End":"04:36.670","Text":"This we count off 1, 2,"},{"Start":"04:36.670 ","End":"04:41.480","Text":"3; 3 times."},{"Start":"04:41.480 ","End":"04:44.870","Text":"I\u0027m just doing these curly braces for educational purposes."},{"Start":"04:44.870 ","End":"04:46.130","Text":"You don\u0027t usually write them,"},{"Start":"04:46.130 ","End":"04:48.395","Text":"but you would write 5*5*5,"},{"Start":"04:48.395 ","End":"04:54.160","Text":"and you would compute it 5*5 is 25 times 5 is 125."},{"Start":"04:54.160 ","End":"05:01.520","Text":"It\u0027s important that you not confuse the exponent with multiplication."},{"Start":"05:01.520 ","End":"05:03.845","Text":"This is not 2 times 6,"},{"Start":"05:03.845 ","End":"05:05.600","Text":"this is not 7 times 2."},{"Start":"05:05.600 ","End":"05:08.684","Text":"Some people get confused."},{"Start":"05:08.684 ","End":"05:10.700","Text":"Please don\u0027t."},{"Start":"05:10.700 ","End":"05:16.265","Text":"Okay, continuing, I almost forgot there\u0027s 1 example I want to give."},{"Start":"05:16.265 ","End":"05:19.975","Text":"The exponent has to be a counting number,"},{"Start":"05:19.975 ","End":"05:21.240","Text":"because we\u0027re counting them."},{"Start":"05:21.240 ","End":"05:23.480","Text":"So it has to be 1, and upwards."},{"Start":"05:23.480 ","End":"05:25.505","Text":"Later on we might extend that."},{"Start":"05:25.505 ","End":"05:30.080","Text":"Meanwhile, assume that the exponent is always a positive whole number."},{"Start":"05:30.080 ","End":"05:31.400","Text":"What about if it\u0027s 1?"},{"Start":"05:31.400 ","End":"05:34.491","Text":"If, suppose I had, I don\u0027t know,"},{"Start":"05:34.491 ","End":"05:36.406","Text":"4 to the power of 1,"},{"Start":"05:36.406 ","End":"05:42.720","Text":"well, we just count with 4s,"},{"Start":"05:42.720 ","End":"05:44.220","Text":"1, that\u0027s it."},{"Start":"05:44.220 ","End":"05:45.675","Text":"We\u0027ve count, we\u0027ve got 2, 1."},{"Start":"05:45.675 ","End":"05:48.680","Text":"It\u0027s just 4, there is nothing to multiply."},{"Start":"05:48.680 ","End":"05:51.065","Text":"Something to the power of 1 is itself,"},{"Start":"05:51.065 ","End":"05:52.425","Text":"because it\u0027s only 1 factor."},{"Start":"05:52.425 ","End":"05:54.485","Text":"I\u0027m counting 1, 2,"},{"Start":"05:54.485 ","End":"05:57.975","Text":"1, 2, 3, 1 and I stop."},{"Start":"05:57.975 ","End":"05:59.880","Text":"It may be confusing,"},{"Start":"05:59.880 ","End":"06:02.785","Text":"but 4 to the power of 1 is just 4."},{"Start":"06:02.785 ","End":"06:09.095","Text":"The next step is I\u0027m going to talk about when the base is negative,"},{"Start":"06:09.095 ","End":"06:13.610","Text":"the exponent will always stay positive whole number because we need to do the counting."},{"Start":"06:13.610 ","End":"06:16.410","Text":"Let\u0027s go on to negative."},{"Start":"06:16.570 ","End":"06:20.435","Text":"I need Some more room, so I\u0027m going to delete all of this."},{"Start":"06:20.435 ","End":"06:23.240","Text":"Already I admitted to being lazy,"},{"Start":"06:23.240 ","End":"06:24.770","Text":"so I did a copy paste here."},{"Start":"06:24.770 ","End":"06:28.590","Text":"Now let\u0027s add Some minuses."},{"Start":"06:30.350 ","End":"06:34.810","Text":"But, I want to put them in parentheses."},{"Start":"06:35.410 ","End":"06:40.590","Text":"The reason I\u0027m putting them in parentheses is that minus 5,"},{"Start":"06:40.670 ","End":"06:43.485","Text":"minus 7, minus 2,"},{"Start":"06:43.485 ","End":"06:45.290","Text":"they\u0027re 1 single number."},{"Start":"06:45.290 ","End":"06:47.690","Text":"It has another meaning if I don\u0027t put the parentheses,"},{"Start":"06:47.690 ","End":"06:49.235","Text":"but we\u0027ll get to that later."},{"Start":"06:49.235 ","End":"06:52.230","Text":"How about I changed the color here?"},{"Start":"06:52.460 ","End":"06:56.210","Text":"Okay, so let\u0027s start computing. The first one."},{"Start":"06:56.210 ","End":"06:59.240","Text":"It\u0027s minus 2 to the power of 6, so minus 2."},{"Start":"06:59.240 ","End":"07:02.465","Text":"It\u0027s a 1, 2, 3,"},{"Start":"07:02.465 ","End":"07:07.080","Text":"4, 5, 6."},{"Start":"07:07.130 ","End":"07:11.221","Text":"Now what will this equal? Let\u0027s see."},{"Start":"07:11.221 ","End":"07:12.620","Text":"We have minus, minus,"},{"Start":"07:12.620 ","End":"07:15.020","Text":"minus, minus, minus, minus,"},{"Start":"07:15.020 ","End":"07:19.610","Text":"and we already know that when we have an even number of minuses,"},{"Start":"07:19.610 ","End":"07:20.855","Text":"it comes out a plus."},{"Start":"07:20.855 ","End":"07:24.770","Text":"Because look, the reason is that this minus and minus cancel,"},{"Start":"07:24.770 ","End":"07:27.050","Text":"each pair of minuses gives me a plus."},{"Start":"07:27.050 ","End":"07:34.770","Text":"So the answer is going to be plus 2 times 2 times 2 times 2 times 2 times 2 is 64."},{"Start":"07:34.770 ","End":"07:39.790","Text":"This is going to be 64, just like here."},{"Start":"07:41.120 ","End":"07:43.335","Text":"Let\u0027s get to something."},{"Start":"07:43.335 ","End":"07:47.340","Text":"I don\u0027t have enough odd examples."},{"Start":"07:47.340 ","End":"07:49.275","Text":"Let\u0027s go down to this one."},{"Start":"07:49.275 ","End":"07:51.525","Text":"Minus 5 to the power of 3."},{"Start":"07:51.525 ","End":"07:59.050","Text":"This time I need to count to 3 minus 5 times minus 5 times minus 5."},{"Start":"07:59.050 ","End":"08:02.204","Text":"Notice also that are dispensed with the multiplication."},{"Start":"08:02.204 ","End":"08:05.330","Text":"When you have brackets, it implies that there\u0027s a multiplication, but you could,"},{"Start":"08:05.330 ","End":"08:10.140","Text":"if you\u0027re not sure about that add multiplication signs everywhere."},{"Start":"08:10.510 ","End":"08:13.655","Text":"I prefer not to because as I said, I\u0027m lazy."},{"Start":"08:13.655 ","End":"08:16.025","Text":"But you can put them in or not."},{"Start":"08:16.025 ","End":"08:17.660","Text":"You know what, I\u0027ll put them in this time."},{"Start":"08:17.660 ","End":"08:20.225","Text":"But, when you have brackets right next to each other,"},{"Start":"08:20.225 ","End":"08:22.760","Text":"it\u0027s multiplication in algebra."},{"Start":"08:22.760 ","End":"08:25.475","Text":"This time we have how many minuses?"},{"Start":"08:25.475 ","End":"08:26.765","Text":"1, 2, 3,"},{"Start":"08:26.765 ","End":"08:28.580","Text":"it\u0027s an odd number of minuses."},{"Start":"08:28.580 ","End":"08:31.609","Text":"Odd number of minuses means it stays minus."},{"Start":"08:31.609 ","End":"08:38.880","Text":"But the numerical answer comes out the Same as if it was plus and so it\u0027s 125."},{"Start":"08:38.980 ","End":"08:43.725","Text":"Similarly, minus 4, just one time."},{"Start":"08:43.725 ","End":"08:45.690","Text":"Well that\u0027s the answer, minus 4."},{"Start":"08:45.690 ","End":"08:51.540","Text":"But also notice that 1 is an odd number and it did come out to be a minus."},{"Start":"08:51.800 ","End":"08:58.500","Text":"Here I get minus 2 times minus 2 times minus 2 times minus 2."},{"Start":"08:58.500 ","End":"09:00.665","Text":"I didn\u0027t write the dot, never mind."},{"Start":"09:00.665 ","End":"09:03.925","Text":"Minus, minus, minus, minus 4 minuses is a plus."},{"Start":"09:03.925 ","End":"09:07.800","Text":"It\u0027s going to be 16, same as here."},{"Start":"09:07.800 ","End":"09:16.460","Text":"This one, minus 7^2 or to the power of 2 minus 7 times minus 7."},{"Start":"09:16.460 ","End":"09:18.560","Text":"Minus times minus is an even number of minuses."},{"Start":"09:18.560 ","End":"09:21.140","Text":"It\u0027s plus, plus 49."},{"Start":"09:21.140 ","End":"09:23.806","Text":"I figured I haven\u0027t given you enough odd ones."},{"Start":"09:23.806 ","End":"09:30.265","Text":"So let\u0027s take minus 2 to the power of 5."},{"Start":"09:30.265 ","End":"09:32.370","Text":"Just think, what is the answer going to be,"},{"Start":"09:32.370 ","End":"09:34.200","Text":"positive or negative,"},{"Start":"09:34.200 ","End":"09:36.030","Text":"5 is even or odd?"},{"Start":"09:36.030 ","End":"09:39.225","Text":"5 is odd. So it\u0027s going to come out negative."},{"Start":"09:39.225 ","End":"09:41.555","Text":"Whatever it is, it\u0027s going to be negative something."},{"Start":"09:41.555 ","End":"09:43.190","Text":"Let us write it as minus 2."},{"Start":"09:43.190 ","End":"09:44.300","Text":"Now can you count to 5?"},{"Start":"09:44.300 ","End":"09:49.390","Text":"It\u0027s 1 already 2, 3 oops."},{"Start":"09:49.640 ","End":"09:51.930","Text":"I meant factor number 3."},{"Start":"09:51.930 ","End":"09:55.155","Text":"3, 4,"},{"Start":"09:55.155 ","End":"09:58.394","Text":"5, and we already said it\u0027s going to be a minus."},{"Start":"09:58.394 ","End":"10:01.115","Text":"Then 2 times 2 times 2 times 2 would be the same"},{"Start":"10:01.115 ","End":"10:04.070","Text":"as we haven\u0027t got the positive case to copy from."},{"Start":"10:04.070 ","End":"10:07.025","Text":"But 2 times 2 times 2 times 2 times 2 is 32."},{"Start":"10:07.025 ","End":"10:08.810","Text":"So it\u0027s minus 32."},{"Start":"10:08.810 ","End":"10:11.810","Text":"The main point I wanted to make here is when we\u0027re dealing"},{"Start":"10:11.810 ","End":"10:15.900","Text":"with positives the power of we\u0027re only going to get positive."},{"Start":"10:22.150 ","End":"10:25.220","Text":"One of the equals turned into a minus."},{"Start":"10:25.220 ","End":"10:26.630","Text":"Now these are all positive 64,"},{"Start":"10:26.630 ","End":"10:28.925","Text":"16, no negatives here at all."},{"Start":"10:28.925 ","End":"10:32.150","Text":"When we take a negative to the power of something,"},{"Start":"10:32.150 ","End":"10:35.300","Text":"it could come out plus and it could come out minus."},{"Start":"10:35.300 ","End":"10:38.105","Text":"It all depends on whether this number is odd or even."},{"Start":"10:38.105 ","End":"10:40.225","Text":"If it\u0027s even, plus."},{"Start":"10:40.225 ","End":"10:43.845","Text":"We don\u0027t see the plus, even, even."},{"Start":"10:43.845 ","End":"10:52.980","Text":"The last ones, actually it just worked out nicely that up to here they were all even."},{"Start":"10:53.540 ","End":"10:57.860","Text":"From here they were all odd and I\u0027m back to blue, never mind."},{"Start":"10:57.860 ","End":"11:02.830","Text":"This is even, and this is odd exponent 4,"},{"Start":"11:02.830 ","End":"11:05.820","Text":"6, 2 all even 5,"},{"Start":"11:05.820 ","End":"11:07.485","Text":"3, 1, they\u0027re all odd."},{"Start":"11:07.485 ","End":"11:09.105","Text":"Then here we get negatives."},{"Start":"11:09.105 ","End":"11:10.520","Text":"I\u0027m not going to write that down,"},{"Start":"11:10.520 ","End":"11:12.170","Text":"but you could write that down as a rule."},{"Start":"11:12.170 ","End":"11:14.929","Text":"That a positive number to an exponent will be positive,"},{"Start":"11:14.929 ","End":"11:16.970","Text":"a negative number to an exponent,"},{"Start":"11:16.970 ","End":"11:20.900","Text":"it depends if the exponent is even as it is here,"},{"Start":"11:20.900 ","End":"11:22.970","Text":"here and here, we\u0027ll get a plus."},{"Start":"11:22.970 ","End":"11:26.030","Text":"If it\u0027s odd, we\u0027ll get a minus here,"},{"Start":"11:26.030 ","End":"11:27.860","Text":"here and here it\u0027s minus because we have 1,"},{"Start":"11:27.860 ","End":"11:30.270","Text":"3, and 5 are odd numbers."},{"Start":"11:31.660 ","End":"11:34.400","Text":"That\u0027s all I want to say about exponents."},{"Start":"11:34.400 ","End":"11:37.139","Text":"Let\u0027s move on to roots."}],"ID":4859},{"Watched":false,"Name":"Exponents and Roots - Part 2","Duration":"11m 46s","ChapterTopicVideoID":4868,"CourseChapterTopicPlaylistID":3855,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.050","Text":"We just did exponents and now onto roots."},{"Start":"00:04.050 ","End":"00:06.345","Text":"Let\u0027s start with an example."},{"Start":"00:06.345 ","End":"00:08.590","Text":"I\u0027m going to write something,"},{"Start":"00:09.230 ","End":"00:12.465","Text":"5 here and a 32 here."},{"Start":"00:12.465 ","End":"00:17.410","Text":"This is pronounced the 5th root of 32."},{"Start":"00:18.890 ","End":"00:22.290","Text":"There\u0027s a number here and there\u0027s a number here."},{"Start":"00:22.290 ","End":"00:26.355","Text":"In some sense this is the reverse of the exponent."},{"Start":"00:26.355 ","End":"00:31.545","Text":"If we want to know what this is we have to answer the question,"},{"Start":"00:31.545 ","End":"00:38.730","Text":"what is it that to the power of 5 gives me 32?"},{"Start":"00:38.730 ","End":"00:44.360","Text":"Now I happen to know what the answer is because in the pattern exponents we"},{"Start":"00:44.360 ","End":"00:52.780","Text":"saw that 2^5 is 32."},{"Start":"00:52.780 ","End":"00:54.620","Text":"To remind you why this is so,"},{"Start":"00:54.620 ","End":"01:00.230","Text":"this is because the 2 times 2 times 2 times 2 times 2 is 32,"},{"Start":"01:00.230 ","End":"01:06.540","Text":"which is 5 factors of 2."},{"Start":"01:06.540 ","End":"01:10.260","Text":"It\u0027s going to be a reverse question,"},{"Start":"01:10.260 ","End":"01:12.780","Text":"and mostly we\u0027ll be doing this on"},{"Start":"01:12.780 ","End":"01:17.990","Text":"the calculator although sometimes we can do it by trial and error guesswork."},{"Start":"01:17.990 ","End":"01:20.015","Text":"I\u0027ll give another example."},{"Start":"01:20.015 ","End":"01:25.320","Text":"The next example will be this."},{"Start":"01:27.350 ","End":"01:31.530","Text":"It\u0027s pronounced not the 3rd root,"},{"Start":"01:31.530 ","End":"01:41.520","Text":"you would think here I would say the 3rd root of 64 but that\u0027s not how it\u0027s said,"},{"Start":"01:41.520 ","End":"01:46.200","Text":"we say the cube root when it\u0027s 3."},{"Start":"01:46.200 ","End":"01:49.365","Text":"There\u0027s also a special name for 2, we\u0027ll come to that."},{"Start":"01:49.365 ","End":"01:53.175","Text":"To answer this we have to ask the question,"},{"Start":"01:53.175 ","End":"01:59.440","Text":"what to the power of 3 gives me 64?"},{"Start":"01:59.570 ","End":"02:03.725","Text":"Now I\u0027m going to give you the answer."},{"Start":"02:03.725 ","End":"02:07.460","Text":"The answer is the question mark is 4,"},{"Start":"02:07.460 ","End":"02:10.370","Text":"4^3 is 64,"},{"Start":"02:10.370 ","End":"02:16.900","Text":"and you can check that 4 times 4 times 4 really is 64."},{"Start":"02:16.900 ","End":"02:21.845","Text":"I\u0027m giving you the answer because there\u0027s no very good computational way of doing it."},{"Start":"02:21.845 ","End":"02:24.590","Text":"The only way you could do it would be guesswork."},{"Start":"02:24.590 ","End":"02:28.120","Text":"You try 1^3, 2^3,"},{"Start":"02:28.120 ","End":"02:30.865","Text":"3^3, 4^3."},{"Start":"02:30.865 ","End":"02:33.260","Text":"Bingo, it fits."},{"Start":"02:33.380 ","End":"02:36.830","Text":"We\u0027re lucky because usually they\u0027re not whole numbers."},{"Start":"02:36.830 ","End":"02:38.540","Text":"In this case anyway,"},{"Start":"02:38.540 ","End":"02:42.405","Text":"we say the cube root of 64 is 4."},{"Start":"02:42.405 ","End":"02:48.860","Text":"The reason is because 4^3 or 4 cubed we say is 64."},{"Start":"02:48.860 ","End":"02:52.150","Text":"Another example."},{"Start":"02:52.150 ","End":"02:55.290","Text":"I need a bit more space here."},{"Start":"02:55.290 ","End":"02:59.620","Text":"Let\u0027s see. What is this?"},{"Start":"03:00.070 ","End":"03:03.440","Text":"First of all, how do we even pronounce it?"},{"Start":"03:03.440 ","End":"03:08.860","Text":"The fact is we do not say the 2nd root,"},{"Start":"03:08.860 ","End":"03:12.700","Text":"it just isn\u0027t said."},{"Start":"03:12.860 ","End":"03:15.240","Text":"You don\u0027t even say 3rd root."},{"Start":"03:15.240 ","End":"03:16.410","Text":"You say 4th root, 5th root,"},{"Start":"03:16.410 ","End":"03:17.670","Text":"6th root, 7th root,"},{"Start":"03:17.670 ","End":"03:23.490","Text":"but for 3 you say cube root and for 2 you say square root."},{"Start":"03:23.490 ","End":"03:28.160","Text":"Just like we raise a number to the power of 2 it\u0027s squared,"},{"Start":"03:28.160 ","End":"03:29.780","Text":"when we raise it to the power of 3 is cubed,"},{"Start":"03:29.780 ","End":"03:34.610","Text":"but it\u0027s much more common to say cube root and square root."},{"Start":"03:34.610 ","End":"03:39.560","Text":"This means, if this is going to be something then we have to"},{"Start":"03:39.560 ","End":"03:45.155","Text":"ask that something to the power of 2 has got to equal 64."},{"Start":"03:45.155 ","End":"03:47.180","Text":"Either calculator or the guesswork approach."},{"Start":"03:47.180 ","End":"03:48.785","Text":"Make a guess: 5,"},{"Start":"03:48.785 ","End":"03:50.640","Text":"5^2, 25, too small."},{"Start":"03:50.640 ","End":"03:52.620","Text":"Let\u0027s try 6^2, 36,"},{"Start":"03:52.620 ","End":"03:54.780","Text":"7^2 is 49, 8^2,"},{"Start":"03:54.780 ","End":"03:56.385","Text":"8 times 8, 64."},{"Start":"03:56.385 ","End":"03:57.900","Text":"Yes. We\u0027ve got it."},{"Start":"03:57.900 ","End":"04:01.815","Text":"The answer here is 8."},{"Start":"04:01.815 ","End":"04:05.925","Text":"We almost never write the 2."},{"Start":"04:05.925 ","End":"04:09.620","Text":"When it\u0027s a square root there\u0027s a convention that we just don\u0027t write"},{"Start":"04:09.620 ","End":"04:14.310","Text":"anything and when there\u0027s nothing on the end of this,"},{"Start":"04:14.310 ","End":"04:15.960","Text":"like here we had a 5, here we had a 3,"},{"Start":"04:15.960 ","End":"04:18.435","Text":"if you don\u0027t see anything it means a 2,"},{"Start":"04:18.435 ","End":"04:19.755","Text":"but we just don\u0027t say it."},{"Start":"04:19.755 ","End":"04:28.140","Text":"We say the square root of 64 is 8 because 8^2 is 64."},{"Start":"04:28.140 ","End":"04:36.195","Text":"The 5th root of 32 is 2 because 2^5 is 32."},{"Start":"04:36.195 ","End":"04:42.045","Text":"The cube root of 64 is 4 because 4^3 is 64."},{"Start":"04:42.045 ","End":"04:46.695","Text":"The square root of 64 is 8 because 8^2 is 64."},{"Start":"04:46.695 ","End":"04:49.780","Text":"This is a reverse process,"},{"Start":"04:50.090 ","End":"04:52.975","Text":"a power is taking a root."},{"Start":"04:52.975 ","End":"04:57.480","Text":"The opposite of taking a number to the power of 3,"},{"Start":"04:57.480 ","End":"05:00.865","Text":"we start with 4, take it to the power of 3 we get 64."},{"Start":"05:00.865 ","End":"05:05.275","Text":"If we take now the cube root we\u0027re back again to where we started with the 4."},{"Start":"05:05.275 ","End":"05:09.160","Text":"So far so good. Now let\u0027s throw in some negatives."},{"Start":"05:09.160 ","End":"05:12.730","Text":"I\u0027m going to erase some stuff. Here we are."},{"Start":"05:12.730 ","End":"05:15.280","Text":"I deleted some and made some small modifications,"},{"Start":"05:15.280 ","End":"05:18.120","Text":"and I think I can move this up a bit."},{"Start":"05:18.120 ","End":"05:24.800","Text":"What I\u0027d like to do is maybe do examples similar to these but with negative numbers."},{"Start":"05:24.800 ","End":"05:31.640","Text":"Let me ask, what is the 5th root of minus 32?"},{"Start":"05:31.640 ","End":"05:34.350","Text":"What is this going to equal?"},{"Start":"05:36.100 ","End":"05:38.610","Text":"I\u0027ve stopped using the question mark,"},{"Start":"05:38.610 ","End":"05:42.425","Text":"let\u0027s use letter a. I have to ask myself what number"},{"Start":"05:42.425 ","End":"05:48.630","Text":"a^5 will give me 32, minus."},{"Start":"05:48.630 ","End":"05:51.615","Text":"Sorry. In other words,"},{"Start":"05:51.615 ","End":"05:56.425","Text":"I\u0027m looking for some number, a,"},{"Start":"05:56.425 ","End":"06:02.675","Text":"such that a times a times a times a times a will be minus 32."},{"Start":"06:02.675 ","End":"06:06.635","Text":"Now, what do you think? Is a going to be positive or negative?"},{"Start":"06:06.635 ","End":"06:10.520","Text":"Well, if a is positive multiplying all positives gives a positive,"},{"Start":"06:10.520 ","End":"06:13.910","Text":"so the only chance we have is negative."},{"Start":"06:13.910 ","End":"06:16.610","Text":"A natural thing to try would be"},{"Start":"06:16.610 ","End":"06:20.160","Text":"negative 2 because we had the 2 work in the positive case."},{"Start":"06:20.160 ","End":"06:23.820","Text":"Let\u0027s see, what is minus 2^5?"},{"Start":"06:23.820 ","End":"06:27.650","Text":"What is minus 2 times minus 2 times minus 2 times minus 2 times minus 2?"},{"Start":"06:27.650 ","End":"06:28.790","Text":"We already have the rule."},{"Start":"06:28.790 ","End":"06:34.485","Text":"This is an odd number and because it\u0027s an odd number we take 2^5,"},{"Start":"06:34.485 ","End":"06:36.500","Text":"then we have that over here is 32,"},{"Start":"06:36.500 ","End":"06:40.139","Text":"but put a minus in front."},{"Start":"06:40.790 ","End":"06:48.025","Text":"The 5th root of minus 32 we can finally say is minus 2."},{"Start":"06:48.025 ","End":"06:51.360","Text":"That is because if we put, instead of a, minus 2,"},{"Start":"06:51.360 ","End":"06:53.970","Text":"minus 2, minus 2, minus 2,"},{"Start":"06:53.970 ","End":"06:55.170","Text":"minus 2, minus 2,"},{"Start":"06:55.170 ","End":"06:57.674","Text":"we will get minus 32."},{"Start":"06:57.674 ","End":"07:03.080","Text":"Next example, I want to know what is the third root,"},{"Start":"07:03.080 ","End":"07:09.540","Text":"remember we say cube root of minus 64."},{"Start":"07:10.120 ","End":"07:15.440","Text":"In other words, I want to know what is this number a,"},{"Start":"07:15.440 ","End":"07:21.975","Text":"such that a^3 will be minus 64,"},{"Start":"07:21.975 ","End":"07:24.645","Text":"a times a times a,"},{"Start":"07:24.645 ","End":"07:27.525","Text":"I\u0027m spelling it until you get used to the concepts,"},{"Start":"07:27.525 ","End":"07:29.850","Text":"has got to equal minus 64."},{"Start":"07:29.850 ","End":"07:34.760","Text":"Well, as before we only have a chance with a negative a and the obvious 1"},{"Start":"07:34.760 ","End":"07:39.740","Text":"to take because I\u0027m mirroring what I did before would be minus 4. Does that work?"},{"Start":"07:39.740 ","End":"07:45.225","Text":"What is minus 4 times minus 4 times minus 4? The answer is yes."},{"Start":"07:45.225 ","End":"07:50.750","Text":"Because the 3 minuses it\u0027s an odd number and that 4 times 4 times 4 is 64,"},{"Start":"07:50.750 ","End":"07:52.910","Text":"so this will be minus 64."},{"Start":"07:52.910 ","End":"07:57.365","Text":"All is well, and so my a is equal to minus 4."},{"Start":"07:57.365 ","End":"07:59.374","Text":"Now the last example,"},{"Start":"07:59.374 ","End":"08:04.970","Text":"I want to know what is the square root of minus 64."},{"Start":"08:04.970 ","End":"08:08.210","Text":"Remember, we don\u0027t have to put the 2 here though I could put the 2 here."},{"Start":"08:08.210 ","End":"08:11.790","Text":"There\u0027s no problem if you want to emphasize it."},{"Start":"08:12.220 ","End":"08:15.335","Text":"If this is equal to something, let\u0027s say a,"},{"Start":"08:15.335 ","End":"08:20.190","Text":"I have to find a such that a^2 is minus 64."},{"Start":"08:20.190 ","End":"08:21.240","Text":"Or in other words,"},{"Start":"08:21.240 ","End":"08:22.755","Text":"a times a,"},{"Start":"08:22.755 ","End":"08:25.625","Text":"these 2 factors has got to equal minus 64."},{"Start":"08:25.625 ","End":"08:28.745","Text":"What do you think a should be plus or minus?"},{"Start":"08:28.745 ","End":"08:32.900","Text":"If we try a as positive then positive times positive is positive,"},{"Start":"08:32.900 ","End":"08:36.139","Text":"so that won\u0027t do, so a must be negative."},{"Start":"08:36.139 ","End":"08:41.540","Text":"But negative times negative is also positive so boy, I don\u0027t know."},{"Start":"08:41.540 ","End":"08:43.610","Text":"Positives don\u0027t work and negatives don\u0027t"},{"Start":"08:43.610 ","End":"08:47.720","Text":"work and the fact is that there is no such number."},{"Start":"08:47.720 ","End":"08:52.810","Text":"There is no number which squared will give us negative 64 and"},{"Start":"08:52.810 ","End":"09:00.310","Text":"so I would just say that there is no answer, no solution."},{"Start":"09:00.310 ","End":"09:03.045","Text":"I want to generalize."},{"Start":"09:03.045 ","End":"09:07.845","Text":"With positive numbers we\u0027re always okay taking any root,"},{"Start":"09:07.845 ","End":"09:09.355","Text":"5th, 3rd,"},{"Start":"09:09.355 ","End":"09:13.815","Text":"square root, 7th root, whatever."},{"Start":"09:13.815 ","End":"09:18.200","Text":"With negative numbers the general rule seems to be that"},{"Start":"09:18.200 ","End":"09:23.120","Text":"if this is even because we see the problem came because 2 was even."},{"Start":"09:23.720 ","End":"09:26.890","Text":"If there\u0027s an odd root,"},{"Start":"09:26.890 ","End":"09:30.490","Text":"a negative number can have an odd order root like"},{"Start":"09:30.490 ","End":"09:34.340","Text":"5th root or 3rd root but it will not have a square root,"},{"Start":"09:34.340 ","End":"09:35.540","Text":"it will not have a 4th root,"},{"Start":"09:35.540 ","End":"09:37.295","Text":"a 6th root and so on."},{"Start":"09:37.295 ","End":"09:40.520","Text":"You can write it down if you want to and I\u0027ll dictate to you"},{"Start":"09:40.520 ","End":"09:46.030","Text":"that a positive number can have a root of any order."},{"Start":"09:46.030 ","End":"09:51.050","Text":"A negative number can only have a root of odd order."},{"Start":"09:51.050 ","End":"09:57.020","Text":"We\u0027re almost done, I just want to leave you with an exercise."},{"Start":"09:57.020 ","End":"10:03.020","Text":"What is 3^4 minus"},{"Start":"10:03.020 ","End":"10:08.905","Text":"the cube root of negative 8?"},{"Start":"10:08.905 ","End":"10:11.600","Text":"It\u0027s made up of 2 parts,"},{"Start":"10:11.600 ","End":"10:13.685","Text":"an exponent and a root."},{"Start":"10:13.685 ","End":"10:17.434","Text":"Let\u0027s do each 1 separately at the side,"},{"Start":"10:17.434 ","End":"10:21.600","Text":"3^4 is 3 times 3 times 3 times 3."},{"Start":"10:21.600 ","End":"10:23.250","Text":"How many 3s are there?"},{"Start":"10:23.250 ","End":"10:28.680","Text":"That\u0027s right, 4. That is 81 if you multiply it out."},{"Start":"10:28.680 ","End":"10:34.530","Text":"The next bit, I\u0027ll just do the cube root of minus 8 equals."},{"Start":"10:34.530 ","End":"10:37.455","Text":"First of all, are we okay it\u0027s a negative number?"},{"Start":"10:37.455 ","End":"10:38.700","Text":"That was yes."},{"Start":"10:38.700 ","End":"10:41.150","Text":"Because this is an odd number root,"},{"Start":"10:41.150 ","End":"10:43.955","Text":"a third root or cube root it is okay."},{"Start":"10:43.955 ","End":"10:49.530","Text":"We\u0027re looking for a negative number such that times itself,"},{"Start":"10:49.530 ","End":"10:51.705","Text":"times itself, altogether 3 of them,"},{"Start":"10:51.705 ","End":"10:53.895","Text":"is going to equal 8."},{"Start":"10:53.895 ","End":"10:55.430","Text":"We\u0027ve seen this thing before,"},{"Start":"10:55.430 ","End":"11:01.605","Text":"it\u0027s minus 2 because minus 2 times minus 2 times minus 2,"},{"Start":"11:01.605 ","End":"11:03.410","Text":"I wrote here 8,"},{"Start":"11:03.410 ","End":"11:05.395","Text":"I meant minus 8, sorry."},{"Start":"11:05.395 ","End":"11:09.360","Text":"There we go. This part is minus 2,"},{"Start":"11:09.360 ","End":"11:13.489","Text":"and so if I put each of these pieces back together here,"},{"Start":"11:13.489 ","End":"11:17.300","Text":"the 3^4 is 81 minus,"},{"Start":"11:17.300 ","End":"11:21.340","Text":"and the cube root of minus 8 is minus 2."},{"Start":"11:21.340 ","End":"11:24.615","Text":"Don\u0027t put minus 1 after another,"},{"Start":"11:24.615 ","End":"11:27.250","Text":"so put this in brackets."},{"Start":"11:27.350 ","End":"11:29.540","Text":"This is a subtraction,"},{"Start":"11:29.540 ","End":"11:31.250","Text":"this is negative 2."},{"Start":"11:31.250 ","End":"11:36.690","Text":"Remember the rule that when you subtract a negative you just add the number,"},{"Start":"11:36.690 ","End":"11:41.805","Text":"say 81 plus 2 and so the answer is 83."},{"Start":"11:41.805 ","End":"11:46.560","Text":"We even did a sample exercise and now we\u0027re really done."}],"ID":4860},{"Watched":false,"Name":"PEDMAS - Part 1","Duration":"4m 48s","ChapterTopicVideoID":4866,"CourseChapterTopicPlaylistID":3855,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.360","Text":"In this clip, I\u0027ll be talking about order of"},{"Start":"00:03.360 ","End":"00:08.865","Text":"operations and I\u0027ll explain what I mean by that through examples."},{"Start":"00:08.865 ","End":"00:15.845","Text":"Normally addition is defined as addition of 2 things like 2 plus 5 and so on."},{"Start":"00:15.845 ","End":"00:19.145","Text":"But, there\u0027s no reason you can\u0027t take more than 2."},{"Start":"00:19.145 ","End":"00:23.225","Text":"I could ask, what is 2 plus 5 plus 4."},{"Start":"00:23.225 ","End":"00:25.130","Text":"But then the question arises,"},{"Start":"00:25.130 ","End":"00:27.650","Text":"how do we do this when we know how to add 2 things?"},{"Start":"00:27.650 ","End":"00:29.915","Text":"Well, we take them 2 at a time."},{"Start":"00:29.915 ","End":"00:34.100","Text":"I could say that this is 2 plus 5 is 7,"},{"Start":"00:34.100 ","End":"00:37.700","Text":"7 plus 4 and that\u0027s 11."},{"Start":"00:37.700 ","End":"00:42.680","Text":"But someone else will say why don\u0027t I do it differently?"},{"Start":"00:42.680 ","End":"00:45.335","Text":"Why don\u0027t I take the 5 plus 4 first,"},{"Start":"00:45.335 ","End":"00:48.327","Text":"so I have 2 plus 9 and,"},{"Start":"00:48.327 ","End":"00:50.570","Text":"again, I get 11."},{"Start":"00:50.570 ","End":"00:53.825","Text":"Now, it\u0027s not a coincidence,"},{"Start":"00:53.825 ","End":"00:56.690","Text":"but it\u0027s not obvious either,"},{"Start":"00:56.690 ","End":"00:58.700","Text":"because let me give you another example."},{"Start":"00:58.700 ","End":"01:01.370","Text":"Suppose I have 7"},{"Start":"01:01.370 ","End":"01:09.320","Text":"minus 3 plus 1."},{"Start":"01:09.320 ","End":"01:12.350","Text":"Now, someone might say,"},{"Start":"01:12.350 ","End":"01:18.010","Text":"let\u0027s take it as 7 minus 3 is 4,"},{"Start":"01:18.010 ","End":"01:23.340","Text":"and then add 1 and we get 5."},{"Start":"01:23.340 ","End":"01:25.180","Text":"This is a plus."},{"Start":"01:25.180 ","End":"01:27.920","Text":"But someone else might say like before,"},{"Start":"01:27.920 ","End":"01:29.950","Text":"let\u0027s do the 3 plus 1 first,"},{"Start":"01:29.950 ","End":"01:34.200","Text":"so you might say that it\u0027s equal to 3 plus 1 is 4,"},{"Start":"01:34.200 ","End":"01:39.150","Text":"so we have 7 minus 4 and get the answer is 3."},{"Start":"01:39.150 ","End":"01:41.295","Text":"This is different."},{"Start":"01:41.295 ","End":"01:43.909","Text":"The question is which is correct?"},{"Start":"01:43.909 ","End":"01:48.350","Text":"Well, turns out that the first 1 is correct and the second 1 is wrong"},{"Start":"01:48.350 ","End":"01:52.925","Text":"because there\u0027s a convention in mathematics that we work from left to right."},{"Start":"01:52.925 ","End":"01:56.000","Text":"Now sometimes it makes no difference, but in general,"},{"Start":"01:56.000 ","End":"02:00.470","Text":"we do each operation when it\u0027s pluses and minuses as it comes from the left to the right."},{"Start":"02:00.470 ","End":"02:03.560","Text":"If you don\u0027t do that, you might get the wrong answer."},{"Start":"02:03.560 ","End":"02:05.540","Text":"With addition, this will never happen when you have"},{"Start":"02:05.540 ","End":"02:08.330","Text":"addition and subtraction mix, it could happen."},{"Start":"02:08.330 ","End":"02:11.555","Text":"Now let me give you another example."},{"Start":"02:11.555 ","End":"02:20.725","Text":"Suppose I say what is 2 plus 3 times 4?"},{"Start":"02:20.725 ","End":"02:22.250","Text":"What does that equal?"},{"Start":"02:22.250 ","End":"02:26.385","Text":"Let me just show you why this is ambiguous possibly."},{"Start":"02:26.385 ","End":"02:28.115","Text":"Once again, someone could say,"},{"Start":"02:28.115 ","End":"02:30.425","Text":"let\u0027s do this 1 from left to right,"},{"Start":"02:30.425 ","End":"02:32.960","Text":"2 plus 3 is 5,"},{"Start":"02:32.960 ","End":"02:35.885","Text":"5 times 4 is 20."},{"Start":"02:35.885 ","End":"02:40.775","Text":"In fact, that\u0027s what my simple calculator gives as the answer but it\u0027s wrong."},{"Start":"02:40.775 ","End":"02:42.530","Text":"Because in mathematics,"},{"Start":"02:42.530 ","End":"02:45.650","Text":"there is a convention that just as here,"},{"Start":"02:45.650 ","End":"02:50.345","Text":"we had a convention of left to right when we have a mixture of plus and times,"},{"Start":"02:50.345 ","End":"02:52.730","Text":"we do the multiplication first."},{"Start":"02:52.730 ","End":"02:56.975","Text":"This is wrong and the correct answer is that it\u0027s equal to,"},{"Start":"02:56.975 ","End":"03:01.149","Text":"we first of all do the 3 times 4 and get 12."},{"Start":"03:01.149 ","End":"03:06.725","Text":"Then 2 plus 12 is 14 and that is the right answer."},{"Start":"03:06.725 ","End":"03:09.980","Text":"But what if I wanted the answer to come out 20?"},{"Start":"03:09.980 ","End":"03:11.930","Text":"I wanted to do the addition first."},{"Start":"03:11.930 ","End":"03:17.825","Text":"Then the way to do that is with brackets or parentheses as they\u0027re also called,"},{"Start":"03:17.825 ","End":"03:19.175","Text":"and you could say,"},{"Start":"03:19.175 ","End":"03:20.885","Text":"2 plus 3,"},{"Start":"03:20.885 ","End":"03:24.170","Text":"put that in parentheses and then multiply by"},{"Start":"03:24.170 ","End":"03:29.014","Text":"4 and the rule is that parentheses are done first."},{"Start":"03:29.014 ","End":"03:33.225","Text":"Then it really is 5 times 4,"},{"Start":"03:33.225 ","End":"03:36.360","Text":"and then it\u0027s equal to 20 and this time it\u0027s right."},{"Start":"03:36.360 ","End":"03:39.210","Text":"But without the parenthesis, it\u0027s 14."},{"Start":"03:39.210 ","End":"03:41.885","Text":"All these rules, and there are more,"},{"Start":"03:41.885 ","End":"03:49.160","Text":"can be summarized and I\u0027m going to give you a list of rules. Here we are."},{"Start":"03:49.160 ","End":"03:52.370","Text":"I have them printed out, the rules."},{"Start":"03:52.370 ","End":"03:55.580","Text":"As you can see, we already talked about addition and subtraction."},{"Start":"03:55.580 ","End":"03:57.410","Text":"We do it from left to right."},{"Start":"03:57.410 ","End":"03:59.210","Text":"Before addition and subtraction,"},{"Start":"03:59.210 ","End":"04:01.835","Text":"we have to do multiplication and division."},{"Start":"04:01.835 ","End":"04:04.010","Text":"Well, there\u0027s a whole hierarchy."},{"Start":"04:04.010 ","End":"04:05.995","Text":"Above multiplication and division,"},{"Start":"04:05.995 ","End":"04:09.110","Text":"there is exponents which means powers,"},{"Start":"04:09.110 ","End":"04:10.340","Text":"and square roots,"},{"Start":"04:10.340 ","End":"04:12.485","Text":"and cube roots and so on,"},{"Start":"04:12.485 ","End":"04:17.885","Text":"and those go before multiplication and also before addition."},{"Start":"04:17.885 ","End":"04:21.410","Text":"But first thing we do are brackets,"},{"Start":"04:21.410 ","End":"04:25.200","Text":"sometimes called parentheses, I prefer to call them brackets."},{"Start":"04:25.240 ","End":"04:28.535","Text":"There could be brackets inside brackets."},{"Start":"04:28.535 ","End":"04:31.705","Text":"Often we have round brackets,"},{"Start":"04:31.705 ","End":"04:36.890","Text":"and that might be part of an expression and around that we might have square brackets."},{"Start":"04:36.890 ","End":"04:38.810","Text":"If we need to put that in brackets,"},{"Start":"04:38.810 ","End":"04:42.740","Text":"we usually use curly brackets or curly braces."},{"Start":"04:42.740 ","End":"04:47.060","Text":"But we do them from the inside out and we\u0027ll see this in the examples,"},{"Start":"04:47.060 ","End":"04:49.200","Text":"so not to worry now."}],"ID":4861},{"Watched":false,"Name":"PEDMAS - Part 2","Duration":"9m 59s","ChapterTopicVideoID":4867,"CourseChapterTopicPlaylistID":3855,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.029","Text":"Let\u0027s start with the first example."},{"Start":"00:03.029 ","End":"00:06.584","Text":"My first exercise will be as follows."},{"Start":"00:06.584 ","End":"00:12.340","Text":"The square root of 196 plus"},{"Start":"00:12.340 ","End":"00:20.340","Text":"5 times 2^2 minus 20 divided by 2."},{"Start":"00:20.740 ","End":"00:25.305","Text":"First, we do parentheses or brackets,"},{"Start":"00:25.305 ","End":"00:27.600","Text":"but I don\u0027t see that we have any here."},{"Start":"00:27.600 ","End":"00:29.790","Text":"Let\u0027s try number 2 exponents,"},{"Start":"00:29.790 ","End":"00:31.410","Text":"which is powers and roots,"},{"Start":"00:31.410 ","End":"00:32.925","Text":"and yes, we have."},{"Start":"00:32.925 ","End":"00:38.295","Text":"Number 2, here we have an exponent,"},{"Start":"00:38.295 ","End":"00:41.175","Text":"and here we have a root."},{"Start":"00:41.175 ","End":"00:48.470","Text":"So we do those first and that will give us the square root of a 196 is 14."},{"Start":"00:48.470 ","End":"00:51.865","Text":"You either do it on the calculator or you just do it by guesswork,"},{"Start":"00:51.865 ","End":"00:53.400","Text":"10 times 10 is 100,"},{"Start":"00:53.400 ","End":"00:55.605","Text":"11 times 11 is 121,"},{"Start":"00:55.605 ","End":"01:01.970","Text":"and so on until you hit 14 times 14 and you get 196."},{"Start":"01:01.970 ","End":"01:05.495","Text":"Plus, now everything else is copied as this,"},{"Start":"01:05.495 ","End":"01:11.110","Text":"the 5 [inaudible] 2^2 is 2 times 2 is 4,"},{"Start":"01:11.110 ","End":"01:15.270","Text":"and everything else, as is."},{"Start":"01:15.270 ","End":"01:19.259","Text":"Next, is multiplication and division,"},{"Start":"01:19.259 ","End":"01:21.225","Text":"and we do have those."},{"Start":"01:21.225 ","End":"01:26.115","Text":"We have 5 times 4 and we have 20/2."},{"Start":"01:26.115 ","End":"01:32.145","Text":"We get 14 plus 5 times 4 is 20,"},{"Start":"01:32.145 ","End":"01:36.015","Text":"20 divided by 2 is 10."},{"Start":"01:36.015 ","End":"01:41.600","Text":"Then we do addition and subtraction left-to-right,"},{"Start":"01:41.600 ","End":"01:43.460","Text":"where we seen that before."},{"Start":"01:43.460 ","End":"01:46.340","Text":"So 14 plus 20,"},{"Start":"01:46.340 ","End":"01:49.775","Text":"the subtotal is 34,"},{"Start":"01:49.775 ","End":"01:56.015","Text":"less 10 is 24 and that is the answer."},{"Start":"01:56.015 ","End":"01:58.520","Text":"On to the next exercise,"},{"Start":"01:58.520 ","End":"02:03.885","Text":"this time it\u0027s minus 2^4"},{"Start":"02:03.885 ","End":"02:12.405","Text":"divided by 2 minus 10 times minus 2^3."},{"Start":"02:12.405 ","End":"02:16.910","Text":"As before, we have exponents first,"},{"Start":"02:16.910 ","End":"02:19.265","Text":"you might say, what about parentheses?"},{"Start":"02:19.265 ","End":"02:23.810","Text":"Well, parentheses is around the single number just to sort of protect it so it\u0027s clearer"},{"Start":"02:23.810 ","End":"02:28.925","Text":"that\u0027s not really parentheses around the single number negative 2."},{"Start":"02:28.925 ","End":"02:32.510","Text":"Exponents first, we have a number to the power of"},{"Start":"02:32.510 ","End":"02:35.840","Text":"4 and we have that same number to the power of 3,"},{"Start":"02:35.840 ","End":"02:38.135","Text":"no roots, but this is what we have."},{"Start":"02:38.135 ","End":"02:40.055","Text":"That\u0027s what we have to do first."},{"Start":"02:40.055 ","End":"02:45.185","Text":"We can do some side exercises like minus 2^4."},{"Start":"02:45.185 ","End":"02:46.580","Text":"I can write it as minus 2,"},{"Start":"02:46.580 ","End":"02:48.290","Text":"minus 2, minus 2, minus 2,"},{"Start":"02:48.290 ","End":"02:55.625","Text":"or we could remember the rule that a minus to an even power is a plus, so it\u0027s plus."},{"Start":"02:55.625 ","End":"03:00.250","Text":"The same thing as 2^4, which is 16."},{"Start":"03:00.250 ","End":"03:02.070","Text":"On the other hand,"},{"Start":"03:02.070 ","End":"03:07.850","Text":"minus 2^3, because 3 is an odd number, is a minus."},{"Start":"03:07.850 ","End":"03:11.300","Text":"Then I just take 2^3, which is 8."},{"Start":"03:11.300 ","End":"03:13.840","Text":"So the answer is minus 8."},{"Start":"03:13.840 ","End":"03:17.600","Text":"Even power of a negative is plus,"},{"Start":"03:17.600 ","End":"03:20.150","Text":"odd power of a negative is minus."},{"Start":"03:20.150 ","End":"03:30.313","Text":"Let\u0027s put those back in here and we have 16 divided by 2 minus 10 times,"},{"Start":"03:30.313 ","End":"03:33.410","Text":"this is like a times here onto the bracket,"},{"Start":"03:33.410 ","End":"03:35.840","Text":"10 times minus 8,"},{"Start":"03:35.840 ","End":"03:37.550","Text":"but we don\u0027t write times minus,"},{"Start":"03:37.550 ","End":"03:40.100","Text":"so we protect the minus 8 with the brackets,"},{"Start":"03:40.100 ","End":"03:42.660","Text":"but it\u0027s not really a brackets."},{"Start":"03:43.490 ","End":"03:49.070","Text":"Now, we have the case that we have some multiplications and divisions."},{"Start":"03:49.070 ","End":"03:53.585","Text":"Here\u0027s a division and here\u0027s a multiplication."},{"Start":"03:53.585 ","End":"03:58.700","Text":"So this equals 16 divided by 2 is 8,"},{"Start":"03:58.700 ","End":"04:00.420","Text":"10 times minus 8,"},{"Start":"04:00.420 ","End":"04:03.470","Text":"it\u0027s a plus times a minus, so it\u0027s minus."},{"Start":"04:03.470 ","End":"04:05.570","Text":"So this is minus 80."},{"Start":"04:05.570 ","End":"04:10.950","Text":"What I have is 8 minus minus 80,"},{"Start":"04:10.950 ","End":"04:14.170","Text":"but we put this in brackets."},{"Start":"04:14.840 ","End":"04:17.220","Text":"This is going to equal."},{"Start":"04:17.220 ","End":"04:21.229","Text":"Now remember that when we subtract a negative,"},{"Start":"04:21.229 ","End":"04:23.720","Text":"it\u0027s the same thing as an addition."},{"Start":"04:23.720 ","End":"04:26.075","Text":"It\u0027s really 8 plus 80,"},{"Start":"04:26.075 ","End":"04:29.853","Text":"and so the final answer is 88."},{"Start":"04:29.853 ","End":"04:35.410","Text":"That\u0027s this exercise and onto the next."},{"Start":"04:35.930 ","End":"04:39.345","Text":"The next exercise is as follows."},{"Start":"04:39.345 ","End":"04:45.180","Text":"Minus 3^2 minus 4 squared brackets,"},{"Start":"04:45.180 ","End":"04:48.705","Text":"5 plus 4 times round brackets,"},{"Start":"04:48.705 ","End":"04:52.290","Text":"7 minus 2,"},{"Start":"04:52.290 ","End":"04:58.600","Text":"and then plus the square root of 900."},{"Start":"04:59.300 ","End":"05:03.705","Text":"This time, we do have parentheses."},{"Start":"05:03.705 ","End":"05:06.215","Text":"I\u0027ll use this color for that."},{"Start":"05:06.215 ","End":"05:08.750","Text":"We have here and here,"},{"Start":"05:08.750 ","End":"05:12.365","Text":"and we also have here and here."},{"Start":"05:12.365 ","End":"05:17.370","Text":"We work them from the inner to the outer so we first do the round ones."},{"Start":"05:17.370 ","End":"05:21.530","Text":"Quite typically, usually the round ones are the inner most ones, and then around that,"},{"Start":"05:21.530 ","End":"05:23.785","Text":"the square brackets around that, the curly ones,"},{"Start":"05:23.785 ","End":"05:25.715","Text":"and if you have another set, I don\u0027t know what,"},{"Start":"05:25.715 ","End":"05:28.055","Text":"we use one of these."},{"Start":"05:28.055 ","End":"05:32.165","Text":"The round brackets, 7 minus 2 is 5."},{"Start":"05:32.165 ","End":"05:37.340","Text":"I just did a copy paste and I\u0027m going to replace the 7 minus 2 by 5."},{"Start":"05:37.340 ","End":"05:41.455","Text":"That\u0027s the round brackets, now we have to figure out these square brackets."},{"Start":"05:41.455 ","End":"05:43.370","Text":"But within the square brackets,"},{"Start":"05:43.370 ","End":"05:47.765","Text":"we have an addition and a multiplication so we also have to do it in the right order."},{"Start":"05:47.765 ","End":"05:50.015","Text":"We\u0027ve already seen this kind of exercise."},{"Start":"05:50.015 ","End":"05:54.440","Text":"What we do is the 4 times 5 first, and that\u0027s 20."},{"Start":"05:54.440 ","End":"06:03.900","Text":"So what I get for the square brackets is altogether 5 plus 20, which is 25."},{"Start":"06:03.900 ","End":"06:05.650","Text":"Let me just copy this."},{"Start":"06:05.650 ","End":"06:08.030","Text":"So here, we have replaced this by 25,"},{"Start":"06:08.030 ","End":"06:16.120","Text":"I forgot the multiplication here."},{"Start":"06:16.120 ","End":"06:23.123","Text":"There, that\u0027s the dot. The next one is the exponents and the roots."},{"Start":"06:23.123 ","End":"06:26.075","Text":"We have an exponent here and we have a root here."},{"Start":"06:26.075 ","End":"06:33.520","Text":"Note that the exponent just applies to the 3 because the minus 3 is not in brackets."},{"Start":"06:34.670 ","End":"06:38.870","Text":"I\u0027ll continue over here and that should really be"},{"Start":"06:38.870 ","End":"06:42.110","Text":"writing equals after each one, equals, equals."},{"Start":"06:42.110 ","End":"06:50.075","Text":"Now this equals minus 3^2 is 9 minus 4 times 25,"},{"Start":"06:50.075 ","End":"06:54.110","Text":"and the square root of 900 is 30."},{"Start":"06:54.110 ","End":"06:57.710","Text":"Next is the multiplication."},{"Start":"06:57.710 ","End":"07:00.940","Text":"I have a multiplication here."},{"Start":"07:00.940 ","End":"07:07.740","Text":"What we have is minus 9 minus 100 plus 30,"},{"Start":"07:07.740 ","End":"07:11.790","Text":"and then we have additions and subtractions left-to-right."},{"Start":"07:12.560 ","End":"07:16.986","Text":"So minus 9 minus a 100,"},{"Start":"07:16.986 ","End":"07:22.790","Text":"that\u0027s two negatives so they strengthen each other."},{"Start":"07:22.790 ","End":"07:25.385","Text":"So we do an addition,"},{"Start":"07:25.385 ","End":"07:26.870","Text":"so it\u0027s 9 plus 100,"},{"Start":"07:26.870 ","End":"07:29.420","Text":"it\u0027s 109, but we take the sign from the minus,"},{"Start":"07:29.420 ","End":"07:32.285","Text":"so it\u0027s minus 109 plus 30."},{"Start":"07:32.285 ","End":"07:33.690","Text":"Now we have a minus and a plus,"},{"Start":"07:33.690 ","End":"07:35.495","Text":"they pull in opposite directions."},{"Start":"07:35.495 ","End":"07:38.075","Text":"I take the difference which is 79,"},{"Start":"07:38.075 ","End":"07:39.890","Text":"but the winner is the negative,"},{"Start":"07:39.890 ","End":"07:42.160","Text":"so it\u0027s minus 79,"},{"Start":"07:42.160 ","End":"07:43.980","Text":"and that\u0027s the answer."},{"Start":"07:43.980 ","End":"07:46.740","Text":"We\u0027re going to do one last exercise,"},{"Start":"07:46.740 ","End":"07:48.270","Text":"one last example,"},{"Start":"07:48.270 ","End":"07:51.265","Text":"and then we\u0027ll be finished with this clip."},{"Start":"07:51.265 ","End":"07:53.420","Text":"Here is the exercise,"},{"Start":"07:53.420 ","End":"07:55.015","Text":"I already wrote it out."},{"Start":"07:55.015 ","End":"07:57.465","Text":"You can see what it is."},{"Start":"07:57.465 ","End":"07:59.880","Text":"We go to parentheses first,"},{"Start":"07:59.880 ","End":"08:03.530","Text":"so this is one set and this is another set."},{"Start":"08:03.530 ","End":"08:07.440","Text":"What we get is the first one,"},{"Start":"08:07.440 ","End":"08:08.550","Text":"minus 4 plus 2,"},{"Start":"08:08.550 ","End":"08:09.740","Text":"you know how to do these already."},{"Start":"08:09.740 ","End":"08:13.190","Text":"This is minus 2 and the second one,"},{"Start":"08:13.190 ","End":"08:17.540","Text":"we have to do also according to the order of things."},{"Start":"08:17.540 ","End":"08:22.975","Text":"I\u0027ll do this at the side minus 3^2 plus 10."},{"Start":"08:22.975 ","End":"08:27.170","Text":"First of all, we have an exponent and then the addition,"},{"Start":"08:27.170 ","End":"08:30.245","Text":"so it\u0027s minus 9 plus 10,"},{"Start":"08:30.245 ","End":"08:33.095","Text":"minus 9 plus 10 is 1."},{"Start":"08:33.095 ","End":"08:35.480","Text":"Opposite signs, the 10 beats the 9,"},{"Start":"08:35.480 ","End":"08:38.460","Text":"so it\u0027s a plus 10 minus minus 1."},{"Start":"08:38.750 ","End":"08:45.175","Text":"This is just 1, and now we have the square root of 64"},{"Start":"08:45.175 ","End":"08:51.990","Text":"divided by minus 2 minus 4^2 times 1."},{"Start":"08:51.990 ","End":"08:54.100","Text":"Next, we go to the exponents,"},{"Start":"08:54.100 ","End":"08:58.600","Text":"which means roots and exponents. That\u0027s these."},{"Start":"08:58.600 ","End":"09:03.260","Text":"We have the square root of 64 is 8 divided by,"},{"Start":"09:03.260 ","End":"09:04.550","Text":"this should be in brackets,"},{"Start":"09:04.550 ","End":"09:06.964","Text":"you don\u0027t put divide and minus usually,"},{"Start":"09:06.964 ","End":"09:11.920","Text":"8 divided by minus 2 and 4^2 is 16."},{"Start":"09:11.920 ","End":"09:13.680","Text":"Squared just applies to the 4,"},{"Start":"09:13.680 ","End":"09:15.090","Text":"the minus is separate."},{"Start":"09:15.090 ","End":"09:17.545","Text":"16 times 1."},{"Start":"09:17.545 ","End":"09:20.300","Text":"Next, multiplication."},{"Start":"09:20.300 ","End":"09:21.770","Text":"We have one of those here,"},{"Start":"09:21.770 ","End":"09:24.965","Text":"and division, that\u0027s the division."},{"Start":"09:24.965 ","End":"09:30.470","Text":"What we get is 8 divided by negative 2,"},{"Start":"09:30.470 ","End":"09:32.495","Text":"a positive divided by a negative is a negative,"},{"Start":"09:32.495 ","End":"09:35.900","Text":"so this is minus 4 and 16 times 1,"},{"Start":"09:35.900 ","End":"09:38.770","Text":"both are pluses, it\u0027s just 16."},{"Start":"09:38.770 ","End":"09:41.870","Text":"We have minus 4 minus 16."},{"Start":"09:41.870 ","End":"09:45.530","Text":"Now, we have two negatives which strengthen each other."},{"Start":"09:45.530 ","End":"09:47.600","Text":"We do an addition in our heads,"},{"Start":"09:47.600 ","End":"09:48.770","Text":"4 and 16 is 20,"},{"Start":"09:48.770 ","End":"09:51.685","Text":"but it stays as minus 20."},{"Start":"09:51.685 ","End":"09:55.245","Text":"I forgot all the equals all along, silly me."},{"Start":"09:55.245 ","End":"09:59.230","Text":"Minus 20 is the answer, and now we\u0027re done."}],"ID":4862},{"Watched":false,"Name":"Exercise 1","Duration":"5m 28s","ChapterTopicVideoID":4855,"CourseChapterTopicPlaylistID":3855,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.500","Text":"In this exercise, we have to calculate the following expressions."},{"Start":"00:04.500 ","End":"00:12.465","Text":"I\u0027d just like to remind you that we have a plus 6 and we have a minus 6,"},{"Start":"00:12.465 ","End":"00:15.540","Text":"but if we just write 6 without anything at all,"},{"Start":"00:15.540 ","End":"00:17.895","Text":"then it means plus 6."},{"Start":"00:17.895 ","End":"00:23.265","Text":"Let\u0027s get started. I\u0027ll copy the exercises below."},{"Start":"00:23.265 ","End":"00:25.290","Text":"Well, I copied the first 4."},{"Start":"00:25.290 ","End":"00:28.410","Text":"Let\u0027s see, 6 plus 1."},{"Start":"00:28.410 ","End":"00:30.600","Text":"We have 2 numbers to add."},{"Start":"00:30.600 ","End":"00:33.600","Text":"The first is a positive 6,"},{"Start":"00:33.600 ","End":"00:34.950","Text":"a plus 6,"},{"Start":"00:34.950 ","End":"00:37.650","Text":"and the second is a plus 1."},{"Start":"00:37.650 ","End":"00:39.450","Text":"I\u0027m adding a plus 6,"},{"Start":"00:39.450 ","End":"00:40.950","Text":"which I put in brackets,"},{"Start":"00:40.950 ","End":"00:42.885","Text":"and a plus 1."},{"Start":"00:42.885 ","End":"00:47.960","Text":"When we have 2 numbers that have the same sign and we add them,"},{"Start":"00:47.960 ","End":"00:50.570","Text":"we add the numbers without the sign,"},{"Start":"00:50.570 ","End":"00:56.165","Text":"that is 6 and 1 is 7 and we take the common sign,"},{"Start":"00:56.165 ","End":"00:58.115","Text":"which is a plus."},{"Start":"00:58.115 ","End":"01:00.440","Text":"We just say 7."},{"Start":"01:00.440 ","End":"01:03.475","Text":"Also notice, I use the European style of 7."},{"Start":"01:03.475 ","End":"01:07.535","Text":"In the next 1, I can also express it as an addition problem,"},{"Start":"01:07.535 ","End":"01:13.610","Text":"plus 6, but this time I\u0027m adding a negative 1, minus 1."},{"Start":"01:13.610 ","End":"01:15.844","Text":"When they have opposite signs,"},{"Start":"01:15.844 ","End":"01:18.785","Text":"I do a subtraction exercise in my head."},{"Start":"01:18.785 ","End":"01:20.705","Text":"The bigger minus the smaller,"},{"Start":"01:20.705 ","End":"01:24.480","Text":"so it\u0027s 5,"},{"Start":"01:24.480 ","End":"01:28.320","Text":"and I decide the sign according to the bigger 1."},{"Start":"01:28.320 ","End":"01:31.725","Text":"6 is bigger than 1 so I take the sign from the 6,"},{"Start":"01:31.725 ","End":"01:33.833","Text":"so it\u0027s a plus 5,"},{"Start":"01:33.833 ","End":"01:35.965","Text":"and, of course, I just write it as 5."},{"Start":"01:35.965 ","End":"01:38.990","Text":"This exercise here is also an addition exercise."},{"Start":"01:38.990 ","End":"01:45.755","Text":"This time I have a negative 6 and I\u0027m adding to it a negative 1."},{"Start":"01:45.755 ","End":"01:50.240","Text":"This is another case where the signs are similar like in the first exercise."},{"Start":"01:50.240 ","End":"01:53.405","Text":"Similar signs means we first do an addition,"},{"Start":"01:53.405 ","End":"01:55.010","Text":"but ignore the sign,"},{"Start":"01:55.010 ","End":"01:59.809","Text":"so 6 plus 1 is 7."},{"Start":"01:59.809 ","End":"02:03.390","Text":"I take the common sign, they\u0027re both minus,"},{"Start":"02:03.390 ","End":"02:04.920","Text":"so it\u0027s a minus 7,"},{"Start":"02:04.920 ","End":"02:07.895","Text":"and this time the minus stays."},{"Start":"02:07.895 ","End":"02:10.040","Text":"Not like before we could throw out the plus,"},{"Start":"02:10.040 ","End":"02:11.825","Text":"minus has to stay."},{"Start":"02:11.825 ","End":"02:13.760","Text":"Last exercise, again,"},{"Start":"02:13.760 ","End":"02:15.650","Text":"I write it as an addition problem."},{"Start":"02:15.650 ","End":"02:20.300","Text":"This time I have a negative 6 and a positive 1."},{"Start":"02:20.300 ","End":"02:22.940","Text":"Because they have opposite signs,"},{"Start":"02:22.940 ","End":"02:24.650","Text":"I take the difference between them."},{"Start":"02:24.650 ","End":"02:26.230","Text":"The bigger less the smaller,"},{"Start":"02:26.230 ","End":"02:30.690","Text":"5, and the sign I take from the larger 1."},{"Start":"02:30.690 ","End":"02:32.250","Text":"Since 6 beats 1,"},{"Start":"02:32.250 ","End":"02:34.035","Text":"I take the sign from the 6,"},{"Start":"02:34.035 ","End":"02:37.380","Text":"so it\u0027s minus 5 is the answer."},{"Start":"02:37.380 ","End":"02:41.380","Text":"I\u0027ll just highlight the answers."},{"Start":"02:42.760 ","End":"02:47.075","Text":"I\u0027d like to do the last exercise next."},{"Start":"02:47.075 ","End":"02:48.739","Text":"When you have more than 2 numbers,"},{"Start":"02:48.739 ","End":"02:51.770","Text":"there is a system that we use where we"},{"Start":"02:51.770 ","End":"02:55.550","Text":"separate the positive numbers from the negative numbers."},{"Start":"02:55.550 ","End":"02:59.780","Text":"This 1 would be positive because it has a plus in front of it,"},{"Start":"02:59.780 ","End":"03:03.505","Text":"5 has nothing in front of it so that\u0027s also positive."},{"Start":"03:03.505 ","End":"03:06.575","Text":"That\u0027s a negative 8 and a negative 12."},{"Start":"03:06.575 ","End":"03:10.250","Text":"What we do is take the pluses separately,"},{"Start":"03:10.250 ","End":"03:13.784","Text":"5 and 17, plus 22."},{"Start":"03:13.784 ","End":"03:17.040","Text":"As for the negatives, ignoring the sign,"},{"Start":"03:17.040 ","End":"03:22.975","Text":"we add 8 and 12 and we get 20, so minus 20."},{"Start":"03:22.975 ","End":"03:27.410","Text":"We have a plus 22 and we have a minus 20,"},{"Start":"03:27.410 ","End":"03:28.985","Text":"and we want to add those."},{"Start":"03:28.985 ","End":"03:30.440","Text":"We have a mixed sign here,"},{"Start":"03:30.440 ","End":"03:35.150","Text":"we have a plus and a minus so we do a subtraction of the larger less the smaller,"},{"Start":"03:35.150 ","End":"03:38.930","Text":"so 22 minus 20 is 2."},{"Start":"03:38.930 ","End":"03:42.425","Text":"The sign I take from the 1 which is the winner,"},{"Start":"03:42.425 ","End":"03:43.920","Text":"22 beats 20,"},{"Start":"03:43.920 ","End":"03:46.970","Text":"so I take the sign from the 22, which is plus."},{"Start":"03:46.970 ","End":"03:48.455","Text":"This is a plus sign,"},{"Start":"03:48.455 ","End":"03:50.120","Text":"we just ignore it,"},{"Start":"03:50.120 ","End":"03:53.320","Text":"and I\u0027ll highlight the answer 2."},{"Start":"03:53.320 ","End":"03:57.530","Text":"Now let me return to E, same method."},{"Start":"03:57.530 ","End":"03:59.450","Text":"I\u0027ll take the pluses,"},{"Start":"03:59.450 ","End":"04:02.760","Text":"then I take the minuses,"},{"Start":"04:03.830 ","End":"04:07.755","Text":"5 and 13 is 18,"},{"Start":"04:07.755 ","End":"04:11.590","Text":"so we have minus 18 for the minus camp."},{"Start":"04:11.590 ","End":"04:13.120","Text":"For the plus camp, well,"},{"Start":"04:13.120 ","End":"04:15.700","Text":"we just have this 1 so it\u0027s a plus 9."},{"Start":"04:15.700 ","End":"04:19.590","Text":"Now I add these 2,"},{"Start":"04:19.590 ","End":"04:21.675","Text":"a minus and a plus."},{"Start":"04:21.675 ","End":"04:23.490","Text":"We do a subtraction,"},{"Start":"04:23.490 ","End":"04:26.355","Text":"18 subtract 9,"},{"Start":"04:26.355 ","End":"04:29.265","Text":"and that leaves me with 9."},{"Start":"04:29.265 ","End":"04:32.470","Text":"But the sign I take from the winner,"},{"Start":"04:32.470 ","End":"04:36.524","Text":"18 beats 9 so I take the sign from the 18."},{"Start":"04:36.524 ","End":"04:40.210","Text":"The answer is minus 9."},{"Start":"04:40.600 ","End":"04:44.000","Text":"Last 1. Let\u0027s take the pluses."},{"Start":"04:44.000 ","End":"04:46.260","Text":"If there\u0027s nothing in front, it means it\u0027s a plus."},{"Start":"04:46.260 ","End":"04:48.665","Text":"If it\u0027s got a plus in front, it\u0027s a plus."},{"Start":"04:48.665 ","End":"04:50.120","Text":"I\u0027ve got another plus here,"},{"Start":"04:50.120 ","End":"04:51.770","Text":"got 3 pluses,"},{"Start":"04:51.770 ","End":"04:54.815","Text":"and I only have 1 minus."},{"Start":"04:54.815 ","End":"04:59.660","Text":"5 and 7 and 1 is plus 13."},{"Start":"04:59.660 ","End":"05:03.650","Text":"Negatives, I just have the minus 23."},{"Start":"05:03.650 ","End":"05:05.750","Text":"Put them in brackets,"},{"Start":"05:05.750 ","End":"05:08.360","Text":"add them together, mixed sign."},{"Start":"05:08.360 ","End":"05:11.965","Text":"We do a subtraction of the bigger minus the smaller,"},{"Start":"05:11.965 ","End":"05:16.410","Text":"so 23 less 13 is 10."},{"Start":"05:16.410 ","End":"05:18.890","Text":"Then we look at which is the larger,"},{"Start":"05:18.890 ","End":"05:22.280","Text":"the larger is the 23 so the sign comes from the 23,"},{"Start":"05:22.280 ","End":"05:24.455","Text":"so we put a minus in front."},{"Start":"05:24.455 ","End":"05:29.280","Text":"We\u0027ve solved all of them and that\u0027s it for this exercise."}],"ID":4863},{"Watched":false,"Name":"Exercise 2","Duration":"3m 43s","ChapterTopicVideoID":4856,"CourseChapterTopicPlaylistID":3855,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.560","Text":"In this exercise, we have to calculate the following expressions."},{"Start":"00:04.560 ","End":"00:09.970","Text":"It\u0027ll be easier if I recopy them to the solution."},{"Start":"00:10.340 ","End":"00:12.920","Text":"There we are."},{"Start":"00:12.920 ","End":"00:15.150","Text":"Let\u0027s start."},{"Start":"00:15.150 ","End":"00:20.235","Text":"This is not 2.5. These dots are, of course, multiplications."},{"Start":"00:20.235 ","End":"00:22.920","Text":"The main thing is to just look at the signs."},{"Start":"00:22.920 ","End":"00:27.089","Text":"What we have here is a plus times a plus."},{"Start":"00:27.089 ","End":"00:32.150","Text":"So the answer will be plus and then 2 times 5 is 10,"},{"Start":"00:32.150 ","End":"00:35.330","Text":"so the answer is plus 10,"},{"Start":"00:35.330 ","End":"00:39.995","Text":"which we write as simply 10."},{"Start":"00:39.995 ","End":"00:44.255","Text":"Here, a number b,"},{"Start":"00:44.255 ","End":"00:48.020","Text":"we have a minus times a minus."},{"Start":"00:48.020 ","End":"00:52.925","Text":"Again, like signs multiplied give us a plus."},{"Start":"00:52.925 ","End":"00:55.310","Text":"So a minus and a minus is also a plus."},{"Start":"00:55.310 ","End":"00:58.040","Text":"Once again, we get plus 10,"},{"Start":"00:58.040 ","End":"01:03.565","Text":"which we can write as simply 10."},{"Start":"01:03.565 ","End":"01:07.464","Text":"Now here we have a minus 2 times a 5."},{"Start":"01:07.464 ","End":"01:09.650","Text":"A 5 without anything is a plus."},{"Start":"01:09.650 ","End":"01:11.600","Text":"So it\u0027s a minus times a plus."},{"Start":"01:11.600 ","End":"01:13.940","Text":"I\u0027m going to stop writing the brackets all the time."},{"Start":"01:13.940 ","End":"01:17.540","Text":"Minus times plus is minus,"},{"Start":"01:17.540 ","End":"01:20.111","Text":"because opposite signs give minus,"},{"Start":"01:20.111 ","End":"01:22.220","Text":"and 2 times 5 is 10,"},{"Start":"01:22.220 ","End":"01:25.830","Text":"so we get minus 10 this time."},{"Start":"01:26.630 ","End":"01:30.315","Text":"Also here, 2 is plus,"},{"Start":"01:30.315 ","End":"01:33.100","Text":"minus 5 is minus,"},{"Start":"01:33.100 ","End":"01:38.130","Text":"plus times minus is minus."},{"Start":"01:38.140 ","End":"01:41.240","Text":"Remember, plus times plus is plus,"},{"Start":"01:41.240 ","End":"01:42.800","Text":"minus times minus is plus."},{"Start":"01:42.800 ","End":"01:44.285","Text":"Then one of each,"},{"Start":"01:44.285 ","End":"01:46.708","Text":"plus minus or minus plus or minus."},{"Start":"01:46.708 ","End":"01:49.050","Text":"Minus 10."},{"Start":"01:50.630 ","End":"01:53.060","Text":"It\u0027s not really a trick question,"},{"Start":"01:53.060 ","End":"01:56.480","Text":"but remember there are numbers besides positive and negative."},{"Start":"01:56.480 ","End":"02:01.310","Text":"There\u0027s also 0, 0 times anything is 0 regardless of whether it\u0027s plus or minus."},{"Start":"02:01.310 ","End":"02:04.535","Text":"So I can write away say that the answer is 0."},{"Start":"02:04.535 ","End":"02:07.490","Text":"Of course, it wouldn\u0027t hurt if you said minus 0,"},{"Start":"02:07.490 ","End":"02:12.000","Text":"but minus 0 is same as 0."},{"Start":"02:12.610 ","End":"02:20.240","Text":"In this one, we have more than 2 terms that are multiplied."},{"Start":"02:20.240 ","End":"02:23.915","Text":"You can either do them 2 at a time."},{"Start":"02:23.915 ","End":"02:27.180","Text":"But the easiest thing is just to count the minuses."},{"Start":"02:27.180 ","End":"02:31.140","Text":"Here we have minus, minus, minus."},{"Start":"02:32.590 ","End":"02:35.626","Text":"Minuses cancel out in pairs,"},{"Start":"02:35.626 ","End":"02:37.263","Text":"because minus times minus is plus."},{"Start":"02:37.263 ","End":"02:39.755","Text":"So this altogether comes out minus."},{"Start":"02:39.755 ","End":"02:44.120","Text":"You could also use the rule that if there\u0027s an odd number of minuses, then it\u0027s minus."},{"Start":"02:44.120 ","End":"02:45.905","Text":"Otherwise, it\u0027s plus."},{"Start":"02:45.905 ","End":"02:51.125","Text":"But then we have to multiply the positive parts, the signless parts."},{"Start":"02:51.125 ","End":"02:54.260","Text":"2 times 3 times 4 is 6 times 4 is 24."},{"Start":"02:54.260 ","End":"02:57.630","Text":"So our answer is minus 24."},{"Start":"02:58.030 ","End":"03:04.250","Text":"Finally we have minus times plus times minus."},{"Start":"03:04.250 ","End":"03:06.350","Text":"Here we have 2 minuses,"},{"Start":"03:06.350 ","End":"03:07.865","Text":"which is an even number."},{"Start":"03:07.865 ","End":"03:12.260","Text":"Or we could just say minus times plus is minus and minus times minus is plus,"},{"Start":"03:12.260 ","End":"03:15.035","Text":"and then the event and even number means plus."},{"Start":"03:15.035 ","End":"03:18.440","Text":"Our answer here is plus 24."},{"Start":"03:18.440 ","End":"03:23.520","Text":"Of course, we drop the plus and we just say 24."},{"Start":"03:24.590 ","End":"03:29.900","Text":"Just to be nice, I\u0027ll highlight the answers."},{"Start":"03:29.900 ","End":"03:31.940","Text":"So this one is 10,"},{"Start":"03:31.940 ","End":"03:34.310","Text":"10, minus 10,"},{"Start":"03:34.310 ","End":"03:37.400","Text":"minus 10, 0,"},{"Start":"03:37.400 ","End":"03:39.635","Text":"minus 24, 24."},{"Start":"03:39.635 ","End":"03:42.660","Text":"We are done."}],"ID":4864},{"Watched":false,"Name":"Exercise 3","Duration":"2m 36s","ChapterTopicVideoID":4857,"CourseChapterTopicPlaylistID":3855,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.005","Text":"In this exercise, we have to calculate the following expressions."},{"Start":"00:04.005 ","End":"00:08.145","Text":"Notice that they\u0027re all the division of signed integers."},{"Start":"00:08.145 ","End":"00:12.840","Text":"For convenience, I just want to copy the exercise down here,"},{"Start":"00:12.840 ","End":"00:16.500","Text":"and then I don\u0027t have to mess with the original."},{"Start":"00:16.500 ","End":"00:19.860","Text":"The first one, the most important thing is the sign."},{"Start":"00:19.860 ","End":"00:21.690","Text":"If there\u0027s nothing, then it\u0027s a plus."},{"Start":"00:21.690 ","End":"00:26.940","Text":"Here I have a plus divided by a plus."},{"Start":"00:26.940 ","End":"00:30.930","Text":"A plus divided by a plus is a plus,"},{"Start":"00:30.930 ","End":"00:34.420","Text":"and so the answer is plus 8 over 4 is 2,"},{"Start":"00:34.420 ","End":"00:38.760","Text":"which we can just write as plain old 2."},{"Start":"00:38.960 ","End":"00:45.035","Text":"In this one, we have a minus divided by a minus,"},{"Start":"00:45.035 ","End":"00:46.715","Text":"which is also a plus."},{"Start":"00:46.715 ","End":"00:48.845","Text":"50 over 10 is 5."},{"Start":"00:48.845 ","End":"00:54.605","Text":"Now as usual, we drop the plus and just write plain old 5."},{"Start":"00:54.605 ","End":"00:57.830","Text":"The next one is a minus."},{"Start":"00:57.830 ","End":"01:00.380","Text":"I\u0027m going to stop writing the brackets, it\u0027s tedious,"},{"Start":"01:00.380 ","End":"01:04.680","Text":"minus divided by a plus, different signs,"},{"Start":"01:04.680 ","End":"01:06.015","Text":"so it\u0027s a minus,"},{"Start":"01:06.015 ","End":"01:09.735","Text":"15 over 3 is 5,"},{"Start":"01:09.735 ","End":"01:12.780","Text":"and so that\u0027s our answer."},{"Start":"01:12.780 ","End":"01:18.585","Text":"Here, we have a minus divided by a minus."},{"Start":"01:18.585 ","End":"01:22.625","Text":"Like signs, so the answer comes out to be a plus."},{"Start":"01:22.625 ","End":"01:26.075","Text":"25 over 5 is 5,"},{"Start":"01:26.075 ","End":"01:30.670","Text":"and as usual, plus 5 is just plain 5."},{"Start":"01:30.670 ","End":"01:33.945","Text":"Minus 30 divided by 3."},{"Start":"01:33.945 ","End":"01:38.760","Text":"A minus, divided by a plus, back to the brackets,"},{"Start":"01:38.760 ","End":"01:44.974","Text":"never mind, that is equal to minus because they\u0027re opposite signs."},{"Start":"01:44.974 ","End":"01:47.255","Text":"30 over 3 is 10,"},{"Start":"01:47.255 ","End":"01:51.815","Text":"and we leave the answer as minus 10."},{"Start":"01:51.815 ","End":"01:56.325","Text":"Next one, 0."},{"Start":"01:56.325 ","End":"01:58.470","Text":"0 divided by anything is 0,"},{"Start":"01:58.470 ","End":"02:04.065","Text":"anything that\u0027s not 0."},{"Start":"02:04.065 ","End":"02:08.925","Text":"This is equal to 0 and that\u0027s just it."},{"Start":"02:08.925 ","End":"02:11.910","Text":"That\u0027s the answer to that one."},{"Start":"02:11.910 ","End":"02:16.330","Text":"Finally, we have also a division,"},{"Start":"02:16.330 ","End":"02:19.930","Text":"but written instead of with the 2 dots with the dividing sign,"},{"Start":"02:19.930 ","End":"02:25.570","Text":"so we have a plus divided by a minus."},{"Start":"02:25.570 ","End":"02:27.690","Text":"Again, different signs,"},{"Start":"02:27.690 ","End":"02:28.980","Text":"so the answer is minus,"},{"Start":"02:28.980 ","End":"02:31.245","Text":"and 32 over 4 is 8,"},{"Start":"02:31.245 ","End":"02:36.580","Text":"so the answer is minus 8. That\u0027s it."}],"ID":4865},{"Watched":false,"Name":"Exercise 4","Duration":"4m 12s","ChapterTopicVideoID":4858,"CourseChapterTopicPlaylistID":3855,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.985","Text":"Here we have to calculate the following expressions and they all use exponents."},{"Start":"00:05.985 ","End":"00:09.765","Text":"Allow me first to copy the exercise here."},{"Start":"00:09.765 ","End":"00:18.520","Text":"I like to keep the original as is. Let\u0027s start."},{"Start":"00:18.740 ","End":"00:22.980","Text":"2^2, we don\u0027t say usually 2^2,"},{"Start":"00:22.980 ","End":"00:28.095","Text":"we pronounce it 2^2 Means that we have 2 factors,"},{"Start":"00:28.095 ","End":"00:29.400","Text":"each of them is 2,"},{"Start":"00:29.400 ","End":"00:33.315","Text":"2 times 2, which is 4."},{"Start":"00:33.315 ","End":"00:37.725","Text":"The 2 at the top means that there are 2 such factors."},{"Start":"00:37.725 ","End":"00:44.880","Text":"Similarly, in B, 2^4 means we have 4 factors,"},{"Start":"00:44.880 ","End":"00:49.920","Text":"i.e 2 times 2 times 2 times 2."},{"Start":"00:49.920 ","End":"00:51.270","Text":"I\u0027m counting 1, 2, 3,"},{"Start":"00:51.270 ","End":"00:54.485","Text":"4, and that\u0027s just 16."},{"Start":"00:54.485 ","End":"00:56.420","Text":"Up till now everything\u0027s positive."},{"Start":"00:56.420 ","End":"00:58.832","Text":"I haven\u0027t related to the fact that it\u0027s plus, plus."},{"Start":"00:58.832 ","End":"01:00.568","Text":"When everything\u0027s positive,"},{"Start":"01:00.568 ","End":"01:03.924","Text":"you don\u0027t even have to worry about the signs."},{"Start":"01:03.924 ","End":"01:06.210","Text":"Here, the same thing 2^6,"},{"Start":"01:06.210 ","End":"01:07.830","Text":"we just have to count higher."},{"Start":"01:07.830 ","End":"01:09.405","Text":"Let\u0027s count them 1,"},{"Start":"01:09.405 ","End":"01:11.250","Text":"2, 3,"},{"Start":"01:11.250 ","End":"01:13.560","Text":"4, 5,"},{"Start":"01:13.560 ","End":"01:16.810","Text":"6 factors, all of them 2."},{"Start":"01:17.090 ","End":"01:21.850","Text":"If I do the math 2 times 2 keep doubling up 16,"},{"Start":"01:21.850 ","End":"01:23.740","Text":"then double up again, 32,"},{"Start":"01:23.740 ","End":"01:28.070","Text":"then 64. That\u0027s what it is."},{"Start":"01:28.520 ","End":"01:33.845","Text":"I just want to remind you that it\u0027s a 1 way of indicating this 6"},{"Start":"01:33.845 ","End":"01:38.860","Text":"here is by placing curly brackets and writing the number 6,"},{"Start":"01:38.860 ","End":"01:43.100","Text":"which means that there are 6 factors and all of them are 2."},{"Start":"01:43.760 ","End":"01:46.120","Text":"Continuing."},{"Start":"01:46.120 ","End":"01:49.909","Text":"Now, when there\u0027s a minus in front,"},{"Start":"01:49.909 ","End":"01:52.400","Text":"the 4 doesn\u0027t relate to the minus."},{"Start":"01:52.400 ","End":"01:54.590","Text":"The 4 only related to the 2,"},{"Start":"01:54.590 ","End":"01:57.650","Text":"minus just stays along for the ride as a minus,"},{"Start":"01:57.650 ","End":"02:02.660","Text":"but it\u0027s 2^4, which means 2 times 2 times 2 times 2."},{"Start":"02:02.660 ","End":"02:06.330","Text":"As usual, I\u0027m counting 4 of these."},{"Start":"02:06.620 ","End":"02:11.100","Text":"2 times 2 times 2 times 2 is 16,"},{"Start":"02:11.100 ","End":"02:14.505","Text":"and the minus just sticks there."},{"Start":"02:14.505 ","End":"02:19.785","Text":"Minus 16 and I\u0027ll highlight that continuing."},{"Start":"02:19.785 ","End":"02:23.370","Text":"This time the minus 2 is in brackets,"},{"Start":"02:23.370 ","End":"02:27.380","Text":"so the 4 has to relate to the minus 2 as a whole."},{"Start":"02:27.380 ","End":"02:33.065","Text":"This time there are 4 factors and each of them is minus 2."},{"Start":"02:33.065 ","End":"02:35.190","Text":"There\u0027s 4 of them, 1, 2, 3,"},{"Start":"02:35.190 ","End":"02:37.645","Text":"4 each of them is minus 2."},{"Start":"02:37.645 ","End":"02:41.915","Text":"As with multiplication, we have to take care of the sign."},{"Start":"02:41.915 ","End":"02:45.050","Text":"We know it\u0027s 2 times 2 times 2 times 2 is 16,"},{"Start":"02:45.050 ","End":"02:47.210","Text":"but is it plus or minus?"},{"Start":"02:47.210 ","End":"02:51.140","Text":"1 way is just to see if it\u0027s an ordinal or even number of minuses,"},{"Start":"02:51.140 ","End":"02:55.115","Text":"1, 2, 3, 4 minuses for is even so it\u0027s plus."},{"Start":"02:55.115 ","End":"02:57.710","Text":"Or you can just say minus and minus is plus."},{"Start":"02:57.710 ","End":"02:59.120","Text":"Minus and minus is plus,"},{"Start":"02:59.120 ","End":"03:00.470","Text":"they\u0027ve canceled each other out."},{"Start":"03:00.470 ","End":"03:05.400","Text":"It\u0027s plus 16, which we just write as 16."},{"Start":"03:08.710 ","End":"03:14.780","Text":"This 1 is a bit of a trick question really."},{"Start":"03:14.780 ","End":"03:19.010","Text":"Because when you put brackets completely around something,"},{"Start":"03:19.010 ","End":"03:21.245","Text":"it doesn\u0027t change anything."},{"Start":"03:21.245 ","End":"03:24.259","Text":"This is just the same as if there were no brackets."},{"Start":"03:24.259 ","End":"03:27.560","Text":"The brackets were not doing anything in the extra operation,"},{"Start":"03:27.560 ","End":"03:29.210","Text":"it\u0027s just minus 2^4,"},{"Start":"03:29.210 ","End":"03:36.840","Text":"and that we\u0027ve solved already in part D. Let me just repeat the answer of minus 16."},{"Start":"03:38.440 ","End":"03:42.710","Text":"Last 1, minus 2^3."},{"Start":"03:42.710 ","End":"03:44.680","Text":"We can say 2^3,"},{"Start":"03:44.680 ","End":"03:46.680","Text":"but I prefer to say 2 cubed,"},{"Start":"03:46.680 ","End":"03:49.830","Text":"just like 2^2 is 2 squared."},{"Start":"03:49.830 ","End":"03:53.840","Text":"The minus is not acted upon by the 3."},{"Start":"03:53.840 ","End":"03:55.745","Text":"The 3 just applies to the 2,"},{"Start":"03:55.745 ","End":"03:57.680","Text":"the minus just sits there."},{"Start":"03:57.680 ","End":"04:02.740","Text":"It\u0027s minus 2 times 2 times 2."},{"Start":"04:02.740 ","End":"04:05.555","Text":"Again, the minus is just there along for the ride,"},{"Start":"04:05.555 ","End":"04:08.870","Text":"2 times 2 times 2 is 8."},{"Start":"04:08.870 ","End":"04:12.660","Text":"That\u0027s the answer, and we\u0027re done."}],"ID":4866},{"Watched":false,"Name":"Exercise 5","Duration":"5m 40s","ChapterTopicVideoID":4859,"CourseChapterTopicPlaylistID":3855,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.260","Text":"In this exercise, we have to calculate the following expressions and they"},{"Start":"00:04.260 ","End":"00:07.800","Text":"all involve roots of one form or another."},{"Start":"00:07.800 ","End":"00:09.735","Text":"Let me just copy it over here."},{"Start":"00:09.735 ","End":"00:13.440","Text":"There we go, and scroll up a bit."},{"Start":"00:13.440 ","End":"00:17.385","Text":"Very well. First one, the square root of 64."},{"Start":"00:17.385 ","End":"00:19.770","Text":"This is called the square root if it doesn\u0027t have"},{"Start":"00:19.770 ","End":"00:22.710","Text":"anything and it\u0027s as if there was a little 2 there."},{"Start":"00:22.710 ","End":"00:27.840","Text":"I need to know what number squared is equal to 64."},{"Start":"00:27.840 ","End":"00:28.979","Text":"A bit of guesswork,"},{"Start":"00:28.979 ","End":"00:30.495","Text":"you try 5^2, 6^2, 7^2,"},{"Start":"00:30.495 ","End":"00:34.230","Text":"finally hit upon 8^2 is 64,"},{"Start":"00:34.230 ","End":"00:37.990","Text":"so the square root of 64 is 8."},{"Start":"00:37.990 ","End":"00:41.810","Text":"Now how about the cube root of 64?"},{"Start":"00:41.810 ","End":"00:44.135","Text":"Once again, we don\u0027t say third root,"},{"Start":"00:44.135 ","End":"00:48.180","Text":"we say cube root, don\u0027t say second root, say square root."},{"Start":"00:48.180 ","End":"00:53.465","Text":"This time we need something cubed to equal 64."},{"Start":"00:53.465 ","End":"00:56.120","Text":"To remind you that means something times itself,"},{"Start":"00:56.120 ","End":"00:59.300","Text":"times itself is 3 of them to equal 64."},{"Start":"00:59.300 ","End":"01:00.530","Text":"Again, you try guesswork,"},{"Start":"01:00.530 ","End":"01:01.730","Text":"you try 1, 2, 3, 4,"},{"Start":"01:01.730 ","End":"01:03.050","Text":"finally hit upon 4 times,"},{"Start":"01:03.050 ","End":"01:05.990","Text":"4 times 4, bingo, it\u0027s 64."},{"Start":"01:05.990 ","End":"01:08.780","Text":"The answer is going to be 4."},{"Start":"01:08.780 ","End":"01:12.535","Text":"I\u0027ll highlight that. Next,"},{"Start":"01:12.535 ","End":"01:14.910","Text":"fifth root of 32."},{"Start":"01:14.910 ","End":"01:18.170","Text":"Fifth root means something to the power of 5,"},{"Start":"01:18.170 ","End":"01:20.825","Text":"has to equal 32."},{"Start":"01:20.825 ","End":"01:22.820","Text":"It\u0027s something, times itself, times itself,"},{"Start":"01:22.820 ","End":"01:26.030","Text":"times itself, times itself to equal 32."},{"Start":"01:26.030 ","End":"01:28.895","Text":"What could possibly be here? Guess what?"},{"Start":"01:28.895 ","End":"01:31.130","Text":"You hit upon it when you get to 2 times 2,"},{"Start":"01:31.130 ","End":"01:33.575","Text":"times 2, times 2, times 2 is 32."},{"Start":"01:33.575 ","End":"01:36.850","Text":"And so the answer is 2."},{"Start":"01:37.030 ","End":"01:40.400","Text":"So far, I haven\u0027t even being worried"},{"Start":"01:40.400 ","End":"01:43.659","Text":"about signs plus or minus because everything has been positive."},{"Start":"01:43.659 ","End":"01:46.730","Text":"In number d, we come to our first negative."},{"Start":"01:46.730 ","End":"01:48.440","Text":"So minus."},{"Start":"01:48.440 ","End":"01:50.480","Text":"What is the square root of minus 16?"},{"Start":"01:50.480 ","End":"01:54.215","Text":"What number squared is going to give me minus 16?"},{"Start":"01:54.215 ","End":"01:58.685","Text":"Or what times itself will equal minus 16?"},{"Start":"01:58.685 ","End":"02:00.860","Text":"The fact is that nothing will work,"},{"Start":"02:00.860 ","End":"02:02.660","Text":"because if I try positive,"},{"Start":"02:02.660 ","End":"02:05.990","Text":"plus times plus, that\u0027s going to give me a plus."},{"Start":"02:05.990 ","End":"02:09.350","Text":"Minus times a minus is also going give me a plus."},{"Start":"02:09.350 ","End":"02:10.565","Text":"Nothing will work."},{"Start":"02:10.565 ","End":"02:14.285","Text":"And so we write that there is no solution,"},{"Start":"02:14.285 ","End":"02:20.375","Text":"no answer and that\u0027s what I\u0027m going to give as my answer is no solution."},{"Start":"02:20.375 ","End":"02:25.025","Text":"Next one, again we\u0027re positive."},{"Start":"02:25.025 ","End":"02:29.200","Text":"Here we run into a different snag, and you\u0027ll see."},{"Start":"02:29.200 ","End":"02:30.740","Text":"Fourth root of 64."},{"Start":"02:30.740 ","End":"02:35.015","Text":"I need to know what number to the power of 4 is equal to 64,"},{"Start":"02:35.015 ","End":"02:36.620","Text":"or if you prefer the long hand way,"},{"Start":"02:36.620 ","End":"02:42.220","Text":"1,2,3,4 I have to put the same number in each of these to get 64."},{"Start":"02:42.220 ","End":"02:46.650","Text":"If I do it by my usual guess work I might try 3 times 3, times 3,"},{"Start":"02:46.650 ","End":"02:51.495","Text":"times 3, that\u0027s too big, 81."},{"Start":"02:51.495 ","End":"02:54.580","Text":"Try 2 times 2, times 2, times 2."},{"Start":"02:54.580 ","End":"02:58.009","Text":"No good, because that gives me 16."},{"Start":"02:58.009 ","End":"02:59.120","Text":"So 2 is too little,"},{"Start":"02:59.120 ","End":"03:00.425","Text":"3 is too big."},{"Start":"03:00.425 ","End":"03:03.580","Text":"Then it\u0027s time to take out your calculator."},{"Start":"03:03.580 ","End":"03:11.480","Text":"On mine it comes out to be 2.828 and a bunch of other stuff."},{"Start":"03:11.480 ","End":"03:16.360","Text":"You could just even leave it to 1 decimal place and write your answer as 2.8."},{"Start":"03:16.360 ","End":"03:19.420","Text":"Next is number f,"},{"Start":"03:19.420 ","End":"03:21.620","Text":"the fourth root of minus 64."},{"Start":"03:21.620 ","End":"03:26.435","Text":"We run into the same problem as we did in d. We have"},{"Start":"03:26.435 ","End":"03:29.390","Text":"an even root of a negative number and there is"},{"Start":"03:29.390 ","End":"03:33.330","Text":"no solution because the fourth root means that something times something,"},{"Start":"03:33.330 ","End":"03:34.800","Text":"times something, times something,"},{"Start":"03:34.800 ","End":"03:37.655","Text":"4 factors would have to equal minus 64."},{"Start":"03:37.655 ","End":"03:39.450","Text":"That\u0027s not possible because plus times plus,"},{"Start":"03:39.450 ","End":"03:41.225","Text":"times plus, times plus is plus."},{"Start":"03:41.225 ","End":"03:43.880","Text":"And if it was minus, minus, minus, minus,"},{"Start":"03:43.880 ","End":"03:46.595","Text":"that\u0027s also plus because these 2 give me a plus,"},{"Start":"03:46.595 ","End":"03:48.140","Text":"these 2 give me a plus."},{"Start":"03:48.140 ","End":"03:49.970","Text":"In other words, once again,"},{"Start":"03:49.970 ","End":"03:52.440","Text":"we have no answer."},{"Start":"03:52.440 ","End":"03:54.000","Text":"You got to watch out for those,"},{"Start":"03:54.000 ","End":"03:59.660","Text":"the even root of a negative number doesn\u0027t have an answer."},{"Start":"03:59.660 ","End":"04:01.520","Text":"Now we come to g,"},{"Start":"04:01.520 ","End":"04:03.410","Text":"which is in 2 parts."},{"Start":"04:03.410 ","End":"04:07.565","Text":"So let me do each part separately."},{"Start":"04:07.565 ","End":"04:09.740","Text":"Let me do the first part,"},{"Start":"04:09.740 ","End":"04:13.595","Text":"which is minus 3^4,"},{"Start":"04:13.595 ","End":"04:18.035","Text":"the 4 applies only to the 3 and not to the minus."},{"Start":"04:18.035 ","End":"04:21.775","Text":"3^4, this part is equal to,"},{"Start":"04:21.775 ","End":"04:23.160","Text":"let me just write it first,"},{"Start":"04:23.160 ","End":"04:24.450","Text":"3 times 3, times 3,"},{"Start":"04:24.450 ","End":"04:27.360","Text":"times 3 and that gives me 81,"},{"Start":"04:27.360 ","End":"04:28.890","Text":"so it\u0027s minus 81,"},{"Start":"04:28.890 ","End":"04:30.890","Text":"the minus just stays for the ride."},{"Start":"04:30.890 ","End":"04:36.050","Text":"The next one, next bit is the cube root of 8."},{"Start":"04:36.050 ","End":"04:39.800","Text":"What I need is something times something,"},{"Start":"04:39.800 ","End":"04:42.980","Text":"times something, I\u0027m sorry, that\u0027s minus 8."},{"Start":"04:42.980 ","End":"04:46.970","Text":"Something times something, times something is equal to minus 8."},{"Start":"04:46.970 ","End":"04:48.050","Text":"Now this is an odd number,"},{"Start":"04:48.050 ","End":"04:49.460","Text":"so there\u0027s no problem."},{"Start":"04:49.460 ","End":"04:53.370","Text":"A quick guess gives us a 2 times 2 times 2 is 8."},{"Start":"04:53.370 ","End":"04:55.710","Text":"So minus 2 times minus 2,"},{"Start":"04:55.710 ","End":"04:58.390","Text":"times minus 2 would be minus 8."},{"Start":"04:58.390 ","End":"05:02.945","Text":"So the answer to the second bit is minus 2."},{"Start":"05:02.945 ","End":"05:08.660","Text":"The first bit is minus 81, that\u0027s this bit."},{"Start":"05:08.660 ","End":"05:13.205","Text":"And the second bit, this is this."},{"Start":"05:13.205 ","End":"05:18.410","Text":"Finally, I want to combine them together and there\u0027s a plus here."},{"Start":"05:18.410 ","End":"05:24.720","Text":"What I need is minus 81 plus minus 2,"},{"Start":"05:24.720 ","End":"05:26.785","Text":"and that comes out to be,"},{"Start":"05:26.785 ","End":"05:28.430","Text":"because they have the same sign,"},{"Start":"05:28.430 ","End":"05:31.880","Text":"I add them without the sign 81 and 2 is 83,"},{"Start":"05:31.880 ","End":"05:34.685","Text":"and then I take the common sign, which is this."},{"Start":"05:34.685 ","End":"05:38.030","Text":"I shall highlight that also."},{"Start":"05:38.030 ","End":"05:41.880","Text":"There we are and that\u0027s the last one. We\u0027re done."}],"ID":4867},{"Watched":false,"Name":"Exercise 6","Duration":"4m 19s","ChapterTopicVideoID":4860,"CourseChapterTopicPlaylistID":3855,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.005","Text":"In this exercise, we have to calculate the following expressions."},{"Start":"00:04.005 ","End":"00:06.720","Text":"Seems to be mostly roots,"},{"Start":"00:06.720 ","End":"00:09.240","Text":"but a few exponents also."},{"Start":"00:09.240 ","End":"00:11.805","Text":"Let me first copy these down here."},{"Start":"00:11.805 ","End":"00:16.095","Text":"There we are. I like to just keep the original as it was."},{"Start":"00:16.095 ","End":"00:20.955","Text":"First 1 is square root of 169."},{"Start":"00:20.955 ","End":"00:22.710","Text":"Don\u0027t need a calculator,"},{"Start":"00:22.710 ","End":"00:24.285","Text":"we just need to ask,"},{"Start":"00:24.285 ","End":"00:28.800","Text":"\u0027\u0027What number squared is equal to 169?\u0027\u0027"},{"Start":"00:28.800 ","End":"00:30.960","Text":"The 2 comes from the fact that it\u0027s a square root,"},{"Start":"00:30.960 ","End":"00:32.220","Text":"which is the second root,"},{"Start":"00:32.220 ","End":"00:34.155","Text":"which we don\u0027t usually say."},{"Start":"00:34.155 ","End":"00:35.825","Text":"You just keep guessing."},{"Start":"00:35.825 ","End":"00:40.285","Text":"Everyone knows that a dozen and a dozen is 144."},{"Start":"00:40.285 ","End":"00:43.395","Text":"It\u0027s going to be bigger than 12 try 13,13^2,"},{"Start":"00:43.395 ","End":"00:46.155","Text":"yes, that\u0027s 169."},{"Start":"00:46.155 ","End":"00:52.010","Text":"That will work. The answer is the square root of 169 is 13,"},{"Start":"00:52.010 ","End":"00:53.725","Text":"and I\u0027ll highlight that."},{"Start":"00:53.725 ","End":"00:57.279","Text":"Next question is just a power,"},{"Start":"00:57.279 ","End":"01:00.515","Text":"4^2 or 4 squared with a minus in front."},{"Start":"01:00.515 ","End":"01:04.175","Text":"Notice that the square root applies just to the 4 and not to the minus."},{"Start":"01:04.175 ","End":"01:11.190","Text":"What this equals, the minus just stays and the 4^2 is 4 times 4, which is 16."},{"Start":"01:11.190 ","End":"01:16.575","Text":"Altogether minus 16 is our answer to that 1."},{"Start":"01:16.575 ","End":"01:20.045","Text":"In the next 1, because it\u0027s in brackets,"},{"Start":"01:20.045 ","End":"01:22.870","Text":"the power of 3 applies to the whole thing."},{"Start":"01:22.870 ","End":"01:25.275","Text":"What we get is the whole minus 3,"},{"Start":"01:25.275 ","End":"01:27.600","Text":"minus 3, minus 3."},{"Start":"01:27.600 ","End":"01:29.990","Text":"The 3 means 1, 2, 3 factors."},{"Start":"01:29.990 ","End":"01:36.675","Text":"An odd number of minuses is minus 3 times 3 times 3 is 27."},{"Start":"01:36.675 ","End":"01:39.460","Text":"That\u0027s our answer to this 1."},{"Start":"01:39.460 ","End":"01:41.765","Text":"To solve d, we have to ask,"},{"Start":"01:41.765 ","End":"01:48.330","Text":"\"What number to the power of 3 is going to be minus 27?\""},{"Start":"01:48.330 ","End":"01:50.480","Text":"We have it right in front of us."},{"Start":"01:50.480 ","End":"01:52.940","Text":"Minus 3 is the answer."},{"Start":"01:52.940 ","End":"01:55.745","Text":"Get that almost for free."},{"Start":"01:55.745 ","End":"01:58.625","Text":"I\u0027ll highlight that."},{"Start":"01:58.625 ","End":"02:03.490","Text":"Now we have the 4th root of 625."},{"Start":"02:03.490 ","End":"02:05.830","Text":"The big numbers from my experience,"},{"Start":"02:05.830 ","End":"02:07.210","Text":"most of them come out whole numbers."},{"Start":"02:07.210 ","End":"02:12.670","Text":"What I have to ask is what number to the fourth is 625."},{"Start":"02:12.670 ","End":"02:21.095","Text":"If you prefer, you could write it as something times itself will be 625."},{"Start":"02:21.095 ","End":"02:23.710","Text":"If you do this, well, guesswork approach,"},{"Start":"02:23.710 ","End":"02:29.040","Text":"we start somewhere around 4 times 4 times 4 times 4 is 256,"},{"Start":"02:29.040 ","End":"02:36.915","Text":"so we try 5 times 5 times 5 times 5, 25,125, 625."},{"Start":"02:36.915 ","End":"02:41.350","Text":"Yes. The answer to this 1 is 5."},{"Start":"02:41.350 ","End":"02:44.690","Text":"Now f, just to see if we\u0027re awake,"},{"Start":"02:44.690 ","End":"02:51.200","Text":"because notice that we have here a 4th root and 4 is an even number of a negative."},{"Start":"02:51.200 ","End":"02:54.200","Text":"We can\u0027t take an even root of a negative."},{"Start":"02:54.200 ","End":"02:55.220","Text":"It says no answer."},{"Start":"02:55.220 ","End":"02:57.635","Text":"But just to explain again why."},{"Start":"02:57.635 ","End":"02:58.940","Text":"Suppose we had something to"},{"Start":"02:58.940 ","End":"03:06.230","Text":"the 4th or something times itself is 4 factors to be minus 16."},{"Start":"03:06.230 ","End":"03:10.190","Text":"There is no way this is going to work out because we can\u0027t take a plus here."},{"Start":"03:10.190 ","End":"03:12.140","Text":"If we take plus for this,"},{"Start":"03:12.140 ","End":"03:14.000","Text":"that it\u0027s going to be plus 16."},{"Start":"03:14.000 ","End":"03:15.785","Text":"If we take minus,"},{"Start":"03:15.785 ","End":"03:20.390","Text":"each pair of minuses is a plus, that\u0027s a plus."},{"Start":"03:20.390 ","End":"03:22.445","Text":"Again, we\u0027ll get a plus."},{"Start":"03:22.445 ","End":"03:26.250","Text":"What we just write is no answer or no solution."},{"Start":"03:29.960 ","End":"03:35.250","Text":"Now the 5th root of minus 32 it\u0027s negative,"},{"Start":"03:35.250 ","End":"03:40.475","Text":"we need something to the 5th to be negative 32."},{"Start":"03:40.475 ","End":"03:49.980","Text":"What we need is something times itself times itself times itself is minus 32."},{"Start":"03:49.980 ","End":"03:53.660","Text":"Obviously, it\u0027s not going to be plus is going to be plus,"},{"Start":"03:53.660 ","End":"03:59.060","Text":"but minus is okay because we have an odd number here and odd number of minuses,"},{"Start":"03:59.060 ","End":"04:00.110","Text":"these 2 give me a plus,"},{"Start":"04:00.110 ","End":"04:02.600","Text":"these 2 give me a plus and I\u0027m left over with a minus."},{"Start":"04:02.600 ","End":"04:04.870","Text":"It has to be minus 2,"},{"Start":"04:04.870 ","End":"04:09.210","Text":"because 2 times 2 times 2 times 2 is 32."},{"Start":"04:09.210 ","End":"04:16.455","Text":"We\u0027ve seen this in previous exercises and so the answer is minus 2, which I highlight."},{"Start":"04:16.455 ","End":"04:19.510","Text":"That\u0027s the last 1. We\u0027re done here."}],"ID":4868},{"Watched":false,"Name":"Exercise 7 Parts a-c","Duration":"6m 47s","ChapterTopicVideoID":4861,"CourseChapterTopicPlaylistID":3855,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.415","Text":"In this exercise, we have to calculate the following expressions and there are 7 of them."},{"Start":"00:05.415 ","End":"00:10.410","Text":"The main thing here is the order of operations. Anyway, let\u0027s start."},{"Start":"00:10.410 ","End":"00:12.120","Text":"I have the solution below."},{"Start":"00:12.120 ","End":"00:14.205","Text":"I\u0027ve copied the exercises again."},{"Start":"00:14.205 ","End":"00:16.755","Text":"Let\u0027s go to the solution."},{"Start":"00:16.755 ","End":"00:18.780","Text":"Start with a."},{"Start":"00:18.780 ","End":"00:25.274","Text":"The first thing we have to do is exponents and roots."},{"Start":"00:25.274 ","End":"00:29.725","Text":"The first thing we\u0027re go to compute is go to be this and this."},{"Start":"00:29.725 ","End":"00:36.470","Text":"The square root of a 196 is 14 and 2^2 squared is 2 times 2 is 4."},{"Start":"00:36.470 ","End":"00:40.410","Text":"Now I get the exercise as"},{"Start":"00:40.410 ","End":"00:49.965","Text":"14 plus 5 times 4 minus 20 divided by 2."},{"Start":"00:49.965 ","End":"00:53.225","Text":"The next thing we do is multiplication and division."},{"Start":"00:53.225 ","End":"00:57.920","Text":"The multiplication here and the division here."},{"Start":"00:57.920 ","End":"01:04.200","Text":"This 5 times 4 comes out to be 20,"},{"Start":"01:04.200 ","End":"01:07.935","Text":"and 20 divided by 2 is 10."},{"Start":"01:07.935 ","End":"01:15.840","Text":"What I get is 14 plus 20 minus 10."},{"Start":"01:15.840 ","End":"01:18.990","Text":"Now I have additions and subtractions."},{"Start":"01:18.990 ","End":"01:22.100","Text":"I could do those just from left to right."},{"Start":"01:22.100 ","End":"01:25.129","Text":"We can do this in our heads without a calculator."},{"Start":"01:25.129 ","End":"01:28.705","Text":"14 plus 20 is 34,"},{"Start":"01:28.705 ","End":"01:33.925","Text":"34 minus 10 is 24."},{"Start":"01:33.925 ","End":"01:36.810","Text":"That\u0027s the answer."},{"Start":"01:37.540 ","End":"01:41.225","Text":"Next, part b."},{"Start":"01:41.225 ","End":"01:43.985","Text":"Here we have brackets,"},{"Start":"01:43.985 ","End":"01:46.430","Text":"but the brackets suggest around a single number,"},{"Start":"01:46.430 ","End":"01:48.260","Text":"so it\u0027s not really bracket."},{"Start":"01:48.260 ","End":"01:51.645","Text":"What we\u0027re go to do first is exponents."},{"Start":"01:51.645 ","End":"01:53.785","Text":"We have this to the power of 4,"},{"Start":"01:53.785 ","End":"01:55.870","Text":"and we have this to the power of 3."},{"Start":"01:55.870 ","End":"01:59.774","Text":"The exponents first, that will give us."},{"Start":"01:59.774 ","End":"02:03.420","Text":"If it was 2^4, it would be 16."},{"Start":"02:03.420 ","End":"02:06.530","Text":"But it\u0027s minus. We have minus, minus, minus, minus."},{"Start":"02:06.530 ","End":"02:09.605","Text":"An even number of minuses, so it\u0027s a plus."},{"Start":"02:09.605 ","End":"02:18.000","Text":"This is just 16. Minus 2 to the power of is 3 is minus 2,"},{"Start":"02:18.000 ","End":"02:19.320","Text":"minus 2, minus 2."},{"Start":"02:19.320 ","End":"02:22.735","Text":"Comes out to be minus 8."},{"Start":"02:22.735 ","End":"02:25.370","Text":"If I write this,"},{"Start":"02:25.370 ","End":"02:29.300","Text":"what it comes out is this part is 16"},{"Start":"02:29.300 ","End":"02:36.065","Text":"divided by 2 minus 10."},{"Start":"02:36.065 ","End":"02:38.570","Text":"Now I can\u0027t just write minus 8,"},{"Start":"02:38.570 ","End":"02:40.940","Text":"I can\u0027t put a multiplication and a minus sign."},{"Start":"02:40.940 ","End":"02:44.880","Text":"The minus 8 will have to go inside brackets."},{"Start":"02:45.380 ","End":"02:51.360","Text":"The next stage is we have to do the divisions."},{"Start":"02:51.360 ","End":"02:55.505","Text":"There\u0027s 1 here and a multiplication, there\u0027s 1 here."},{"Start":"02:55.505 ","End":"02:59.755","Text":"What we get is 16 divided by 2."},{"Start":"02:59.755 ","End":"03:05.385","Text":"That\u0027s 8, and 10 times minus 8."},{"Start":"03:05.385 ","End":"03:07.850","Text":"Positive times negative is negative,"},{"Start":"03:07.850 ","End":"03:12.160","Text":"so this is minus 80."},{"Start":"03:12.160 ","End":"03:19.080","Text":"I end up getting, now we\u0027re going to do the subtraction."},{"Start":"03:19.980 ","End":"03:23.305","Text":"I\u0027m getting a bit confused with the coloring. Forgive me."},{"Start":"03:23.305 ","End":"03:25.450","Text":"What we get here is,"},{"Start":"03:25.450 ","End":"03:28.240","Text":"I\u0027ll first of all write it as 8 minus."},{"Start":"03:28.240 ","End":"03:31.705","Text":"Now again, I need brackets because you can\u0027t just write minus minus."},{"Start":"03:31.705 ","End":"03:33.780","Text":"2 operations, 1 after the other."},{"Start":"03:33.780 ","End":"03:35.605","Text":"This will have to go in brackets."},{"Start":"03:35.605 ","End":"03:39.385","Text":"It\u0027s 8 minus minus 80."},{"Start":"03:39.385 ","End":"03:43.285","Text":"Because it\u0027s minus of a negative,"},{"Start":"03:43.285 ","End":"03:45.560","Text":"this becomes a plus."},{"Start":"03:45.560 ","End":"03:50.775","Text":"This becomes 8 plus 80."},{"Start":"03:50.775 ","End":"03:56.110","Text":"Finally, it\u0027s 88 is the answer to this 1."},{"Start":"03:56.180 ","End":"03:59.455","Text":"In this 1, we have brackets."},{"Start":"03:59.455 ","End":"04:02.609","Text":"In fact, we have round brackets and square brackets."},{"Start":"04:02.609 ","End":"04:04.880","Text":"Brackets are done first,"},{"Start":"04:04.880 ","End":"04:07.010","Text":"but they\u0027re done from inside out."},{"Start":"04:07.010 ","End":"04:11.200","Text":"The very first thing we would do is the 7 minus 2,"},{"Start":"04:11.200 ","End":"04:14.800","Text":"and that would come out to be 9."},{"Start":"04:14.800 ","End":"04:17.450","Text":"I don\u0027t want to copy the whole thing again,"},{"Start":"04:17.450 ","End":"04:21.400","Text":"because now we have only 1 set of brackets."},{"Start":"04:21.400 ","End":"04:25.535","Text":"Imagine the 7 minus 2 disappeared and we have 9 instead of it."},{"Start":"04:25.535 ","End":"04:28.070","Text":"What we have is really just,"},{"Start":"04:28.070 ","End":"04:30.200","Text":"I\u0027m going to do it in a thicker line."},{"Start":"04:30.200 ","End":"04:32.585","Text":"This brackets here."},{"Start":"04:32.585 ","End":"04:35.288","Text":"This brackets here,"},{"Start":"04:35.288 ","End":"04:45.435","Text":"what it basically says is 5 plus 4 times, sorry, not 9."},{"Start":"04:45.435 ","End":"04:49.060","Text":"It\u0027s 5. Silly me."},{"Start":"04:50.090 ","End":"04:54.265","Text":"I\u0027d like to confirm that 7 minus 2 is 5."},{"Start":"04:54.265 ","End":"04:55.850","Text":"Check it on your calculators."},{"Start":"04:55.850 ","End":"04:57.913","Text":"Yes, sorry about that."},{"Start":"04:57.913 ","End":"05:01.930","Text":"I\u0027m going to write this out as"},{"Start":"05:01.930 ","End":"05:08.900","Text":"minus 3^2 minus 4 times."},{"Start":"05:08.900 ","End":"05:12.755","Text":"I didn\u0027t actually do the calculation."},{"Start":"05:12.755 ","End":"05:15.890","Text":"Let me go back to this, actually."},{"Start":"05:15.890 ","End":"05:19.325","Text":"I want to compute this 5 plus 4 times 5."},{"Start":"05:19.325 ","End":"05:22.430","Text":"Don\u0027t make the mistake of doing 5 plus 4 is 9 times,"},{"Start":"05:22.430 ","End":"05:23.930","Text":"and then times 5."},{"Start":"05:23.930 ","End":"05:27.875","Text":"The multiplication comes before the addition."},{"Start":"05:27.875 ","End":"05:30.505","Text":"I do 4 times 5 is 20,"},{"Start":"05:30.505 ","End":"05:34.595","Text":"20 plus 5 is 25."},{"Start":"05:34.595 ","End":"05:38.250","Text":"This whole thing is 25."},{"Start":"05:38.250 ","End":"05:40.080","Text":"I\u0027ll put that 25 here."},{"Start":"05:40.080 ","End":"05:43.395","Text":"This 25 is 5 plus 4 times 5."},{"Start":"05:43.395 ","End":"05:48.625","Text":"Finally, plus the square root of 900."},{"Start":"05:48.625 ","End":"05:52.670","Text":"Now we have no more brackets,"},{"Start":"05:52.670 ","End":"05:57.725","Text":"but we have powers or exponents and roots."},{"Start":"05:57.725 ","End":"06:02.370","Text":"Those I\u0027m going to highlight in yellow."},{"Start":"06:02.370 ","End":"06:06.515","Text":"We have to do this square root and we have to do the exponent."},{"Start":"06:06.515 ","End":"06:09.470","Text":"Notice that the minus is not covered under the exponent."},{"Start":"06:09.470 ","End":"06:12.235","Text":"The 2 applies just to the 3."},{"Start":"06:12.235 ","End":"06:17.753","Text":"What we get is 3 squared is 9,"},{"Start":"06:17.753 ","End":"06:20.070","Text":"and the square root of 900 is 30."},{"Start":"06:20.070 ","End":"06:22.305","Text":"30 times 30 is 900."},{"Start":"06:22.305 ","End":"06:26.370","Text":"Now what we have is minus 9,"},{"Start":"06:26.370 ","End":"06:33.175","Text":"4 times 25 is 100 and plus 30."},{"Start":"06:33.175 ","End":"06:35.535","Text":"This we do from left to right,"},{"Start":"06:35.535 ","End":"06:38.720","Text":"minus 9 minus 100 is minus 109, plus 30."},{"Start":"06:38.720 ","End":"06:47.670","Text":"Minus 79. That\u0027s the answer to part c. Let\u0027s take a break and continue in the next clip."}],"ID":4869},{"Watched":false,"Name":"Exercise 7 Parts d-g","Duration":"8m 12s","ChapterTopicVideoID":4862,"CourseChapterTopicPlaylistID":3855,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.380 ","End":"00:06.790","Text":"Moving on to D and I think I\u0027ll scroll down a bit."},{"Start":"00:07.070 ","End":"00:10.170","Text":"We take care of brackets first,"},{"Start":"00:10.170 ","End":"00:11.730","Text":"and I\u0027ll just highlight those."},{"Start":"00:11.730 ","End":"00:12.930","Text":"We have brackets here,"},{"Start":"00:12.930 ","End":"00:15.090","Text":"and we have brackets here."},{"Start":"00:15.090 ","End":"00:20.010","Text":"The first one, minus 4 plus 2 is easy that\u0027s"},{"Start":"00:20.010 ","End":"00:26.370","Text":"-2 and here we have -3^2."},{"Start":"00:26.370 ","End":"00:28.095","Text":"Remember the 2,"},{"Start":"00:28.095 ","End":"00:30.330","Text":"the power acts only on the 3,"},{"Start":"00:30.330 ","End":"00:32.835","Text":"the minus is separate."},{"Start":"00:32.835 ","End":"00:36.700","Text":"This is actually -9+10."},{"Start":"00:38.030 ","End":"00:41.505","Text":"If I write this out."},{"Start":"00:41.505 ","End":"00:47.235","Text":"I now have square root of 64 divided by,"},{"Start":"00:47.235 ","End":"00:52.670","Text":"I need brackets around this -2 because can\u0027t have two operations,"},{"Start":"00:52.670 ","End":"00:54.200","Text":"one after another, division,"},{"Start":"00:54.200 ","End":"01:00.035","Text":"and subtraction, -4^2 times,"},{"Start":"01:00.035 ","End":"01:03.030","Text":"and then, yes,"},{"Start":"01:03.030 ","End":"01:04.845","Text":"I\u0027ll just compute this."},{"Start":"01:04.845 ","End":"01:07.300","Text":"-9+10 is 1."},{"Start":"01:07.300 ","End":"01:09.470","Text":"The next thing we do,"},{"Start":"01:09.470 ","End":"01:11.600","Text":"now that we no longer have brackets."},{"Start":"01:11.600 ","End":"01:12.920","Text":"This doesn\u0027t count as brackets."},{"Start":"01:12.920 ","End":"01:15.290","Text":"This just separates the number."},{"Start":"01:15.290 ","End":"01:18.335","Text":"Next thing we do is the exponents, and the roots."},{"Start":"01:18.335 ","End":"01:21.845","Text":"That will be this here and this here,"},{"Start":"01:21.845 ","End":"01:25.135","Text":"this one gives us 8."},{"Start":"01:25.135 ","End":"01:27.470","Text":"We\u0027ve done this more than once."},{"Start":"01:27.470 ","End":"01:28.910","Text":"8 times 8 is 64."},{"Start":"01:28.910 ","End":"01:32.990","Text":"4^2 is 4^2."},{"Start":"01:32.990 ","End":"01:36.650","Text":"4 times 4 is 16."},{"Start":"01:36.650 ","End":"01:45.080","Text":"Now we have 8 divided by -2,"},{"Start":"01:45.080 ","End":"01:51.475","Text":"and then -16 times 1."},{"Start":"01:51.475 ","End":"01:55.430","Text":"Next is the multiplication and the division."},{"Start":"01:55.430 ","End":"01:57.500","Text":"We have the division here,"},{"Start":"01:57.500 ","End":"01:59.480","Text":"the multiplication here."},{"Start":"01:59.480 ","End":"02:01.525","Text":"I\u0027ll continue here."},{"Start":"02:01.525 ","End":"02:06.900","Text":"8 divided by -2."},{"Start":"02:06.900 ","End":"02:09.695","Text":"A positive divided by a negative is a negative."},{"Start":"02:09.695 ","End":"02:12.610","Text":"8 divided by 2 is 4."},{"Start":"02:12.610 ","End":"02:15.255","Text":"Then minus."},{"Start":"02:15.255 ","End":"02:17.205","Text":"16 times 1."},{"Start":"02:17.205 ","End":"02:21.025","Text":"Anything times 1 is itself so it\u0027s 16."},{"Start":"02:21.025 ","End":"02:25.505","Text":"Now I just have straightforward subtraction here."},{"Start":"02:25.505 ","End":"02:35.850","Text":"We can look at this as like a -4+16 or we can think of it as -4+ or -16."},{"Start":"02:35.850 ","End":"02:37.860","Text":"Either way, you look at it,"},{"Start":"02:37.860 ","End":"02:42.850","Text":"what you get is -20."},{"Start":"02:43.100 ","End":"02:46.180","Text":"That\u0027s the answer to this one."},{"Start":"02:46.180 ","End":"02:48.440","Text":"Moving on to the next,"},{"Start":"02:48.440 ","End":"02:49.820","Text":"we don\u0027t have brackets."},{"Start":"02:49.820 ","End":"02:52.400","Text":"Like I said, a single number with a minus,"},{"Start":"02:52.400 ","End":"02:53.690","Text":"that doesn\u0027t count as brackets."},{"Start":"02:53.690 ","End":"02:57.050","Text":"It\u0027s just to separate the minus from the multiplication."},{"Start":"02:57.050 ","End":"02:58.790","Text":"We have exponents,"},{"Start":"02:58.790 ","End":"03:01.420","Text":"and square roots and roots in general, I mean."},{"Start":"03:01.420 ","End":"03:02.855","Text":"Here we have a root,"},{"Start":"03:02.855 ","End":"03:04.280","Text":"and here we have an exponent,"},{"Start":"03:04.280 ","End":"03:07.355","Text":"the power. Do those first."},{"Start":"03:07.355 ","End":"03:13.570","Text":"What we get here is 12 and -2^2."},{"Start":"03:13.570 ","End":"03:16.310","Text":"The 2 applies to the minus as well."},{"Start":"03:16.310 ","End":"03:17.990","Text":"So it\u0027s -2 times -2,"},{"Start":"03:17.990 ","End":"03:19.610","Text":"which gives us +4,"},{"Start":"03:19.610 ","End":"03:21.755","Text":"and I don\u0027t need to write the plus."},{"Start":"03:21.755 ","End":"03:25.320","Text":"This brings us to 12-20"},{"Start":"03:27.190 ","End":"03:35.585","Text":"divided by 4+3 times 4."},{"Start":"03:35.585 ","End":"03:40.310","Text":"We need to do next the multiplications and divisions."},{"Start":"03:40.310 ","End":"03:41.750","Text":"We have a division here,"},{"Start":"03:41.750 ","End":"03:44.275","Text":"we have a multiplication here,"},{"Start":"03:44.275 ","End":"03:48.350","Text":"and that means that what we get is 12."},{"Start":"03:48.350 ","End":"03:50.375","Text":"Let\u0027s just do this in our heads."},{"Start":"03:50.375 ","End":"03:53.885","Text":"20 divided by 4 is 5."},{"Start":"03:53.885 ","End":"03:56.990","Text":"3 times 4 is 12."},{"Start":"03:56.990 ","End":"03:58.310","Text":"Do it from left to right."},{"Start":"03:58.310 ","End":"04:01.810","Text":"12-5 is 7."},{"Start":"04:01.810 ","End":"04:06.350","Text":"7+12 is 19."},{"Start":"04:06.350 ","End":"04:07.655","Text":"Onto the next."},{"Start":"04:07.655 ","End":"04:09.755","Text":"Let\u0027s go for brackets first."},{"Start":"04:09.755 ","End":"04:13.470","Text":"We have brackets here, the square brackets."},{"Start":"04:14.260 ","End":"04:17.960","Text":"But, there is another place where there\u0027s brackets."},{"Start":"04:17.960 ","End":"04:20.150","Text":"Not explicitly but implicitly."},{"Start":"04:20.150 ","End":"04:25.320","Text":"Under a square root sign that\u0027s counted as if it was in brackets."},{"Start":"04:25.540 ","End":"04:29.325","Text":"We get. Let\u0027s do the easier one first."},{"Start":"04:29.325 ","End":"04:33.045","Text":"The 10+6 is 16."},{"Start":"04:33.045 ","End":"04:39.080","Text":"This one is, let\u0027s just compute this as a side exercise."},{"Start":"04:39.080 ","End":"04:45.240","Text":"We have -3+4 times -2."},{"Start":"04:45.240 ","End":"04:47.220","Text":"It\u0027s like an expression in itself."},{"Start":"04:47.220 ","End":"04:50.365","Text":"We have a plus, and we have a,"},{"Start":"04:50.365 ","End":"04:52.865","Text":"what happened to my little dot there?"},{"Start":"04:52.865 ","End":"04:54.800","Text":"Make it a bit darker there."},{"Start":"04:54.800 ","End":"05:00.605","Text":"Yeah. We have a multiplication that we have to do before addition."},{"Start":"05:00.605 ","End":"05:02.375","Text":"-8."},{"Start":"05:02.375 ","End":"05:04.130","Text":"I can\u0027t just write -8,"},{"Start":"05:04.130 ","End":"05:08.300","Text":"I need to put it in brackets to separate plus from the minus."},{"Start":"05:08.300 ","End":"05:13.370","Text":"This is just =-11."},{"Start":"05:13.370 ","End":"05:19.890","Text":"Now I could rewrite this as 3+4 times,"},{"Start":"05:19.890 ","End":"05:23.540","Text":"and instead of all these square brackets, I\u0027ve got -11."},{"Start":"05:23.540 ","End":"05:25.250","Text":"There\u0027s a multiplication here,"},{"Start":"05:25.250 ","End":"05:27.985","Text":"so the -11 needs to go in brackets."},{"Start":"05:27.985 ","End":"05:32.740","Text":"Then plus the square root of 16."},{"Start":"05:32.740 ","End":"05:35.885","Text":"I\u0027m sneaking in the powers, and roots here."},{"Start":"05:35.885 ","End":"05:37.655","Text":"Yeah, it\u0027s 4."},{"Start":"05:37.655 ","End":"05:40.040","Text":"So +4,"},{"Start":"05:40.040 ","End":"05:43.170","Text":"because the square root of 16 is 4."},{"Start":"05:43.340 ","End":"05:48.365","Text":"We just have a multiplication to do first."},{"Start":"05:48.365 ","End":"05:49.639","Text":"4 times"},{"Start":"05:49.639 ","End":"06:00.850","Text":"-11 is"},{"Start":"06:00.860 ","End":"06:04.410","Text":"-44."},{"Start":"06:04.410 ","End":"06:06.150","Text":"3+(-44)+4. We do it from left to right."},{"Start":"06:06.150 ","End":"06:11.430","Text":"3 and -44 is -41."},{"Start":"06:11.430 ","End":"06:22.210","Text":"Let\u0027s just remember that, -41+4 is -37. That\u0027s the answer."},{"Start":"06:22.310 ","End":"06:25.585","Text":"Now, the last one,"},{"Start":"06:25.585 ","End":"06:28.980","Text":"I don\u0027t actually have brackets here."},{"Start":"06:28.980 ","End":"06:31.910","Text":"Like I said, when it\u0027s a negative number that\u0027s in brackets"},{"Start":"06:31.910 ","End":"06:35.075","Text":"just to protect the minus that doesn\u0027t count as brackets."},{"Start":"06:35.075 ","End":"06:40.850","Text":"So we go on to the phase of the exponents and roots,"},{"Start":"06:40.850 ","End":"06:42.995","Text":"no roots, but there are exponents."},{"Start":"06:42.995 ","End":"06:48.674","Text":"There\u0027s this, and there is this."},{"Start":"06:48.674 ","End":"06:52.830","Text":"The 4 applies to the whole of the -3 because of the brackets,"},{"Start":"06:52.830 ","End":"06:55.590","Text":"and the 3 applies to the -2."},{"Start":"06:55.590 ","End":"07:02.360","Text":"What we get over there is -3^4 because it\u0027s an even number."},{"Start":"07:02.360 ","End":"07:04.595","Text":"Minus minus, minus, minus."},{"Start":"07:04.595 ","End":"07:07.610","Text":"-^4 is plus, and 3^4 is 81."},{"Start":"07:07.610 ","End":"07:13.435","Text":"So it\u0027s +81, and minus -2^3."},{"Start":"07:13.435 ","End":"07:14.780","Text":"Minus, minus, minus."},{"Start":"07:14.780 ","End":"07:17.000","Text":"It\u0027s an odd number, so it stays minus."},{"Start":"07:17.000 ","End":"07:19.280","Text":"2 times 2 times 2 is 8."},{"Start":"07:19.280 ","End":"07:24.940","Text":"What I get is 81 divided by"},{"Start":"07:24.940 ","End":"07:31.530","Text":"-9, -5 times -8."},{"Start":"07:31.530 ","End":"07:33.870","Text":"I need to put the -8 in brackets."},{"Start":"07:33.870 ","End":"07:35.390","Text":"We apply the next rule,"},{"Start":"07:35.390 ","End":"07:38.450","Text":"which is the division, and multiplication."},{"Start":"07:38.450 ","End":"07:44.480","Text":"What we get is positive divided by negative is negative,"},{"Start":"07:44.480 ","End":"07:47.120","Text":"and 81 over 9 is 9."},{"Start":"07:47.120 ","End":"07:51.440","Text":"But it\u0027s negative, minus."},{"Start":"07:51.440 ","End":"07:56.910","Text":"Let\u0027s see, 5 times -8 is -40,"},{"Start":"07:56.910 ","End":"08:00.290","Text":"and when we subtract a negative it\u0027s like adding."},{"Start":"08:00.290 ","End":"08:05.730","Text":"This is the same as"},{"Start":"08:05.730 ","End":"08:13.320","Text":"-9+40 and that\u0027s +31. That\u0027s it."}],"ID":4870}],"Thumbnail":null,"ID":3855},{"Name":"Fractions","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Fractions","Duration":"14m 48s","ChapterTopicVideoID":4891,"CourseChapterTopicPlaylistID":3856,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.545","Text":"Now we come to the section on fractions,"},{"Start":"00:04.545 ","End":"00:08.010","Text":"and you will have a general idea what a fraction is."},{"Start":"00:08.010 ","End":"00:13.090","Text":"I mean, you\u0027ve seen the most famous fraction would be 1/2,"},{"Start":"00:13.090 ","End":"00:16.838","Text":"but we could have 5/6,"},{"Start":"00:16.838 ","End":"00:22.390","Text":"and even negative fractions minus 3/4."},{"Start":"00:23.570 ","End":"00:30.068","Text":"Which is actually the same thing as put the minus 3 on the top,"},{"Start":"00:30.068 ","End":"00:31.110","Text":"and the 4 on the bottom."},{"Start":"00:31.110 ","End":"00:35.720","Text":"The reason I did that is because I want to say that the fraction has a top and a bottom,"},{"Start":"00:35.720 ","End":"00:39.685","Text":"which we call the numerator,"},{"Start":"00:39.685 ","End":"00:44.680","Text":"and this one, the denominator."},{"Start":"00:45.340 ","End":"00:48.380","Text":"Somehow these are unfriendly words,"},{"Start":"00:48.380 ","End":"00:50.810","Text":"but that\u0027s the words that we have."},{"Start":"00:50.810 ","End":"00:53.510","Text":"Now, in these examples,"},{"Start":"00:53.510 ","End":"00:57.650","Text":"you\u0027ll notice that the top is less than the bottom."},{"Start":"00:57.650 ","End":"01:00.170","Text":"I say top, and bottom instead of numerator,"},{"Start":"01:00.170 ","End":"01:02.585","Text":"and denominator, just easier."},{"Start":"01:02.585 ","End":"01:05.555","Text":"Here, the top is less than the bottom."},{"Start":"01:05.555 ","End":"01:09.600","Text":"Here also the top is less than the bottom."},{"Start":"01:10.270 ","End":"01:16.715","Text":"This is the case. These are called proper fractions."},{"Start":"01:16.715 ","End":"01:27.305","Text":"But sometimes, we also have cases where the top is greater than the bottom, like 3/2,"},{"Start":"01:27.305 ","End":"01:35.520","Text":"and negative 8/5,"},{"Start":"01:35.520 ","End":"01:38.690","Text":"and these are called improper fractions because at"},{"Start":"01:38.690 ","End":"01:42.350","Text":"one time it was considered that if the top is greater than the bottom,"},{"Start":"01:42.350 ","End":"01:43.895","Text":"then you\u0027ve got more than a whole,"},{"Start":"01:43.895 ","End":"01:47.120","Text":"like 3/2 is 1 1/2,"},{"Start":"01:47.120 ","End":"01:49.370","Text":"and that\u0027s not really a fraction,"},{"Start":"01:49.370 ","End":"01:51.665","Text":"but we say that yes it is,"},{"Start":"01:51.665 ","End":"01:54.335","Text":"and I\u0027m going to give a definition in a moment,"},{"Start":"01:54.335 ","End":"01:59.315","Text":"and I\u0027d also like to write this same as here,"},{"Start":"01:59.315 ","End":"02:02.120","Text":"as minus 8/5,"},{"Start":"02:02.120 ","End":"02:03.950","Text":"put the minus in front."},{"Start":"02:03.950 ","End":"02:06.860","Text":"Now, what I\u0027d like to say is that"},{"Start":"02:06.860 ","End":"02:11.330","Text":"a fraction is improper if the top is bigger than the bottom."},{"Start":"02:11.330 ","End":"02:17.750","Text":"The reason I prefer to write the minus in front is that someone could say,"},{"Start":"02:17.750 ","End":"02:20.390","Text":"in this case, here,"},{"Start":"02:20.390 ","End":"02:23.450","Text":"someone could say the top is less than the bottom"},{"Start":"02:23.450 ","End":"02:27.585","Text":"because negative 8 is less than 5 on the number line."},{"Start":"02:27.585 ","End":"02:30.725","Text":"When I talk about the top being greater than the bottom,"},{"Start":"02:30.725 ","End":"02:32.090","Text":"I\u0027m ignoring the minus,"},{"Start":"02:32.090 ","End":"02:34.340","Text":"or pull the minus out in front."},{"Start":"02:34.340 ","End":"02:37.500","Text":"Let me just write that down."},{"Start":"02:37.870 ","End":"02:44.400","Text":"In general. All of these are called simple fractions."},{"Start":"02:46.330 ","End":"02:52.010","Text":"Simple fraction is just a whole number over a whole number,"},{"Start":"02:52.010 ","End":"02:54.035","Text":"a numerator over a denominator."},{"Start":"02:54.035 ","End":"02:56.450","Text":"But I sub-divide them into 2 kinds,"},{"Start":"02:56.450 ","End":"02:59.165","Text":"into proper and improper."},{"Start":"02:59.165 ","End":"03:01.820","Text":"I won\u0027t write the word fractions again,"},{"Start":"03:01.820 ","End":"03:04.400","Text":"I\u0027ll just write proper and improper."},{"Start":"03:04.400 ","End":"03:09.680","Text":"Now, the proper is the one where the top is less than the bottom."},{"Start":"03:09.680 ","End":"03:12.030","Text":"Maybe I\u0027ll color them."},{"Start":"03:12.190 ","End":"03:15.515","Text":"The proper ones would be,"},{"Start":"03:15.515 ","End":"03:17.255","Text":"this is a proper fraction,"},{"Start":"03:17.255 ","End":"03:19.340","Text":"this is the proper fraction,"},{"Start":"03:19.340 ","End":"03:23.555","Text":"and this is a proper fraction."},{"Start":"03:23.555 ","End":"03:27.770","Text":"Because like I said, if the top is less than the bottom,"},{"Start":"03:27.770 ","End":"03:31.520","Text":"it doesn\u0027t matter if the minus is in front or the minus is on top,"},{"Start":"03:31.520 ","End":"03:33.635","Text":"we ignore the minus for this purpose,"},{"Start":"03:33.635 ","End":"03:35.690","Text":"and the improper,"},{"Start":"03:35.690 ","End":"03:37.610","Text":"there I\u0027ll use this color,"},{"Start":"03:37.610 ","End":"03:41.765","Text":"will be the ones where the top is greater than the bottom."},{"Start":"03:41.765 ","End":"03:45.725","Text":"This is improper, and this is improper,"},{"Start":"03:45.725 ","End":"03:47.690","Text":"and like I said,"},{"Start":"03:47.690 ","End":"03:49.950","Text":"we ignore the minus."},{"Start":"03:50.300 ","End":"03:54.020","Text":"That\u0027s 3 examples of this,"},{"Start":"03:54.020 ","End":"03:56.015","Text":"and 2 examples of that."},{"Start":"03:56.015 ","End":"04:00.300","Text":"But, there is another fraction also."},{"Start":"04:00.730 ","End":"04:10.610","Text":"I\u0027ll give examples. You\u0027ve seen numbers like 2 1/2 or minus 1 and 2/3,"},{"Start":"04:10.610 ","End":"04:13.510","Text":"and they have a whole number part,"},{"Start":"04:13.510 ","End":"04:15.780","Text":"and the fraction part,"},{"Start":"04:15.780 ","End":"04:23.240","Text":"and these are called mixed numbers, or mixed fractions,"},{"Start":"04:23.240 ","End":"04:26.672","Text":"I\u0027ll say mixed numbers,"},{"Start":"04:26.672 ","End":"04:30.770","Text":"but you could also call them mixed fractions."},{"Start":"04:30.770 ","End":"04:32.990","Text":"Both are acceptable."},{"Start":"04:32.990 ","End":"04:35.880","Text":"I will color those as well."},{"Start":"04:36.170 ","End":"04:38.565","Text":"Mixed numbers."},{"Start":"04:38.565 ","End":"04:41.675","Text":"That\u0027s this, and that\u0027s this."},{"Start":"04:41.675 ","End":"04:49.070","Text":"Notice that the fraction part should be a proper fraction."},{"Start":"04:49.070 ","End":"04:52.498","Text":"What I mean is if someone said 3,"},{"Start":"04:52.498 ","End":"04:59.030","Text":"and 8/5, that would not be a good mixed number."},{"Start":"04:59.030 ","End":"05:03.274","Text":"You should have this part here as being a proper fraction."},{"Start":"05:03.274 ","End":"05:04.400","Text":"This is the whole part,"},{"Start":"05:04.400 ","End":"05:07.685","Text":"this is the fraction part, and so on."},{"Start":"05:07.685 ","End":"05:11.450","Text":"Here I copied the number line from a previous clip,"},{"Start":"05:11.450 ","End":"05:15.515","Text":"but just want to show you how fractions fit on here."},{"Start":"05:15.515 ","End":"05:18.160","Text":"Not going to do all of them."},{"Start":"05:18.160 ","End":"05:22.745","Text":"1/2 of course, would be halfway between 0, and 1."},{"Start":"05:22.745 ","End":"05:25.625","Text":"This might be 1/2."},{"Start":"05:25.625 ","End":"05:33.980","Text":"Let\u0027s say minus 3/4 would be 3/4 of the way from 0 to minus 1,"},{"Start":"05:33.980 ","End":"05:38.670","Text":"somewhere around here, I\u0027d have minus 3/4."},{"Start":"05:39.230 ","End":"05:45.210","Text":"2 1/2 would be somewhere between 2 and 3 and"},{"Start":"05:45.210 ","End":"05:50.540","Text":"minus 1 and 2/3 would be somewhere over here."},{"Start":"05:50.540 ","End":"05:52.505","Text":"Minus 1 and 2/3."},{"Start":"05:52.505 ","End":"05:55.415","Text":"I just wanted to give you a general idea,"},{"Start":"05:55.415 ","End":"05:59.330","Text":"and I wanted to erase this because this was a bad example,"},{"Start":"05:59.330 ","End":"06:03.740","Text":"and the next thing I wanted to discuss was"},{"Start":"06:03.740 ","End":"06:09.845","Text":"that it turns out that there\u0027s a close connection between mixed numbers,"},{"Start":"06:09.845 ","End":"06:14.780","Text":"and improper fractions that each can be converted into the other."},{"Start":"06:14.780 ","End":"06:17.585","Text":"I\u0027d like to show you some examples."},{"Start":"06:17.585 ","End":"06:20.245","Text":"I think I\u0027ll just squeezed them in over here."},{"Start":"06:20.245 ","End":"06:22.940","Text":"The first example, I\u0027ll take this one,"},{"Start":"06:22.940 ","End":"06:24.995","Text":"definitely an improper fraction."},{"Start":"06:24.995 ","End":"06:28.020","Text":"Let\u0027s take 3/2."},{"Start":"06:28.780 ","End":"06:33.070","Text":"I think it\u0027s clear that if we have 3/2,"},{"Start":"06:33.070 ","End":"06:35.300","Text":"whether it\u0027s pizzas or whatever,"},{"Start":"06:35.300 ","End":"06:38.960","Text":"that we have a whole 1 1/2 leftover."},{"Start":"06:38.960 ","End":"06:41.240","Text":"This is 1 1/2."},{"Start":"06:41.240 ","End":"06:45.920","Text":"Now, the mathematical way of doing it is doing division with remainder."},{"Start":"06:45.920 ","End":"06:47.750","Text":"We say 2 goes into 3,"},{"Start":"06:47.750 ","End":"06:51.305","Text":"1 time and the remainder is 1."},{"Start":"06:51.305 ","End":"06:56.300","Text":"This is the remainder and then we just put it over the same denominator here."},{"Start":"06:56.300 ","End":"07:02.660","Text":"I\u0027ll go with the other one, minus 8/5."},{"Start":"07:02.660 ","End":"07:05.270","Text":"Well the minus just gets copied,"},{"Start":"07:05.270 ","End":"07:08.600","Text":"so don\u0027t worry about the minus, and then 8/5."},{"Start":"07:08.600 ","End":"07:12.670","Text":"Once again, it goes in once."},{"Start":"07:12.670 ","End":"07:18.785","Text":"There\u0027s 3 leftover, so it\u0027s minus 1 and 3/5."},{"Start":"07:18.785 ","End":"07:21.580","Text":"That does this direction of the arrow"},{"Start":"07:21.580 ","End":"07:24.535","Text":"converting from improper fractions to mixed numbers."},{"Start":"07:24.535 ","End":"07:28.105","Text":"Now let\u0027s take some examples of the other way round."},{"Start":"07:28.105 ","End":"07:33.055","Text":"I\u0027ll take first of all 2 and 3/8,"},{"Start":"07:33.055 ","End":"07:36.115","Text":"and see what that is."},{"Start":"07:36.115 ","End":"07:37.990","Text":"Well, if you think about it,"},{"Start":"07:37.990 ","End":"07:39.610","Text":"if you use the pizza model,"},{"Start":"07:39.610 ","End":"07:42.520","Text":"2 pizzas and you divide them into 8."},{"Start":"07:42.520 ","End":"07:50.030","Text":"We\u0027ve got the 16 slices and another 3 slices is 19 slices."},{"Start":"07:50.070 ","End":"07:52.720","Text":"Each slice is an 1/8 of a pizza,"},{"Start":"07:52.720 ","End":"07:54.010","Text":"so it\u0027s 19 over 8."},{"Start":"07:54.010 ","End":"07:58.855","Text":"Now there was a technical way of doing this that is as follows."},{"Start":"07:58.855 ","End":"08:02.110","Text":"Whenever you have a mixed number like this,"},{"Start":"08:02.110 ","End":"08:05.920","Text":"it could put a little arrow here and an arrow here."},{"Start":"08:05.920 ","End":"08:07.915","Text":"Here I put times,"},{"Start":"08:07.915 ","End":"08:10.825","Text":"I\u0027ll put it as a cross-modal dot and a plus."},{"Start":"08:10.825 ","End":"08:14.635","Text":"What we do is like this 8 times 2 plus 3."},{"Start":"08:14.635 ","End":"08:21.925","Text":"8 times 2 is 16 plus 3 is 19 and that\u0027s what I put here on this always gets copied here."},{"Start":"08:21.925 ","End":"08:24.610","Text":"I\u0027ll show you another example."},{"Start":"08:24.610 ","End":"08:28.991","Text":"Let\u0027s say 12 and 2/5, you know what?"},{"Start":"08:28.991 ","End":"08:31.960","Text":"I\u0027ll make it minus also."},{"Start":"08:31.960 ","End":"08:34.690","Text":"With the same scheme,"},{"Start":"08:34.690 ","End":"08:37.795","Text":"I put here,"},{"Start":"08:37.795 ","End":"08:43.696","Text":"and here and times and plus 5 times 12 is 60,"},{"Start":"08:43.696 ","End":"08:46.990","Text":"60 plus 2 is 62."},{"Start":"08:46.990 ","End":"08:52.030","Text":"It\u0027s 62 and then the 5 by copy,"},{"Start":"08:52.030 ","End":"08:54.340","Text":"but I still have to keep the minus."},{"Start":"08:54.340 ","End":"08:58.620","Text":"This is equal to minus 62 over 5."},{"Start":"08:58.620 ","End":"09:05.365","Text":"That completes the other direction from mixed numbers to improper."},{"Start":"09:05.365 ","End":"09:08.570","Text":"Onto the next topic."},{"Start":"09:08.790 ","End":"09:13.810","Text":"We\u0027re going to discuss expanding and reducing fractions."},{"Start":"09:13.810 ","End":"09:17.185","Text":"I\u0027ll start with an example."},{"Start":"09:17.185 ","End":"09:24.130","Text":"If I have a fraction which is 1/2 and I\u0027m going to use the pizza module."},{"Start":"09:24.130 ","End":"09:29.800","Text":"We\u0027ll imagine that I had pizza that maybe a personal"},{"Start":"09:29.800 ","End":"09:36.250","Text":"Size 1 that was cut into 2 pieces and I took 1 piece, I would have a 1/2."},{"Start":"09:36.250 ","End":"09:41.520","Text":"But if I cut the pizza into 4 and I took 2 of the pieces,"},{"Start":"09:41.520 ","End":"09:43.980","Text":"that would be the same amount of pizza."},{"Start":"09:43.980 ","End":"09:46.275","Text":"That\u0027s the same as 2/4."},{"Start":"09:46.275 ","End":"09:49.690","Text":"Or if I took a medium-size that was sliced"},{"Start":"09:49.690 ","End":"09:53.245","Text":"into 6 pieces and I took 3 out of those 6 pieces,"},{"Start":"09:53.245 ","End":"09:56.280","Text":"I\u0027d still get half the pizza and so on."},{"Start":"09:56.280 ","End":"10:03.045","Text":"If it was divided into 8 and I took 4 and so on."},{"Start":"10:03.045 ","End":"10:06.495","Text":"What will be your last one 5/10,"},{"Start":"10:06.495 ","End":"10:10.225","Text":"if take 5 out of 10, again, that\u0027s a 1/2."},{"Start":"10:10.225 ","End":"10:13.630","Text":"The conclusion is in this case,"},{"Start":"10:13.630 ","End":"10:16.240","Text":"but also in general that if I multiply the top"},{"Start":"10:16.240 ","End":"10:19.405","Text":"and the bottom of a fraction by the same number,"},{"Start":"10:19.405 ","End":"10:22.315","Text":"I\u0027ll get an equivalent fraction."},{"Start":"10:22.315 ","End":"10:29.110","Text":"For example, if I took 1 over 2 and multiply top and bottom both by 3,"},{"Start":"10:29.110 ","End":"10:31.585","Text":"I\u0027d get 3 over 6 and so on."},{"Start":"10:31.585 ","End":"10:35.425","Text":"This is a general principle,"},{"Start":"10:35.425 ","End":"10:38.740","Text":"and this is the expanding part."},{"Start":"10:38.740 ","End":"10:42.220","Text":"The reducing part is the other way is starting with"},{"Start":"10:42.220 ","End":"10:46.000","Text":"a fraction and dividing top and bottom by the same number."},{"Start":"10:46.000 ","End":"10:48.940","Text":"I\u0027ll just illustrate in one of the cases,"},{"Start":"10:48.940 ","End":"10:55.030","Text":"suppose that had the 4 over 8 and if I divide top and bottom by 4,"},{"Start":"10:55.030 ","End":"11:03.070","Text":"we often put a line through a strike out and divide this by 4, I get 1."},{"Start":"11:03.070 ","End":"11:04.480","Text":"If I divide this by 4,"},{"Start":"11:04.480 ","End":"11:07.000","Text":"I get 2 and so this is equal to a 1/2."},{"Start":"11:07.000 ","End":"11:12.715","Text":"From here to here it\u0027s expanding and from here back to here it\u0027s reducing."},{"Start":"11:12.715 ","End":"11:15.190","Text":"I won\u0027t spend more time on this."},{"Start":"11:15.190 ","End":"11:20.245","Text":"I\u0027ll go straight to the next topic which is comparing fractions."},{"Start":"11:20.245 ","End":"11:21.445","Text":"How do we compare?"},{"Start":"11:21.445 ","End":"11:24.775","Text":"How do we know which fraction is bigger than the other?"},{"Start":"11:24.775 ","End":"11:26.965","Text":"Let\u0027s take an example."},{"Start":"11:26.965 ","End":"11:32.945","Text":"Suppose I have 5/8 and 7/8."},{"Start":"11:32.945 ","End":"11:38.503","Text":"What I want to do, is put either a less than or a greater than symbol between them."},{"Start":"11:38.503 ","End":"11:41.960","Text":"Hope you remember which is which."},{"Start":"11:42.270 ","End":"11:44.785","Text":"It\u0027s fairly clear here."},{"Start":"11:44.785 ","End":"11:48.160","Text":"If I think of pizzas that were divided into slices,"},{"Start":"11:48.160 ","End":"11:50.980","Text":"into 8 slices, someone got 5 slices,"},{"Start":"11:50.980 ","End":"11:52.795","Text":"someone got 7 slices."},{"Start":"11:52.795 ","End":"11:56.485","Text":"Because 5 is less than 7,"},{"Start":"11:56.485 ","End":"11:59.695","Text":"then 5/8 is is less than 7/8."},{"Start":"11:59.695 ","End":"12:03.460","Text":"This is a particular case when the denominators are equal."},{"Start":"12:03.460 ","End":"12:04.690","Text":"That\u0027s always the rule."},{"Start":"12:04.690 ","End":"12:06.295","Text":"When the denominators are equal,"},{"Start":"12:06.295 ","End":"12:08.965","Text":"you just compare the numerators."},{"Start":"12:08.965 ","End":"12:11.800","Text":"Now let\u0027s take another example."},{"Start":"12:11.800 ","End":"12:14.840","Text":"This time I\u0027m going to compare,"},{"Start":"12:15.510 ","End":"12:25.450","Text":"let\u0027s say, 3/5 and 3/7."},{"Start":"12:25.450 ","End":"12:26.680","Text":"Now, which is less?"},{"Start":"12:26.680 ","End":"12:28.015","Text":"Well, if you think about it,"},{"Start":"12:28.015 ","End":"12:30.460","Text":"in each case I had 3,"},{"Start":"12:30.460 ","End":"12:36.174","Text":"and I divided in 1 case into 5 pieces and in the other case into 7 pieces."},{"Start":"12:36.174 ","End":"12:40.615","Text":"Seems clear that the more I divide into the smaller the pieces are."},{"Start":"12:40.615 ","End":"12:45.380","Text":"Although once again, 5 is less than 7,"},{"Start":"12:45.380 ","End":"12:49.625","Text":"I put the sign greater than here."},{"Start":"12:49.625 ","End":"12:51.985","Text":"That\u0027s another rule."},{"Start":"12:51.985 ","End":"12:54.460","Text":"If they have the same numerators,"},{"Start":"12:54.460 ","End":"12:58.930","Text":"you compare the denominators and see which one of these two it is,"},{"Start":"12:58.930 ","End":"13:00.985","Text":"but take the opposite."},{"Start":"13:00.985 ","End":"13:07.810","Text":"The third case is the more general cases where nothing is equal to anything."},{"Start":"13:07.810 ","End":"13:14.830","Text":"As an example, I\u0027ll take 2/3 and 3/4."},{"Start":"13:14.830 ","End":"13:18.980","Text":"Not immediately obvious which way it goes."},{"Start":"13:19.080 ","End":"13:23.050","Text":"In this case, what we\u0027re going to do is use the"},{"Start":"13:23.050 ","End":"13:28.630","Text":"expanding to get them both to have the same denominator."},{"Start":"13:28.630 ","End":"13:33.175","Text":"I\u0027m going to have to multiply 3 by something in 4 by something to get that,"},{"Start":"13:33.175 ","End":"13:35.380","Text":"to get something common."},{"Start":"13:35.380 ","End":"13:40.610","Text":"It\u0027s hard to see that if we took 12 here,"},{"Start":"13:40.650 ","End":"13:43.180","Text":"that 3 goes into 12 and 4,"},{"Start":"13:43.180 ","End":"13:47.575","Text":"goes into 12, I\u0027m multiplying the denominator by"},{"Start":"13:47.575 ","End":"13:52.390","Text":"4 so that I multiply the numerator by 4 also."},{"Start":"13:52.390 ","End":"13:54.220","Text":"Then it\u0027ll be okay,"},{"Start":"13:54.220 ","End":"13:56.560","Text":"it\u0027ll be the expanding."},{"Start":"13:56.560 ","End":"13:58.945","Text":"Anyway, 2 times 4 is 8."},{"Start":"13:58.945 ","End":"14:03.700","Text":"What I\u0027ve done now is rewritten 2/3 as as 8/12."},{"Start":"14:03.700 ","End":"14:10.180","Text":"Similarly, with the case of 3/4 if I multiply top and bottom by 3,"},{"Start":"14:10.180 ","End":"14:13.585","Text":"this I multiply by 3 and this I multiply by 3."},{"Start":"14:13.585 ","End":"14:18.760","Text":"Then I\u0027m going to get 9 over 12."},{"Start":"14:18.760 ","End":"14:21.340","Text":"Now when they have the same denominator,"},{"Start":"14:21.340 ","End":"14:23.390","Text":"we use the first case to say,"},{"Start":"14:23.390 ","End":"14:26.095","Text":"we just have to compare numerators."},{"Start":"14:26.095 ","End":"14:29.780","Text":"8/12 is certainly less than 9/12."},{"Start":"14:30.240 ","End":"14:33.760","Text":"This is going to be less than this."},{"Start":"14:33.760 ","End":"14:37.535","Text":"This is an important concept and we\u0027ll see it again later."},{"Start":"14:37.535 ","End":"14:42.590","Text":"The idea of taking two different fractions with different denominators and"},{"Start":"14:42.590 ","End":"14:48.810","Text":"converting them to have the same denominator. I\u0027m done for now."}],"ID":4871},{"Watched":false,"Name":"Decimals - Part 1","Duration":"6m 20s","ChapterTopicVideoID":4871,"CourseChapterTopicPlaylistID":3856,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.085","Text":"In this clip, I\u0027m going to be talking about decimal fractions and decimals,"},{"Start":"00:05.085 ","End":"00:08.205","Text":"very closely related concepts."},{"Start":"00:08.205 ","End":"00:12.705","Text":"Before I start, I\u0027ll just give you an example of where I\u0027m heading with this."},{"Start":"00:12.705 ","End":"00:17.760","Text":"I would like to say that 1/4 is the same as"},{"Start":"00:17.760 ","End":"00:25.800","Text":"25 over 100 and this is equal to 0.25."},{"Start":"00:25.800 ","End":"00:30.360","Text":"If you understand this, basically you\u0027ve understood the rest of the lesson."},{"Start":"00:30.360 ","End":"00:33.525","Text":"This is a fraction,"},{"Start":"00:33.525 ","End":"00:35.480","Text":"this is a decimal fraction,"},{"Start":"00:35.480 ","End":"00:37.655","Text":"and this is a decimal."},{"Start":"00:37.655 ","End":"00:41.662","Text":"Now let me just talk a bit more about decimal fractions,"},{"Start":"00:41.662 ","End":"00:45.740","Text":"but before that, I wanted to remind you of powers of 10."},{"Start":"00:45.740 ","End":"00:48.250","Text":"If I take 10^1,"},{"Start":"00:48.250 ","End":"00:50.220","Text":"that will just be 10."},{"Start":"00:50.220 ","End":"00:52.890","Text":"If I take 10^2,"},{"Start":"00:52.890 ","End":"00:57.644","Text":"that will be 10 times 10, which is 100."},{"Start":"00:57.644 ","End":"01:00.510","Text":"If I take 10 to the 3,"},{"Start":"01:00.510 ","End":"01:03.810","Text":"10 times 10 times 10 is 1,000."},{"Start":"01:03.810 ","End":"01:07.005","Text":"I\u0027ll do 1 more. You know what?"},{"Start":"01:07.005 ","End":"01:10.980","Text":"We\u0027ll just do 10^6,"},{"Start":"01:10.980 ","End":"01:12.344","Text":"that\u0027s actually a million,"},{"Start":"01:12.344 ","End":"01:17.409","Text":"10 times 10 times 10 times 10 times 10 times 10 is actually a million."},{"Start":"01:17.409 ","End":"01:22.010","Text":"Optionally, you can put commas every 3 from the right."},{"Start":"01:22.010 ","End":"01:26.105","Text":"Sometimes you put a comma, totally optional."},{"Start":"01:26.105 ","End":"01:29.480","Text":"Now notice that, for example,"},{"Start":"01:29.480 ","End":"01:32.750","Text":"in this 1 there\u0027s a 3 here and there\u0027s 3 0s."},{"Start":"01:32.750 ","End":"01:34.580","Text":"This is not a coincidence."},{"Start":"01:34.580 ","End":"01:37.880","Text":"In other words, this 3 here is the same as 1, 2,"},{"Start":"01:37.880 ","End":"01:43.180","Text":"3 places after the 1, 3, 0s."},{"Start":"01:43.210 ","End":"01:47.869","Text":"This leads me to the concept of a decimal fraction."},{"Start":"01:47.869 ","End":"01:54.080","Text":"Anything which has a denominator that\u0027s a power of 10 is called a decimal fraction."},{"Start":"01:54.080 ","End":"01:57.290","Text":"This is an example of a decimal fraction."},{"Start":"01:57.290 ","End":"01:59.375","Text":"I\u0027ll give you some more examples,"},{"Start":"01:59.375 ","End":"02:02.185","Text":"3/10 is a decimal fraction,"},{"Start":"02:02.185 ","End":"02:05.820","Text":"1/10 is a decimal fraction,"},{"Start":"02:05.820 ","End":"02:08.690","Text":"25/100, we already said,"},{"Start":"02:08.690 ","End":"02:13.474","Text":"but let\u0027s say 12/100 is a decimal fraction,"},{"Start":"02:13.474 ","End":"02:21.565","Text":"and 27/1,000 is a decimal fraction."},{"Start":"02:21.565 ","End":"02:23.885","Text":"I\u0027ll do 1 more."},{"Start":"02:23.885 ","End":"02:33.395","Text":"Let\u0027s say 113/1,000."},{"Start":"02:33.395 ","End":"02:36.110","Text":"Let\u0027s not go higher than 1,000."},{"Start":"02:36.110 ","End":"02:40.235","Text":"Now all these decimal fractions"},{"Start":"02:40.235 ","End":"02:47.360","Text":"and these are very convenient for converting into what we call a decimal."},{"Start":"02:47.360 ","End":"02:51.920","Text":"In fact, a decimal is just another way of writing a decimal fraction."},{"Start":"02:51.920 ","End":"02:54.605","Text":"I\u0027ll just show you by example how it works."},{"Start":"02:54.605 ","End":"02:59.150","Text":"We, first of all, write a 0 and a dot here."},{"Start":"02:59.150 ","End":"03:02.323","Text":"Then we count the number of 0s here,"},{"Start":"03:02.323 ","End":"03:05.585","Text":"there\u0027s only 1, so I count 1 place."},{"Start":"03:05.585 ","End":"03:07.265","Text":"Then what\u0027s on the top,"},{"Start":"03:07.265 ","End":"03:09.630","Text":"I put in here."},{"Start":"03:09.710 ","End":"03:16.500","Text":"I\u0027ll give you another example, 12/100."},{"Start":"03:16.500 ","End":"03:19.125","Text":"I put 0 point,"},{"Start":"03:19.125 ","End":"03:21.970","Text":"it\u0027s called not dot, 0 point,"},{"Start":"03:21.970 ","End":"03:23.030","Text":"and it\u0027s not in the middle,"},{"Start":"03:23.030 ","End":"03:24.980","Text":"it\u0027s at the bottom part."},{"Start":"03:24.980 ","End":"03:27.530","Text":"Because the 100 has 2 0s,"},{"Start":"03:27.530 ","End":"03:28.700","Text":"I count 1,"},{"Start":"03:28.700 ","End":"03:32.305","Text":"2 or because 100 is 10^2,"},{"Start":"03:32.305 ","End":"03:37.275","Text":"either way I count 2 and then I fill it in with the 12."},{"Start":"03:37.275 ","End":"03:40.890","Text":"Now, let\u0027s take this example."},{"Start":"03:40.890 ","End":"03:42.755","Text":"Here it\u0027s a bit different."},{"Start":"03:42.755 ","End":"03:44.540","Text":"This example is different."},{"Start":"03:44.540 ","End":"03:49.130","Text":"I start with 0 point and I see 1,000 has 3 zeros,"},{"Start":"03:49.130 ","End":"03:50.690","Text":"so I count 1, 2,"},{"Start":"03:50.690 ","End":"03:54.170","Text":"3, and 27 only has 2 places,"},{"Start":"03:54.170 ","End":"03:57.055","Text":"not 3, but you fill it in from the right."},{"Start":"03:57.055 ","End":"04:00.075","Text":"I put the 7 and the 2 here."},{"Start":"04:00.075 ","End":"04:01.819","Text":"If there\u0027s a blank,"},{"Start":"04:01.819 ","End":"04:04.189","Text":"you fill it in with an extra 0."},{"Start":"04:04.189 ","End":"04:09.725","Text":"In actual fact, 27 is 0.027."},{"Start":"04:09.725 ","End":"04:11.875","Text":"As for the others,"},{"Start":"04:11.875 ","End":"04:15.480","Text":"1/10 is just the same as 0.1,"},{"Start":"04:15.480 ","End":"04:16.590","Text":"just like this,"},{"Start":"04:16.590 ","End":"04:24.225","Text":"and 113/1,000 also, I would be taking 3 places and then filling them in with 1,"},{"Start":"04:24.225 ","End":"04:26.095","Text":"1, and 3."},{"Start":"04:26.095 ","End":"04:31.040","Text":"But this is the interesting example because it shows you"},{"Start":"04:31.040 ","End":"04:36.245","Text":"that you might have 0s after this and you have to fill from right to left."},{"Start":"04:36.245 ","End":"04:39.560","Text":"Perhaps I could have done another example."},{"Start":"04:39.560 ","End":"04:49.720","Text":"Let me replace that last example with another instructive example, like 7/100."},{"Start":"04:49.720 ","End":"04:59.790","Text":"Then once again, I take 0 point and then I count 2 places and I have to put the 7 in,"},{"Start":"04:59.790 ","End":"05:01.620","Text":"but I put it in on the right,"},{"Start":"05:01.620 ","End":"05:04.470","Text":"I fill from the right and there\u0027s a blank here."},{"Start":"05:04.470 ","End":"05:08.085","Text":"This is 0.07."},{"Start":"05:08.085 ","End":"05:14.270","Text":"I\u0027d like to add that we\u0027ve only discussed simple fractions so far."},{"Start":"05:14.270 ","End":"05:22.185","Text":"It works also with the mixed numbers."},{"Start":"05:22.185 ","End":"05:25.210","Text":"I\u0027ll just give an example,"},{"Start":"05:25.460 ","End":"05:36.880","Text":"3 1/4 is 3 and 25/100 and that is equal to 3.25."},{"Start":"05:37.070 ","End":"05:43.550","Text":"Basically the only real use for decimal fractions as far as you\u0027re concerned is like"},{"Start":"05:43.550 ","End":"05:46.759","Text":"a halfway house because we\u0027re going to learn to convert"},{"Start":"05:46.759 ","End":"05:51.515","Text":"from fractions to decimals and from decimals to fractions."},{"Start":"05:51.515 ","End":"05:56.540","Text":"In each case, we\u0027ll go through an intermediate decimal fraction."},{"Start":"05:56.540 ","End":"06:01.370","Text":"Just note the anomaly that when it\u0027s 3.25,"},{"Start":"06:01.370 ","End":"06:03.035","Text":"we write it as 3 1/4,"},{"Start":"06:03.035 ","End":"06:11.495","Text":"but when it\u0027s 0.25, we don\u0027t write it as 0 1/4."},{"Start":"06:11.495 ","End":"06:13.730","Text":"That\u0027s all I want to say in the theory part."},{"Start":"06:13.730 ","End":"06:20.580","Text":"Let\u0027s just do some sample exercises of converting from each of these forms to another."}],"ID":4872},{"Watched":false,"Name":"Decimals - Part 2","Duration":"8m 20s","ChapterTopicVideoID":4872,"CourseChapterTopicPlaylistID":3856,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:01.280 ","End":"00:06.235","Text":"Now I\u0027m going to continue with some exercises for conversion."},{"Start":"00:06.235 ","End":"00:09.370","Text":"Note that what we\u0027ve done basically is we did"},{"Start":"00:09.370 ","End":"00:14.050","Text":"the conversions from decimal fractions to decimals."},{"Start":"00:14.050 ","End":"00:16.495","Text":"This is the direction that we did."},{"Start":"00:16.495 ","End":"00:21.380","Text":"Now let\u0027s go and do some other conversions."},{"Start":"00:21.870 ","End":"00:24.820","Text":"Something I should have mentioned."},{"Start":"00:24.820 ","End":"00:28.435","Text":"We\u0027re going to mostly just be working with positive numbers."},{"Start":"00:28.435 ","End":"00:31.255","Text":"If this happens to be negative,"},{"Start":"00:31.255 ","End":"00:32.980","Text":"in negative just stays."},{"Start":"00:32.980 ","End":"00:35.880","Text":"For example, if I had minus 3 and 1/4,"},{"Start":"00:35.880 ","End":"00:39.780","Text":"it would be minus 3.25."},{"Start":"00:39.780 ","End":"00:41.830","Text":"Here also I didn\u0027t leave much room,"},{"Start":"00:41.830 ","End":"00:43.960","Text":"so I\u0027ll do it in a different color."},{"Start":"00:43.960 ","End":"00:47.570","Text":"The minuses don\u0027t really play a big part here,"},{"Start":"00:47.570 ","End":"00:51.950","Text":"so we\u0027ll just work on the positive case and the next case I\u0027m going to do"},{"Start":"00:51.950 ","End":"00:56.475","Text":"is converting from decimals to decimal fractions."},{"Start":"00:56.475 ","End":"00:58.605","Text":"Let\u0027s take some examples."},{"Start":"00:58.605 ","End":"01:05.510","Text":"Let\u0027s take 0.3 it\u0027s pretty much the same as 1 going this way."},{"Start":"01:05.510 ","End":"01:10.740","Text":"We count and we see there is 1 decimal place."},{"Start":"01:10.900 ","End":"01:15.700","Text":"This is going to be a decimal fraction with 10^1."},{"Start":"01:15.700 ","End":"01:20.735","Text":"Or just a single place, single 0."},{"Start":"01:20.735 ","End":"01:22.970","Text":"Then we just copy this bit on the top,"},{"Start":"01:22.970 ","End":"01:24.155","Text":"so it\u0027s 3/10,"},{"Start":"01:24.155 ","End":"01:25.970","Text":"so that was easy."},{"Start":"01:25.970 ","End":"01:31.530","Text":"Next example will be 0.02,"},{"Start":"01:31.530 ","End":"01:33.345","Text":"and this is equal to,"},{"Start":"01:33.345 ","End":"01:35.625","Text":"we count 1 2,"},{"Start":"01:35.625 ","End":"01:42.315","Text":"so I need a 1 with 2 0s and I put 0.02,"},{"Start":"01:42.315 ","End":"01:44.455","Text":"I\u0027ll just put the 2 on the top."},{"Start":"01:44.455 ","End":"01:47.000","Text":"Of course, if we want to cancel,"},{"Start":"01:47.000 ","End":"01:48.260","Text":"we can continue all the way,"},{"Start":"01:48.260 ","End":"01:50.720","Text":"we should really put the arrow all the way here because"},{"Start":"01:50.720 ","End":"01:53.270","Text":"we can continue if we can reduce the fraction."},{"Start":"01:53.270 ","End":"01:56.120","Text":"In this case, I can divide top and bottom by"},{"Start":"01:56.120 ","End":"02:02.270","Text":"2 and 1/50 units customer to continue and reduce."},{"Start":"02:02.270 ","End":"02:06.330","Text":"Next example, 1.012."},{"Start":"02:08.840 ","End":"02:11.295","Text":"This is equal to,"},{"Start":"02:11.295 ","End":"02:15.675","Text":"the 1 just stays and we have here,"},{"Start":"02:15.675 ","End":"02:17.715","Text":"1 2 3,"},{"Start":"02:17.715 ","End":"02:22.050","Text":"so it\u0027s 3 zeros or 1 with 3 zeros."},{"Start":"02:22.050 ","End":"02:27.630","Text":"Here we put 0.012, just 12."},{"Start":"02:27.630 ","End":"02:33.045","Text":"This is as a decimal fraction and then we continue to cancel."},{"Start":"02:33.045 ","End":"02:35.920","Text":"I can cancel top and bottom by 4,"},{"Start":"02:35.920 ","End":"02:41.390","Text":"4 will go into 12 and 4 will go into 1,000 so it will be 3 over 250."},{"Start":"02:44.810 ","End":"02:49.020","Text":"Last example 2.75."},{"Start":"02:49.020 ","End":"02:50.360","Text":"The 2 stays."},{"Start":"02:50.360 ","End":"02:54.620","Text":"We convert 0.75 to a decimal fraction."},{"Start":"02:54.620 ","End":"02:56.120","Text":"We count 2 places,"},{"Start":"02:56.120 ","End":"02:58.550","Text":"so it\u0027s 2 zeros and there\u0027s a 1 there,"},{"Start":"02:58.550 ","End":"03:01.370","Text":"of course, and the 75 goes on top."},{"Start":"03:01.370 ","End":"03:04.175","Text":"That\u0027s the answer as a decimal fraction,"},{"Start":"03:04.175 ","End":"03:07.165","Text":"we\u0027d like to reduce it."},{"Start":"03:07.165 ","End":"03:09.950","Text":"You can see the 25 will go into each of them,"},{"Start":"03:09.950 ","End":"03:14.645","Text":"25 into 75 goes 3 times,"},{"Start":"03:14.645 ","End":"03:22.100","Text":"and 25 into 100 goes 4 times so the answer is 2 and 3/4."},{"Start":"03:22.100 ","End":"03:30.230","Text":"Next we\u0027ll fill in the gap we\u0027ll convert from fractions to decimals."},{"Start":"03:30.230 ","End":"03:36.455","Text":"But we\u0027ll go through decimal fractions from here to here, through here."},{"Start":"03:36.455 ","End":"03:40.800","Text":"Out with the old and in with the new."},{"Start":"03:40.910 ","End":"03:45.380","Text":"The first 1, look at the denominator."},{"Start":"03:45.380 ","End":"03:48.770","Text":"It already is a power of 10,"},{"Start":"03:48.770 ","End":"03:51.230","Text":"so it already is a decimal fraction."},{"Start":"03:51.230 ","End":"03:53.900","Text":"We just do like we already learned."},{"Start":"03:53.900 ","End":"03:55.550","Text":"We take the 1,"},{"Start":"03:55.550 ","End":"03:57.200","Text":"put a decimal point,"},{"Start":"03:57.200 ","End":"03:59.520","Text":"and then count 1 2 3,"},{"Start":"03:59.520 ","End":"04:00.890","Text":"1 2 3,"},{"Start":"04:00.890 ","End":"04:05.880","Text":"and put the 12 in and fill in with a 0."},{"Start":"04:06.350 ","End":"04:09.105","Text":"It can dispense with this."},{"Start":"04:09.105 ","End":"04:13.759","Text":"Next 1. The denominator here is 50."},{"Start":"04:13.759 ","End":"04:17.575","Text":"I have to convert it first to a decimal fraction,"},{"Start":"04:17.575 ","End":"04:25.440","Text":"so I have to multiply it by something that will make it a power of 10."},{"Start":"04:25.540 ","End":"04:32.940","Text":"The next power of 10 is 100 and 50 does go into 100 twice,"},{"Start":"04:32.940 ","End":"04:35.950","Text":"so I multiply top and bottom by 2."},{"Start":"04:35.950 ","End":"04:38.440","Text":"Remember this is called expanding the fraction,"},{"Start":"04:38.440 ","End":"04:41.105","Text":"multiply top and bottom by 2."},{"Start":"04:41.105 ","End":"04:42.630","Text":"This times 2,"},{"Start":"04:42.630 ","End":"04:47.970","Text":"this times 2 and I get 6 over 100."},{"Start":"04:47.970 ","End":"04:53.355","Text":"This becomes 0 point so we count 2 places,"},{"Start":"04:53.355 ","End":"04:57.865","Text":"there\u0027s 6 here and there\u0027s a blank which I fill in with a 0."},{"Start":"04:57.865 ","End":"05:00.250","Text":"In this 1, I\u0027m changing my mind."},{"Start":"05:00.250 ","End":"05:02.570","Text":"I\u0027m going to change it to 7/8,"},{"Start":"05:02.570 ","End":"05:05.310","Text":"that get a little bit more challenging."},{"Start":"05:05.310 ","End":"05:09.360","Text":"Once again, I copy the whole number part"},{"Start":"05:09.360 ","End":"05:14.580","Text":"and for 7/8 I have to see if 8 will go into some power of 10."},{"Start":"05:14.580 ","End":"05:17.235","Text":"Well, it won\u0027t go into 10 and in fact,"},{"Start":"05:17.235 ","End":"05:19.464","Text":"8 doesn\u0027t even go into 100."},{"Start":"05:19.464 ","End":"05:22.170","Text":"But if we try the next 1,000,"},{"Start":"05:22.170 ","End":"05:24.765","Text":"that already works, 8 goes into 1,000,"},{"Start":"05:24.765 ","End":"05:30.285","Text":"125 times so I multiply this by 125,"},{"Start":"05:30.285 ","End":"05:33.060","Text":"and this by 125."},{"Start":"05:33.060 ","End":"05:39.075","Text":"Then I get 8 times 125 is 1,000,"},{"Start":"05:39.075 ","End":"05:48.670","Text":"7 times 125 is 875 and so our answer will be 5.875."},{"Start":"05:50.470 ","End":"05:53.525","Text":"Now finally, this 1,"},{"Start":"05:53.525 ","End":"05:55.250","Text":"which is going to be problematic,"},{"Start":"05:55.250 ","End":"05:57.064","Text":"I can tell you already,"},{"Start":"05:57.064 ","End":"06:00.595","Text":"because in the case of 8,"},{"Start":"06:00.595 ","End":"06:01.640","Text":"we went high enough."},{"Start":"06:01.640 ","End":"06:03.470","Text":"We\u0027ve got to 1,000 divided into,"},{"Start":"06:03.470 ","End":"06:04.850","Text":"in the case of 6,"},{"Start":"06:04.850 ","End":"06:08.450","Text":"no power of 10 is divisible by 6."},{"Start":"06:08.450 ","End":"06:13.295","Text":"It just doesn\u0027t go into any power of 10."},{"Start":"06:13.295 ","End":"06:15.410","Text":"We have 2 choices,"},{"Start":"06:15.410 ","End":"06:16.930","Text":"we either use the calculator,"},{"Start":"06:16.930 ","End":"06:18.430","Text":"which is the modern approach."},{"Start":"06:18.430 ","End":"06:22.690","Text":"But I feel that you should learn the classic how to do"},{"Start":"06:22.690 ","End":"06:27.700","Text":"it pen and paper style without a calculator. I\u0027m going to show you."},{"Start":"06:27.700 ","End":"06:32.154","Text":"Let me do it at the side here put a 1 of these."},{"Start":"06:32.154 ","End":"06:36.475","Text":"Sometimes this is written in around way."},{"Start":"06:36.475 ","End":"06:38.529","Text":"That\u0027s not important."},{"Start":"06:38.529 ","End":"06:39.790","Text":"Here we put the number,"},{"Start":"06:39.790 ","End":"06:42.715","Text":"we\u0027re going to divide, that\u0027s the 5."},{"Start":"06:42.715 ","End":"06:44.800","Text":"But since we\u0027re working in decimals,"},{"Start":"06:44.800 ","End":"06:50.180","Text":"we write it as 5.00 and as many zeros as we need, we can keep adding."},{"Start":"06:50.900 ","End":"06:58.435","Text":"Here I put the 6 number that\u0027s on the denominator and I say 6 into 5 doesn\u0027t go,"},{"Start":"06:58.435 ","End":"06:59.815","Text":"so it\u0027s a 0,"},{"Start":"06:59.815 ","End":"07:03.440","Text":"and the decimal point always goes above also."},{"Start":"07:03.440 ","End":"07:07.320","Text":"I ask what about 6 into 50,"},{"Start":"07:07.320 ","End":"07:11.760","Text":"6 into 50 goes 8 times,"},{"Start":"07:11.760 ","End":"07:14.100","Text":"8 times 6 is 48,"},{"Start":"07:14.100 ","End":"07:17.205","Text":"and the remainder is 2."},{"Start":"07:17.205 ","End":"07:21.035","Text":"6 doesn\u0027t go into to drop down another 0."},{"Start":"07:21.035 ","End":"07:23.815","Text":"6 goes into 20 3 times,"},{"Start":"07:23.815 ","End":"07:30.025","Text":"3 times 6 is 18, remainder is 2."},{"Start":"07:30.025 ","End":"07:33.375","Text":"Once again dropped down to 0,"},{"Start":"07:33.375 ","End":"07:38.310","Text":"6 into 20 goes 3 times"},{"Start":"07:38.310 ","End":"07:45.000","Text":"and again 3 times 6 is again 18 and so on and so on."},{"Start":"07:45.000 ","End":"07:49.295","Text":"You can see that this pattern starts repeating itself, 20,18 20 18."},{"Start":"07:49.295 ","End":"07:54.285","Text":"It\u0027s 6 in short, 0.833,"},{"Start":"07:54.285 ","End":"08:03.480","Text":"but it keeps on being 3 to infinity theoretically and that\u0027s just what happens sometimes."},{"Start":"08:03.480 ","End":"08:06.830","Text":"You can try it on the calculator you won\u0027t get an infinite number of 3s,"},{"Start":"08:06.830 ","End":"08:10.150","Text":"but it\u0027s however many is your calculator can hold."},{"Start":"08:10.150 ","End":"08:16.050","Text":"Sometimes decimals or just non-terminating and all we can do is approximate them."},{"Start":"08:17.080 ","End":"08:21.390","Text":"I am done for this lesson."}],"ID":4873},{"Watched":false,"Name":"Percent","Duration":"3m 54s","ChapterTopicVideoID":4892,"CourseChapterTopicPlaylistID":3856,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.230 ","End":"00:02.550","Text":"No, you\u0027re not in the wrong lesson."},{"Start":"00:02.550 ","End":"00:05.225","Text":"This is the lesson on percentages."},{"Start":"00:05.225 ","End":"00:07.285","Text":"Yes, percentages,"},{"Start":"00:07.285 ","End":"00:12.165","Text":"but I just thought I would reuse the examples here."},{"Start":"00:12.165 ","End":"00:14.715","Text":"I guess I don\u0027t really need this."},{"Start":"00:14.715 ","End":"00:17.820","Text":"I\u0027ll add what this is in percentages,"},{"Start":"00:17.820 ","End":"00:20.580","Text":"so you\u0027ll see where I\u0027m going in this exercise."},{"Start":"00:20.580 ","End":"00:26.280","Text":"Now, we have still another form of a fraction and it\u0027s called a percentage."},{"Start":"00:26.280 ","End":"00:32.625","Text":"A percentage is really a fraction that has a denominator of 100."},{"Start":"00:32.625 ","End":"00:35.245","Text":"25 percent, 25/100."},{"Start":"00:35.245 ","End":"00:36.800","Text":"That\u0027s the end of the lesson really,"},{"Start":"00:36.800 ","End":"00:38.210","Text":"but no, you should stay,"},{"Start":"00:38.210 ","End":"00:40.329","Text":"I\u0027m going to give some more examples."},{"Start":"00:40.329 ","End":"00:42.140","Text":"It\u0027s not really relevant,"},{"Start":"00:42.140 ","End":"00:45.860","Text":"but I\u0027ll give you a historical reason why the percent is written this way."},{"Start":"00:45.860 ","End":"00:50.705","Text":"It\u0027s an abbreviation of the word cento, which is 100."},{"Start":"00:50.705 ","End":"00:53.000","Text":"I think it\u0027s in Spanish, but I\u0027m not sure,"},{"Start":"00:53.000 ","End":"00:58.020","Text":"which was written in short cto and then it became in script,"},{"Start":"00:58.020 ","End":"01:00.300","Text":"you join it, cto."},{"Start":"01:00.300 ","End":"01:04.010","Text":"If you didn\u0027t have the little line here,"},{"Start":"01:04.010 ","End":"01:06.365","Text":"it became like this, and eventually,"},{"Start":"01:06.365 ","End":"01:08.955","Text":"just started looking like this."},{"Start":"01:08.955 ","End":"01:10.640","Text":"That\u0027s the origin of it,"},{"Start":"01:10.640 ","End":"01:13.070","Text":"but that\u0027s neither here nor there."},{"Start":"01:13.070 ","End":"01:16.800","Text":"In fact, I\u0027m going to erase it."},{"Start":"01:17.530 ","End":"01:22.075","Text":"We\u0027re going to be talking very briefly about percentages."},{"Start":"01:22.075 ","End":"01:24.220","Text":"We\u0027re not going to talk about conversion from"},{"Start":"01:24.220 ","End":"01:26.380","Text":"percent to decimal even though it\u0027s very easy,"},{"Start":"01:26.380 ","End":"01:28.324","Text":"it\u0027s not in the syllabus."},{"Start":"01:28.324 ","End":"01:31.560","Text":"That\u0027s out. Really,"},{"Start":"01:31.560 ","End":"01:35.050","Text":"I\u0027ll do a couple of examples of fraction to"},{"Start":"01:35.050 ","End":"01:38.980","Text":"percentage and a couple of examples from percentage to fraction."},{"Start":"01:38.980 ","End":"01:44.375","Text":"Here we go. Here are the 4 exercises."},{"Start":"01:44.375 ","End":"01:46.030","Text":"First 1, 50 percent,"},{"Start":"01:46.030 ","End":"01:47.485","Text":"you want it as a fraction."},{"Start":"01:47.485 ","End":"01:50.725","Text":"The first step is percentage is always over 100."},{"Start":"01:50.725 ","End":"01:52.870","Text":"In decimal, it could be any power of 10,"},{"Start":"01:52.870 ","End":"01:55.360","Text":"and percentages is specifically 100."},{"Start":"01:55.360 ","End":"01:57.700","Text":"It\u0027s 50/100."},{"Start":"01:57.700 ","End":"02:01.325","Text":"Then we do some reduction."},{"Start":"02:01.325 ","End":"02:06.370","Text":"What I see is that both top and bottom are divisible by 50."},{"Start":"02:06.370 ","End":"02:08.100","Text":"It goes into 50 once."},{"Start":"02:08.100 ","End":"02:10.750","Text":"It goes into 100 twice,"},{"Start":"02:10.750 ","End":"02:14.035","Text":"and so the answer is 1/2. Next 1."},{"Start":"02:14.035 ","End":"02:20.830","Text":"This is equal to, it\u0027s always 100 on the denominator, 75/100."},{"Start":"02:20.830 ","End":"02:23.170","Text":"Then we start canceling."},{"Start":"02:23.170 ","End":"02:27.850","Text":"This time, I see that 25 goes into top and bottom."},{"Start":"02:27.850 ","End":"02:30.840","Text":"25 goes into 75, 3 times,"},{"Start":"02:30.840 ","End":"02:33.450","Text":"25 goes into 100, 4 times,"},{"Start":"02:33.450 ","End":"02:37.020","Text":"and so the answer is 3/4."},{"Start":"02:37.020 ","End":"02:42.460","Text":"Now, the other way, from fractions to percentages."},{"Start":"02:42.980 ","End":"02:45.220","Text":"To convert to percent I,"},{"Start":"02:45.220 ","End":"02:47.290","Text":"first of all, have to go through a fraction,"},{"Start":"02:47.290 ","End":"02:50.955","Text":"which is 100 on the denominator."},{"Start":"02:50.955 ","End":"02:55.870","Text":"I say 10 into 100 goes 10 times."},{"Start":"02:55.870 ","End":"02:59.020","Text":"If I multiply the denominator by 10,"},{"Start":"02:59.020 ","End":"03:02.420","Text":"I have to also multiply the numerator by 10."},{"Start":"03:02.420 ","End":"03:05.380","Text":"That gives me 40/100,"},{"Start":"03:05.380 ","End":"03:08.515","Text":"and that means 40 percent."},{"Start":"03:08.515 ","End":"03:14.245","Text":"In the next 1, I also have to write it as something over 100."},{"Start":"03:14.245 ","End":"03:18.020","Text":"20 goes into 100 5 times."},{"Start":"03:18.710 ","End":"03:21.220","Text":"If I multiply the bottom,"},{"Start":"03:21.220 ","End":"03:23.170","Text":"I also have to multiply the top,"},{"Start":"03:23.170 ","End":"03:28.405","Text":"and 5 times 5 is 25, of course."},{"Start":"03:28.405 ","End":"03:35.000","Text":"That means that it is 25 percent."},{"Start":"03:35.000 ","End":"03:39.310","Text":"That\u0027s the answer, and we are done."},{"Start":"03:39.310 ","End":"03:43.820","Text":"Except that I forgot to mention that percentages are mostly used in word problems."},{"Start":"03:43.820 ","End":"03:49.175","Text":"For example, price is marked up or there\u0027s a discount and so on."},{"Start":"03:49.175 ","End":"03:51.410","Text":"We\u0027ll see this later on in word problems."},{"Start":"03:51.410 ","End":"03:53.820","Text":"Now I am done."}],"ID":4874},{"Watched":false,"Name":"Fractions: Addition and Subtraction - Part 1","Duration":"4m 38s","ChapterTopicVideoID":4888,"CourseChapterTopicPlaylistID":3856,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.060","Text":"In this clip, I\u0027ll be talking about the addition and subtraction of fractions."},{"Start":"00:06.060 ","End":"00:09.465","Text":"I\u0027m going to do it mostly by means of examples."},{"Start":"00:09.465 ","End":"00:14.070","Text":"I want to take a simple case when the 2 fractions have the same denominator."},{"Start":"00:14.070 ","End":"00:23.970","Text":"Let\u0027s say for example that I had 2/7 plus 4/7."},{"Start":"00:23.970 ","End":"00:25.845","Text":"If they\u0027re both 7ths,"},{"Start":"00:25.845 ","End":"00:28.545","Text":"then I just say, I have 2 of them and 4 of them."},{"Start":"00:28.545 ","End":"00:32.340","Text":"So I have 2 plus 4, 7ths."},{"Start":"00:32.340 ","End":"00:35.485","Text":"And then it\u0027s 6/7."},{"Start":"00:35.485 ","End":"00:39.545","Text":"In other words, if they have the same denominator,"},{"Start":"00:39.545 ","End":"00:43.940","Text":"we just keep that denominator and add the numerators."},{"Start":"00:43.940 ","End":"00:48.020","Text":"Now let\u0027s see what happens if they don\u0027t have the same denominator."},{"Start":"00:48.020 ","End":"00:55.250","Text":"I\u0027ll give an example, 3/4 plus 1/6."},{"Start":"00:55.250 ","End":"00:57.635","Text":"If they don\u0027t have the same denominator,"},{"Start":"00:57.635 ","End":"01:02.060","Text":"we\u0027ll have to find the denominator that we can convert them both into."},{"Start":"01:02.060 ","End":"01:07.565","Text":"Very similar to what we did when we studied the comparison of fractions,"},{"Start":"01:07.565 ","End":"01:13.130","Text":"and I suggest you review that clip and it might help here."},{"Start":"01:13.130 ","End":"01:20.029","Text":"What we\u0027re going to do is find some number that 4 goes into and 6 goes into."},{"Start":"01:20.030 ","End":"01:28.685","Text":"Now, I can see that 12 will work and 4 and 6 both go into 12."},{"Start":"01:28.685 ","End":"01:33.920","Text":"And later I\u0027ll discuss what you can do if you can\u0027t immediately see a common denominator."},{"Start":"01:33.920 ","End":"01:35.720","Text":"But let\u0027s, first of all, take 12."},{"Start":"01:35.720 ","End":"01:40.445","Text":"So 3/4 I\u0027m going to write as something over 12 and 1/6,"},{"Start":"01:40.445 ","End":"01:43.740","Text":"I\u0027m going to write as something over 12,"},{"Start":"01:43.740 ","End":"01:48.345","Text":"and then I\u0027m going to let these 2."},{"Start":"01:48.345 ","End":"01:54.720","Text":"Why don\u0027t I just straightaway erase this?"},{"Start":"01:54.720 ","End":"02:00.600","Text":"In general, we cut out a step of doing this, plus this is this."},{"Start":"02:00.600 ","End":"02:03.815","Text":"We straightaway go to saying it\u0027s something over 12,"},{"Start":"02:03.815 ","End":"02:06.410","Text":"and then it\u0027s going to be something plus something."},{"Start":"02:06.410 ","End":"02:11.330","Text":"Now 3/4 is how many 12ths?"},{"Start":"02:11.330 ","End":"02:15.500","Text":"The way we do that is this is a method I use."},{"Start":"02:15.500 ","End":"02:18.980","Text":"I say, 4 into 12 goes 3 times,"},{"Start":"02:18.980 ","End":"02:22.805","Text":"so I\u0027m also going to have to multiply the numerator by 3."},{"Start":"02:22.805 ","End":"02:25.925","Text":"6 into 12 goes twice,"},{"Start":"02:25.925 ","End":"02:29.180","Text":"so I\u0027m also going to have to multiply this by 2."},{"Start":"02:29.180 ","End":"02:31.640","Text":"Because remember, if you want to get an equivalent fraction,"},{"Start":"02:31.640 ","End":"02:35.525","Text":"you have to multiply top and bottom by the same thing."},{"Start":"02:35.525 ","End":"02:39.545","Text":"So 3 times 3 is 9,"},{"Start":"02:39.545 ","End":"02:42.740","Text":"2 times 1 is 2."},{"Start":"02:42.740 ","End":"02:46.190","Text":"Just to show to explain what this means,"},{"Start":"02:46.190 ","End":"02:51.620","Text":"this means that 3/4 is 9/12 and 1/6 is 2/12,"},{"Start":"02:51.620 ","End":"02:54.185","Text":"and we\u0027re adding them now like we did above."},{"Start":"02:54.185 ","End":"02:58.645","Text":"Our answer will be 11 over 12."},{"Start":"02:58.645 ","End":"03:06.035","Text":"Now I said that I\u0027d help you if case you can\u0027t find the smallest common denominator."},{"Start":"03:06.035 ","End":"03:09.330","Text":"Let\u0027s start the same exercise again."},{"Start":"03:10.810 ","End":"03:15.530","Text":"Now if you can\u0027t find some number that they both go into"},{"Start":"03:15.530 ","End":"03:20.795","Text":"this 1 surefire foolproof way of doing it is to multiply these 2."},{"Start":"03:20.795 ","End":"03:25.140","Text":"Can always say 4 times 6 is 24."},{"Start":"03:25.700 ","End":"03:29.090","Text":"You might ask, why don\u0027t I always do that?"},{"Start":"03:29.090 ","End":"03:32.690","Text":"Well, you\u0027ll see because we basically work with larger numbers and"},{"Start":"03:32.690 ","End":"03:36.380","Text":"more chance of making a mistake is sometimes this can be much bigger than this."},{"Start":"03:36.380 ","End":"03:39.140","Text":"This time it\u0027s just twice as much as it needed to be."},{"Start":"03:39.140 ","End":"03:42.560","Text":"In this case, we use the same system as above."},{"Start":"03:42.560 ","End":"03:44.965","Text":"We\u0027d say 4 into 24,"},{"Start":"03:44.965 ","End":"03:49.845","Text":"6 times; 6 into 24, 4 times;"},{"Start":"03:49.845 ","End":"03:54.330","Text":"6 times 3 is 18;"},{"Start":"03:54.330 ","End":"03:57.060","Text":"4 times 1 is 4,"},{"Start":"03:57.060 ","End":"04:04.250","Text":"and then we add these and we get 22 over 24."},{"Start":"04:04.250 ","End":"04:07.700","Text":"But this fraction is not reduced."},{"Start":"04:07.700 ","End":"04:12.815","Text":"I can see that top and bottom are both even so I can certainly at least divide by 2."},{"Start":"04:12.815 ","End":"04:14.750","Text":"If I divide this by 2,"},{"Start":"04:14.750 ","End":"04:19.040","Text":"I get 11 and this by 2, I get 12."},{"Start":"04:19.040 ","End":"04:22.550","Text":"This is also equal to 11 over 12."},{"Start":"04:22.550 ","End":"04:27.140","Text":"We\u0027d be in trouble if the same exercise gave 2 different answers."},{"Start":"04:27.140 ","End":"04:30.335","Text":"You can do either."},{"Start":"04:30.335 ","End":"04:35.435","Text":"I recommend in general trying to find the smallest common denominator as possible."},{"Start":"04:35.435 ","End":"04:38.430","Text":"Now let\u0027s do some exercises."}],"ID":4875},{"Watched":false,"Name":"Fractions: Addition and Subtraction - Part 2","Duration":"5m 19s","ChapterTopicVideoID":4889,"CourseChapterTopicPlaylistID":3856,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.535","Text":"I wrote the exercises in advance to save time."},{"Start":"00:05.535 ","End":"00:07.260","Text":"There\u0027s 4 here,"},{"Start":"00:07.260 ","End":"00:09.710","Text":"but there will be 8 in all, will be another 4."},{"Start":"00:09.710 ","End":"00:12.750","Text":"You need the practice."},{"Start":"00:12.750 ","End":"00:15.105","Text":"Let\u0027s start with the first."},{"Start":"00:15.105 ","End":"00:19.740","Text":"Here is the lucky case that they both have the same denominator,"},{"Start":"00:19.740 ","End":"00:23.985","Text":"so all we do is put them over that denominator,"},{"Start":"00:23.985 ","End":"00:27.810","Text":"which is 4, and then 1 plus 3."},{"Start":"00:27.810 ","End":"00:31.155","Text":"1 plus 3 is 4,"},{"Start":"00:31.155 ","End":"00:35.930","Text":"4 over 4, and at the end we cancel."},{"Start":"00:35.930 ","End":"00:38.390","Text":"You can easily see that we cancel both by 4,"},{"Start":"00:38.390 ","End":"00:41.480","Text":"that this is just equal to 1 over 1, which is 1."},{"Start":"00:41.480 ","End":"00:44.510","Text":"I mean, 4 quarters is 1 whole."},{"Start":"00:44.510 ","End":"00:49.020","Text":"The next 1,"},{"Start":"00:49.880 ","End":"00:53.954","Text":"we want to look for a common denominator of 2 and 4."},{"Start":"00:53.954 ","End":"00:56.495","Text":"Now, like I said,"},{"Start":"00:56.495 ","End":"00:58.250","Text":"if you don\u0027t want to think at all,"},{"Start":"00:58.250 ","End":"01:00.980","Text":"you can always say 2 times 4 is 8,"},{"Start":"01:00.980 ","End":"01:06.350","Text":"but I do recommend looking for the smallest common denominator."},{"Start":"01:06.350 ","End":"01:08.207","Text":"We have 1 possibility which is 8,"},{"Start":"01:08.207 ","End":"01:10.075","Text":"but let\u0027s see if we can improve on 8."},{"Start":"01:10.075 ","End":"01:13.000","Text":"Now, whenever you get 1 in hand,"},{"Start":"01:13.000 ","End":"01:15.280","Text":"if you want to improve it, you just divide it by something."},{"Start":"01:15.280 ","End":"01:16.900","Text":"What can we divide 8 by?"},{"Start":"01:16.900 ","End":"01:19.345","Text":"Let\u0027s try dividing it by 2 and then we get 4."},{"Start":"01:19.345 ","End":"01:21.292","Text":"Yes, 4 will work."},{"Start":"01:21.292 ","End":"01:23.260","Text":"Because 4 goes into 4, that\u0027s obvious,"},{"Start":"01:23.260 ","End":"01:25.240","Text":"and 2 goes into 4, so we\u0027re going to take 4."},{"Start":"01:25.240 ","End":"01:28.060","Text":"But if you used 8,"},{"Start":"01:28.060 ","End":"01:30.430","Text":"you should get the same answer in the end."},{"Start":"01:30.430 ","End":"01:34.765","Text":"You just have some canceling to do and a bit of larger numbers in the middle."},{"Start":"01:34.765 ","End":"01:39.123","Text":"I\u0027m going to use the system and even the same color as before."},{"Start":"01:39.123 ","End":"01:41.710","Text":"2 into 4 goes twice,"},{"Start":"01:41.710 ","End":"01:43.735","Text":"so I write that 2 over here,"},{"Start":"01:43.735 ","End":"01:45.610","Text":"4 into 4 goes once,"},{"Start":"01:45.610 ","End":"01:47.855","Text":"so I write that 1 over here,"},{"Start":"01:47.855 ","End":"01:50.880","Text":"2 times 5 is 10,"},{"Start":"01:50.880 ","End":"01:54.040","Text":"1 times 7 is 7."},{"Start":"01:54.040 ","End":"01:59.450","Text":"I add the numbers in the numerator 17 over 4."},{"Start":"01:59.450 ","End":"02:02.465","Text":"I can\u0027t reduce it,"},{"Start":"02:02.465 ","End":"02:07.339","Text":"but it is an improper fraction and so I\u0027d like to convert it to a mixed number."},{"Start":"02:07.339 ","End":"02:08.550","Text":"What do we do?"},{"Start":"02:08.550 ","End":"02:11.700","Text":"4 into 17 goes 4 times,"},{"Start":"02:11.700 ","End":"02:13.188","Text":"remainder is 1,"},{"Start":"02:13.188 ","End":"02:15.045","Text":"because 4 times 4 is 16,"},{"Start":"02:15.045 ","End":"02:17.790","Text":"and then I put it over the same 4 as here,"},{"Start":"02:17.790 ","End":"02:20.110","Text":"so 4 and 1/4."},{"Start":"02:20.990 ","End":"02:29.393","Text":"Next 1. I think you get the idea that we don\u0027t have to multiply them."},{"Start":"02:29.393 ","End":"02:31.215","Text":"That 8, in fact,"},{"Start":"02:31.215 ","End":"02:33.980","Text":"it will serve as a common denominator for all of them,"},{"Start":"02:33.980 ","End":"02:35.112","Text":"because they all go into 8,"},{"Start":"02:35.112 ","End":"02:37.096","Text":"so let\u0027s put this over 8."},{"Start":"02:37.096 ","End":"02:40.220","Text":"There\u0027s 3 terms here,"},{"Start":"02:40.220 ","End":"02:44.180","Text":"but not much more difficult. Same idea."},{"Start":"02:44.180 ","End":"02:48.470","Text":"2 into 8 goes 4 times,"},{"Start":"02:48.470 ","End":"02:52.160","Text":"4 into 8 goes twice,"},{"Start":"02:52.160 ","End":"02:55.130","Text":"8 into 8 goes once."},{"Start":"02:55.130 ","End":"02:59.250","Text":"Let\u0027s see, 4 times 3 is 12,"},{"Start":"02:59.250 ","End":"03:02.850","Text":"2 times 1 is 2, it\u0027s a plus,"},{"Start":"03:02.850 ","End":"03:06.450","Text":"1 times 5 is 5 but it\u0027s a minus,"},{"Start":"03:06.450 ","End":"03:08.820","Text":"so 12 plus 2 is 14,"},{"Start":"03:08.820 ","End":"03:10.335","Text":"minus 5 is 9,"},{"Start":"03:10.335 ","End":"03:12.840","Text":"it\u0027s 9 over 8."},{"Start":"03:12.840 ","End":"03:14.580","Text":"I would like to reduce,"},{"Start":"03:14.580 ","End":"03:18.010","Text":"and so this is 1 and 1/8."},{"Start":"03:18.540 ","End":"03:24.640","Text":"In this 1, the art is to find the common denominator."},{"Start":"03:24.640 ","End":"03:30.370","Text":"Now, the worst-case scenario is you multiply all 3 together."},{"Start":"03:30.370 ","End":"03:34.310","Text":"The best case is if you can find the smallest common denominator,"},{"Start":"03:34.310 ","End":"03:39.595","Text":"and I can see because I have experienced that 18 will be the smallest."},{"Start":"03:39.595 ","End":"03:41.920","Text":"But you could go for a middle approach."},{"Start":"03:41.920 ","End":"03:43.540","Text":"Suppose you said, well,"},{"Start":"03:43.540 ","End":"03:45.130","Text":"3 goes into 9 and 3 goes into 6."},{"Start":"03:45.130 ","End":"03:48.660","Text":"So 3 is already taken care of,"},{"Start":"03:48.660 ","End":"03:52.885","Text":"and you could just multiply 9 times 6 and get 54."},{"Start":"03:52.885 ","End":"03:58.900","Text":"That would certainly be better than 54 times 3 is 162,"},{"Start":"03:58.900 ","End":"04:01.640","Text":"but 54 is not the best you can do."},{"Start":"04:01.640 ","End":"04:04.700","Text":"Even if you have 54,"},{"Start":"04:04.700 ","End":"04:08.480","Text":"you can try dividing it by 2 is 27, no, that won\u0027t work,"},{"Start":"04:08.480 ","End":"04:10.340","Text":"54 over 3 is 18, yeah,"},{"Start":"04:10.340 ","End":"04:12.860","Text":"that\u0027ll work, whatever."},{"Start":"04:12.860 ","End":"04:16.685","Text":"But if you chose 54 or even if you chose 162,"},{"Start":"04:16.685 ","End":"04:18.875","Text":"you should get to the same answer."},{"Start":"04:18.875 ","End":"04:20.690","Text":"I\u0027m sticking with the smallest."},{"Start":"04:20.690 ","End":"04:25.385","Text":"It\u0027s worth spending a bit of time thinking about getting its smallest here."},{"Start":"04:25.385 ","End":"04:31.385","Text":"Let\u0027s do our usual thing of seeing how many times each goes into 18,"},{"Start":"04:31.385 ","End":"04:33.530","Text":"3 into 18,"},{"Start":"04:33.530 ","End":"04:37.415","Text":"6, 9 into 18,"},{"Start":"04:37.415 ","End":"04:43.545","Text":"twice, 6 into 18, 3 times."},{"Start":"04:43.545 ","End":"04:46.290","Text":"6 times 2 is 12,"},{"Start":"04:46.290 ","End":"04:49.245","Text":"2 times 5 is 10,"},{"Start":"04:49.245 ","End":"04:51.300","Text":"3 times 1 is 3."},{"Start":"04:51.300 ","End":"04:53.670","Text":"Here\u0027s the plus, here\u0027s the minus,"},{"Start":"04:53.670 ","End":"04:56.655","Text":"let\u0027s see, 12 plus 10 is 22,"},{"Start":"04:56.655 ","End":"05:00.795","Text":"22 minus 3 is 19,"},{"Start":"05:00.795 ","End":"05:04.500","Text":"and it\u0027s still over 18."},{"Start":"05:04.500 ","End":"05:09.830","Text":"Finally, let\u0027s make it a mixed number,"},{"Start":"05:09.830 ","End":"05:15.485","Text":"18 into 19 goes once and 1 left over, 1 and 18th."},{"Start":"05:15.485 ","End":"05:17.405","Text":"That\u0027s these 4 done,"},{"Start":"05:17.405 ","End":"05:19.710","Text":"on to the next 4."}],"ID":4876},{"Watched":false,"Name":"Fractions: Addition and Subtraction - Part 3","Duration":"12s","ChapterTopicVideoID":4890,"CourseChapterTopicPlaylistID":3856,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.875","Text":"You know what, guys?"},{"Start":"00:01.875 ","End":"00:04.095","Text":"I think 4 exercises were enough."},{"Start":"00:04.095 ","End":"00:05.340","Text":"I\u0027ll skip these."},{"Start":"00:05.340 ","End":"00:09.090","Text":"I\u0027ll throw them into the exercises that follow the tutorial."},{"Start":"00:09.090 ","End":"00:12.220","Text":"We\u0027re done, and that\u0027s it."}],"ID":4877},{"Watched":false,"Name":"Exercise 1","Duration":"3m 1s","ChapterTopicVideoID":4873,"CourseChapterTopicPlaylistID":3856,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.360","Text":"In this exercise, I basically have to convert from improper fractions to mixed numbers."},{"Start":"00:06.360 ","End":"00:09.180","Text":"Each of these is an improper fraction because you can see that"},{"Start":"00:09.180 ","End":"00:13.020","Text":"the numerator is bigger than the denominator in each case."},{"Start":"00:13.020 ","End":"00:19.155","Text":"I\u0027m going to leave the original and just work on the solution,"},{"Start":"00:19.155 ","End":"00:21.910","Text":"which I copied here."},{"Start":"00:22.160 ","End":"00:26.475","Text":"I\u0027d like to start with B. I have a reason."},{"Start":"00:26.475 ","End":"00:31.890","Text":"In B, the way we do the conversion is we do a division with remainder."},{"Start":"00:31.890 ","End":"00:39.410","Text":"What I do is I say that 8 divided by 5 is how many times and what is the remainder?"},{"Start":"00:39.410 ","End":"00:41.915","Text":"Well, it goes in only 1 times,"},{"Start":"00:41.915 ","End":"00:49.145","Text":"and the remainder is 3, 3 leftover."},{"Start":"00:49.145 ","End":"00:52.055","Text":"Now, I take these numbers,"},{"Start":"00:52.055 ","End":"00:55.033","Text":"so I write the 1 here,"},{"Start":"00:55.033 ","End":"00:58.049","Text":"and then I take the remainder and write it here,"},{"Start":"00:58.049 ","End":"00:59.240","Text":"and over here,"},{"Start":"00:59.240 ","End":"01:01.520","Text":"I write the original denominator."},{"Start":"01:01.520 ","End":"01:09.620","Text":"This is the process that I take whatever is written here, I write here."},{"Start":"01:09.620 ","End":"01:16.325","Text":"Whatever is here, goes to the denominator,"},{"Start":"01:16.325 ","End":"01:21.680","Text":"and the remainder is what goes here."},{"Start":"01:21.680 ","End":"01:23.353","Text":"This works like that every time."},{"Start":"01:23.353 ","End":"01:25.850","Text":"I\u0027m not going to be highlighting every time."},{"Start":"01:25.850 ","End":"01:28.415","Text":"Let\u0027s go and do A now."},{"Start":"01:28.415 ","End":"01:33.705","Text":"What I do is 3 divided by 2,"},{"Start":"01:33.705 ","End":"01:35.550","Text":"2 goes into 3,"},{"Start":"01:35.550 ","End":"01:40.660","Text":"1 time, and remainder is 1."},{"Start":"01:40.660 ","End":"01:44.270","Text":"The reason I did B first is because all the numbers are different here."},{"Start":"01:44.270 ","End":"01:45.470","Text":"I get the 1 twice. You know what,"},{"Start":"01:45.470 ","End":"01:48.950","Text":"I\u0027ll just highlight this 1 so we can identify it."},{"Start":"01:48.950 ","End":"01:50.660","Text":"Because the answer now,"},{"Start":"01:50.660 ","End":"01:52.700","Text":"just like here is we take this,"},{"Start":"01:52.700 ","End":"01:55.450","Text":"the 1 that was in green, 1,"},{"Start":"01:55.450 ","End":"01:58.595","Text":"and then we take the remainder,"},{"Start":"01:58.595 ","End":"02:00.260","Text":"which is the other 1."},{"Start":"02:00.260 ","End":"02:01.310","Text":"It appears in the same order."},{"Start":"02:01.310 ","End":"02:03.170","Text":"It\u0027s like 1 and 3, 1 and 3,"},{"Start":"02:03.170 ","End":"02:04.520","Text":"but here, it\u0027s 1 and 1."},{"Start":"02:04.520 ","End":"02:08.210","Text":"Here, the 2 from here goes here."},{"Start":"02:08.210 ","End":"02:11.225","Text":"This is 1 and 1/2, this 1 is this 1,"},{"Start":"02:11.225 ","End":"02:13.180","Text":"and the other 1 is the other 1."},{"Start":"02:13.180 ","End":"02:17.685","Text":"Number C, 13 divided by 2."},{"Start":"02:17.685 ","End":"02:20.490","Text":"How many times this 2 go into 13?"},{"Start":"02:20.490 ","End":"02:22.890","Text":"Goes in 6 times."},{"Start":"02:22.890 ","End":"02:24.900","Text":"6 times, that makes it 12,"},{"Start":"02:24.900 ","End":"02:29.130","Text":"there\u0027s 1 leftover, so the remainder is 1."},{"Start":"02:29.130 ","End":"02:32.730","Text":"Now, it\u0027s easy. This 6 is this 6."},{"Start":"02:32.730 ","End":"02:37.670","Text":"This 1 goes here and what goes in the denominator is what was here,"},{"Start":"02:37.670 ","End":"02:40.785","Text":"or that\u0027s the original denominator, 6 and 1/2."},{"Start":"02:40.785 ","End":"02:44.475","Text":"Next 1, 18 divided by 4,"},{"Start":"02:44.475 ","End":"02:46.170","Text":"4 goes into 18."},{"Start":"02:46.170 ","End":"02:47.360","Text":"It goes 4 times,"},{"Start":"02:47.360 ","End":"02:51.500","Text":"that gives me 16, and the remainder is 2."},{"Start":"02:51.500 ","End":"02:53.735","Text":"It\u0027s this 4 here,"},{"Start":"02:53.735 ","End":"02:58.690","Text":"this 2 here, and the 4 here."},{"Start":"02:58.690 ","End":"03:02.700","Text":"So it\u0027s 4 and 2/4. We\u0027re done."}],"ID":4878},{"Watched":false,"Name":"Exercise 2","Duration":"2m 4s","ChapterTopicVideoID":4874,"CourseChapterTopicPlaylistID":3856,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.860","Text":"This exercise is a reverse of the previous exercise."},{"Start":"00:04.860 ","End":"00:09.810","Text":"Here we have to convert mixed numbers to improper fractions."},{"Start":"00:09.810 ","End":"00:12.495","Text":"As usual, I like to work on a copy,"},{"Start":"00:12.495 ","End":"00:15.645","Text":"so here I am down here."},{"Start":"00:15.645 ","End":"00:19.035","Text":"I\u0027m just going to show you the how and not the why."},{"Start":"00:19.035 ","End":"00:22.800","Text":"Basically, I will draw a little diagram here."},{"Start":"00:22.800 ","End":"00:24.315","Text":"I connect these two,"},{"Start":"00:24.315 ","End":"00:25.815","Text":"and then these two."},{"Start":"00:25.815 ","End":"00:30.105","Text":"What we\u0027re going to do is take this and multiply,"},{"Start":"00:30.105 ","End":"00:34.560","Text":"I\u0027ll mark that with a dot, by this,"},{"Start":"00:34.560 ","End":"00:39.045","Text":"and then I\u0027m going to add an arrow here and then add this."},{"Start":"00:39.045 ","End":"00:40.970","Text":"I\u0027ll show you what I mean. In our case,"},{"Start":"00:40.970 ","End":"00:50.760","Text":"what we get is 8 times 2 plus 3,"},{"Start":"00:50.760 ","End":"00:53.675","Text":"that comes out, in our case, to be 19."},{"Start":"00:53.675 ","End":"00:58.700","Text":"I write the 19 down and then on the denominator,"},{"Start":"00:58.700 ","End":"01:00.350","Text":"I put the original denominator,"},{"Start":"01:00.350 ","End":"01:01.670","Text":"which is 8,"},{"Start":"01:01.670 ","End":"01:03.649","Text":"and that\u0027s the answer,"},{"Start":"01:03.649 ","End":"01:05.585","Text":"which I will highlight."},{"Start":"01:05.585 ","End":"01:08.930","Text":"Let\u0027s try it again on the next one,"},{"Start":"01:08.930 ","End":"01:10.230","Text":"keeping the same pattern."},{"Start":"01:10.230 ","End":"01:12.180","Text":"This times this plus this,"},{"Start":"01:12.180 ","End":"01:16.815","Text":"5 times 12 plus 2,"},{"Start":"01:16.815 ","End":"01:20.130","Text":"5 times 12 is 60 plus 2 is 62,"},{"Start":"01:20.130 ","End":"01:23.700","Text":"I write the 62 on the top of the fraction in the numerator,"},{"Start":"01:23.700 ","End":"01:25.880","Text":"and on the denominator,"},{"Start":"01:25.880 ","End":"01:28.430","Text":"the same as it was there,"},{"Start":"01:28.430 ","End":"01:30.820","Text":"that\u0027s another one solved."},{"Start":"01:30.820 ","End":"01:34.740","Text":"Next one, once again,"},{"Start":"01:34.740 ","End":"01:36.930","Text":"this times this plus this,"},{"Start":"01:36.930 ","End":"01:40.455","Text":"2 times 6 plus 1,"},{"Start":"01:40.455 ","End":"01:42.615","Text":"12 plus 1 is 13,"},{"Start":"01:42.615 ","End":"01:43.980","Text":"and on the bottom,"},{"Start":"01:43.980 ","End":"01:46.590","Text":"the same 2 as it was there,"},{"Start":"01:46.590 ","End":"01:48.750","Text":"that\u0027s the third one solved."},{"Start":"01:48.750 ","End":"01:50.220","Text":"Onto the last one,"},{"Start":"01:50.220 ","End":"01:54.615","Text":"4 times 8 plus 1,"},{"Start":"01:54.615 ","End":"01:56.595","Text":"which is 33,"},{"Start":"01:56.595 ","End":"01:59.895","Text":"over the same 4 that was here,"},{"Start":"01:59.895 ","End":"02:02.010","Text":"and that\u0027s this one done,"},{"Start":"02:02.010 ","End":"02:04.840","Text":"so we\u0027re altogether done."}],"ID":4879},{"Watched":false,"Name":"Exercise 3","Duration":"3m 39s","ChapterTopicVideoID":4875,"CourseChapterTopicPlaylistID":3856,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.730","Text":"In this exercise, we\u0027re given a set of pairs of fractions and in each pair,"},{"Start":"00:05.730 ","End":"00:08.340","Text":"we have to say which one is bigger."},{"Start":"00:08.340 ","End":"00:11.939","Text":"I copied the exercises down here,"},{"Start":"00:11.939 ","End":"00:13.725","Text":"I just removed the word or,"},{"Start":"00:13.725 ","End":"00:15.120","Text":"so at the end,"},{"Start":"00:15.120 ","End":"00:19.540","Text":"I can put a greater than or less than sign in between."},{"Start":"00:19.540 ","End":"00:22.020","Text":"Let\u0027s take the first one."},{"Start":"00:22.020 ","End":"00:26.325","Text":"The first one has the same denominator,"},{"Start":"00:26.325 ","End":"00:30.090","Text":"and the rule is that if they have the same denominator,"},{"Start":"00:30.090 ","End":"00:32.505","Text":"then you just compare numerators."},{"Start":"00:32.505 ","End":"00:37.920","Text":"Now, since 5 is most certainly bigger than 3,"},{"Start":"00:37.920 ","End":"00:42.120","Text":"here, we also put a bigger than sign."},{"Start":"00:42.120 ","End":"00:46.175","Text":"You\u0027re lucky if you have the same denominator, not much work."},{"Start":"00:46.175 ","End":"00:50.735","Text":"This is not the case in b and c, however. Let\u0027s see."},{"Start":"00:50.735 ","End":"00:54.365","Text":"In b, however, we do have the same numerator."},{"Start":"00:54.365 ","End":"00:57.594","Text":"Then it\u0027s kind of a reverse rule,"},{"Start":"00:57.594 ","End":"00:59.629","Text":"you compare the denominators,"},{"Start":"00:59.629 ","End":"01:04.025","Text":"and certainly, 5 is less than 7."},{"Start":"01:04.025 ","End":"01:07.970","Text":"But because it\u0027s a denominator comparison,"},{"Start":"01:07.970 ","End":"01:09.875","Text":"the rule is that you reverse;"},{"Start":"01:09.875 ","End":"01:12.925","Text":"not less than, but greater than."},{"Start":"01:12.925 ","End":"01:15.030","Text":"That\u0027s not a typographical error,"},{"Start":"01:15.030 ","End":"01:17.055","Text":"that\u0027s a deliberate reversal."},{"Start":"01:17.055 ","End":"01:20.765","Text":"If you compare numerators with same denominator, same sign."},{"Start":"01:20.765 ","End":"01:25.174","Text":"Compare denominators with same numerator, you reverse."},{"Start":"01:25.174 ","End":"01:27.585","Text":"In number c,"},{"Start":"01:27.585 ","End":"01:32.420","Text":"neither numerators are different and denominators are different."},{"Start":"01:32.420 ","End":"01:36.380","Text":"So what we do is we convert, well,"},{"Start":"01:36.380 ","End":"01:43.940","Text":"we do an expansion of fractions to put them both on the same denominator."},{"Start":"01:43.940 ","End":"01:51.710","Text":"Let\u0027s see if we can convert 3/4 and 4/5 to have the same denominator."},{"Start":"01:51.710 ","End":"01:55.895","Text":"One way to get a common denominator is just to multiply them."},{"Start":"01:55.895 ","End":"01:57.830","Text":"So I\u0027m going to write a 20 here,"},{"Start":"01:57.830 ","End":"01:59.390","Text":"which is 4 times 5,"},{"Start":"01:59.390 ","End":"02:01.650","Text":"and a 20 here."},{"Start":"02:02.120 ","End":"02:09.285","Text":"Then what we do is we say 4 into 20 goes how many times?"},{"Start":"02:09.285 ","End":"02:12.690","Text":"To get from this 4 to this 20,"},{"Start":"02:12.690 ","End":"02:15.120","Text":"I have to multiply by 5."},{"Start":"02:15.120 ","End":"02:17.680","Text":"Remember, from here to here is 5,"},{"Start":"02:17.680 ","End":"02:20.810","Text":"so from here to here also has to be 5."},{"Start":"02:20.810 ","End":"02:24.790","Text":"I do 3 times 5 and I get 15."},{"Start":"02:24.790 ","End":"02:27.620","Text":"3/4 is the same as 15 over 20,"},{"Start":"02:27.620 ","End":"02:33.755","Text":"just expanded, multiplying numerator and denominator by the same number, 5."},{"Start":"02:33.755 ","End":"02:38.360","Text":"In the second one, we have to see what we do to 5 to get 20."},{"Start":"02:38.360 ","End":"02:40.790","Text":"Well, we multiply it by 4, clearly."},{"Start":"02:40.790 ","End":"02:43.445","Text":"From here to here, I multiply by 4."},{"Start":"02:43.445 ","End":"02:46.445","Text":"From here to here, I also have to multiply by 4."},{"Start":"02:46.445 ","End":"02:48.740","Text":"4 times 4 is 16."},{"Start":"02:48.740 ","End":"02:52.570","Text":"That\u0027s a rewrite of the second fraction by expanding it."},{"Start":"02:52.570 ","End":"02:58.530","Text":"Now, I have 2 fractions with the same denominator and it\u0027s"},{"Start":"02:58.530 ","End":"03:00.915","Text":"easy to see which is"},{"Start":"03:00.915 ","End":"03:05.180","Text":"bigger or which is smaller because I just look at the numerators and I say,"},{"Start":"03:05.180 ","End":"03:08.990","Text":"well, 15 is less than 16."},{"Start":"03:08.990 ","End":"03:12.040","Text":"That must mean that this is less than this."},{"Start":"03:12.040 ","End":"03:14.160","Text":"Because these are equal,"},{"Start":"03:14.160 ","End":"03:16.750","Text":"this is less than this."},{"Start":"03:16.750 ","End":"03:20.810","Text":"Now, finally, I just want to answer the question."},{"Start":"03:20.810 ","End":"03:25.380","Text":"The question says which of them is biggest?"},{"Start":"03:25.380 ","End":"03:28.190","Text":"In this case, because this is bigger than this,"},{"Start":"03:28.190 ","End":"03:29.590","Text":"this is the biggest."},{"Start":"03:29.590 ","End":"03:33.697","Text":"Here also, the first one\u0027s the biggest."},{"Start":"03:33.697 ","End":"03:34.730","Text":"In the third one,"},{"Start":"03:34.730 ","End":"03:36.065","Text":"the second is the biggest."},{"Start":"03:36.065 ","End":"03:39.510","Text":"This is the biggest in each case, and we\u0027re done."}],"ID":4880},{"Watched":false,"Name":"Exercise 4","Duration":"4m 1s","ChapterTopicVideoID":4876,"CourseChapterTopicPlaylistID":3856,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.895","Text":"In this exercise, we have to convert the following decimals to fractions."},{"Start":"00:05.895 ","End":"00:10.125","Text":"As usual, I like to copy the questions."},{"Start":"00:10.125 ","End":"00:12.285","Text":"Here we are,"},{"Start":"00:12.285 ","End":"00:14.505","Text":"have some space now."},{"Start":"00:14.505 ","End":"00:17.250","Text":"If the decimal begins with a 0,"},{"Start":"00:17.250 ","End":"00:19.350","Text":"it\u0027s 0 point something,"},{"Start":"00:19.350 ","End":"00:21.535","Text":"then it\u0027s even easier."},{"Start":"00:21.535 ","End":"00:25.920","Text":"Let me start with the second one to make a point."},{"Start":"00:25.920 ","End":"00:30.750","Text":"What we do is we count the number of places after the decimal."},{"Start":"00:30.750 ","End":"00:33.420","Text":"I would count here 1, 2."},{"Start":"00:33.420 ","End":"00:35.870","Text":"Now that I know that there\u0027s 2 places,"},{"Start":"00:35.870 ","End":"00:40.155","Text":"I take a fraction and I put a 1 with 2 zeros,"},{"Start":"00:40.155 ","End":"00:43.830","Text":"that actually 10^2 and that\u0027s where the 2 comes in,"},{"Start":"00:43.830 ","End":"00:46.010","Text":"not this 2, the 2 that I\u0027m counting."},{"Start":"00:46.010 ","End":"00:49.610","Text":"Then, I just take the number that appears here,"},{"Start":"00:49.610 ","End":"00:51.520","Text":"which is 2 and put it here,"},{"Start":"00:51.520 ","End":"00:53.405","Text":"and that\u0027s all there is to it."},{"Start":"00:53.405 ","End":"00:55.445","Text":"Let\u0027s go back to the first one."},{"Start":"00:55.445 ","End":"00:59.720","Text":"In this case, there\u0027s only 1 place after the decimal."},{"Start":"00:59.720 ","End":"01:01.250","Text":"So on the denominator,"},{"Start":"01:01.250 ","End":"01:05.615","Text":"I put a 1 and only 1 zero, that\u0027s 10^1."},{"Start":"01:05.615 ","End":"01:07.955","Text":"On the numerator, I put a 3."},{"Start":"01:07.955 ","End":"01:10.045","Text":"That\u0027s the answer."},{"Start":"01:10.045 ","End":"01:15.860","Text":"I\u0027ll just highlight them, this and this."},{"Start":"01:15.860 ","End":"01:18.785","Text":"Now, here I have a 1."},{"Start":"01:18.785 ","End":"01:24.155","Text":"The only difference is is that I do this as a mixed fraction and I write 1."},{"Start":"01:24.155 ","End":"01:25.850","Text":"Then, I use the same trick,"},{"Start":"01:25.850 ","End":"01:27.865","Text":"1, 2, 3."},{"Start":"01:27.865 ","End":"01:33.210","Text":"3 means 1 with 3 zeros and the 12,"},{"Start":"01:33.210 ","End":"01:38.805","Text":"I copy here, and that\u0027s the answer. I\u0027ll highlight it."},{"Start":"01:38.805 ","End":"01:43.830","Text":"Last one. Here we have the 2,"},{"Start":"01:43.830 ","End":"01:45.585","Text":"we copy as the 2,"},{"Start":"01:45.585 ","End":"01:47.805","Text":"and then the decimal part, 1, 2."},{"Start":"01:47.805 ","End":"01:52.140","Text":"So it\u0027s over 2 zeros and a 1."},{"Start":"01:52.140 ","End":"01:56.195","Text":"Then I copy the 75 above."},{"Start":"01:56.195 ","End":"01:59.375","Text":"I don\u0027t copy the 2 or the 1 here, just the decimal part."},{"Start":"01:59.375 ","End":"02:02.940","Text":"That\u0027s the answer to the last 1."},{"Start":"02:02.940 ","End":"02:05.145","Text":"But not so fast."},{"Start":"02:05.145 ","End":"02:08.300","Text":"Usually, you\u0027re expected to simplify,"},{"Start":"02:08.300 ","End":"02:16.986","Text":"that is to reduce the fraction to lower terms."},{"Start":"02:16.986 ","End":"02:21.935","Text":"What we\u0027re going to do is take a look at each one and see if we can cancel anything."},{"Start":"02:21.935 ","End":"02:24.005","Text":"This, no. This,"},{"Start":"02:24.005 ","End":"02:27.215","Text":"yes, 2 is a common factor."},{"Start":"02:27.215 ","End":"02:32.120","Text":"If I want to cancel it,"},{"Start":"02:32.120 ","End":"02:38.365","Text":"I can say, \"2 goes into 100 50 times.\""},{"Start":"02:38.365 ","End":"02:46.720","Text":"What we get is 1/50."},{"Start":"02:46.720 ","End":"02:48.550","Text":"I divided top and bottom by 2."},{"Start":"02:48.550 ","End":"02:52.270","Text":"You can divide or multiply top and bottom by the same thing."},{"Start":"02:52.270 ","End":"02:57.780","Text":"Here, also I can reduce."},{"Start":"02:57.780 ","End":"03:02.520","Text":"I can say that 4 goes into 12 3 times,"},{"Start":"03:02.520 ","End":"03:06.585","Text":"4 goes into 1,000 250 times,"},{"Start":"03:06.585 ","End":"03:13.720","Text":"and so I write my answer as 1 and 3/250."},{"Start":"03:15.290 ","End":"03:18.495","Text":"In the case of the last one,"},{"Start":"03:18.495 ","End":"03:21.060","Text":"certainly we can, 25."},{"Start":"03:21.060 ","End":"03:22.470","Text":"This is 3 times 25."},{"Start":"03:22.470 ","End":"03:24.290","Text":"This is 4 times 25."},{"Start":"03:24.290 ","End":"03:27.395","Text":"If I take out 25, I have 3 and 4,"},{"Start":"03:27.395 ","End":"03:34.605","Text":"and so the answer is 2 and 3/4."},{"Start":"03:34.605 ","End":"03:36.945","Text":"This one just I leave it as is,"},{"Start":"03:36.945 ","End":"03:38.531","Text":"3/10, nothing to cancel."},{"Start":"03:38.531 ","End":"03:43.160","Text":"That\u0027s another set of answers after I\u0027ve reduced."},{"Start":"03:43.160 ","End":"03:44.750","Text":"I\u0027ll just those put those in green."},{"Start":"03:44.750 ","End":"03:48.350","Text":"Those are better. In fact,"},{"Start":"03:48.350 ","End":"03:51.620","Text":"really, we expect to have the answers reduced."},{"Start":"03:51.620 ","End":"03:55.760","Text":"I\u0027m going to just erase the highlighting of"},{"Start":"03:55.760 ","End":"04:01.650","Text":"these and say that these are the answers, and we\u0027re done."}],"ID":4881},{"Watched":false,"Name":"Exercise 5","Duration":"2m 35s","ChapterTopicVideoID":4877,"CourseChapterTopicPlaylistID":3856,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.170","Text":"This exercise is like the previous one, but in reverse."},{"Start":"00:04.170 ","End":"00:08.940","Text":"Here we have conversion from fractions to decimals."},{"Start":"00:08.940 ","End":"00:13.635","Text":"I\u0027ll just copy the exercises to this solution part."},{"Start":"00:13.635 ","End":"00:17.879","Text":"Get some more space. Let\u0027s begin."},{"Start":"00:17.879 ","End":"00:20.280","Text":"I\u0027ll start at b,"},{"Start":"00:20.280 ","End":"00:25.350","Text":"it\u0027s more typical. Let\u0027s see."},{"Start":"00:25.350 ","End":"00:29.955","Text":"We have a 100 in the denominator and all of them we have powers of 10,"},{"Start":"00:29.955 ","End":"00:33.090","Text":"10, 100, 1000 and again 1000."},{"Start":"00:33.090 ","End":"00:34.890","Text":"The 100 has 2 zeros,"},{"Start":"00:34.890 ","End":"00:36.600","Text":"that\u0027s the thing to do is you count the number of"},{"Start":"00:36.600 ","End":"00:38.965","Text":"zeros and I see that there\u0027s two of them."},{"Start":"00:38.965 ","End":"00:45.660","Text":"What I do is I start off with a decimal point and I mark off two places,"},{"Start":"00:45.660 ","End":"00:49.770","Text":"and then I write what\u0027s in the numerator,"},{"Start":"00:49.880 ","End":"00:53.910","Text":"starting in this place here."},{"Start":"00:53.910 ","End":"00:55.145","Text":"If there\u0027s any blanks,"},{"Start":"00:55.145 ","End":"00:56.840","Text":"I fill them in with 0,"},{"Start":"00:56.840 ","End":"01:02.810","Text":"so the answer is .01 and it\u0027s okay to leave like this,"},{"Start":"01:02.810 ","End":"01:05.915","Text":"but it\u0027s customary to write a leading 0 as well."},{"Start":"01:05.915 ","End":"01:07.385","Text":"It does nothing in front,"},{"Start":"01:07.385 ","End":"01:12.870","Text":"so 0.01, and I\u0027ll just highlight that."},{"Start":"01:13.640 ","End":"01:17.265","Text":"Now back to the first exercise."},{"Start":"01:17.265 ","End":"01:22.800","Text":"This time, there\u0027s only one 0 here,"},{"Start":"01:22.800 ","End":"01:29.870","Text":"so what I do is I put the decimal point and only go one place from the decimal point,"},{"Start":"01:29.870 ","End":"01:31.735","Text":"and that\u0027s where I put the 1."},{"Start":"01:31.735 ","End":"01:36.310","Text":"So the answer will be 0.1."},{"Start":"01:38.150 ","End":"01:42.150","Text":"Here, we count again 1,"},{"Start":"01:42.150 ","End":"01:44.270","Text":"2, 3 zeros."},{"Start":"01:44.270 ","End":"01:49.466","Text":"So I take a decimal point and count 1, 2,"},{"Start":"01:49.466 ","End":"01:55.910","Text":"3, and this is where I put the 3 and blanks I fill in with 0."},{"Start":"01:55.910 ","End":"01:59.900","Text":"So the answer, some people write it as .003,"},{"Start":"01:59.900 ","End":"02:04.260","Text":"but we\u0027ll write it as 0.003."},{"Start":"02:05.440 ","End":"02:10.610","Text":"Finally, we have the same denominators,"},{"Start":"02:10.610 ","End":"02:13.250","Text":"this 1000, which means 3 places."},{"Start":"02:13.250 ","End":"02:17.050","Text":"So here\u0027s the decimal point and 1, 2, 3."},{"Start":"02:17.050 ","End":"02:20.780","Text":"Now the 12 fits in so the last digit,"},{"Start":"02:20.780 ","End":"02:21.830","Text":"goes in the last place,"},{"Start":"02:21.830 ","End":"02:23.480","Text":"the 12 goes in here."},{"Start":"02:23.480 ","End":"02:27.812","Text":"This blank gets a 0 and then we put a leading 0."},{"Start":"02:27.812 ","End":"02:32.975","Text":"The answer is 0.012."},{"Start":"02:32.975 ","End":"02:36.660","Text":"We\u0027re done."}],"ID":4882},{"Watched":false,"Name":"Exercise 6","Duration":"2m 53s","ChapterTopicVideoID":4878,"CourseChapterTopicPlaylistID":3856,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.050 ","End":"00:09.406","Text":"In this exercise,"},{"Start":"00:09.406 ","End":"00:13.443","Text":"we have to convert the following fractions to decimals."},{"Start":"00:13.443 ","End":"00:14.571","Text":"This is sort of a fraction. It\u0027s a mixed fraction, it\u0027s a mixed number. The way"},{"Start":"00:14.571 ","End":"00:16.770","Text":"we convert it as we keep the leading 1"},{"Start":"00:16.770 ","End":"00:22.520","Text":"and we convert 12 over 1,000 to decimal just like we used to do,"},{"Start":"00:22.520 ","End":"00:24.270","Text":"so we count 1, 2,"},{"Start":"00:24.270 ","End":"00:26.640","Text":"3 zeros, so we count 1,"},{"Start":"00:26.640 ","End":"00:28.335","Text":"2, 3 places."},{"Start":"00:28.335 ","End":"00:31.995","Text":"We squeeze the 12 in so it just fits in at the end,"},{"Start":"00:31.995 ","End":"00:33.540","Text":"and if there\u0027s any blanks,"},{"Start":"00:33.540 ","End":"00:36.078","Text":"then they get filled in with a 0,"},{"Start":"00:36.078 ","End":"00:41.085","Text":"and so the answer is 1.012,"},{"Start":"00:41.085 ","End":"00:44.120","Text":"and I\u0027ll highlight it."},{"Start":"00:44.120 ","End":"00:47.120","Text":"The next one is something else."},{"Start":"00:47.120 ","End":"00:51.680","Text":"The problem here is that the denominator should be a power of"},{"Start":"00:51.680 ","End":"00:55.790","Text":"10 like a 100 or a 1,000 and so on, but it isn\u0027t,"},{"Start":"00:55.790 ","End":"01:00.080","Text":"so we\u0027re just going to expand the fraction by multiplying top and"},{"Start":"01:00.080 ","End":"01:05.170","Text":"bottom by the same thing to make this denominator a power of 10."},{"Start":"01:05.170 ","End":"01:08.247","Text":"The closest is 100,"},{"Start":"01:08.247 ","End":"01:11.135","Text":"and we can multiply top and bottom by 2."},{"Start":"01:11.135 ","End":"01:13.700","Text":"If we do that, on the bottom,"},{"Start":"01:13.700 ","End":"01:18.860","Text":"we get 100 because we multiplied it by 2 and multiply the top, we get 6."},{"Start":"01:18.860 ","End":"01:21.320","Text":"Now that we have it in this form,"},{"Start":"01:21.320 ","End":"01:24.395","Text":"we can write it as, there is no whole part,"},{"Start":"01:24.395 ","End":"01:28.850","Text":"so it\u0027s just 0 point and then there\u0027s two places, 1, 2,"},{"Start":"01:28.850 ","End":"01:30.455","Text":"we put the 6 in here,"},{"Start":"01:30.455 ","End":"01:36.720","Text":"fill the blank with 0, so this is just 0.06."},{"Start":"01:37.300 ","End":"01:40.655","Text":"Similarly, in this one,"},{"Start":"01:40.655 ","End":"01:43.385","Text":"the denominator is not a power of 10,"},{"Start":"01:43.385 ","End":"01:47.480","Text":"but again we can make it to 100 by multiplying by 5."},{"Start":"01:47.480 ","End":"01:50.120","Text":"So multiply top and bottom by 5,"},{"Start":"01:50.120 ","End":"01:51.470","Text":"100 on the bottom,"},{"Start":"01:51.470 ","End":"01:55.150","Text":"5 times 7 is 35 on the top."},{"Start":"01:55.150 ","End":"01:59.025","Text":"0 point, there\u0027s two places, 1, 2."},{"Start":"01:59.025 ","End":"02:01.410","Text":"Put the 35 in."},{"Start":"02:01.410 ","End":"02:02.760","Text":"There are no blanks."},{"Start":"02:02.760 ","End":"02:04.440","Text":"So that\u0027s just the answer,"},{"Start":"02:04.440 ","End":"02:12.375","Text":"0.35 and highlight it."},{"Start":"02:12.375 ","End":"02:17.070","Text":"Last one, 12 over 25."},{"Start":"02:17.070 ","End":"02:19.130","Text":"It looks like for the third time,"},{"Start":"02:19.130 ","End":"02:21.650","Text":"we\u0027re going to have to multiply it to get to 100,"},{"Start":"02:21.650 ","End":"02:26.360","Text":"but don\u0027t go thinking it\u0027s always 100, it could be 1,000."},{"Start":"02:26.360 ","End":"02:28.850","Text":"Anyway, to make it 100,"},{"Start":"02:28.850 ","End":"02:30.815","Text":"we multiply it by 4,"},{"Start":"02:30.815 ","End":"02:33.680","Text":"so we have to multiply the top also by 4."},{"Start":"02:33.680 ","End":"02:36.785","Text":"That gives us 12 times 4 is 48."},{"Start":"02:36.785 ","End":"02:42.410","Text":"Once again, two decimal places because of the two notes here,"},{"Start":"02:42.410 ","End":"02:44.855","Text":"48 goes in here,"},{"Start":"02:44.855 ","End":"02:52.500","Text":"so 0.48, and that\u0027s it."}],"ID":4883},{"Watched":false,"Name":"Exercise 7","Duration":"2m 12s","ChapterTopicVideoID":4879,"CourseChapterTopicPlaylistID":3856,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.370","Text":"In this exercise, we have to convert some percentages to fractions."},{"Start":"00:05.370 ","End":"00:08.295","Text":"As usual, I like to work on a copy."},{"Start":"00:08.295 ","End":"00:12.730","Text":"Here it is, and so we have more room."},{"Start":"00:12.740 ","End":"00:16.860","Text":"Remember, a percentage means part of 100."},{"Start":"00:16.860 ","End":"00:24.370","Text":"What we do in each of them is we say that this is equal to 50 out of 100."},{"Start":"00:25.340 ","End":"00:28.260","Text":"Then we can leave this as the answer,"},{"Start":"00:28.260 ","End":"00:30.120","Text":"but it\u0027s customary to cancel."},{"Start":"00:30.120 ","End":"00:33.230","Text":"Let\u0027s see if we can reduce the fraction."},{"Start":"00:33.230 ","End":"00:35.855","Text":"Well, 50 goes into the top and the bottom,"},{"Start":"00:35.855 ","End":"00:41.580","Text":"so 50 over 50 is 1 and 100 over 50 is 2."},{"Start":"00:41.580 ","End":"00:44.795","Text":"The answer is simply 1/2."},{"Start":"00:44.795 ","End":"00:47.565","Text":"Just let me highlight that, and that\u0027s it."},{"Start":"00:47.565 ","End":"00:51.310","Text":"Now next one, 25 percent,"},{"Start":"00:51.310 ","End":"00:56.600","Text":"so we just write it as 25 over 100."},{"Start":"00:56.600 ","End":"00:58.565","Text":"Then again, we cancel."},{"Start":"00:58.565 ","End":"01:00.500","Text":"100 is 4 times 25,"},{"Start":"01:00.500 ","End":"01:02.990","Text":"everyone knows, so I can cancel top and bottom."},{"Start":"01:02.990 ","End":"01:04.775","Text":"I reduce it by 25,"},{"Start":"01:04.775 ","End":"01:07.235","Text":"and I get 1 over 4."},{"Start":"01:07.235 ","End":"01:09.290","Text":"Which is the way I prefer to give the answer."},{"Start":"01:09.290 ","End":"01:11.015","Text":"Although technically, this is correct,"},{"Start":"01:11.015 ","End":"01:14.615","Text":"it\u0027s just like not going all the way."},{"Start":"01:14.615 ","End":"01:20.120","Text":"75 percent, 75 over 100."},{"Start":"01:20.120 ","End":"01:25.640","Text":"Once again, 25 goes into numerator and denominator."},{"Start":"01:25.640 ","End":"01:27.470","Text":"If we divide everything by 25,"},{"Start":"01:27.470 ","End":"01:31.145","Text":"it goes 3 times into that and 4 times into that."},{"Start":"01:31.145 ","End":"01:34.270","Text":"The answer is just 3/4."},{"Start":"01:34.270 ","End":"01:37.180","Text":"Lastly, we have 15 percent,"},{"Start":"01:37.180 ","End":"01:39.830","Text":"which is 15 over 100."},{"Start":"01:39.830 ","End":"01:42.530","Text":"It\u0027s always over 100 if it\u0027s percent."},{"Start":"01:42.530 ","End":"01:44.705","Text":"That\u0027s equal to now,"},{"Start":"01:44.705 ","End":"01:46.385","Text":"what can I divide by?"},{"Start":"01:46.385 ","End":"01:51.185","Text":"If you think about it, 5 will go into both of them, but no bigger."},{"Start":"01:51.185 ","End":"01:57.775","Text":"If I divide by 5 and I sometimes like to show cancellation when with a line through it,"},{"Start":"01:57.775 ","End":"02:01.080","Text":"5 goes into this 3 times,"},{"Start":"02:01.080 ","End":"02:03.485","Text":"5 goes into this 20 times,"},{"Start":"02:03.485 ","End":"02:08.960","Text":"and we end up with 3 over 20,"},{"Start":"02:08.960 ","End":"02:12.990","Text":"which is the answer to the last part, and we\u0027re done."}],"ID":4884},{"Watched":false,"Name":"Exercise 8","Duration":"1m 20s","ChapterTopicVideoID":4880,"CourseChapterTopicPlaylistID":3856,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.980","Text":"In this exercise, we have to convert these fractions to percentages."},{"Start":"00:04.980 ","End":"00:09.610","Text":"I generally like to work on a copy."},{"Start":"00:09.770 ","End":"00:13.230","Text":"Percentages means part of a 100,"},{"Start":"00:13.230 ","End":"00:17.490","Text":"so I\u0027d like to change this fraction to be something over a 100."},{"Start":"00:17.490 ","End":"00:20.910","Text":"I already know that it\u0027s equal to something over a 100,"},{"Start":"00:20.910 ","End":"00:23.265","Text":"I just have to figure out what to put here."},{"Start":"00:23.265 ","End":"00:25.500","Text":"By expansion of fractions,"},{"Start":"00:25.500 ","End":"00:29.100","Text":"I see that 10 has to be multiplied by 10 to get 100,"},{"Start":"00:29.100 ","End":"00:30.930","Text":"that\u0027s easy to see."},{"Start":"00:30.930 ","End":"00:33.300","Text":"If I multiply the denominator by 10,"},{"Start":"00:33.300 ","End":"00:38.275","Text":"I have to also multiply the numerator by 10 so 4 times 10 is 40."},{"Start":"00:38.275 ","End":"00:42.920","Text":"Once I have a fraction whose denominator is 100, that\u0027s the percentage."},{"Start":"00:42.920 ","End":"00:48.600","Text":"That is just 40 percent and I\u0027ll highlight that."},{"Start":"00:48.680 ","End":"00:52.519","Text":"Next, I have 5 over 20."},{"Start":"00:52.519 ","End":"00:56.980","Text":"Again, I\u0027d like to write this fraction as something over a 100."},{"Start":"00:56.980 ","End":"01:01.020","Text":"What I say to myself is how many times does 20 go into a 100."},{"Start":"01:01.020 ","End":"01:03.010","Text":"That\u0027s easy to see, that\u0027s 5."},{"Start":"01:03.010 ","End":"01:06.200","Text":"This 5 I have to also multiply it by the numerator here,"},{"Start":"01:06.200 ","End":"01:09.335","Text":"so 5 times 5 is 25."},{"Start":"01:09.335 ","End":"01:11.795","Text":"Once I have a fraction over a 100,"},{"Start":"01:11.795 ","End":"01:15.080","Text":"then I just take that number 25 and write"},{"Start":"01:15.080 ","End":"01:20.490","Text":"the percentage sign after it and that\u0027s the answer to this one, and we\u0027re done."}],"ID":4885},{"Watched":false,"Name":"Exercise 9","Duration":"5m 32s","ChapterTopicVideoID":4881,"CourseChapterTopicPlaylistID":3856,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.150","Text":"Here, we have to calculate the following expressions."},{"Start":"00:03.150 ","End":"00:06.809","Text":"As you can see, they will involve addition and subtraction of fractions."},{"Start":"00:06.809 ","End":"00:10.480","Text":"I just would like to copy the exercise down here."},{"Start":"00:11.180 ","End":"00:13.830","Text":"Let\u0027s get started."},{"Start":"00:13.830 ","End":"00:17.025","Text":"We have to find a common denominator."},{"Start":"00:17.025 ","End":"00:19.530","Text":"If all the denominators are the same,"},{"Start":"00:19.530 ","End":"00:20.775","Text":"then that\u0027s the simplest."},{"Start":"00:20.775 ","End":"00:24.900","Text":"In this case, that is the common denominator and the way to do it is"},{"Start":"00:24.900 ","End":"00:29.100","Text":"make a dividing line and put that common 4 here,"},{"Start":"00:29.100 ","End":"00:33.915","Text":"and then we just copy the numerator 1 plus 3."},{"Start":"00:33.915 ","End":"00:40.835","Text":"The answer is 1 plus 3, which is 4/4."},{"Start":"00:40.835 ","End":"00:44.660","Text":"Although technically I could leave the answer like that,"},{"Start":"00:44.660 ","End":"00:47.675","Text":"it\u0027s usual to reduce the fraction."},{"Start":"00:47.675 ","End":"00:52.410","Text":"We see that the 4 is a common factor to both."},{"Start":"00:52.410 ","End":"00:59.335","Text":"We can divide top and bottom by 4 and get 1/1 and 1/1 is just 1."},{"Start":"00:59.335 ","End":"01:02.059","Text":"On to the next 1."},{"Start":"01:02.059 ","End":"01:04.550","Text":"This time it\u0027s not the same denominator,"},{"Start":"01:04.550 ","End":"01:06.860","Text":"so we have to find a common denominator,"},{"Start":"01:06.860 ","End":"01:12.535","Text":"which is a number that both 2 and 4 will go into, but the least."},{"Start":"01:12.535 ","End":"01:17.720","Text":"Although 8 is a possible common denominator, it\u0027s not the smallest."},{"Start":"01:17.720 ","End":"01:19.550","Text":"In fact, 4 if you look at it,"},{"Start":"01:19.550 ","End":"01:23.000","Text":"will actually work because both 2 and 4 go into 4."},{"Start":"01:23.000 ","End":"01:26.705","Text":"When we\u0027ve chosen that as a common denominator,"},{"Start":"01:26.705 ","End":"01:29.850","Text":"what we have to do is make adjustments."},{"Start":"01:30.230 ","End":"01:35.660","Text":"What we do is ask ourselves 2 into 4 goes how many times?"},{"Start":"01:35.660 ","End":"01:36.890","Text":"The answer to that is 2,"},{"Start":"01:36.890 ","End":"01:39.140","Text":"so I write a little 2 somewhere here."},{"Start":"01:39.140 ","End":"01:42.435","Text":"4 into 4 goes once,"},{"Start":"01:42.435 ","End":"01:44.065","Text":"so I write a little 1."},{"Start":"01:44.065 ","End":"01:50.645","Text":"Then what I do is on the numerator here I say 2 times 5 is 10,"},{"Start":"01:50.645 ","End":"01:56.130","Text":"1 times 7 is 7 and the plus is just the plus."},{"Start":"01:56.130 ","End":"02:01.020","Text":"What I have now is 17/4."},{"Start":"02:01.020 ","End":"02:03.240","Text":"This is the answer,"},{"Start":"02:03.240 ","End":"02:05.180","Text":"but we\u0027re going to work on it further because this is"},{"Start":"02:05.180 ","End":"02:09.590","Text":"an improper fraction and I\u0027d like to leave my answer as a mixed number."},{"Start":"02:09.590 ","End":"02:18.280","Text":"As usual, what I do is division with remainder and I say 4 into 17 goes 4 times."},{"Start":"02:18.280 ","End":"02:22.555","Text":"That gives me 16 already and there\u0027s still 1 leftover,"},{"Start":"02:22.555 ","End":"02:24.390","Text":"so it\u0027s 4 and 1/4."},{"Start":"02:24.390 ","End":"02:25.670","Text":"If you\u0027re not sure about this,"},{"Start":"02:25.670 ","End":"02:33.710","Text":"I suggest you review the section on converting improper fractions to mixed numbers."},{"Start":"02:33.710 ","End":"02:36.065","Text":"This is our answer,"},{"Start":"02:36.065 ","End":"02:38.614","Text":"and on to the next 1."},{"Start":"02:38.614 ","End":"02:46.115","Text":"Once again, the main thing is to find the least common denominator."},{"Start":"02:46.115 ","End":"02:48.950","Text":"In this case, if you think about it,"},{"Start":"02:48.950 ","End":"02:54.155","Text":"8 is good enough for all of them because 2 and 4 already go into 8."},{"Start":"02:54.155 ","End":"02:55.490","Text":"Just like in the previous 1,"},{"Start":"02:55.490 ","End":"03:00.330","Text":"we were lucky that 1 of them actually is divisible by all the rest."},{"Start":"03:00.330 ","End":"03:06.680","Text":"Denominator will be 8 and now using that same accounting method as before,"},{"Start":"03:06.680 ","End":"03:07.970","Text":"what I do is for each of them,"},{"Start":"03:07.970 ","End":"03:12.525","Text":"I say 2 into 8 goes 4 times and I record that,"},{"Start":"03:12.525 ","End":"03:15.420","Text":"4 into 8 goes twice,"},{"Start":"03:15.420 ","End":"03:18.825","Text":"8 into 8 goes just once."},{"Start":"03:18.825 ","End":"03:26.045","Text":"What we get now is we multiply 4 times 3 is 12."},{"Start":"03:26.045 ","End":"03:29.715","Text":"I\u0027ll just stop to explain why I just do the 4 times 3,"},{"Start":"03:29.715 ","End":"03:34.205","Text":"you don\u0027t have to know why, but it\u0027s because 2 into 8 goes 4 times."},{"Start":"03:34.205 ","End":"03:36.560","Text":"If I multiply the denominator by 4,"},{"Start":"03:36.560 ","End":"03:38.795","Text":"I also have to multiply the numerator by 4."},{"Start":"03:38.795 ","End":"03:39.995","Text":"That\u0027s the reasoning, but,"},{"Start":"03:39.995 ","End":"03:41.824","Text":"of course, you could just do it mechanically."},{"Start":"03:41.824 ","End":"03:44.430","Text":"2 times 1 is 2,"},{"Start":"03:44.430 ","End":"03:47.535","Text":"1 times 5 is 5."},{"Start":"03:47.535 ","End":"03:51.485","Text":"I miscopied, it should be a minus."},{"Start":"03:51.485 ","End":"03:54.559","Text":"This arithmetic is simple enough not to need a calculator."},{"Start":"03:54.559 ","End":"03:56.192","Text":"12 plus 2 is 14,"},{"Start":"03:56.192 ","End":"04:01.265","Text":"14 minus 5 is 9 so our answer is 9/8."},{"Start":"04:01.265 ","End":"04:06.230","Text":"Once again, it\u0027s an improper fraction and I would like it as a mixed number,"},{"Start":"04:06.230 ","End":"04:13.330","Text":"so 8 into 9 goes once and there\u0027s 1 leftover so we get 1 and 1/8."},{"Start":"04:13.330 ","End":"04:16.910","Text":"In the last 1, I just realized I miscopied the exercise,"},{"Start":"04:16.910 ","End":"04:21.210","Text":"it should be a 3. Excuse me for that."},{"Start":"04:21.210 ","End":"04:25.600","Text":"Anyway, we have to find the common denominator, preferably the least."},{"Start":"04:25.600 ","End":"04:27.880","Text":"If you look at these numbers,"},{"Start":"04:27.880 ","End":"04:30.730","Text":"easy to see or maybe not so easy,"},{"Start":"04:30.730 ","End":"04:33.820","Text":"but if you think about it, 18 will be the smallest that they\u0027ll all go"},{"Start":"04:33.820 ","End":"04:37.890","Text":"into so I put this over 18."},{"Start":"04:37.890 ","End":"04:44.625","Text":"As usual, we have a system where we say 3 into 18 goes 6 times."},{"Start":"04:44.625 ","End":"04:48.690","Text":"9 into 18 goes twice,"},{"Start":"04:48.690 ","End":"04:53.250","Text":"6 into 18 goes 3 times,"},{"Start":"04:53.250 ","End":"04:57.210","Text":"and then we do some multiplications and addings,"},{"Start":"04:57.210 ","End":"05:01.695","Text":"6 times 2 is 12."},{"Start":"05:01.695 ","End":"05:05.955","Text":"Then there\u0027s a plus. 2 times 5 is 10,"},{"Start":"05:05.955 ","End":"05:07.440","Text":"and there\u0027s a minus,"},{"Start":"05:07.440 ","End":"05:10.635","Text":"3 times 1 is 3,"},{"Start":"05:10.635 ","End":"05:17.200","Text":"12 plus 10 is 22 minus 3 gives us 19/18,"},{"Start":"05:17.200 ","End":"05:19.490","Text":"which is an improper fraction."},{"Start":"05:19.490 ","End":"05:21.590","Text":"We see that 18 goes into 19,"},{"Start":"05:21.590 ","End":"05:28.340","Text":"1 and there\u0027s a remainder of 1 so the answer is 1 and 1/18,"},{"Start":"05:28.340 ","End":"05:32.160","Text":"1 and an 18th if you like and we\u0027re done."}],"ID":4886},{"Watched":false,"Name":"Exercise 10","Duration":"8m 34s","ChapterTopicVideoID":4882,"CourseChapterTopicPlaylistID":3856,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.405","Text":"This is another set of exercises involving addition and subtraction of fractions."},{"Start":"00:06.405 ","End":"00:14.715","Text":"But this time, we have also mixed numbers and not just simple fractions."},{"Start":"00:14.715 ","End":"00:20.170","Text":"As usual, I like to copy the exercises down here."},{"Start":"00:20.170 ","End":"00:25.100","Text":"Here we are. We have more space now."},{"Start":"00:25.100 ","End":"00:28.205","Text":"Let\u0027s start with the first 1."},{"Start":"00:28.205 ","End":"00:35.790","Text":"As usual, the tricky part is to find the common denominator."},{"Start":"00:36.010 ","End":"00:38.450","Text":"If you think about it,"},{"Start":"00:38.450 ","End":"00:44.840","Text":"you\u0027ll see that 60 will serve as the common denominator and less won\u0027t do."},{"Start":"00:44.840 ","End":"00:50.230","Text":"We put a 60 on the denominator and what we do,"},{"Start":"00:50.230 ","End":"00:54.019","Text":"if you recall, is we ask ourselves,"},{"Start":"00:54.019 ","End":"00:57.200","Text":"4 into 60 goes how many times?"},{"Start":"00:57.200 ","End":"00:58.980","Text":"The answer is 15,"},{"Start":"00:58.980 ","End":"01:04.395","Text":"and 6 into 60 goes 10 times,"},{"Start":"01:04.395 ","End":"01:08.730","Text":"5 into 60, 12 times."},{"Start":"01:08.730 ","End":"01:11.280","Text":"Now, we just do some multiplications,"},{"Start":"01:11.280 ","End":"01:15.105","Text":"15 times 3 is 45."},{"Start":"01:15.105 ","End":"01:16.680","Text":"Then there\u0027s a minus here."},{"Start":"01:16.680 ","End":"01:18.420","Text":"I put a minus here,"},{"Start":"01:18.420 ","End":"01:21.390","Text":"10 times 5 is 50,"},{"Start":"01:21.390 ","End":"01:30.670","Text":"and 12 times 7 is 84, so plus 84."},{"Start":"01:32.000 ","End":"01:34.815","Text":"Now some arithmetic."},{"Start":"01:34.815 ","End":"01:42.935","Text":"If I do this computation I get 79 over 60 but this is an improper fraction."},{"Start":"01:42.935 ","End":"01:44.900","Text":"The numerator is bigger than the denominator"},{"Start":"01:44.900 ","End":"01:47.030","Text":"and I\u0027d like to convert it to a mixed number."},{"Start":"01:47.030 ","End":"01:52.195","Text":"The usual technique, 60 into 79 only goes once."},{"Start":"01:52.195 ","End":"01:55.950","Text":"The leftover, the remainder is 19."},{"Start":"01:55.950 ","End":"02:00.055","Text":"The answer is 1 and 19/60."},{"Start":"02:00.055 ","End":"02:03.575","Text":"That\u0027s our answer for the first exercise."},{"Start":"02:03.575 ","End":"02:06.230","Text":"Now, on to the second."},{"Start":"02:06.230 ","End":"02:09.890","Text":"In this 1, we have a subtraction of fractions,"},{"Start":"02:09.890 ","End":"02:12.260","Text":"but the first 1 is a mixed number."},{"Start":"02:12.260 ","End":"02:16.144","Text":"We\u0027ll convert that to a fraction, an improper fraction."},{"Start":"02:16.144 ","End":"02:18.100","Text":"What we say is the following,"},{"Start":"02:18.100 ","End":"02:21.660","Text":"8 times 1 plus 1 is 9."},{"Start":"02:21.660 ","End":"02:23.880","Text":"The first 1 is 9/8."},{"Start":"02:23.880 ","End":"02:28.235","Text":"Then the other 1 already is a fraction, so it\u0027s 11/12."},{"Start":"02:28.235 ","End":"02:33.685","Text":"Now, we\u0027re back to the usual situation and we have to find a common denominator."},{"Start":"02:33.685 ","End":"02:38.005","Text":"Of course, you could multiply it by 12 but we would like the least common denominator."},{"Start":"02:38.005 ","End":"02:42.950","Text":"If you look at it a bit, you\u0027ll see that 24 is certainly going to be good enough."},{"Start":"02:42.950 ","End":"02:45.980","Text":"We\u0027ll put everything over 24."},{"Start":"02:45.980 ","End":"02:51.455","Text":"Now we start with the technique of asking how many times does 8 go into 24?"},{"Start":"02:51.455 ","End":"02:53.005","Text":"That is 3 times."},{"Start":"02:53.005 ","End":"02:55.435","Text":"How many times does 12 go into 24?"},{"Start":"02:55.435 ","End":"02:57.445","Text":"The answer is twice."},{"Start":"02:57.445 ","End":"02:59.530","Text":"Now, we have some multiplications,"},{"Start":"02:59.530 ","End":"03:06.700","Text":"3 times 9 is 27 and 2 times 11 is 22,"},{"Start":"03:06.700 ","End":"03:08.560","Text":"but there\u0027s a minus in the middle,"},{"Start":"03:08.560 ","End":"03:11.935","Text":"27 minus 22 is 5."},{"Start":"03:11.935 ","End":"03:16.924","Text":"The answer is 5/24 for this 1."},{"Start":"03:16.924 ","End":"03:22.460","Text":"In this 1, we have a mixed number and we also have a whole number."},{"Start":"03:22.460 ","End":"03:25.040","Text":"Let\u0027s see what we do about those."},{"Start":"03:25.040 ","End":"03:27.080","Text":"The mixed number, like in the previous,"},{"Start":"03:27.080 ","End":"03:28.670","Text":"we convert to an improper fraction,"},{"Start":"03:28.670 ","End":"03:32.340","Text":"9 times 1 plus 1 is 10."},{"Start":"03:32.340 ","End":"03:34.665","Text":"We have 10/9."},{"Start":"03:34.665 ","End":"03:37.215","Text":"Next 1 is just as is,"},{"Start":"03:37.215 ","End":"03:40.775","Text":"it\u0027s fraction already, 23/27."},{"Start":"03:40.775 ","End":"03:42.260","Text":"When we have a whole number,"},{"Start":"03:42.260 ","End":"03:44.225","Text":"we can just write it as over 1."},{"Start":"03:44.225 ","End":"03:47.180","Text":"I\u0027ll write 2, is 2 divided by 1."},{"Start":"03:47.180 ","End":"03:52.880","Text":"Now, the usual technique is we have to find a least common denominator."},{"Start":"03:52.880 ","End":"03:55.070","Text":"Everything goes into 27,"},{"Start":"03:55.070 ","End":"03:59.060","Text":"so we\u0027ll choose 27 as our least common denominator."},{"Start":"03:59.060 ","End":"04:01.970","Text":"Let\u0027s see how many times each goes into 27."},{"Start":"04:01.970 ","End":"04:05.860","Text":"Well, 9 goes into 27 3 times,"},{"Start":"04:05.860 ","End":"04:09.045","Text":"27 goes into 27 just once,"},{"Start":"04:09.045 ","End":"04:13.780","Text":"and 1 goes into 27 27 times."},{"Start":"04:14.600 ","End":"04:18.785","Text":"What we get now is set of multiplications,"},{"Start":"04:18.785 ","End":"04:22.730","Text":"3 times 10 is 30,"},{"Start":"04:22.730 ","End":"04:25.460","Text":"1 times 23 is 23,"},{"Start":"04:25.460 ","End":"04:26.480","Text":"and there\u0027s a minus there."},{"Start":"04:26.480 ","End":"04:28.490","Text":"Next there\u0027s a plus,"},{"Start":"04:28.490 ","End":"04:32.515","Text":"27 times 2 is 54."},{"Start":"04:32.515 ","End":"04:34.940","Text":"Let\u0027s do the arithmetic."},{"Start":"04:34.940 ","End":"04:40.880","Text":"30 minus 23 is 7,"},{"Start":"04:40.880 ","End":"04:44.845","Text":"7 plus 54 is 61."},{"Start":"04:44.845 ","End":"04:48.620","Text":"So it\u0027s 61 over 27."},{"Start":"04:48.620 ","End":"04:51.155","Text":"Although this is an answer,"},{"Start":"04:51.155 ","End":"04:55.130","Text":"I would like to convert the improper fraction to a mixed number."},{"Start":"04:55.130 ","End":"04:56.885","Text":"Let\u0027s see, we can do it in our heads."},{"Start":"04:56.885 ","End":"04:59.345","Text":"27 goes into 61,"},{"Start":"04:59.345 ","End":"05:03.725","Text":"it will go in twice because 54 will be twice 27."},{"Start":"05:03.725 ","End":"05:06.500","Text":"It\u0027s twice, and from the 54,"},{"Start":"05:06.500 ","End":"05:08.075","Text":"I still have another 7,"},{"Start":"05:08.075 ","End":"05:11.740","Text":"61 minus 54 is 7, 7 leftover."},{"Start":"05:11.740 ","End":"05:20.020","Text":"The answer is 2 and 7/27. Last 1."},{"Start":"05:20.500 ","End":"05:25.355","Text":"Here, we have a mixed number,"},{"Start":"05:25.355 ","End":"05:28.590","Text":"a fraction, and a whole number just like before."},{"Start":"05:28.590 ","End":"05:30.160","Text":"Let\u0027s do the same technique."},{"Start":"05:30.160 ","End":"05:34.669","Text":"First of all, let\u0027s convert this mixed number into an improper fraction."},{"Start":"05:34.669 ","End":"05:38.770","Text":"21 times 1 plus 2 is 23."},{"Start":"05:38.770 ","End":"05:41.460","Text":"This is 23/21,"},{"Start":"05:41.460 ","End":"05:44.730","Text":"minus 3/14 is okay as is,"},{"Start":"05:44.730 ","End":"05:48.775","Text":"and 3 I write as 3/1."},{"Start":"05:48.775 ","End":"05:51.425","Text":"Next, we ask ourselves,"},{"Start":"05:51.425 ","End":"05:53.750","Text":"what is the common denominator?"},{"Start":"05:53.750 ","End":"05:56.355","Text":"A common denominator,"},{"Start":"05:56.355 ","End":"06:01.490","Text":"I know it\u0027s not going to be 21 times 14 because I see they have a 7 in common."},{"Start":"06:01.490 ","End":"06:03.155","Text":"If you think about it a bit,"},{"Start":"06:03.155 ","End":"06:10.970","Text":"you\u0027ll see that the common denominator would be 42."},{"Start":"06:10.970 ","End":"06:14.015","Text":"You need to play with it a bit."},{"Start":"06:14.015 ","End":"06:18.424","Text":"Because 21 goes into 42 obviously and so does 14."},{"Start":"06:18.424 ","End":"06:26.835","Text":"What we actually get is as follows: 21 into 42 goes twice,"},{"Start":"06:26.835 ","End":"06:31.694","Text":"14 into 42 goes 3 times,"},{"Start":"06:31.694 ","End":"06:36.460","Text":"and 1 into 42 is 42 times."},{"Start":"06:36.890 ","End":"06:40.185","Text":"Next thing we get is doing the multiplication,"},{"Start":"06:40.185 ","End":"06:42.525","Text":"2 times 23 is 46,"},{"Start":"06:42.525 ","End":"06:44.816","Text":"there is a minus,"},{"Start":"06:44.816 ","End":"06:47.085","Text":"3 times 3 is 9,"},{"Start":"06:47.085 ","End":"06:53.920","Text":"another minus, 3 times 42 is 126."},{"Start":"06:55.940 ","End":"06:58.380","Text":"Now the arithmetic,"},{"Start":"06:58.380 ","End":"07:09.640","Text":"46 minus 9 is 37, 37 minus 126."},{"Start":"07:10.430 ","End":"07:13.125","Text":"This is worth doing at the side."},{"Start":"07:13.125 ","End":"07:18.300","Text":"We said 37 minus 126."},{"Start":"07:18.300 ","End":"07:21.850","Text":"This is a plus 37,"},{"Start":"07:23.360 ","End":"07:29.985","Text":"37 minus 126,"},{"Start":"07:29.985 ","End":"07:33.455","Text":"it\u0027s a plus and a minus, opposite signs."},{"Start":"07:33.455 ","End":"07:38.420","Text":"Remember what we do. If we have a plus 37 and a minus 126,"},{"Start":"07:38.420 ","End":"07:41.420","Text":"we do a subtraction of the larger minus the smaller,"},{"Start":"07:41.420 ","End":"07:43.025","Text":"the larger and absolute value."},{"Start":"07:43.025 ","End":"07:44.480","Text":"We do the other way round,"},{"Start":"07:44.480 ","End":"07:49.905","Text":"126 minus 37 and that\u0027s"},{"Start":"07:49.905 ","End":"07:55.390","Text":"89 but because the dominating sign,"},{"Start":"07:55.390 ","End":"07:57.414","Text":"the sign of the bigger 1 was a minus,"},{"Start":"07:57.414 ","End":"08:00.110","Text":"it gets to be a minus."},{"Start":"08:00.690 ","End":"08:07.250","Text":"What we have here is minus 89 over 42."},{"Start":"08:07.340 ","End":"08:10.630","Text":"Again, we don\u0027t want to leave improper fractions."},{"Start":"08:10.630 ","End":"08:14.155","Text":"We could, but it\u0027s customary to write the answer as a mixed number."},{"Start":"08:14.155 ","End":"08:16.810","Text":"I know it\u0027s going to be minus something."},{"Start":"08:16.810 ","End":"08:21.140","Text":"Let\u0027s see, 42 into 89 goes twice."},{"Start":"08:21.140 ","End":"08:23.100","Text":"Twice this is 84."},{"Start":"08:23.100 ","End":"08:24.900","Text":"We have 5 leftover."},{"Start":"08:24.900 ","End":"08:29.840","Text":"It\u0027s 2 and 5/42."},{"Start":"08:29.840 ","End":"08:34.940","Text":"That\u0027s the answer to the last 1, and we\u0027re done"}],"ID":4887},{"Watched":false,"Name":"Exercise 11","Duration":"3m 49s","ChapterTopicVideoID":4883,"CourseChapterTopicPlaylistID":3856,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.355","Text":"In this exercise, we have to calculate the following expressions."},{"Start":"00:04.355 ","End":"00:07.395","Text":"They all involve multiplication of fractions."},{"Start":"00:07.395 ","End":"00:12.555","Text":"Luckily, multiplication is the easiest of all the fraction operations."},{"Start":"00:12.555 ","End":"00:17.340","Text":"Let me just copy the exercises down here."},{"Start":"00:17.340 ","End":"00:24.030","Text":"There we are, and we have more space now. Let\u0027s begin."},{"Start":"00:24.030 ","End":"00:26.370","Text":"Multiplication of fractions,"},{"Start":"00:26.370 ","End":"00:30.390","Text":"we just multiply the numerators and multiply the denominators."},{"Start":"00:30.390 ","End":"00:37.440","Text":"What we get is 2 times 2 over 3 times 5."},{"Start":"00:37.440 ","End":"00:43.440","Text":"We just evaluate and it\u0027s 4 over 15,"},{"Start":"00:43.440 ","End":"00:46.480","Text":"and that\u0027s the answer."},{"Start":"00:47.030 ","End":"00:50.610","Text":"In the second one,"},{"Start":"00:50.610 ","End":"00:53.225","Text":"the first one is a whole number,"},{"Start":"00:53.225 ","End":"00:56.090","Text":"so we first have to convert it to a fraction,"},{"Start":"00:56.090 ","End":"00:59.275","Text":"but 4 is just 4 over 1."},{"Start":"00:59.275 ","End":"01:03.244","Text":"Now we\u0027ve got multiplication of 2 fractions,"},{"Start":"01:03.244 ","End":"01:05.314","Text":"we proceed as before."},{"Start":"01:05.314 ","End":"01:07.850","Text":"We put a dividing line and on the top,"},{"Start":"01:07.850 ","End":"01:10.160","Text":"we put 4 times 2, on the bottom,"},{"Start":"01:10.160 ","End":"01:11.825","Text":"1 times 5,"},{"Start":"01:11.825 ","End":"01:18.290","Text":"which is 8 over 5."},{"Start":"01:18.290 ","End":"01:20.965","Text":"We don\u0027t leave the answer as an improper fraction."},{"Start":"01:20.965 ","End":"01:24.100","Text":"5 into 8 goes 1, remainder 3,"},{"Start":"01:24.100 ","End":"01:28.850","Text":"so the answer is 1 and 3/5."},{"Start":"01:29.480 ","End":"01:33.490","Text":"In the next one, they\u0027re both giving us"},{"Start":"01:33.490 ","End":"01:37.420","Text":"mixed numbers and we want to convert those to fractions the usual way,"},{"Start":"01:37.420 ","End":"01:42.720","Text":"3 times 2 plus 1 is 6 plus 1 is 7."},{"Start":"01:42.720 ","End":"01:46.035","Text":"The first fraction is 7 over 3,"},{"Start":"01:46.035 ","End":"01:47.850","Text":"and the second fraction,"},{"Start":"01:47.850 ","End":"01:49.770","Text":"4 times 1 plus 1 is 5,"},{"Start":"01:49.770 ","End":"01:53.190","Text":"so it\u0027s 5 over the 4 from here."},{"Start":"01:53.190 ","End":"02:01.630","Text":"Now we multiply the tops and the bottom so we get 7 times 5 over 3 times 4."},{"Start":"02:01.630 ","End":"02:08.870","Text":"Of course, I could have skipped this step and said straightaway that it was 35 over 12."},{"Start":"02:08.870 ","End":"02:12.590","Text":"All that\u0027s left to do is to convert the improper fraction to a mixed fraction."},{"Start":"02:12.590 ","End":"02:14.540","Text":"12 goes into 35, well, not quite,"},{"Start":"02:14.540 ","End":"02:16.729","Text":"3 times anyway, it goes in twice."},{"Start":"02:16.729 ","End":"02:19.354","Text":"That\u0027s 24 and there\u0027s 11 leftover."},{"Start":"02:19.354 ","End":"02:23.890","Text":"It\u0027s 2 and 11 over 12 and that\u0027s the answer to this one."},{"Start":"02:23.890 ","End":"02:26.690","Text":"Now the last one, once again,"},{"Start":"02:26.690 ","End":"02:32.340","Text":"we have product of two mixed fractions or mixed numbers."},{"Start":"02:32.340 ","End":"02:36.955","Text":"Here, 3 times 3 plus 1 is 10 over 3 times,"},{"Start":"02:36.955 ","End":"02:44.510","Text":"5 times 2 plus 2 is 12, 12 over 5."},{"Start":"02:45.470 ","End":"02:50.585","Text":"This last example, I introduce a new element,"},{"Start":"02:50.585 ","End":"02:54.425","Text":"is that we should really reduce or cancel."},{"Start":"02:54.425 ","End":"02:56.090","Text":"Well, on the next step."},{"Start":"02:56.090 ","End":"02:59.780","Text":"We have 10 times 12 over 3 times 5."},{"Start":"02:59.780 ","End":"03:02.225","Text":"Before we just go blindly multiplying,"},{"Start":"03:02.225 ","End":"03:08.360","Text":"if we could, we should reduce the fraction first."},{"Start":"03:08.360 ","End":"03:13.505","Text":"For example, 5 goes into 10 twice."},{"Start":"03:13.505 ","End":"03:15.320","Text":"If I divide each of these by 5,"},{"Start":"03:15.320 ","End":"03:18.310","Text":"this is 1 and this is 2."},{"Start":"03:18.310 ","End":"03:22.830","Text":"Similarly here, 3 and 12 have a common factor of 3,"},{"Start":"03:22.830 ","End":"03:28.415","Text":"so if I divide this by 3 and this by 3, getting 4 here,"},{"Start":"03:28.415 ","End":"03:36.810","Text":"what I get finally is 2 times 4 is 8,"},{"Start":"03:36.810 ","End":"03:39.210","Text":"1 times 1 is 1,"},{"Start":"03:39.210 ","End":"03:40.950","Text":"so it\u0027s 8 over 1."},{"Start":"03:40.950 ","End":"03:44.025","Text":"The answer, something over 1 is just itself."},{"Start":"03:44.025 ","End":"03:49.770","Text":"It\u0027s a whole number this time and the answer is 8 and we are done."}],"ID":4888},{"Watched":false,"Name":"Exercise 12","Duration":"4m ","ChapterTopicVideoID":4884,"CourseChapterTopicPlaylistID":3856,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.190","Text":"In this exercise we have to calculate"},{"Start":"00:02.190 ","End":"00:05.955","Text":"the following expressions and they\u0027re all division of fractions."},{"Start":"00:05.955 ","End":"00:08.685","Text":"I\u0027d like to just copy them down here."},{"Start":"00:08.685 ","End":"00:11.925","Text":"That\u0027ll give us a bit more space."},{"Start":"00:11.925 ","End":"00:17.159","Text":"There we are. Let\u0027s start with the first 1."},{"Start":"00:17.159 ","End":"00:21.105","Text":"The thing to do with division is to convert it into a multiplication."},{"Start":"00:21.105 ","End":"00:22.380","Text":"The method is this,"},{"Start":"00:22.380 ","End":"00:25.980","Text":"the first 1 you write as is, it\u0027s 2/3."},{"Start":"00:25.980 ","End":"00:32.295","Text":"The second 1 you convert the division into a multiplication and invert the fraction."},{"Start":"00:32.295 ","End":"00:37.255","Text":"Instead of 5/6, it\u0027s 6/5."},{"Start":"00:37.255 ","End":"00:40.940","Text":"Now, we use our usual technique for multiplying."},{"Start":"00:40.940 ","End":"00:45.135","Text":"It\u0027s 2 times 6 over 3 times 5."},{"Start":"00:45.135 ","End":"00:46.595","Text":"Let\u0027s see if anything cancels."},{"Start":"00:46.595 ","End":"00:48.575","Text":"Yes, 3 will go into 6,"},{"Start":"00:48.575 ","End":"00:52.140","Text":"so 3 into 6,"},{"Start":"00:52.140 ","End":"00:56.220","Text":"this is 1 and this is 2 when we divide by 3,"},{"Start":"00:56.220 ","End":"00:57.840","Text":"and nothing else will cancel."},{"Start":"00:57.840 ","End":"01:02.115","Text":"2 times 2/5 times 1,"},{"Start":"01:02.115 ","End":"01:05.190","Text":"which is 4/5,"},{"Start":"01:05.190 ","End":"01:09.360","Text":"and that\u0027s the answer."},{"Start":"01:09.360 ","End":"01:12.430","Text":"On to the next 1."},{"Start":"01:13.220 ","End":"01:17.780","Text":"Here, the second 1 is a whole number not a fraction,"},{"Start":"01:17.780 ","End":"01:19.725","Text":"so we write it as a fraction."},{"Start":"01:19.725 ","End":"01:22.510","Text":"2/5 divided by,"},{"Start":"01:22.510 ","End":"01:25.275","Text":"and 4 is 4/1."},{"Start":"01:25.275 ","End":"01:28.390","Text":"As before, the first 1 stays the same."},{"Start":"01:28.390 ","End":"01:32.800","Text":"The division changes into a multiplication and we invert the fraction."},{"Start":"01:32.800 ","End":"01:35.845","Text":"The inverse of 4/1 is 1/4."},{"Start":"01:35.845 ","End":"01:38.420","Text":"Now, we just multiply."},{"Start":"01:39.140 ","End":"01:43.250","Text":"In fact, we can even cancel at this level."},{"Start":"01:44.010 ","End":"01:52.200","Text":"We can say that 2 goes into 4 twice."},{"Start":"01:52.200 ","End":"01:55.275","Text":"If I divide by 2, this is 1 and this is 2."},{"Start":"01:55.275 ","End":"01:57.150","Text":"Then when we multiply out,"},{"Start":"01:57.150 ","End":"01:59.535","Text":"we get 1 times 1 is 1,"},{"Start":"01:59.535 ","End":"02:02.550","Text":"and 5 times 2 is 10."},{"Start":"02:02.550 ","End":"02:05.945","Text":"The answer is 1/10."},{"Start":"02:05.945 ","End":"02:10.660","Text":"Here also, we have a whole number,"},{"Start":"02:10.660 ","End":"02:14.970","Text":"so we need to convert that into a fraction,"},{"Start":"02:14.970 ","End":"02:19.660","Text":"so that\u0027s 6/1 divided by 3/4."},{"Start":"02:19.660 ","End":"02:22.090","Text":"The first 1 stays as is,"},{"Start":"02:22.090 ","End":"02:27.560","Text":"it\u0027s the second that we invert and multiply 4/3."},{"Start":"02:27.560 ","End":"02:30.105","Text":"Canceling a little bit,"},{"Start":"02:30.105 ","End":"02:32.400","Text":"3 and 6,"},{"Start":"02:32.400 ","End":"02:33.930","Text":"that\u0027s 1 and 2."},{"Start":"02:33.930 ","End":"02:35.250","Text":"We divide by 3."},{"Start":"02:35.250 ","End":"02:42.765","Text":"2 times 4/1 times 1 is 8/1,"},{"Start":"02:42.765 ","End":"02:44.655","Text":"which is 8,"},{"Start":"02:44.655 ","End":"02:47.320","Text":"and that\u0027s our answer."},{"Start":"02:48.220 ","End":"02:50.300","Text":"In this last 1,"},{"Start":"02:50.300 ","End":"02:51.710","Text":"they\u0027re both mixed fractions,"},{"Start":"02:51.710 ","End":"02:55.385","Text":"so let\u0027s convert them to improper fractions."},{"Start":"02:55.385 ","End":"02:57.255","Text":"3 times 2 plus 2,"},{"Start":"02:57.255 ","End":"03:01.080","Text":"is 6 plus 2 is 8/3."},{"Start":"03:01.080 ","End":"03:02.850","Text":"Then the second 1,"},{"Start":"03:02.850 ","End":"03:09.150","Text":"5 times 1 plus 1 is 6/5."},{"Start":"03:09.150 ","End":"03:11.600","Text":"Our exercise is this divided by this,"},{"Start":"03:11.600 ","End":"03:14.645","Text":"and now we convert it to multiplication the usual way."},{"Start":"03:14.645 ","End":"03:19.870","Text":"We get 8/3 times 5/6,"},{"Start":"03:19.870 ","End":"03:22.395","Text":"which equals, does anything cancel?"},{"Start":"03:22.395 ","End":"03:25.425","Text":"Yes, 2 will go into 6 and 8."},{"Start":"03:25.425 ","End":"03:29.490","Text":"If I divide each of these by 2,"},{"Start":"03:29.490 ","End":"03:35.085","Text":"then I get here 3 and here 4,"},{"Start":"03:35.085 ","End":"03:40.545","Text":"and so what I get is 4 times 5 is 20."},{"Start":"03:40.545 ","End":"03:43.635","Text":"3 times 3 is 9."},{"Start":"03:43.635 ","End":"03:45.570","Text":"But this is an improper fraction."},{"Start":"03:45.570 ","End":"03:49.380","Text":"Let\u0027s see, 9 goes into 20 twice."},{"Start":"03:49.380 ","End":"03:52.395","Text":"Twice is 18, and the leftover with 2,"},{"Start":"03:52.395 ","End":"03:54.645","Text":"so it\u0027s 2 and 2/9."},{"Start":"03:54.645 ","End":"04:01.090","Text":"That\u0027s the answer to the last 1 and we are done."}],"ID":4889},{"Watched":false,"Name":"Exercise 13","Duration":"5m 57s","ChapterTopicVideoID":4885,"CourseChapterTopicPlaylistID":3856,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.550","Text":"In this exercise, we have to calculate"},{"Start":"00:02.550 ","End":"00:05.780","Text":"the following expressions which use fractions or some division,"},{"Start":"00:05.780 ","End":"00:07.623","Text":"some multiplication, addition,"},{"Start":"00:07.623 ","End":"00:11.235","Text":"exponents, mixed exercises with fractions."},{"Start":"00:11.235 ","End":"00:13.770","Text":"I\u0027d like to work on a copy,"},{"Start":"00:13.770 ","End":"00:17.010","Text":"and that way we\u0027ll have a bit more room,"},{"Start":"00:17.010 ","End":"00:21.180","Text":"here we go, and let\u0027s start with the first 1."},{"Start":"00:21.180 ","End":"00:23.055","Text":"The first 1 is a division,"},{"Start":"00:23.055 ","End":"00:26.580","Text":"but it\u0027s not simple fractions,"},{"Start":"00:26.580 ","End":"00:28.125","Text":"here we have a mixed number,"},{"Start":"00:28.125 ","End":"00:30.135","Text":"so we have to convert that 1,"},{"Start":"00:30.135 ","End":"00:35.415","Text":"so we get 5/9 divided by now 3 and 1/3,"},{"Start":"00:35.415 ","End":"00:40.075","Text":"we do 3 times 3 plus 1 is 10, it\u0027s 10/3."},{"Start":"00:40.075 ","End":"00:42.800","Text":"If you remember, when we divide fractions,"},{"Start":"00:42.800 ","End":"00:44.630","Text":"we multiply by the inverse,"},{"Start":"00:44.630 ","End":"00:48.990","Text":"we have to multiplied by 3/10,"},{"Start":"00:49.180 ","End":"00:58.140","Text":"then we can do some canceling because 5 and 10 are both divisible by 5,"},{"Start":"00:58.140 ","End":"00:59.625","Text":"it\u0027s 1 and 2,"},{"Start":"00:59.625 ","End":"01:02.550","Text":"and also 3 and 9 are both divisible by 3,"},{"Start":"01:02.550 ","End":"01:05.860","Text":"here we get 1 and here we get 3."},{"Start":"01:05.860 ","End":"01:11.345","Text":"In the end, we get 1 times 1 is 1,"},{"Start":"01:11.345 ","End":"01:14.015","Text":"and 3 times 2 is 6,"},{"Start":"01:14.015 ","End":"01:17.455","Text":"and that would be the answer to this 1."},{"Start":"01:17.455 ","End":"01:21.540","Text":"Onto the next, here we have an exponent,"},{"Start":"01:21.540 ","End":"01:23.240","Text":"the fraction is in brackets,"},{"Start":"01:23.240 ","End":"01:26.285","Text":"which means that the exponent applies to all of it,"},{"Start":"01:26.285 ","End":"01:29.150","Text":"and the rule is that we apply the exponent separately to"},{"Start":"01:29.150 ","End":"01:32.165","Text":"the numerator and to the denominator,"},{"Start":"01:32.165 ","End":"01:35.050","Text":"so each 1 is raised to the power of 3."},{"Start":"01:35.050 ","End":"01:41.280","Text":"Now we just evaluated 3^3 is 3 times 3 times 3 is 27,"},{"Start":"01:41.280 ","End":"01:45.870","Text":"and here we have 4 times 4 times 4 is 64,"},{"Start":"01:45.870 ","End":"01:47.940","Text":"and there\u0027s nothing further to do,"},{"Start":"01:47.940 ","End":"01:49.470","Text":"this is the answer."},{"Start":"01:49.470 ","End":"01:51.735","Text":"On to the next 1."},{"Start":"01:51.735 ","End":"01:55.410","Text":"Here, the exponent is just a numerator,"},{"Start":"01:55.410 ","End":"01:58.579","Text":"all we have to do is evaluate this 1 separately,"},{"Start":"01:58.579 ","End":"02:01.825","Text":"3^3, we already know what it is from the previous,"},{"Start":"02:01.825 ","End":"02:05.235","Text":"that\u0027s 27 and the 4 remains."},{"Start":"02:05.235 ","End":"02:07.080","Text":"This is an improper fraction,"},{"Start":"02:07.080 ","End":"02:09.300","Text":"you convert it to a mixed number as follows,"},{"Start":"02:09.300 ","End":"02:13.440","Text":"4 into 27 it goes 6 times,"},{"Start":"02:13.440 ","End":"02:15.900","Text":"because 6 times 4 is 24,"},{"Start":"02:15.900 ","End":"02:18.585","Text":"which means that we have 3 leftover,"},{"Start":"02:18.585 ","End":"02:23.075","Text":"our answer is 6 and 3/4."},{"Start":"02:23.075 ","End":"02:29.170","Text":"Another last 1 has a mixture of stuff in it, let\u0027s see."},{"Start":"02:29.170 ","End":"02:34.370","Text":"Let\u0027s, first of all, convert the mixed numbers to improper fractions,"},{"Start":"02:34.370 ","End":"02:39.375","Text":"what we get is 9/20 as is,"},{"Start":"02:39.375 ","End":"02:42.730","Text":"1 and 1/3 comes out to be 4/3,"},{"Start":"02:44.750 ","End":"02:50.310","Text":"then we have a plus 1 and 1/4 is 5/4,"},{"Start":"02:50.310 ","End":"02:52.860","Text":"4 times 1 plus 1,"},{"Start":"02:52.860 ","End":"02:58.660","Text":"and divide it by 1/2."},{"Start":"02:59.190 ","End":"03:05.470","Text":"What we have here has some different arithmetical operations plus,"},{"Start":"03:05.470 ","End":"03:08.095","Text":"times, divided by,"},{"Start":"03:08.095 ","End":"03:10.910","Text":"but remember what takes precedence,"},{"Start":"03:10.910 ","End":"03:14.845","Text":"the multiplication and division come before the addition,"},{"Start":"03:14.845 ","End":"03:19.165","Text":"what we\u0027re going to do is this first and then this,"},{"Start":"03:19.165 ","End":"03:23.825","Text":"and then we\u0027ll add the 2 answers, let\u0027s see."},{"Start":"03:23.825 ","End":"03:27.505","Text":"I could do these aside exercises and in fact I will,"},{"Start":"03:27.505 ","End":"03:30.050","Text":"let\u0027s do this bit here,"},{"Start":"03:30.050 ","End":"03:33.720","Text":"I\u0027ll do it over here where there is plenty of room,"},{"Start":"03:33.720 ","End":"03:38.250","Text":"9/20 times 4/3,"},{"Start":"03:38.250 ","End":"03:40.320","Text":"that\u0027s this first bit."},{"Start":"03:40.320 ","End":"03:45.000","Text":"Now we can cancel this and this,"},{"Start":"03:45.000 ","End":"03:47.010","Text":"we can divide each of them by 3,"},{"Start":"03:47.010 ","End":"03:48.960","Text":"this is 3 and this is 1,"},{"Start":"03:48.960 ","End":"03:52.010","Text":"and this and this are both divisible by 4,"},{"Start":"03:52.010 ","End":"03:54.500","Text":"here we have 1 and here we have 5,"},{"Start":"03:54.500 ","End":"04:00.420","Text":"which means that this bit comes out to be 3 times 1 is 3,"},{"Start":"04:00.420 ","End":"04:02.040","Text":"5 times 1 is 5,"},{"Start":"04:02.040 ","End":"04:04.950","Text":"this is 3/5, that\u0027s this bit."},{"Start":"04:04.950 ","End":"04:09.180","Text":"In fact, maybe I\u0027ll write it down here,"},{"Start":"04:09.180 ","End":"04:11.590","Text":"this bit is 3/5."},{"Start":"04:11.690 ","End":"04:17.190","Text":"Now the second 1 is a division,"},{"Start":"04:17.190 ","End":"04:25.510","Text":"we convert the division into a multiplication by inverting the second 1,"},{"Start":"04:25.510 ","End":"04:29.760","Text":"this is 5/4 times 2/1."},{"Start":"04:29.960 ","End":"04:33.950","Text":"Once again, we prefer to cancel that with smaller numbers,"},{"Start":"04:33.950 ","End":"04:37.745","Text":"so 2 into 4 goes twice,"},{"Start":"04:37.745 ","End":"04:39.350","Text":"what we\u0027re left here"},{"Start":"04:39.350 ","End":"04:48.920","Text":"with is 5/2,"},{"Start":"04:48.920 ","End":"04:51.710","Text":"and I\u0027ll record that here,"},{"Start":"04:51.710 ","End":"04:59.470","Text":"5/2, what we have is 3/5 plus 5/2."},{"Start":"04:59.470 ","End":"05:00.900","Text":"Addition of fractions."},{"Start":"05:00.900 ","End":"05:03.950","Text":"Remember what we do in the case of addition,"},{"Start":"05:03.950 ","End":"05:07.790","Text":"is we put everything over a common denominator,"},{"Start":"05:07.790 ","End":"05:11.210","Text":"the common denominator of 5 and 2 is 10."},{"Start":"05:11.210 ","End":"05:15.945","Text":"The first 1 is something 1/10,"},{"Start":"05:15.945 ","End":"05:17.595","Text":"5 into 10 goes 2,"},{"Start":"05:17.595 ","End":"05:19.050","Text":"2 times 3 is 6,"},{"Start":"05:19.050 ","End":"05:20.655","Text":"this is 6/10,"},{"Start":"05:20.655 ","End":"05:24.500","Text":"the second 1, multiply top and bottom by 5,"},{"Start":"05:24.500 ","End":"05:26.900","Text":"I get 25/10,"},{"Start":"05:26.900 ","End":"05:37.995","Text":"altogether 6 and 25 is 31 and we put that over that common denominator."},{"Start":"05:37.995 ","End":"05:43.670","Text":"Finally, the improper fraction as a mixed fraction 10 into 31,"},{"Start":"05:43.670 ","End":"05:48.395","Text":"goes 3 times and a remainder of 1,"},{"Start":"05:48.395 ","End":"05:52.010","Text":"so it\u0027s 3 and 1/10,"},{"Start":"05:52.010 ","End":"05:57.240","Text":"and that\u0027s the answer and we are done."}],"ID":4890},{"Watched":false,"Name":"Exercise 14","Duration":"3m 33s","ChapterTopicVideoID":4886,"CourseChapterTopicPlaylistID":3856,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.190","Text":"In this exercise, we have to calculate the following expressions involving fractions,"},{"Start":"00:05.190 ","End":"00:06.885","Text":"multiplication, and division."},{"Start":"00:06.885 ","End":"00:10.210","Text":"I prefer to work on a copy."},{"Start":"00:11.420 ","End":"00:17.610","Text":"Here we are. The first 1 is a product,"},{"Start":"00:17.610 ","End":"00:21.600","Text":"multiplication of fractions, so as usual,"},{"Start":"00:21.600 ","End":"00:25.815","Text":"we multiply the numerators and multiply the denominators."},{"Start":"00:25.815 ","End":"00:31.365","Text":"We have 4 times 2 over 3 times 7 I can\u0027t see anything to cancel."},{"Start":"00:31.365 ","End":"00:37.090","Text":"The answer will be 8/21."},{"Start":"00:37.820 ","End":"00:41.665","Text":"In the second exercise,"},{"Start":"00:41.665 ","End":"00:46.955","Text":"we have to convert the mixed number to a fraction first."},{"Start":"00:46.955 ","End":"00:51.770","Text":"This case we do 3 times 5 plus 1 is 16,"},{"Start":"00:51.770 ","End":"00:59.870","Text":"so we have 16/3 divided by 1/6 and remember for a division,"},{"Start":"00:59.870 ","End":"01:02.504","Text":"we multiply by the inverse,"},{"Start":"01:02.504 ","End":"01:08.810","Text":"so it\u0027s 16/3 multiplied by 6/1."},{"Start":"01:08.810 ","End":"01:11.000","Text":"We can cancel something,"},{"Start":"01:11.000 ","End":"01:12.635","Text":"3 into 6 goes,"},{"Start":"01:12.635 ","End":"01:14.750","Text":"so if we divide each of them by 3,"},{"Start":"01:14.750 ","End":"01:18.065","Text":"this is 1 and this is 2."},{"Start":"01:18.065 ","End":"01:22.625","Text":"That brings us to 16 times 2/1,"},{"Start":"01:22.625 ","End":"01:29.015","Text":"32/1, which is just 32, a whole number."},{"Start":"01:29.015 ","End":"01:31.700","Text":"This 1, I\u0027m sorry,"},{"Start":"01:31.700 ","End":"01:33.275","Text":"I made a mistake in copying."},{"Start":"01:33.275 ","End":"01:37.280","Text":"This should be a 5. Sorry about that."},{"Start":"01:37.280 ","End":"01:41.015","Text":"Anyway, with the product of 3 or more,"},{"Start":"01:41.015 ","End":"01:43.760","Text":"it works the same way as the product of 2."},{"Start":"01:43.760 ","End":"01:48.860","Text":"You just take the product of the denominators separately"},{"Start":"01:48.860 ","End":"01:54.460","Text":"and the product of the numerators separately and,"},{"Start":"01:54.460 ","End":"01:56.795","Text":"to see can we cancel anything?"},{"Start":"01:56.795 ","End":"02:04.350","Text":"I think so, 2 goes into 4 twice."},{"Start":"02:04.350 ","End":"02:07.455","Text":"We don\u0027t bother with writing the ones anymore."},{"Start":"02:07.455 ","End":"02:13.770","Text":"Now 2 goes into 6, 3 times."},{"Start":"02:13.770 ","End":"02:15.375","Text":"What are we left with?"},{"Start":"02:15.375 ","End":"02:18.045","Text":"This 3 goes with this 3."},{"Start":"02:18.045 ","End":"02:27.045","Text":"I am left with simply just 9 on the top and 5 on the bottom, 9/5."},{"Start":"02:27.045 ","End":"02:29.760","Text":"But that\u0027s an improper fraction,"},{"Start":"02:29.760 ","End":"02:32.715","Text":"so 5 into 9 is 1 remainder 4,"},{"Start":"02:32.715 ","End":"02:37.660","Text":"so this is 1 and 4/5 and that\u0027s the answer."},{"Start":"02:37.660 ","End":"02:41.690","Text":"The last 1, both are mixed numbers,"},{"Start":"02:41.690 ","End":"02:47.115","Text":"so let\u0027s convert them to improper fractions."},{"Start":"02:47.115 ","End":"02:50.190","Text":"2 times 3 plus 1 is 7,"},{"Start":"02:50.190 ","End":"02:57.040","Text":"7/2 times 5 times 4 plus 2 is 22/5."},{"Start":"02:58.010 ","End":"03:00.060","Text":"See if we can cancel."},{"Start":"03:00.060 ","End":"03:05.210","Text":"Yes, 2 goes into 22 11 times, can\u0027t do anymore."},{"Start":"03:05.210 ","End":"03:09.630","Text":"7 times 11 is 77/5."},{"Start":"03:10.940 ","End":"03:19.800","Text":"Let\u0027s see. 5 goes into 75 15 times so it\u0027s 77,"},{"Start":"03:19.800 ","End":"03:21.630","Text":"there\u0027s a remainder of 2."},{"Start":"03:21.630 ","End":"03:29.790","Text":"So the answer is 15 and 2/5 and that\u0027s the answer to this 1."},{"Start":"03:29.790 ","End":"03:32.620","Text":"We are done."}],"ID":4891},{"Watched":false,"Name":"Exercise 15","Duration":"3m 31s","ChapterTopicVideoID":4887,"CourseChapterTopicPlaylistID":3856,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.200 ","End":"00:03.750","Text":"Here we have to calculate the following expressions."},{"Start":"00:03.750 ","End":"00:07.455","Text":"It\u0027s, again, multiplication and division of fractions."},{"Start":"00:07.455 ","End":"00:09.690","Text":"Let\u0027s start with the first 1."},{"Start":"00:09.690 ","End":"00:11.174","Text":"This is a fraction."},{"Start":"00:11.174 ","End":"00:14.490","Text":"This 1 has to be converted to a fraction because it\u0027s a whole number."},{"Start":"00:14.490 ","End":"00:23.980","Text":"We write this as 5/6 divided by 3/1."},{"Start":"00:24.530 ","End":"00:26.700","Text":"Division of fractions,"},{"Start":"00:26.700 ","End":"00:30.540","Text":"we just make the second 1 inverted."},{"Start":"00:30.540 ","End":"00:39.430","Text":"It\u0027s 5/6 times 1/3 and the division changes to a multiplication."},{"Start":"00:39.500 ","End":"00:43.640","Text":"5 times 1 is 5,"},{"Start":"00:43.640 ","End":"00:51.360","Text":"6 times 3 is 18 and the answer is 5/18."},{"Start":"00:51.360 ","End":"00:56.040","Text":"Next 1. Both are mixed fractions,"},{"Start":"00:56.040 ","End":"01:00.240","Text":"we have to convert them to improper fractions."},{"Start":"01:00.240 ","End":"01:05.415","Text":"4 times 3 plus 3 is 12 plus 3 is 15/4,"},{"Start":"01:05.415 ","End":"01:13.770","Text":"divided by 8 times 5 plus 5 is 45/8"},{"Start":"01:13.770 ","End":"01:18.735","Text":"and this changes to"},{"Start":"01:18.735 ","End":"01:27.545","Text":"15/4 division becomes multiplication of the inverse 8/45."},{"Start":"01:27.545 ","End":"01:29.990","Text":"Let\u0027s do some canceling."},{"Start":"01:29.990 ","End":"01:34.380","Text":"15 goes into 45 3 times,"},{"Start":"01:35.350 ","End":"01:40.190","Text":"4 goes into 8 twice."},{"Start":"01:40.190 ","End":"01:45.065","Text":"What we\u0027re left with is simply 2/3."},{"Start":"01:45.065 ","End":"01:47.570","Text":"That\u0027s it for that 1."},{"Start":"01:47.570 ","End":"01:49.565","Text":"Now to the last 1."},{"Start":"01:49.565 ","End":"01:51.697","Text":"The last 1,"},{"Start":"01:51.697 ","End":"01:56.940","Text":"we just have to write the 8 as a fraction,"},{"Start":"01:56.940 ","End":"02:05.540","Text":"so it\u0027s 8/1 times 3/2 divided by 12/20."},{"Start":"02:05.540 ","End":"02:09.280","Text":"Now here we have mixture multiplication and division."},{"Start":"02:09.280 ","End":"02:11.920","Text":"We do it from left to right."},{"Start":"02:11.920 ","End":"02:17.915","Text":"Let\u0027s, first of all, multiply these 2 and see what we get."},{"Start":"02:17.915 ","End":"02:20.490","Text":"I\u0027ll do that at the side."},{"Start":"02:20.490 ","End":"02:26.565","Text":"8/1 times 3/2 equals,"},{"Start":"02:26.565 ","End":"02:30.390","Text":"let\u0027s cancel 2 with 8."},{"Start":"02:30.390 ","End":"02:31.980","Text":"It goes in 4 times,"},{"Start":"02:31.980 ","End":"02:38.835","Text":"4 times 3 is 12/1,"},{"Start":"02:38.835 ","End":"02:43.332","Text":"which is just 12."},{"Start":"02:43.332 ","End":"02:48.080","Text":"Yes, that\u0027s right."},{"Start":"02:48.080 ","End":"02:53.185","Text":"Now, when I put that 12 back here, this is 12."},{"Start":"02:53.185 ","End":"03:02.215","Text":"I\u0027d like to rewrite it as 12/1 because I\u0027m starting a division exercise now, 12/20."},{"Start":"03:02.215 ","End":"03:08.785","Text":"Remember, division is done by multiplying by the inverse."},{"Start":"03:08.785 ","End":"03:16.680","Text":"It\u0027s 12/1 times 20/12 and look how lucky we are."},{"Start":"03:16.680 ","End":"03:19.480","Text":"This whole 12 cancels."},{"Start":"03:19.480 ","End":"03:25.485","Text":"What we\u0027re left with now is 20/1,"},{"Start":"03:25.485 ","End":"03:31.780","Text":"20 and that\u0027s the answer. We are done."}],"ID":4892}],"Thumbnail":null,"ID":3856},{"Name":"Algebraic Expressions","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Introduction, Substitution","Duration":"8m 20s","ChapterTopicVideoID":4904,"CourseChapterTopicPlaylistID":3857,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.630","Text":"In this clip, we\u0027ll introduce algebraic expressions,"},{"Start":"00:03.630 ","End":"00:06.645","Text":"and we want to know what one is."},{"Start":"00:06.645 ","End":"00:16.170","Text":"Basically, it\u0027s just an expression which involves 3 things mixed together."},{"Start":"00:16.170 ","End":"00:23.325","Text":"One of them is numbers or constants,"},{"Start":"00:23.325 ","End":"00:27.930","Text":"the other is variables or letters like x and a,"},{"Start":"00:27.930 ","End":"00:29.490","Text":"that\u0027s the variables,"},{"Start":"00:29.490 ","End":"00:31.170","Text":"and this combined together with"},{"Start":"00:31.170 ","End":"00:36.470","Text":"the arithmetic operations which are the ones we usually know, addition,"},{"Start":"00:36.470 ","End":"00:43.200","Text":"subtraction, multiplication, division, exponents,"},{"Start":"00:43.200 ","End":"00:45.405","Text":"and roots all combined together."},{"Start":"00:45.405 ","End":"00:48.860","Text":"The best thing to do is just to give some examples."},{"Start":"00:48.860 ","End":"00:51.755","Text":"I\u0027ll give you 5 of them that should do."},{"Start":"00:51.755 ","End":"00:54.386","Text":"I\u0027ll write them down and few words of explanation,"},{"Start":"00:54.386 ","End":"00:56.390","Text":"and then we\u0027ll learn what they\u0027re good for."},{"Start":"00:56.390 ","End":"00:59.175","Text":"The first one I\u0027ll give you just be 5x."},{"Start":"00:59.175 ","End":"01:03.220","Text":"It has a number, which is 5,"},{"Start":"01:03.220 ","End":"01:05.360","Text":"it has a variable x,"},{"Start":"01:05.360 ","End":"01:08.135","Text":"and it has an arithmetic operation, though you don\u0027t see it."},{"Start":"01:08.135 ","End":"01:11.510","Text":"There\u0027s an implicit multiplication here,"},{"Start":"01:11.510 ","End":"01:15.030","Text":"which we usually don\u0027t write but you got to remember, it\u0027s there."},{"Start":"01:15.460 ","End":"01:18.455","Text":"Likely come more complicated?"},{"Start":"01:18.455 ","End":"01:20.120","Text":"5x minus 7,"},{"Start":"01:20.120 ","End":"01:22.010","Text":"I have some constants or numbers,"},{"Start":"01:22.010 ","End":"01:23.360","Text":"and I have a letter,"},{"Start":"01:23.360 ","End":"01:24.620","Text":"and I have multiplication,"},{"Start":"01:24.620 ","End":"01:26.495","Text":"subtraction, and so on."},{"Start":"01:26.495 ","End":"01:36.320","Text":"What about exponents? We could have something like minus x^2 plus x^3 minus 9."},{"Start":"01:36.320 ","End":"01:39.980","Text":"I have addition and subtraction and exponents here,"},{"Start":"01:39.980 ","End":"01:42.065","Text":"and still just one letter."},{"Start":"01:42.065 ","End":"01:44.090","Text":"x, of course, it could be more than one letter."},{"Start":"01:44.090 ","End":"01:48.350","Text":"For example, I could have 4 times a times x,"},{"Start":"01:48.350 ","End":"01:53.930","Text":"then I could take away 8 times y^2 times"},{"Start":"01:53.930 ","End":"01:59.660","Text":"b times t. Here\u0027s one with some division."},{"Start":"01:59.660 ","End":"02:03.770","Text":"Division could be written like a fraction with the numerator,"},{"Start":"02:03.770 ","End":"02:05.720","Text":"a^2 minus b^2,"},{"Start":"02:05.720 ","End":"02:07.430","Text":"and a denominator,"},{"Start":"02:07.430 ","End":"02:14.500","Text":"a minus b. I think I\u0027ve been stingy with 5, I\u0027ll give you another one."},{"Start":"02:15.560 ","End":"02:18.140","Text":"We haven\u0027t used the square root yet,"},{"Start":"02:18.140 ","End":"02:24.320","Text":"so how about square root of a times x times y?"},{"Start":"02:24.320 ","End":"02:27.830","Text":"Here, we have some algebraic expressions."},{"Start":"02:27.830 ","End":"02:31.805","Text":"Notice that what is not present is an equality sign."},{"Start":"02:31.805 ","End":"02:33.620","Text":"These are not equations,"},{"Start":"02:33.620 ","End":"02:34.820","Text":"they are just expressions."},{"Start":"02:34.820 ","End":"02:37.570","Text":"We\u0027ll use them in equations later,"},{"Start":"02:37.570 ","End":"02:39.560","Text":"just in case you\u0027re wondering,"},{"Start":"02:39.560 ","End":"02:42.005","Text":"we know the algebra is equations."},{"Start":"02:42.005 ","End":"02:44.270","Text":"These are just expressions, and you know what?"},{"Start":"02:44.270 ","End":"02:45.440","Text":"I\u0027ll show you how to use them,"},{"Start":"02:45.440 ","End":"02:48.595","Text":"but I\u0027ll use some fresh examples."},{"Start":"02:48.595 ","End":"02:51.600","Text":"Just need to correct a small error,"},{"Start":"02:51.600 ","End":"02:53.905","Text":"I meant to write variables."},{"Start":"02:53.905 ","End":"03:00.445","Text":"That\u0027s better. One of the main things expressions are good for is substitution."},{"Start":"03:00.445 ","End":"03:04.025","Text":"I\u0027ll show you what I mean with examples."},{"Start":"03:04.025 ","End":"03:08.410","Text":"I\u0027ll give you 4 examples."},{"Start":"03:08.410 ","End":"03:10.785","Text":"Well, maybe 4 is a lot."},{"Start":"03:10.785 ","End":"03:13.140","Text":"Just for example, I\u0027ll give 3."},{"Start":"03:13.140 ","End":"03:14.940","Text":"I\u0027ll give Number 1,"},{"Start":"03:14.940 ","End":"03:16.755","Text":"I\u0027ll give Number 2,"},{"Start":"03:16.755 ","End":"03:19.170","Text":"and I\u0027ll give Number 3. Let\u0027s see."},{"Start":"03:19.170 ","End":"03:25.280","Text":"The first one will be x+y^3."},{"Start":"03:25.280 ","End":"03:26.690","Text":"Now, that\u0027s the expression."},{"Start":"03:26.690 ","End":"03:28.938","Text":"I have to tell you what to substitute."},{"Start":"03:28.938 ","End":"03:31.520","Text":"For each of the variables or letters,"},{"Start":"03:31.520 ","End":"03:33.200","Text":"I have to tell you what it is,"},{"Start":"03:33.200 ","End":"03:35.890","Text":"I need to say what is x and what is y."},{"Start":"03:35.890 ","End":"03:40.500","Text":"Let us say that x is 4,"},{"Start":"03:40.500 ","End":"03:44.235","Text":"and y is negative 3."},{"Start":"03:44.235 ","End":"03:47.260","Text":"Now, how do we go about this?"},{"Start":"03:47.260 ","End":"03:50.600","Text":"I looked at this and everywhere I see x,"},{"Start":"03:50.600 ","End":"03:52.010","Text":"I replace it with 4,"},{"Start":"03:52.010 ","End":"03:53.090","Text":"and everywhere I see y,"},{"Start":"03:53.090 ","End":"03:56.735","Text":"I replace it with negative 3. Let\u0027s start."},{"Start":"03:56.735 ","End":"04:01.080","Text":"What I have is x is 4,"},{"Start":"04:01.080 ","End":"04:03.390","Text":"I\u0027m just copying from here with replacement,"},{"Start":"04:03.390 ","End":"04:05.700","Text":"y is negative 3,"},{"Start":"04:05.700 ","End":"04:12.540","Text":"but I\u0027d better protect it with brackets, cubed."},{"Start":"04:12.540 ","End":"04:19.635","Text":"There\u0027s x and there\u0027s y. I\u0027ve substituted them both and now we do our arithmetic."},{"Start":"04:19.635 ","End":"04:21.380","Text":"We first take care of the brackets."},{"Start":"04:21.380 ","End":"04:22.700","Text":"These are not brackets,"},{"Start":"04:22.700 ","End":"04:24.575","Text":"just protect the minus."},{"Start":"04:24.575 ","End":"04:27.860","Text":"4 and negative 3 is positive 1,"},{"Start":"04:27.860 ","End":"04:29.600","Text":"the plus beats the minus,"},{"Start":"04:29.600 ","End":"04:32.210","Text":"so it\u0027s 1^3,"},{"Start":"04:32.210 ","End":"04:34.055","Text":"which is equal to 1."},{"Start":"04:34.055 ","End":"04:37.730","Text":"That\u0027s the answer of the substitution of x is this,"},{"Start":"04:37.730 ","End":"04:41.800","Text":"and y is this, and this expression involving x and y."},{"Start":"04:41.800 ","End":"04:46.320","Text":"Next one, and I wrote it while you weren\u0027t looking,"},{"Start":"04:46.320 ","End":"04:51.160","Text":"2a^5 minus 4a^4 minus a^3 plus 9."},{"Start":"04:51.160 ","End":"04:56.525","Text":"We\u0027re going to evaluate what this is when a is minus 1."},{"Start":"04:56.525 ","End":"04:59.970","Text":"Let\u0027s see. Here, we have plenty of exponents."},{"Start":"04:59.970 ","End":"05:02.360","Text":"We get twice,"},{"Start":"05:02.360 ","End":"05:03.930","Text":"a is minus 1,"},{"Start":"05:03.930 ","End":"05:06.450","Text":"but we have to protect it in brackets,"},{"Start":"05:06.450 ","End":"05:09.735","Text":"to the power of 5 or to the 5th,"},{"Start":"05:09.735 ","End":"05:16.920","Text":"minus 4 times minus 1^4,"},{"Start":"05:16.920 ","End":"05:23.855","Text":"the power of 4, minus minus 1^3 plus 9."},{"Start":"05:23.855 ","End":"05:28.093","Text":"Now, remember about powers of a negative number?"},{"Start":"05:28.093 ","End":"05:30.010","Text":"If it\u0027s an even number,"},{"Start":"05:30.010 ","End":"05:33.520","Text":"it\u0027s positive, and if it\u0027s an odd number, it\u0027s negative."},{"Start":"05:33.520 ","End":"05:36.420","Text":"What I get is 2 times."},{"Start":"05:36.420 ","End":"05:41.828","Text":"Minus 1^5 is going to come up minus 1."},{"Start":"05:41.828 ","End":"05:43.610","Text":"You can also just multiply."},{"Start":"05:43.610 ","End":"05:48.310","Text":"Minus 1 times minus 1 times minus 1 minus 1, it\u0027s minus 1."},{"Start":"05:48.310 ","End":"05:53.055","Text":"Minus 4, but minus 1^4 comes up plus 1,"},{"Start":"05:53.055 ","End":"05:55.695","Text":"so I\u0027m just going to write 1."},{"Start":"05:55.695 ","End":"05:59.450","Text":"Minus 1^3, again, 3 is odd,"},{"Start":"05:59.450 ","End":"06:05.635","Text":"so this comes out to be minus 1 plus 9."},{"Start":"06:05.635 ","End":"06:07.835","Text":"Now, let\u0027s see."},{"Start":"06:07.835 ","End":"06:10.685","Text":"We have here minus 2,"},{"Start":"06:10.685 ","End":"06:16.440","Text":"minus 4, minus minus 1 is plus 1, plus 9."},{"Start":"06:16.440 ","End":"06:20.390","Text":"One way to do this is to take all the positives,"},{"Start":"06:20.390 ","End":"06:24.860","Text":"that\u0027s 10, all the negatives, that\u0027s minus 6."},{"Start":"06:24.860 ","End":"06:27.235","Text":"10 minus 6,"},{"Start":"06:27.235 ","End":"06:31.340","Text":"so the answer is 4."},{"Start":"06:31.340 ","End":"06:35.935","Text":"Let\u0027s underline it and onto the next one."},{"Start":"06:35.935 ","End":"06:45.365","Text":"Here it is, 9m^2 minus 16n^2 when m is 1/3 and n is negative 1/2."},{"Start":"06:45.365 ","End":"06:47.370","Text":"Let\u0027s get to it."},{"Start":"06:47.370 ","End":"06:51.596","Text":"What we have is 9, m^2."},{"Start":"06:51.596 ","End":"06:56.710","Text":"Now, we put 1/3 in brackets because the whole thing is raised to the power of 2."},{"Start":"06:56.710 ","End":"06:58.480","Text":"The brackets is like a negative,"},{"Start":"06:58.480 ","End":"07:01.510","Text":"you need to protect it to see it\u0027s one whole thing."},{"Start":"07:01.510 ","End":"07:03.939","Text":"Minus 16, same thing here,"},{"Start":"07:03.939 ","End":"07:09.850","Text":"we put the minus 1/2 in brackets, squared."},{"Start":"07:09.950 ","End":"07:12.840","Text":"This is going to equal,"},{"Start":"07:12.840 ","End":"07:16.100","Text":"let\u0027s see, 1/3^2,"},{"Start":"07:16.100 ","End":"07:17.660","Text":"we\u0027ve already learned the exponent,"},{"Start":"07:17.660 ","End":"07:20.900","Text":"is 1/3 times 1/3, it\u0027s 1/9,"},{"Start":"07:20.900 ","End":"07:22.834","Text":"multiply the tops, multiply the bottoms,"},{"Start":"07:22.834 ","End":"07:24.560","Text":"or you can say 1^2 over 3^2."},{"Start":"07:24.560 ","End":"07:26.960","Text":"Either way, it\u0027s 1/9."},{"Start":"07:26.960 ","End":"07:28.805","Text":"We have 9 times"},{"Start":"07:28.805 ","End":"07:38.270","Text":"1/9 minus minus 1/2^2 this is an even power,"},{"Start":"07:38.270 ","End":"07:40.300","Text":"so it\u0027s going to be positive,"},{"Start":"07:40.300 ","End":"07:41.835","Text":"plus a 1/4,"},{"Start":"07:41.835 ","End":"07:46.065","Text":"so we get 16 times 1/4."},{"Start":"07:46.065 ","End":"07:52.425","Text":"This is equal to 9 times a 9th is 1,"},{"Start":"07:52.425 ","End":"07:56.194","Text":"16 times a quarter is, remember,"},{"Start":"07:56.194 ","End":"07:58.275","Text":"we just multiply the top,"},{"Start":"07:58.275 ","End":"08:01.215","Text":"minus 16 over 4."},{"Start":"08:01.215 ","End":"08:05.025","Text":"But 16 over 4 is 4,"},{"Start":"08:05.025 ","End":"08:09.360","Text":"so I\u0027ve got 1 minus 4,"},{"Start":"08:09.360 ","End":"08:14.850","Text":"and finally, see if I make it, yes, minus 3."},{"Start":"08:14.850 ","End":"08:20.620","Text":"Just enough space there. We\u0027re done."}],"ID":4893},{"Watched":false,"Name":"Combining Like Terms","Duration":"7m 43s","ChapterTopicVideoID":4893,"CourseChapterTopicPlaylistID":3857,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.800","Text":"In this clip I\u0027m going to talk about an algebraic technique called combining like terms,"},{"Start":"00:07.800 ","End":"00:09.990","Text":"or you can say collecting like terms,"},{"Start":"00:09.990 ","End":"00:13.000","Text":"I actually prefer the term collecting."},{"Start":"00:13.490 ","End":"00:22.140","Text":"I\u0027ll start with something outside of mathematics just to show you what the idea is."},{"Start":"00:22.140 ","End":"00:25.845","Text":"They say you can\u0027t compare apples and oranges,"},{"Start":"00:25.845 ","End":"00:29.415","Text":"though some wise guy here tried to do just that."},{"Start":"00:29.415 ","End":"00:33.840","Text":"The other slogan or saying is actually from"},{"Start":"00:33.840 ","End":"00:40.760","Text":"the Bible and it\u0027s the concept of separating the sheep from the goats."},{"Start":"00:40.760 ","End":"00:44.645","Text":"These 2 images may help you or may not,"},{"Start":"00:44.645 ","End":"00:47.435","Text":"when I talk about collecting like terms."},{"Start":"00:47.435 ","End":"00:49.890","Text":"Let\u0027s start with an example."},{"Start":"00:49.910 ","End":"00:55.290","Text":"I can say that 2 apples and 3 apples is 5 apples,"},{"Start":"00:55.290 ","End":"01:00.675","Text":"and I can even say that 4 oranges less 1 orange is 3 oranges."},{"Start":"01:00.675 ","End":"01:04.095","Text":"But how much is 2 oranges plus 3 apples?"},{"Start":"01:04.095 ","End":"01:06.000","Text":"I can\u0027t say."},{"Start":"01:06.000 ","End":"01:08.265","Text":"It\u0027s like that in algebra."},{"Start":"01:08.265 ","End":"01:16.455","Text":"If I see an expression like 4x+3x-10x,"},{"Start":"01:16.455 ","End":"01:21.060","Text":"they all involve x so x is like apples,"},{"Start":"01:21.060 ","End":"01:26.675","Text":"and I say, how much is 4 apples plus 3 apples minus 10 apples."},{"Start":"01:26.675 ","End":"01:28.550","Text":"Let\u0027s say the minus is a debt."},{"Start":"01:28.550 ","End":"01:32.120","Text":"I have 4 apples from 1 place and 3 apples from another place,"},{"Start":"01:32.120 ","End":"01:33.875","Text":"but I owe 10 apples."},{"Start":"01:33.875 ","End":"01:37.615","Text":"So altogether I have 4 plus 3 minus 10,"},{"Start":"01:37.615 ","End":"01:40.920","Text":"is 7 minus 10, is minus 3."},{"Start":"01:40.920 ","End":"01:43.530","Text":"I still owe 3 apples."},{"Start":"01:43.530 ","End":"01:48.030","Text":"That\u0027s one example. I\u0027m going to do 7 examples in all."},{"Start":"01:48.030 ","End":"01:56.040","Text":"If I have a^5 plus 2a^5,"},{"Start":"01:56.040 ","End":"01:59.245","Text":"if it just says a^5, it\u0027s one a^5."},{"Start":"01:59.245 ","End":"02:05.130","Text":"This could be let\u0027s say an orange and another 2 oranges and then I"},{"Start":"02:05.130 ","End":"02:11.280","Text":"would have 3 oranges, 3a^5."},{"Start":"02:11.280 ","End":"02:21.595","Text":"I could have 9m minus 11 minus 7m plus 3."},{"Start":"02:21.595 ","End":"02:25.970","Text":"This time I really do have apples and oranges,"},{"Start":"02:25.970 ","End":"02:27.980","Text":"or sheep and goats."},{"Start":"02:27.980 ","End":"02:36.340","Text":"I have things that contain m and I have terms that are just numbers."},{"Start":"02:36.340 ","End":"02:39.600","Text":"Let\u0027s say the m are the sheep and"},{"Start":"02:39.600 ","End":"02:42.970","Text":"the numbers are the goats so I\u0027m going to put the sheep on the right,"},{"Start":"02:42.970 ","End":"02:44.140","Text":"no, it\u0027s the left, never mind."},{"Start":"02:44.140 ","End":"02:47.950","Text":"What I do is I say, okay,"},{"Start":"02:47.950 ","End":"02:54.940","Text":"I have 9m minus 7m and so that comes out to be 2m,"},{"Start":"02:54.940 ","End":"02:57.570","Text":"and that\u0027s the sheep."},{"Start":"02:57.570 ","End":"03:01.365","Text":"Numbers, I have minus 11 plus 3,"},{"Start":"03:01.365 ","End":"03:03.120","Text":"and that\u0027s minus 8,"},{"Start":"03:03.120 ","End":"03:05.590","Text":"and that\u0027s separating the sheep from the goats."},{"Start":"03:05.590 ","End":"03:08.830","Text":"Perhaps it\u0027s enough with these analogies."},{"Start":"03:08.830 ","End":"03:12.167","Text":"There are other fruits besides apples and oranges,"},{"Start":"03:12.167 ","End":"03:15.920","Text":"we also have bananas and we\u0027ll see that in the following example."},{"Start":"03:15.920 ","End":"03:26.880","Text":"I\u0027ll take 1 plus b^2 minus 3b minus 4 minus 5b^2."},{"Start":"03:27.670 ","End":"03:31.430","Text":"Now, I have here numbers,"},{"Start":"03:31.430 ","End":"03:32.960","Text":"that\u0027s this one, and this one."},{"Start":"03:32.960 ","End":"03:36.320","Text":"I have terms containing b, that\u0027s this one,"},{"Start":"03:36.320 ","End":"03:38.610","Text":"and I have terms containing b^2,"},{"Start":"03:38.610 ","End":"03:39.945","Text":"that\u0027s this and this."},{"Start":"03:39.945 ","End":"03:43.355","Text":"Now b and b^2 are not to be mixed."},{"Start":"03:43.355 ","End":"03:50.435","Text":"Let\u0027s say the numbers are like apples,"},{"Start":"03:50.435 ","End":"03:51.710","Text":"that\u0027s this and this,"},{"Start":"03:51.710 ","End":"03:55.460","Text":"the b can be like oranges,"},{"Start":"03:55.460 ","End":"04:02.315","Text":"and the b^2 can be like the bananas,"},{"Start":"04:02.315 ","End":"04:04.405","Text":"so that\u0027s this and this."},{"Start":"04:04.405 ","End":"04:07.150","Text":"What we do is when we collect like terms,"},{"Start":"04:07.150 ","End":"04:12.460","Text":"we collect the same kinds of fruits colors together."},{"Start":"04:12.460 ","End":"04:16.920","Text":"So what we\u0027ll get is first the apples,"},{"Start":"04:16.920 ","End":"04:22.540","Text":"1 minus 4, minus 3."},{"Start":"04:22.700 ","End":"04:25.740","Text":"Oranges, there\u0027s only one,"},{"Start":"04:25.740 ","End":"04:28.830","Text":"that\u0027s this one, that\u0027s minus 3b."},{"Start":"04:28.830 ","End":"04:32.170","Text":"Then bananas, which is the b^2 terms,"},{"Start":"04:32.170 ","End":"04:34.260","Text":"I have plus 1 of them,"},{"Start":"04:34.260 ","End":"04:36.045","Text":"it\u0027s like there was a 1 here."},{"Start":"04:36.045 ","End":"04:38.295","Text":"Plus 1 minus 5,"},{"Start":"04:38.295 ","End":"04:45.880","Text":"altogether minus 4b^2, and there we are."},{"Start":"04:45.880 ","End":"04:50.280","Text":"Next example is the exercise,"},{"Start":"04:50.280 ","End":"04:52.185","Text":"I wrote it out already."},{"Start":"04:52.185 ","End":"04:54.560","Text":"Let\u0027s take a look at what we have here."},{"Start":"04:54.560 ","End":"04:58.040","Text":"We have an xy, we have x^2y,"},{"Start":"04:58.040 ","End":"05:01.550","Text":"we have y^2x, and we have xy."},{"Start":"05:01.550 ","End":"05:06.270","Text":"Now the xy and the xy obviously can be combined."},{"Start":"05:06.270 ","End":"05:14.495","Text":"But x^2y and y^2x are different so one of them"},{"Start":"05:14.495 ","End":"05:18.965","Text":"will be one kind of fruit and the other will be"},{"Start":"05:18.965 ","End":"05:23.480","Text":"another kind of fruit using the fruit analogy."},{"Start":"05:23.480 ","End":"05:26.510","Text":"All we can combine the apples,"},{"Start":"05:26.510 ","End":"05:27.950","Text":"or whatever you want to call them,"},{"Start":"05:27.950 ","End":"05:31.720","Text":"and we have 4 minus 5 of them,"},{"Start":"05:31.720 ","End":"05:34.170","Text":"so we end up with minus 1 of them."},{"Start":"05:34.170 ","End":"05:35.520","Text":"I don\u0027t write the 1,"},{"Start":"05:35.520 ","End":"05:39.490","Text":"I just write minus xy, it\u0027s minus 1xy."},{"Start":"05:39.490 ","End":"05:44.390","Text":"Then this I just have to take as is, minus 3x^2y,"},{"Start":"05:44.390 ","End":"05:51.739","Text":"and this as is, 3y^2x."},{"Start":"05:51.739 ","End":"05:55.140","Text":"Next example, and here it is."},{"Start":"05:55.140 ","End":"05:57.230","Text":"Let\u0027s see what we can collect."},{"Start":"05:57.230 ","End":"05:59.270","Text":"Let\u0027s start with the first one, a^2b,"},{"Start":"05:59.270 ","End":"06:02.765","Text":"are there any more of those, a^2b?"},{"Start":"06:02.765 ","End":"06:04.550","Text":"No, I don\u0027t see any more of those,"},{"Start":"06:04.550 ","End":"06:06.080","Text":"so that\u0027s on its own."},{"Start":"06:06.080 ","End":"06:09.065","Text":"The next one, ab."},{"Start":"06:09.065 ","End":"06:14.795","Text":"Well, yes, there\u0027s one here and there\u0027s another one here."},{"Start":"06:14.795 ","End":"06:18.590","Text":"Next, b^2a."},{"Start":"06:18.590 ","End":"06:22.335","Text":"There\u0027s this one here, this one."},{"Start":"06:22.335 ","End":"06:24.945","Text":"Now its a trick question kind of,"},{"Start":"06:24.945 ","End":"06:29.160","Text":"because multiplication, the order isn\u0027t important,"},{"Start":"06:29.160 ","End":"06:35.540","Text":"b^2a and ab^2 are surely the same thing so this doesn\u0027t look like the same kind of fruit,"},{"Start":"06:35.540 ","End":"06:38.480","Text":"but it actually is."},{"Start":"06:38.480 ","End":"06:45.035","Text":"Let\u0027s combine them and we get"},{"Start":"06:45.035 ","End":"06:53.105","Text":"a^2b is on its own, so that\u0027s a^2b."},{"Start":"06:53.105 ","End":"07:02.325","Text":"Then these two combine minus 1ab plus 9ab, altogether plus 8ab."},{"Start":"07:02.325 ","End":"07:06.405","Text":"Here we have 3 of these minus 5 of these,"},{"Start":"07:06.405 ","End":"07:08.830","Text":"so we have minus 2 of them."},{"Start":"07:08.830 ","End":"07:13.825","Text":"Then you have to just take your pick whether you want to write ab^2, or b^2a."},{"Start":"07:13.825 ","End":"07:19.565","Text":"Well, let\u0027s keep alphabetical order and I\u0027ll go with this one, ab^2."},{"Start":"07:19.565 ","End":"07:22.600","Text":"They are the same but they just look different."},{"Start":"07:22.600 ","End":"07:25.820","Text":"Last example."},{"Start":"07:26.520 ","End":"07:29.120","Text":"I was going to do this exercise,"},{"Start":"07:29.120 ","End":"07:32.630","Text":"but I changed my mind because I realized that you haven\u0027t yet"},{"Start":"07:32.630 ","End":"07:36.170","Text":"learned how to expand brackets,"},{"Start":"07:36.170 ","End":"07:43.380","Text":"and that will be the subject of the next lesson so I\u0027m done here for now."}],"ID":4894},{"Watched":false,"Name":"Expansion - Part 1","Duration":"11m 23s","ChapterTopicVideoID":4902,"CourseChapterTopicPlaylistID":3857,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.100","Text":"In this clip, I\u0027m continuing with"},{"Start":"00:02.100 ","End":"00:07.050","Text":"algebraic expressions and this time the topic is expanding brackets."},{"Start":"00:07.050 ","End":"00:09.270","Text":"Now, I\u0027m going to give"},{"Start":"00:09.270 ","End":"00:14.430","Text":"a numerical example before I write something called the distributive law."},{"Start":"00:14.430 ","End":"00:15.675","Text":"But if I just write it,"},{"Start":"00:15.675 ","End":"00:17.786","Text":"you\u0027ll be scared of it,"},{"Start":"00:17.786 ","End":"00:20.100","Text":"so I\u0027ll start with an example."},{"Start":"00:20.100 ","End":"00:24.900","Text":"I have a friend, and I want to buy him an apple and an orange."},{"Start":"00:24.900 ","End":"00:32.190","Text":"Let\u0027s say an apple costs 40 cents,"},{"Start":"00:32.190 ","End":"00:40.030","Text":"and let\u0027s say that an orange costs 50 cents."},{"Start":"00:41.620 ","End":"00:44.630","Text":"I gave it a bit of color here."},{"Start":"00:44.630 ","End":"00:49.505","Text":"I would say that the total cost would be 40,"},{"Start":"00:49.505 ","End":"00:50.960","Text":"lets work in cents,"},{"Start":"00:50.960 ","End":"00:55.000","Text":"plus 50, which would equal 90."},{"Start":"00:55.000 ","End":"01:01.275","Text":"Now, suppose that I had 4 friends."},{"Start":"01:01.275 ","End":"01:04.860","Text":"Everything is times 4,"},{"Start":"01:04.860 ","End":"01:07.725","Text":"how much would it cost me now?"},{"Start":"01:07.725 ","End":"01:11.780","Text":"Well, and I wanted to buy an apple and orange for each,"},{"Start":"01:11.780 ","End":"01:13.790","Text":"I meant, sorry if I didn\u0027t say that."},{"Start":"01:13.790 ","End":"01:16.235","Text":"There\u0027s 2 ways of computing it."},{"Start":"01:16.235 ","End":"01:20.555","Text":"I noticed that when I go to the grocer,"},{"Start":"01:20.555 ","End":"01:24.305","Text":"that he does it the long way, what I call,"},{"Start":"01:24.305 ","End":"01:27.065","Text":"because 1 way of computing it is to say,"},{"Start":"01:27.065 ","End":"01:32.960","Text":"\"If I have 4 apples and 40 cents times 4, that\u0027s 160 cents.\""},{"Start":"01:32.960 ","End":"01:35.230","Text":"Yeah, I know it\u0027s a $1.60."},{"Start":"01:35.230 ","End":"01:37.500","Text":"Which working in sense."},{"Start":"01:37.500 ","End":"01:42.950","Text":"If I say an orange is 50 cents and I take 4 oranges and I get 4 times 50,"},{"Start":"01:42.950 ","End":"01:45.725","Text":"that\u0027s 200 cents, yes, 2 dollars."},{"Start":"01:45.725 ","End":"01:54.450","Text":"Then I would add these 2 together and say 160 plus 200 is equal to 360."},{"Start":"01:54.450 ","End":"01:59.060","Text":"My cost is 360 cents, so $3.60."},{"Start":"01:59.060 ","End":"02:03.515","Text":"But I always figured that the easier way to do it was to say, well,"},{"Start":"02:03.515 ","End":"02:06.000","Text":"each friend has an apple or an orange,"},{"Start":"02:06.000 ","End":"02:07.110","Text":"and is 90 cents,"},{"Start":"02:07.110 ","End":"02:12.150","Text":"and why not just say 90 times 4 and get 360 straightaway."},{"Start":"02:12.150 ","End":"02:15.094","Text":"This way I have 1 addition and 1 multiplication."},{"Start":"02:15.094 ","End":"02:19.310","Text":"The other way I have 2 multiplications and 1 addition."},{"Start":"02:20.210 ","End":"02:22.700","Text":"The answer comes out the same,"},{"Start":"02:22.700 ","End":"02:23.930","Text":"but there\u0027s 2 different ways."},{"Start":"02:23.930 ","End":"02:28.325","Text":"Now, the way to abstract this first of all with numbers,"},{"Start":"02:28.325 ","End":"02:29.870","Text":"you can say that"},{"Start":"02:29.870 ","End":"02:39.680","Text":"4(40+50) is 4 times"},{"Start":"02:39.680 ","End":"02:44.165","Text":"40 plus 4 times 50."},{"Start":"02:44.165 ","End":"02:50.375","Text":"They both give the same answer because this 1 gives me 4 times 90,"},{"Start":"02:50.375 ","End":"02:56.460","Text":"and this 1 gives me 160 plus 200,"},{"Start":"02:56.460 ","End":"02:58.850","Text":"and that 1 looks like decimal point."},{"Start":"02:58.850 ","End":"03:02.555","Text":"Let\u0027s change that to multiplication sign."},{"Start":"03:02.555 ","End":"03:07.745","Text":"I don\u0027t usually use the cross sign anymore for multiplication, we use the dot."},{"Start":"03:07.745 ","End":"03:13.010","Text":"But this is what you would call the distributive law."},{"Start":"03:13.010 ","End":"03:19.325","Text":"If I was to state it in letters,"},{"Start":"03:19.325 ","End":"03:29.030","Text":"then I would say that a(b+c)=ab+ac."},{"Start":"03:29.030 ","End":"03:32.675","Text":"Maybe we\u0027ll just emphasize it by putting a dot here."},{"Start":"03:32.675 ","End":"03:38.130","Text":"This is what\u0027s called the distributive law."},{"Start":"03:40.390 ","End":"03:44.450","Text":"All it says is that if I take 2 numbers and add them,"},{"Start":"03:44.450 ","End":"03:45.830","Text":"it could be subtraction also,"},{"Start":"03:45.830 ","End":"03:48.990","Text":"the same thing would work if I had a minus here."},{"Start":"03:50.050 ","End":"03:53.600","Text":"Multiply a number by the sum of 2 numbers,"},{"Start":"03:53.600 ","End":"03:57.695","Text":"I can multiply it by each 1 separately and then add."},{"Start":"03:57.695 ","End":"04:00.660","Text":"Just like if I wanted to do 4(40+50),"},{"Start":"04:00.850 ","End":"04:06.980","Text":"I could multiply 4 by each 1 of them separately and then add."},{"Start":"04:06.980 ","End":"04:12.000","Text":"That\u0027s the distributive law and that\u0027s the key to expanding brackets."},{"Start":"04:12.200 ","End":"04:14.420","Text":"I\u0027m going to erase most of this,"},{"Start":"04:14.420 ","End":"04:16.685","Text":"then we\u0027ll do some examples."},{"Start":"04:16.685 ","End":"04:18.725","Text":"That\u0027s to clear the board."},{"Start":"04:18.725 ","End":"04:21.810","Text":"Let me put this out of the way here,"},{"Start":"04:21.810 ","End":"04:25.160","Text":"and I think some people are scared of words like distributive,"},{"Start":"04:25.160 ","End":"04:31.790","Text":"so let\u0027s just get rid of that and instead what I\u0027ll do is I\u0027ll explain it with arrows."},{"Start":"04:31.790 ","End":"04:35.925","Text":"So that when I multiply a(b+c),"},{"Start":"04:35.925 ","End":"04:38.645","Text":"I multiply it by the first,"},{"Start":"04:38.645 ","End":"04:41.045","Text":"and then I multiply it by the second."},{"Start":"04:41.045 ","End":"04:42.770","Text":"So a times b is here,"},{"Start":"04:42.770 ","End":"04:44.360","Text":"and a times c is here."},{"Start":"04:44.360 ","End":"04:47.195","Text":"Same thing if there was a minus here."},{"Start":"04:47.195 ","End":"04:51.095","Text":"Let me just add that it would work for more than 2 things."},{"Start":"04:51.095 ","End":"04:53.990","Text":"For example, if I"},{"Start":"04:53.990 ","End":"05:02.600","Text":"had a(b+c-d) then this"},{"Start":"05:02.600 ","End":"05:10.365","Text":"would equal a times b plus a times c plus a times d,"},{"Start":"05:10.365 ","End":"05:16.110","Text":"and I\u0027m dispensing with the dot ab+ac-ad."},{"Start":"05:20.680 ","End":"05:23.435","Text":"Let\u0027s take an example."},{"Start":"05:23.435 ","End":"05:26.790","Text":"If I have 3(x+5),"},{"Start":"05:28.880 ","End":"05:31.780","Text":"this time we have some letters."},{"Start":"05:31.780 ","End":"05:34.675","Text":"I mean, not like with the apples and oranges."},{"Start":"05:34.675 ","End":"05:37.240","Text":"What we do is we say,"},{"Start":"05:37.240 ","End":"05:39.760","Text":"I\u0027m not going to put red arrows all the time,"},{"Start":"05:39.760 ","End":"05:41.770","Text":"maybe once I will say"},{"Start":"05:41.770 ","End":"05:50.020","Text":"3(x+5)=3x+15,"},{"Start":"05:50.020 ","End":"05:52.610","Text":"3 times 5 is 15."},{"Start":"05:52.910 ","End":"05:58.360","Text":"If I have something like y(y-4),"},{"Start":"06:00.410 ","End":"06:06.165","Text":"that will equal y times y is y^2,"},{"Start":"06:06.165 ","End":"06:09.645","Text":"y times 4,"},{"Start":"06:09.645 ","End":"06:11.545","Text":"I could write y4."},{"Start":"06:11.545 ","End":"06:12.770","Text":"We don\u0027t usually do that."},{"Start":"06:12.770 ","End":"06:15.860","Text":"We say 4y, the order of multiplication doesn\u0027t matter."},{"Start":"06:15.860 ","End":"06:18.990","Text":"You put the number before the letter."},{"Start":"06:19.600 ","End":"06:21.940","Text":"Another example,"},{"Start":"06:21.940 ","End":"06:32.355","Text":"-2(b-3x)."},{"Start":"06:32.355 ","End":"06:37.665","Text":"What I get is -2 times b is -2b."},{"Start":"06:37.665 ","End":"06:41.190","Text":"Now, -2 times -3x,"},{"Start":"06:41.190 ","End":"06:43.515","Text":"minus with minus is plus."},{"Start":"06:43.515 ","End":"06:48.850","Text":"So it\u0027s 2 times 3x is 6x."},{"Start":"06:48.950 ","End":"06:51.090","Text":"Another"},{"Start":"06:51.090 ","End":"07:00.840","Text":"x(x-2y+3z)"},{"Start":"07:00.840 ","End":"07:03.900","Text":"if you\u0027re in the UK,"},{"Start":"07:03.900 ","End":"07:10.950","Text":"actually say z, but mostly it\u0027s Americans, so it\u0027s z."},{"Start":"07:11.180 ","End":"07:17.865","Text":"This equals x with x is x^2."},{"Start":"07:17.865 ","End":"07:20.070","Text":"Remember x times x is x^2,"},{"Start":"07:20.070 ","End":"07:22.080","Text":"x times 2y,"},{"Start":"07:22.080 ","End":"07:23.670","Text":"but it\u0027s a minus,"},{"Start":"07:23.670 ","End":"07:25.620","Text":"I don\u0027t write x2y,"},{"Start":"07:25.620 ","End":"07:28.000","Text":"I write 2xy,"},{"Start":"07:28.340 ","End":"07:37.980","Text":"and then plus 3xz."},{"Start":"07:37.980 ","End":"07:39.390","Text":"In case you\u0027re wondering the reason"},{"Start":"07:39.390 ","End":"07:39.930","Text":"I"},{"Start":"07:39.930 ","End":"07:41.650","Text":"crossed"},{"Start":"07:49.400 ","End":"07:50.820","Text":"it,"},{"Start":"07:50.820 ","End":"07:53.638","Text":"it\u0027s commonly done so it\u0027s not confused with the 2"},{"Start":"07:53.638 ","End":"07:57.410","Text":"because you can write this and you can write this,"},{"Start":"07:57.410 ","End":"08:00.245","Text":"anyway, cross to run cross,"},{"Start":"08:00.245 ","End":"08:02.755","Text":"z or z same thing."},{"Start":"08:02.755 ","End":"08:05.739","Text":"Now, let\u0027s try another example."},{"Start":"08:05.739 ","End":"08:06.815","Text":"Let\u0027s"},{"Start":"08:06.815 ","End":"08:08.250","Text":"take"},{"Start":"08:11.030 ","End":"08:21.250","Text":"-(7x-5)."},{"Start":"08:21.250 ","End":"08:25.085","Text":"When you have a minus in front of a bracket,"},{"Start":"08:25.085 ","End":"08:29.390","Text":"it really means -1 times this thing."},{"Start":"08:29.390 ","End":"08:31.430","Text":"If it\u0027s -1,"},{"Start":"08:31.430 ","End":"08:34.830","Text":"I multiply by each of these."},{"Start":"08:35.270 ","End":"08:39.780","Text":"I like the red arcs, so I kept them."},{"Start":"08:39.780 ","End":"08:47.090","Text":"I multiply by each of them so -1 times 7x minus times plus is minus,"},{"Start":"08:47.090 ","End":"08:49.220","Text":"so it\u0027s -7x,"},{"Start":"08:49.220 ","End":"08:51.590","Text":"but minus times minus is plus,"},{"Start":"08:51.590 ","End":"08:55.565","Text":"so -1 times -5 is 5."},{"Start":"08:55.565 ","End":"09:00.905","Text":"Basically, there\u0027s a rule of thumb that if you have minus in front of a bracket,"},{"Start":"09:00.905 ","End":"09:03.680","Text":"you can get rid of the brackets and the minus,"},{"Start":"09:03.680 ","End":"09:06.170","Text":"but you have to reverse all the signs; where there was plus,"},{"Start":"09:06.170 ","End":"09:09.660","Text":"there will be minus and where there\u0027s minus there\u0027s plus."},{"Start":"09:10.630 ","End":"09:18.335","Text":"Another example, 2/3 of,"},{"Start":"09:18.335 ","End":"09:26.930","Text":"let\u0027s say 9a-6b."},{"Start":"09:26.930 ","End":"09:28.670","Text":"This time we have a fraction,"},{"Start":"09:28.670 ","End":"09:30.815","Text":"but it makes no difference."},{"Start":"09:30.815 ","End":"09:33.455","Text":"I can\u0027t be bothered to put the arcs in."},{"Start":"09:33.455 ","End":"09:36.620","Text":"It\u0027s 2/3 times 9a,"},{"Start":"09:36.620 ","End":"09:40.640","Text":"but I might have to do a side exercise if you\u0027re not good in fractions in your head,"},{"Start":"09:40.640 ","End":"09:46.040","Text":"we could say 2/3 times 9 is when we multiply by a whole number,"},{"Start":"09:46.040 ","End":"09:50.775","Text":"put it on the top, 2 times 9 over 3."},{"Start":"09:50.775 ","End":"09:55.050","Text":"The 3 cancels, it comes out to be 6."},{"Start":"09:55.050 ","End":"10:02.400","Text":"Then we want to do 2/3 times 6 also will need that."},{"Start":"10:02.400 ","End":"10:07.560","Text":"That will be 2 times 6 over 3."},{"Start":"10:07.560 ","End":"10:09.555","Text":"Whether you cancel or not,"},{"Start":"10:09.555 ","End":"10:10.800","Text":"comes out to be 4."},{"Start":"10:10.800 ","End":"10:16.230","Text":"So back here, 2/3 times 9a is 2/3 times 9 times a,"},{"Start":"10:16.230 ","End":"10:18.585","Text":"which is 6a,"},{"Start":"10:18.585 ","End":"10:27.660","Text":"and minus 2/3 times 6 times b,"},{"Start":"10:27.660 ","End":"10:31.495","Text":"and 2/3 times 6 is 4, so it\u0027s -4b."},{"Start":"10:31.495 ","End":"10:39.540","Text":"I\u0027ll do 1 more exercise because I have room for it, (2x+3y)4."},{"Start":"10:44.240 ","End":"10:49.225","Text":"Here they put the number on the right."},{"Start":"10:49.225 ","End":"10:54.290","Text":"Since multiplication doesn\u0027t depend on the order,"},{"Start":"10:54.290 ","End":"10:56.210","Text":"I can put the 4 in front."},{"Start":"10:56.210 ","End":"11:02.005","Text":"I would first of all say this is 4(2x+3y),"},{"Start":"11:02.005 ","End":"11:04.110","Text":"and then proceed as usual,"},{"Start":"11:04.110 ","End":"11:06.765","Text":"4 times 2x is 8x,"},{"Start":"11:06.765 ","End":"11:10.935","Text":"4 times 3y is 12y."},{"Start":"11:10.935 ","End":"11:12.960","Text":"I think this is enough,"},{"Start":"11:12.960 ","End":"11:16.925","Text":"because they\u0027re also solved exercises besides this tutorial."},{"Start":"11:16.925 ","End":"11:23.970","Text":"But we will continue this topic in the next clip. Bye for now."}],"ID":4895},{"Watched":false,"Name":"Expansion - Part 2","Duration":"9m 10s","ChapterTopicVideoID":4903,"CourseChapterTopicPlaylistID":3857,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.185","Text":"Hi, recycled some of the stuff from the previous clip."},{"Start":"00:04.185 ","End":"00:08.010","Text":"In fact, we are going to be continuing,"},{"Start":"00:08.010 ","End":"00:10.425","Text":"so this is continued."},{"Start":"00:10.425 ","End":"00:13.515","Text":"We talked about the distributive law."},{"Start":"00:13.515 ","End":"00:18.870","Text":"We took an example with an apple and an orange."},{"Start":"00:18.870 ","End":"00:21.425","Text":"I want to just generalize this a bit,"},{"Start":"00:21.425 ","End":"00:24.410","Text":"and then write a more general form where I have the sum of"},{"Start":"00:24.410 ","End":"00:28.280","Text":"two things or just use the letters of the alphabet in order,"},{"Start":"00:28.280 ","End":"00:31.530","Text":"a plus b times c plus"},{"Start":"00:31.530 ","End":"00:37.535","Text":"d. This tunnel going to write it first and then explain it again with apples and oranges."},{"Start":"00:37.535 ","End":"00:41.630","Text":"This is going to equal ac plus"},{"Start":"00:41.630 ","End":"00:49.345","Text":"ad plus bc plus bd."},{"Start":"00:49.345 ","End":"00:59.875","Text":"You want to buy, someone may be 2 apples and 3 oranges."},{"Start":"00:59.875 ","End":"01:02.905","Text":"This is a bad letter because it looked like a zero."},{"Start":"01:02.905 ","End":"01:04.645","Text":"Change it into a pair."},{"Start":"01:04.645 ","End":"01:11.380","Text":"So we\u0027re saying you want to get like this may be a gift pack of 2 apples and 3 pears."},{"Start":"01:11.380 ","End":"01:15.460","Text":"We\u0027re going to first of all buy them for,"},{"Start":"01:15.460 ","End":"01:20.525","Text":"I don\u0027t know, 4 relatives and for 5 friends."},{"Start":"01:20.525 ","End":"01:23.215","Text":"How much is it going to cost me?"},{"Start":"01:23.215 ","End":"01:26.020","Text":"Wait a minute, this is getting to be too expensive."},{"Start":"01:26.020 ","End":"01:29.935","Text":"I\u0027m just going to get 1 apple and 1 pear for each,"},{"Start":"01:29.935 ","End":"01:34.695","Text":"just 2 relatives and 3 friends."},{"Start":"01:34.695 ","End":"01:39.720","Text":"One way of doing it is to say,"},{"Start":"01:39.720 ","End":"01:44.855","Text":"each apple is $0.40."},{"Start":"01:44.855 ","End":"01:47.540","Text":"Each pair is $0.50,"},{"Start":"01:47.540 ","End":"01:53.135","Text":"so each apple and pear package is 40 plus $0.50."},{"Start":"01:53.135 ","End":"01:56.240","Text":"Now I\u0027m going to have to multiply it by the number of people,"},{"Start":"01:56.240 ","End":"02:01.114","Text":"but the number of people is 2 relatives plus 3 friends."},{"Start":"02:01.114 ","End":"02:06.279","Text":"I could say an apple and a pear is a $0.90 package."},{"Start":"02:06.279 ","End":"02:12.300","Text":"I have altogether 5 friends and I could multiply 5 times"},{"Start":"02:12.300 ","End":"02:19.995","Text":"$0.90 and get $0.450 and that\u0027s $4.50,"},{"Start":"02:19.995 ","End":"02:23.360","Text":"but I could do it another way,"},{"Start":"02:23.360 ","End":"02:28.345","Text":"I have to say I could say that I have"},{"Start":"02:28.345 ","End":"02:34.880","Text":"2 relatives and I\u0027m going to buy each of them an apple."},{"Start":"02:34.880 ","End":"02:39.245","Text":"I could say there\u0027s 2 apples,"},{"Start":"02:39.245 ","End":"02:48.920","Text":"and also I have 2 pears that\u0027s twice $0.50 plus I"},{"Start":"02:48.920 ","End":"02:59.195","Text":"have also another 3 apples for the friend and 3 pears."},{"Start":"02:59.195 ","End":"03:01.205","Text":"Now if I do it this way,"},{"Start":"03:01.205 ","End":"03:05.080","Text":"what I\u0027ll get will be, let\u0027s see,"},{"Start":"03:05.080 ","End":"03:07.695","Text":"80 and 100 is 180,"},{"Start":"03:07.695 ","End":"03:17.230","Text":"180 plus 120 is 300 and it comes out to be 450 also."},{"Start":"03:17.230 ","End":"03:27.395","Text":"Anyway, I\u0027m going to highlight this and let\u0027s just do some more abstract examples."},{"Start":"03:27.395 ","End":"03:29.960","Text":"This was just a story."},{"Start":"03:29.960 ","End":"03:33.095","Text":"If you remember in the previous clip,"},{"Start":"03:33.095 ","End":"03:35.945","Text":"I put some arcs here to explain it."},{"Start":"03:35.945 ","End":"03:38.495","Text":"Basically what we do is this,"},{"Start":"03:38.495 ","End":"03:41.720","Text":"If I see a sum of 2 things times the sum of 2 things,"},{"Start":"03:41.720 ","End":"03:43.280","Text":"could be more than 2 things,"},{"Start":"03:43.280 ","End":"03:45.515","Text":"but meanwhile, let\u0027s leave it like that."},{"Start":"03:45.515 ","End":"03:48.890","Text":"I take each one in here and multiply it by each one in here,"},{"Start":"03:48.890 ","End":"03:50.675","Text":"so this with this,"},{"Start":"03:50.675 ","End":"03:52.385","Text":"that\u0027s my ac,"},{"Start":"03:52.385 ","End":"03:57.460","Text":"this with this, that\u0027s my ad."},{"Start":"03:57.460 ","End":"03:59.645","Text":"Then this one with this one,"},{"Start":"03:59.645 ","End":"04:01.070","Text":"that\u0027s the bc,"},{"Start":"04:01.070 ","End":"04:03.680","Text":"and this one with this one that\u0027s the bd,"},{"Start":"04:03.680 ","End":"04:05.030","Text":"each one in here,"},{"Start":"04:05.030 ","End":"04:06.800","Text":"with each one in here."},{"Start":"04:06.800 ","End":"04:11.405","Text":"I\u0027ll just do a couple of examples because there\u0027s plenty more in the exercises."},{"Start":"04:11.405 ","End":"04:15.455","Text":"Let\u0027s try 2x plus 3"},{"Start":"04:15.455 ","End":"04:22.100","Text":"times x minus 4 and see what we get."},{"Start":"04:22.100 ","End":"04:27.170","Text":"Remember the good for pluses or minuses, the distributive law."},{"Start":"04:27.170 ","End":"04:34.640","Text":"First thing I do while I\u0027m new at it is to draw a little arcs this with this,"},{"Start":"04:34.640 ","End":"04:39.030","Text":"this, with this, this with this and this with this."},{"Start":"04:39.760 ","End":"04:45.830","Text":"Then I say 2x times x is 2x^2."},{"Start":"04:45.830 ","End":"04:54.810","Text":"2x times 4 is 2 times 4 times x is 8x,"},{"Start":"04:54.810 ","End":"04:56.160","Text":"but it\u0027s minus,"},{"Start":"04:56.160 ","End":"05:00.570","Text":"so it\u0027s minus 8x."},{"Start":"05:00.570 ","End":"05:05.535","Text":"Then this with this is plus 3x."},{"Start":"05:05.535 ","End":"05:11.840","Text":"Then this with this 3 times minus 4 is minus 12."},{"Start":"05:11.840 ","End":"05:14.875","Text":"But I don\u0027t leave the answer like this."},{"Start":"05:14.875 ","End":"05:20.830","Text":"I collect like terms here I have a term with x and here term with x. I would write"},{"Start":"05:20.830 ","End":"05:30.535","Text":"the answer as 2x^2 minus 8 plus 3 is minus 5x minus 12."},{"Start":"05:30.535 ","End":"05:32.350","Text":"I\u0027ll just do one more,"},{"Start":"05:32.350 ","End":"05:36.010","Text":"but we\u0027ll make it a good one."},{"Start":"05:36.010 ","End":"05:38.635","Text":"Here it is. I wrote it out."},{"Start":"05:38.635 ","End":"05:41.530","Text":"Let\u0027s see how we approach this."},{"Start":"05:41.530 ","End":"05:46.450","Text":"This is a more complex problem with different elements."},{"Start":"05:46.450 ","End":"05:51.860","Text":"Now, we do order of operations."},{"Start":"05:52.110 ","End":"05:56.350","Text":"I have to do multiplication before subtraction."},{"Start":"05:56.350 ","End":"05:58.900","Text":"I\u0027ll do this multiplication first,"},{"Start":"05:58.900 ","End":"06:00.940","Text":"and I used to do this multiplication."},{"Start":"06:00.940 ","End":"06:05.320","Text":"I\u0027ll use the previous distributive law, the simple one,"},{"Start":"06:05.320 ","End":"06:12.745","Text":"where I just need to do this with this and this with this."},{"Start":"06:12.745 ","End":"06:16.210","Text":"Then in anticipation of what\u0027s later on,"},{"Start":"06:16.210 ","End":"06:18.430","Text":"I\u0027ll put the box here,"},{"Start":"06:18.430 ","End":"06:19.510","Text":"this with this, this,"},{"Start":"06:19.510 ","End":"06:21.985","Text":"with this, this, with this, this with this."},{"Start":"06:21.985 ","End":"06:24.370","Text":"But meanwhile we\u0027re here,"},{"Start":"06:24.370 ","End":"06:34.770","Text":"and so we get 4 times 3a is 4 times 3 times a is 12a,"},{"Start":"06:34.770 ","End":"06:39.595","Text":"4 times minus 2 is minus 8 minus."},{"Start":"06:39.595 ","End":"06:41.600","Text":"Now I\u0027m going to multiply this out,"},{"Start":"06:41.600 ","End":"06:44.030","Text":"but the minus has to apply to all of this."},{"Start":"06:44.030 ","End":"06:45.515","Text":"I\u0027ll put a bracket,"},{"Start":"06:45.515 ","End":"06:48.089","Text":"let\u0027s make it a square bracket."},{"Start":"06:49.520 ","End":"06:57.135","Text":"Here, we\u0027ll do, let\u0027s see this with this 2a times 3a."},{"Start":"06:57.135 ","End":"07:03.060","Text":"I can change the order 2 times 3 times a times a 6a^2,"},{"Start":"07:03.060 ","End":"07:05.850","Text":"2a with 5 is 2 times 5 times"},{"Start":"07:05.850 ","End":"07:14.790","Text":"a is 10a minus 1 times 3a is minus 3a."},{"Start":"07:14.790 ","End":"07:18.880","Text":"Minus 1 times 5 is minus 5."},{"Start":"07:18.880 ","End":"07:23.075","Text":"Then need to collect like terms."},{"Start":"07:23.075 ","End":"07:27.665","Text":"But I\u0027ll save a step if I just say that I\u0027m going to rewrite this"},{"Start":"07:27.665 ","End":"07:34.920","Text":"as 7a just instead of that, I\u0027ll put 7a."},{"Start":"07:34.920 ","End":"07:37.180","Text":"Just save a step."},{"Start":"07:39.500 ","End":"07:43.455","Text":"12a minus 8."},{"Start":"07:43.455 ","End":"07:48.910","Text":"Now, remember what I said previously about minus in front of the brackets."},{"Start":"07:48.910 ","End":"07:53.034","Text":"All you have to do is reverse the sign of everything."},{"Start":"07:53.034 ","End":"07:58.798","Text":"Wherever it was plus, so it was plus 6a^2 is minus 6a^2."},{"Start":"07:58.798 ","End":"08:02.070","Text":"It was plus 7a, it becomes minus 7a."},{"Start":"08:02.070 ","End":"08:03.780","Text":"It was minus 5,"},{"Start":"08:03.780 ","End":"08:06.620","Text":"it becomes plus 5."},{"Start":"08:06.620 ","End":"08:12.115","Text":"Then finally, we collect like terms."},{"Start":"08:12.115 ","End":"08:17.140","Text":"We have terms containing a squared."},{"Start":"08:17.140 ","End":"08:20.365","Text":"Remember apples, oranges, and bananas."},{"Start":"08:20.365 ","End":"08:25.810","Text":"Then the a, that will be my oranges here and here."},{"Start":"08:25.810 ","End":"08:30.220","Text":"And the numbers will be,"},{"Start":"08:30.220 ","End":"08:32.575","Text":"I don\u0027t know, lemons, bananas."},{"Start":"08:32.575 ","End":"08:36.170","Text":"Then we collect like terms. Remember that?"},{"Start":"08:36.510 ","End":"08:40.420","Text":"We get, let\u0027s collect them in,"},{"Start":"08:40.420 ","End":"08:46.600","Text":"usually do them in order of decreasing exponent first a^2,"},{"Start":"08:46.600 ","End":"08:49.650","Text":"that would be minus 6a^2."},{"Start":"08:49.650 ","End":"08:56.165","Text":"Then the a\u0027s, that\u0027s the oranges 12 minus 7 is 5."},{"Start":"08:56.165 ","End":"09:02.220","Text":"Then just the numbers minus 8 plus 5 is minus 3,"},{"Start":"09:02.220 ","End":"09:05.220","Text":"and that\u0027s the answer to this one."},{"Start":"09:05.220 ","End":"09:11.450","Text":"We are done with the tutorial on expanding brackets."}],"ID":4896},{"Watched":false,"Name":"Exercise 1","Duration":"9m 3s","ChapterTopicVideoID":4894,"CourseChapterTopicPlaylistID":3857,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.970","Text":"In this exercise, we have to evaluate"},{"Start":"00:02.970 ","End":"00:08.385","Text":"the following algebraic expressions using substitution or plugging in."},{"Start":"00:08.385 ","End":"00:11.590","Text":"I prefer to work on a copy."},{"Start":"00:13.070 ","End":"00:17.055","Text":"Let\u0027s start with the first one."},{"Start":"00:17.055 ","End":"00:22.785","Text":"In this expression we have x and y and we\u0027re given the value of each."},{"Start":"00:22.785 ","End":"00:25.590","Text":"What we have to do is to,"},{"Start":"00:25.590 ","End":"00:27.480","Text":"instead of x put 5,"},{"Start":"00:27.480 ","End":"00:33.120","Text":"and instead of y put minus 4 or plug-in 5 and minus 4."},{"Start":"00:33.120 ","End":"00:39.060","Text":"What we get is 5 instead of x"},{"Start":"00:39.060 ","End":"00:45.640","Text":"plus instead of y minus 4^3."},{"Start":"00:45.640 ","End":"00:49.760","Text":"Now it\u0027s not customary to have plus and minus next to each other."},{"Start":"00:49.760 ","End":"00:52.130","Text":"Usually when there\u0027s a minus,"},{"Start":"00:52.130 ","End":"00:58.445","Text":"we wrap it in brackets just to separate the operations plus and minus."},{"Start":"00:58.445 ","End":"01:03.680","Text":"Continuing. We do brackets first,"},{"Start":"01:03.680 ","End":"01:04.880","Text":"remember order of operations,"},{"Start":"01:04.880 ","End":"01:06.350","Text":"this is not really a bracket,"},{"Start":"01:06.350 ","End":"01:10.355","Text":"this is just to protect the minus from the plus."},{"Start":"01:10.355 ","End":"01:14.955","Text":"We first do the addition inside the brackets,"},{"Start":"01:14.955 ","End":"01:20.085","Text":"so 5 plus negative 4 is 1."},{"Start":"01:20.085 ","End":"01:26.400","Text":"We have 1 or plus 1^3 and that\u0027s 1 times 1 times 1,"},{"Start":"01:26.400 ","End":"01:30.960","Text":"which is 1, and that\u0027s the answer to the first one."},{"Start":"01:30.960 ","End":"01:40.910","Text":"Continuing. Here, we have the expression a^5 minus 3a^4 minus a^3 plus 7."},{"Start":"01:40.910 ","End":"01:42.380","Text":"There\u0027s only one variable a,"},{"Start":"01:42.380 ","End":"01:44.960","Text":"and we\u0027re told to substitute minus 1."},{"Start":"01:44.960 ","End":"01:46.670","Text":"Once again because of the minus,"},{"Start":"01:46.670 ","End":"01:48.860","Text":"I prefer to put it in brackets."},{"Start":"01:48.860 ","End":"01:55.080","Text":"I have minus 1^5 minus 3, and again,"},{"Start":"01:55.080 ","End":"02:00.724","Text":"a is minus 1^4 minus,"},{"Start":"02:00.724 ","End":"02:02.975","Text":"every time I see a,"},{"Start":"02:02.975 ","End":"02:05.344","Text":"I put minus 1 in brackets."},{"Start":"02:05.344 ","End":"02:10.250","Text":"Minus 1^3 plus 7."},{"Start":"02:10.250 ","End":"02:13.300","Text":"Exponents come first,"},{"Start":"02:13.300 ","End":"02:16.190","Text":"but there are several of them we\u0027ll do each one separately."},{"Start":"02:16.190 ","End":"02:20.720","Text":"Minus 1^5, because 5 is an odd number,"},{"Start":"02:20.720 ","End":"02:22.460","Text":"the answer will be minus,"},{"Start":"02:22.460 ","End":"02:25.260","Text":"so this is minus 1."},{"Start":"02:25.310 ","End":"02:31.725","Text":"Then minus 1^4 is plus 1 because this is an even number,"},{"Start":"02:31.725 ","End":"02:35.565","Text":"so it\u0027s minus 3 times"},{"Start":"02:35.565 ","End":"02:44.115","Text":"1 or let\u0027s just say times 1, it\u0027s possibly better."},{"Start":"02:44.115 ","End":"02:47.925","Text":"Minus, minus 1^3,"},{"Start":"02:47.925 ","End":"02:49.160","Text":"because 3 is an odd number,"},{"Start":"02:49.160 ","End":"02:51.415","Text":"is also minus 1,"},{"Start":"02:51.415 ","End":"02:54.070","Text":"and then plus 7."},{"Start":"02:54.070 ","End":"02:56.450","Text":"Then we do the multiplication,"},{"Start":"02:56.450 ","End":"02:58.910","Text":"but I\u0027m not going to waste a whole step on that."},{"Start":"02:58.910 ","End":"03:03.630","Text":"I\u0027ll just say that 3 times 1 is 3."},{"Start":"03:04.010 ","End":"03:06.900","Text":"What we get is minus 1,"},{"Start":"03:06.900 ","End":"03:09.730","Text":"minus 3, minus minus 1 plus 7."},{"Start":"03:09.730 ","End":"03:13.865","Text":"We\u0027re just adding signed numbers and we can do it from left to right."},{"Start":"03:13.865 ","End":"03:16.940","Text":"Minus 1, minus 3 is minus 4,"},{"Start":"03:16.940 ","End":"03:19.060","Text":"this becomes plus 1,"},{"Start":"03:19.060 ","End":"03:26.565","Text":"so that\u0027s minus 3 plus 7 gives us 4,"},{"Start":"03:26.565 ","End":"03:29.590","Text":"and that\u0027s the answer to that one."},{"Start":"03:29.830 ","End":"03:32.255","Text":"Continuing."},{"Start":"03:32.255 ","End":"03:39.505","Text":"Here we have m and n. We\u0027re substituting fractions this time."},{"Start":"03:39.505 ","End":"03:42.420","Text":"16 and instead of m,"},{"Start":"03:42.420 ","End":"03:47.090","Text":"I put 1/2 and also for fractions,"},{"Start":"03:47.090 ","End":"03:48.445","Text":"as well as the negatives,"},{"Start":"03:48.445 ","End":"03:53.690","Text":"in general, it\u0027s best to put the number you\u0027re substituting in brackets."},{"Start":"03:53.690 ","End":"03:56.850","Text":"If in doubt, put it in brackets."},{"Start":"03:56.950 ","End":"04:01.525","Text":"Minus 9n is minus a 1/3,"},{"Start":"04:01.525 ","End":"04:04.715","Text":"negative under fraction squared."},{"Start":"04:04.715 ","End":"04:07.284","Text":"Brackets will always be okay."},{"Start":"04:07.284 ","End":"04:11.300","Text":"It\u0027s never a mistake to put them in that is."},{"Start":"04:11.910 ","End":"04:14.619","Text":"We have to do the order of operations."},{"Start":"04:14.619 ","End":"04:19.465","Text":"First of all, the squares and the powers I mean."},{"Start":"04:19.465 ","End":"04:22.210","Text":"In this case we have 16."},{"Start":"04:22.210 ","End":"04:25.060","Text":"Now notice that this is a product,"},{"Start":"04:25.060 ","End":"04:27.160","Text":"even though there\u0027s no multiplication sign,"},{"Start":"04:27.160 ","End":"04:29.590","Text":"we mustn\u0027t forget that it\u0027s a product."},{"Start":"04:29.590 ","End":"04:32.860","Text":"That\u0027s in fact what we had in the previous exercise."},{"Start":"04:32.860 ","End":"04:35.290","Text":"If you note that this here,"},{"Start":"04:35.290 ","End":"04:41.270","Text":"there is a product here between the 3 and the a^4, there\u0027s a product."},{"Start":"04:41.980 ","End":"04:45.020","Text":"Similarly here there\u0027s a product here."},{"Start":"04:45.020 ","End":"04:47.390","Text":"I don\u0027t put the dot, but there\u0027s a product here,"},{"Start":"04:47.390 ","End":"04:48.985","Text":"and there\u0027s a product here,"},{"Start":"04:48.985 ","End":"04:50.885","Text":"so we just get used to that."},{"Start":"04:50.885 ","End":"04:54.474","Text":"That when you have letters, we just don\u0027t put anything, we mean multiplication."},{"Start":"04:54.474 ","End":"04:58.600","Text":"16 times 1/2 squared."},{"Start":"04:58.600 ","End":"05:01.595","Text":"We square the numerator and the denominator,"},{"Start":"05:01.595 ","End":"05:04.520","Text":"1^2 over 2^2 that\u0027s a 1/4."},{"Start":"05:04.520 ","End":"05:07.550","Text":"If in doubt I can leave it in brackets or not."},{"Start":"05:07.550 ","End":"05:09.755","Text":"I think there\u0027s no reason here."},{"Start":"05:09.755 ","End":"05:16.470","Text":"Minus 1/3 squared is minus 1/3 times minus 1/3,"},{"Start":"05:16.470 ","End":"05:20.415","Text":"which is plus 1/3 times 1/3."},{"Start":"05:20.415 ","End":"05:25.240","Text":"That becomes the 9 stays out becomes 1/9."},{"Start":"05:25.330 ","End":"05:29.195","Text":"Now we do the multiplications."},{"Start":"05:29.195 ","End":"05:32.885","Text":"This is effectively a multiplication."},{"Start":"05:32.885 ","End":"05:35.240","Text":"16 times 1/4,"},{"Start":"05:35.240 ","End":"05:36.755","Text":"we can cancel the 4,"},{"Start":"05:36.755 ","End":"05:38.870","Text":"it just comes out to be 4."},{"Start":"05:38.870 ","End":"05:42.060","Text":"I\u0027m assuming, you know multiplication already."},{"Start":"05:46.990 ","End":"05:57.825","Text":"9 times 1/9 is 9/1 times 1/9 because it\u0027s a whole number,"},{"Start":"05:57.825 ","End":"06:05.445","Text":"and 9/1 times 1/9 is just 9 times 1 is 9/9 and then we cancel."},{"Start":"06:05.445 ","End":"06:08.025","Text":"This becomes equal to 1."},{"Start":"06:08.025 ","End":"06:13.095","Text":"Back to here we have 4 minus 1,"},{"Start":"06:13.095 ","End":"06:23.370","Text":"which equals 3 and that\u0027s the answer to c. Now the last one."},{"Start":"06:23.370 ","End":"06:27.290","Text":"Here we have 3 variables to substitute a, b,"},{"Start":"06:27.290 ","End":"06:30.635","Text":"and c. No problem,"},{"Start":"06:30.635 ","End":"06:32.240","Text":"let\u0027s start doing it."},{"Start":"06:32.240 ","End":"06:33.886","Text":"What we get is 4,"},{"Start":"06:33.886 ","End":"06:37.890","Text":"a negative should go in brackets and as I said,"},{"Start":"06:37.890 ","End":"06:41.850","Text":"you can always put round brackets."},{"Start":"06:41.850 ","End":"06:48.265","Text":"Squared minus 3 times b is 2."},{"Start":"06:48.265 ","End":"06:53.585","Text":"In fact, putting it in brackets means that you don\u0027t have to write the multiplication."},{"Start":"06:53.585 ","End":"06:55.610","Text":"Because when you have things next to each other without"},{"Start":"06:55.610 ","End":"06:59.065","Text":"anything in mathematics that\u0027s multiplication,"},{"Start":"06:59.065 ","End":"07:03.300","Text":"over c, which is minus 4."},{"Start":"07:03.300 ","End":"07:05.985","Text":"Again order of operations."},{"Start":"07:05.985 ","End":"07:07.875","Text":"We do the product first,"},{"Start":"07:07.875 ","End":"07:11.595","Text":"minus 1 squared is 1."},{"Start":"07:11.595 ","End":"07:15.800","Text":"What we get is 4 times 1"},{"Start":"07:15.800 ","End":"07:20.380","Text":"and I put the dot in because there\u0027s no more brackets I took the brackets out minus,"},{"Start":"07:20.380 ","End":"07:28.710","Text":"I can remove the brackets if I put a multiplication 3 times 2 over minus 4."},{"Start":"07:29.020 ","End":"07:32.719","Text":"The next thing is,"},{"Start":"07:32.719 ","End":"07:36.720","Text":"we have to do the multiplications."},{"Start":"07:37.240 ","End":"07:40.670","Text":"Actually there\u0027s a brackets here, you don\u0027t see it,"},{"Start":"07:40.670 ","End":"07:42.530","Text":"but a dividing line is,"},{"Start":"07:42.530 ","End":"07:47.375","Text":"whatever is above or below a dividing line is considered as if it\u0027s in brackets."},{"Start":"07:47.375 ","End":"07:53.465","Text":"Let\u0027s do the top, the numerator of this fraction as if it were in brackets."},{"Start":"07:53.465 ","End":"07:59.255","Text":"We evaluate this again, multiplication before subtraction."},{"Start":"07:59.255 ","End":"08:02.015","Text":"What we get is, I\u0027ll continue over here,"},{"Start":"08:02.015 ","End":"08:05.670","Text":"4 times 1 is 4."},{"Start":"08:05.950 ","End":"08:11.770","Text":"Minus 3 times 2 is 6."},{"Start":"08:11.770 ","End":"08:14.650","Text":"I\u0027m still over minus 4."},{"Start":"08:14.650 ","End":"08:16.760","Text":"I\u0027m still working on the top which was brackets,"},{"Start":"08:16.760 ","End":"08:21.580","Text":"4 minus 6 is minus 2."},{"Start":"08:21.580 ","End":"08:23.720","Text":"I\u0027ve evaluated the numerator,"},{"Start":"08:23.720 ","End":"08:25.655","Text":"we\u0027ve got the denominator."},{"Start":"08:25.655 ","End":"08:27.350","Text":"Now I have a division,"},{"Start":"08:27.350 ","End":"08:32.190","Text":"this is minus 2 divided by minus 4."},{"Start":"08:33.130 ","End":"08:36.240","Text":"Minus over minus makes it plus,"},{"Start":"08:36.240 ","End":"08:39.005","Text":"and actually we can use cancellation."},{"Start":"08:39.005 ","End":"08:42.260","Text":"We can say that it\u0027s a fraction,"},{"Start":"08:42.260 ","End":"08:44.240","Text":"that 2 into 4,"},{"Start":"08:44.240 ","End":"08:47.870","Text":"this is 1 and this is 2 divide each of them by 2."},{"Start":"08:47.870 ","End":"08:53.635","Text":"Then what we say is that minus over minus is plus and we\u0027re just left with 1/2."},{"Start":"08:53.635 ","End":"08:59.390","Text":"It\u0027s plus 1/2, which we just write it as 1/2, and that\u0027s it."},{"Start":"08:59.390 ","End":"09:02.850","Text":"We\u0027re done with this set of exercises."}],"ID":4897},{"Watched":false,"Name":"Exercise 2","Duration":"7m 26s","ChapterTopicVideoID":4895,"CourseChapterTopicPlaylistID":3857,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.620","Text":"In this exercise, we have to evaluate"},{"Start":"00:02.620 ","End":"00:06.865","Text":"the following algebraic expressions and we\u0027re going to use substitution,"},{"Start":"00:06.865 ","End":"00:08.635","Text":"also known as plugging in."},{"Start":"00:08.635 ","End":"00:10.015","Text":"Start with the first,"},{"Start":"00:10.015 ","End":"00:12.190","Text":"but actually I\u0027d like to work on a copy,"},{"Start":"00:12.190 ","End":"00:16.660","Text":"and then we\u0027ll have a bit more room to work in."},{"Start":"00:16.660 ","End":"00:20.665","Text":"In the first one we have two variables,"},{"Start":"00:20.665 ","End":"00:22.000","Text":"a and c,"},{"Start":"00:22.000 ","End":"00:26.620","Text":"and we have to substitute a as 2 and c as minus 2."},{"Start":"00:26.620 ","End":"00:31.705","Text":"We have a bracket, then we have a which is 2, minus 2."},{"Start":"00:31.705 ","End":"00:33.325","Text":"Now, c is minus 2,"},{"Start":"00:33.325 ","End":"00:35.660","Text":"and I\u0027ll put that in brackets."},{"Start":"00:35.660 ","End":"00:41.760","Text":"All this to the power of 4 over a,"},{"Start":"00:41.760 ","End":"00:43.350","Text":"which is 2,"},{"Start":"00:43.350 ","End":"00:45.695","Text":"and then minus a^2,"},{"Start":"00:45.695 ","End":"00:49.460","Text":"which is minus 2^2, doesn\u0027t need brackets."},{"Start":"00:49.460 ","End":"00:51.410","Text":"Let\u0027s see what this is equal to."},{"Start":"00:51.410 ","End":"00:54.230","Text":"We have to do brackets first."},{"Start":"00:54.230 ","End":"00:56.855","Text":"Let\u0027s do that as a side exercise."},{"Start":"00:56.855 ","End":"00:59.560","Text":"Let\u0027s see, 2 minus,"},{"Start":"00:59.560 ","End":"01:02.700","Text":"twice minus 2,"},{"Start":"01:02.700 ","End":"01:07.275","Text":"this is equal to 2."},{"Start":"01:07.275 ","End":"01:10.074","Text":"Then we do the multiplication first."},{"Start":"01:10.074 ","End":"01:14.130","Text":"2 times minus 2 is minus 4,"},{"Start":"01:14.130 ","End":"01:16.130","Text":"so we have 2 minus minus 4."},{"Start":"01:16.130 ","End":"01:17.750","Text":"You can\u0027t just leave minus minus,"},{"Start":"01:17.750 ","End":"01:19.340","Text":"put this in brackets."},{"Start":"01:19.340 ","End":"01:25.640","Text":"Subtracting a negative is like an addition and so what we get is 6."},{"Start":"01:25.640 ","End":"01:27.290","Text":"Going back here,"},{"Start":"01:27.290 ","End":"01:28.950","Text":"well all this brackets is 6,"},{"Start":"01:28.950 ","End":"01:38.165","Text":"so what we have is 6^4 over 2 minus 2^2."},{"Start":"01:38.165 ","End":"01:40.685","Text":"Next thing we do our exponents;"},{"Start":"01:40.685 ","End":"01:48.350","Text":"6^4 is"},{"Start":"01:48.350 ","End":"01:51.090","Text":"1,296 over"},{"Start":"01:51.090 ","End":"01:55.420","Text":"2 minus 2^2 is 4."},{"Start":"01:55.420 ","End":"01:57.785","Text":"Then we do the division."},{"Start":"01:57.785 ","End":"02:00.570","Text":"Division gives us,"},{"Start":"02:00.770 ","End":"02:09.945","Text":"move a bit down here, 648 minus 4."},{"Start":"02:09.945 ","End":"02:15.375","Text":"Finally, that gives us 644,"},{"Start":"02:15.375 ","End":"02:19.625","Text":"and that\u0027s the answer, and I\u0027ll highlight it."},{"Start":"02:19.625 ","End":"02:22.355","Text":"Onto the next."},{"Start":"02:22.355 ","End":"02:27.860","Text":"Here we have an expression with a and b and we substitute a and b. a^2"},{"Start":"02:27.860 ","End":"02:34.535","Text":"is 3^2 plus twice,"},{"Start":"02:34.535 ","End":"02:39.285","Text":"put a multiplication here, 3 times,"},{"Start":"02:39.285 ","End":"02:40.905","Text":"here I need a bracket,"},{"Start":"02:40.905 ","End":"02:44.780","Text":"minus 5 plus b^2."},{"Start":"02:44.780 ","End":"02:48.785","Text":"Again, I\u0027ll need brackets because the whole thing is squared."},{"Start":"02:48.785 ","End":"02:51.305","Text":"Order of operations."},{"Start":"02:51.305 ","End":"02:52.895","Text":"I\u0027ll do the exponents,"},{"Start":"02:52.895 ","End":"02:54.800","Text":"so 3^2 is 9,"},{"Start":"02:54.800 ","End":"02:59.740","Text":"and I\u0027ll leave the multiplication."},{"Start":"03:00.340 ","End":"03:02.690","Text":"I\u0027ll do this exponent."},{"Start":"03:02.690 ","End":"03:05.135","Text":"Now, a negative squared is plus,"},{"Start":"03:05.135 ","End":"03:08.630","Text":"it\u0027s an even number, so it\u0027s plus 5^2,"},{"Start":"03:08.630 ","End":"03:11.380","Text":"so it\u0027s plus 25."},{"Start":"03:11.380 ","End":"03:15.950","Text":"Now, I can do the product,"},{"Start":"03:15.950 ","End":"03:17.990","Text":"2 times 3 times minus 5,"},{"Start":"03:17.990 ","End":"03:20.240","Text":"plus times plus times minus is going to come"},{"Start":"03:20.240 ","End":"03:22.640","Text":"out a minus because there\u0027s an odd number of minuses,"},{"Start":"03:22.640 ","End":"03:25.485","Text":"so it\u0027s going to be 9 minus something."},{"Start":"03:25.485 ","End":"03:27.750","Text":"Now, 2 times 3 times 5 we can do in our heads,"},{"Start":"03:27.750 ","End":"03:31.610","Text":"is 30 plus 25."},{"Start":"03:31.610 ","End":"03:34.400","Text":"Now, it\u0027s just addition and subtraction from left to right,"},{"Start":"03:34.400 ","End":"03:39.480","Text":"9 minus 30 is minus 21 plus 25,"},{"Start":"03:39.480 ","End":"03:41.235","Text":"so it\u0027s plus 4,"},{"Start":"03:41.235 ","End":"03:44.555","Text":"and that\u0027s the answer to the second one."},{"Start":"03:44.555 ","End":"03:46.640","Text":"Onto the third,"},{"Start":"03:46.640 ","End":"03:52.190","Text":"we have here two variables, x and b."},{"Start":"03:52.190 ","End":"03:56.090","Text":"Let\u0027s start substituting or plugging in."},{"Start":"03:56.090 ","End":"03:58.310","Text":"x is 5,"},{"Start":"03:58.310 ","End":"04:03.780","Text":"so 5 minus 3^2 plus 3 times,"},{"Start":"04:03.780 ","End":"04:06.070","Text":"I\u0027ll put a dot, 5^2,"},{"Start":"04:06.070 ","End":"04:07.640","Text":"I don\u0027t need a bracket here,"},{"Start":"04:07.640 ","End":"04:09.365","Text":"I put a dot there."},{"Start":"04:09.365 ","End":"04:12.290","Text":"In the case of a negative number,"},{"Start":"04:12.290 ","End":"04:15.710","Text":"I will need brackets minus 1."},{"Start":"04:15.710 ","End":"04:19.460","Text":"Now lets start doing order of operations."},{"Start":"04:19.460 ","End":"04:22.220","Text":"Brackets first and this is not really a bracket,"},{"Start":"04:22.220 ","End":"04:24.155","Text":"it\u0027s around a single number, like I said."},{"Start":"04:24.155 ","End":"04:26.180","Text":"This is a real bracket,"},{"Start":"04:26.180 ","End":"04:28.295","Text":"so 5 minus 3 is 2."},{"Start":"04:28.295 ","End":"04:38.200","Text":"What we get is 2^2 plus 3 times 5^2 times minus 1."},{"Start":"04:38.200 ","End":"04:41.210","Text":"Next thing to do is exponents."},{"Start":"04:41.210 ","End":"04:46.295","Text":"So 2^2 is 4 plus 3,"},{"Start":"04:46.295 ","End":"04:51.180","Text":"5^2 is 25 times minus 1."},{"Start":"04:51.180 ","End":"04:53.735","Text":"Optionally, you can put a dot here."},{"Start":"04:53.735 ","End":"04:58.115","Text":"Next thing is multiplication before addition,"},{"Start":"04:58.115 ","End":"05:01.225","Text":"so this is 4."},{"Start":"05:01.225 ","End":"05:04.655","Text":"Now the multiplication here is plus plus minus,"},{"Start":"05:04.655 ","End":"05:05.975","Text":"so we get a minus."},{"Start":"05:05.975 ","End":"05:10.070","Text":"3 times 25 times 1 is 75,"},{"Start":"05:10.070 ","End":"05:15.880","Text":"4 minus 75 is minus 71."},{"Start":"05:16.700 ","End":"05:19.905","Text":"That deserves to be highlighted."},{"Start":"05:19.905 ","End":"05:22.040","Text":"Onto the last one."},{"Start":"05:22.040 ","End":"05:25.290","Text":"Here we have x and y."},{"Start":"05:25.570 ","End":"05:29.810","Text":"They\u0027re both negative, we\u0027ll need to use little brackets around each."},{"Start":"05:29.810 ","End":"05:32.510","Text":"We have minus, that\u0027s this minus."},{"Start":"05:32.510 ","End":"05:40.845","Text":"Now the x itself is minus 2 and it\u0027s cubed."},{"Start":"05:40.845 ","End":"05:44.445","Text":"Then we have minus 2,"},{"Start":"05:44.445 ","End":"05:46.860","Text":"x is minus 2,"},{"Start":"05:46.860 ","End":"05:49.755","Text":"y is minus 1,"},{"Start":"05:49.755 ","End":"05:53.580","Text":"and plus y to the 4th,"},{"Start":"05:53.580 ","End":"05:57.580","Text":"minus 1 to the power of 4."},{"Start":"05:57.700 ","End":"06:00.425","Text":"Order of operations."},{"Start":"06:00.425 ","End":"06:03.365","Text":"Let\u0027s do the exponents first."},{"Start":"06:03.365 ","End":"06:06.380","Text":"We have minus, now minus 2^3,"},{"Start":"06:06.380 ","End":"06:07.970","Text":"minus minus, minus is minus,"},{"Start":"06:07.970 ","End":"06:10.470","Text":"so it gets minus 8."},{"Start":"06:11.120 ","End":"06:14.970","Text":"Then minus 2 minus 2,"},{"Start":"06:14.970 ","End":"06:16.590","Text":"minus 1 as is."},{"Start":"06:16.590 ","End":"06:19.875","Text":"Now, another exponent, minus 1^4,"},{"Start":"06:19.875 ","End":"06:23.960","Text":"4 is even, it comes up plus, it\u0027s plus 1."},{"Start":"06:23.960 ","End":"06:30.600","Text":"Then we want to do products."},{"Start":"06:30.600 ","End":"06:37.109","Text":"Product here is 2 times minus 2 times minus 1, I\u0027ll just write that to the side,"},{"Start":"06:37.109 ","End":"06:41.890","Text":"2 times minus 2 times minus 1 is,"},{"Start":"06:41.890 ","End":"06:47.015","Text":"positive, negative, negative is positive, so that\u0027s 4."},{"Start":"06:47.015 ","End":"06:48.590","Text":"I left the minus here,"},{"Start":"06:48.590 ","End":"06:50.285","Text":"the subtraction, I didn\u0027t include it there."},{"Start":"06:50.285 ","End":"06:53.365","Text":"What I have is minus,"},{"Start":"06:53.365 ","End":"06:56.405","Text":"minus 8 minus,"},{"Start":"06:56.405 ","End":"06:58.730","Text":"and this part was the product I did up there,"},{"Start":"06:58.730 ","End":"07:03.300","Text":"which is 4 and plus 1."},{"Start":"07:03.800 ","End":"07:07.170","Text":"What I get is minus,"},{"Start":"07:07.170 ","End":"07:11.370","Text":"minus 8 is plus 8,"},{"Start":"07:11.370 ","End":"07:14.835","Text":"minus 4 plus 1,"},{"Start":"07:14.835 ","End":"07:17.100","Text":"that\u0027s just straightforward from left to right,"},{"Start":"07:17.100 ","End":"07:20.580","Text":"8 minus 4 plus 1 is 5."},{"Start":"07:20.580 ","End":"07:22.170","Text":"That\u0027s our answer,"},{"Start":"07:22.170 ","End":"07:23.865","Text":"which I shall highlight,"},{"Start":"07:23.865 ","End":"07:26.380","Text":"and we are done."}],"ID":4898},{"Watched":false,"Name":"Exercise 3","Duration":"4m 12s","ChapterTopicVideoID":4896,"CourseChapterTopicPlaylistID":3857,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.270","Text":"In this exercise, we have to combine like terms."},{"Start":"00:03.270 ","End":"00:05.745","Text":"Some people say collect like terms."},{"Start":"00:05.745 ","End":"00:08.475","Text":"But I prefer to work on a copy,"},{"Start":"00:08.475 ","End":"00:12.370","Text":"and then we\u0027ll have a bit more room to work."},{"Start":"00:12.770 ","End":"00:16.260","Text":"Let\u0027s start with the first 1,"},{"Start":"00:16.260 ","End":"00:20.610","Text":"which is 5x plus 3x minus 12x."},{"Start":"00:20.610 ","End":"00:25.490","Text":"Now x is like in our example of fruit apples,"},{"Start":"00:25.490 ","End":"00:29.270","Text":"I have 5 apples plus 3 apples less 12 apples."},{"Start":"00:29.270 ","End":"00:32.119","Text":"I know the answer is going to be in x."},{"Start":"00:32.119 ","End":"00:35.795","Text":"All I have to do is 5 plus 3 minus 12."},{"Start":"00:35.795 ","End":"00:37.930","Text":"At the side I think to myself,"},{"Start":"00:37.930 ","End":"00:40.865","Text":"5 plus 3 minus 12."},{"Start":"00:40.865 ","End":"00:42.050","Text":"We could do it in our heads."},{"Start":"00:42.050 ","End":"00:46.500","Text":"5 plus 3 is 8 minus 12 is minus 4."},{"Start":"00:46.610 ","End":"00:49.680","Text":"When I get back here, I say,"},{"Start":"00:49.680 ","End":"00:53.440","Text":"okay, I have minus 4 apples, minus 4x."},{"Start":"00:54.770 ","End":"01:00.225","Text":"In the next 1, we have a^5 and a^5."},{"Start":"01:00.225 ","End":"01:01.740","Text":"All the a^5,"},{"Start":"01:01.740 ","End":"01:03.675","Text":"again, it\u0027s like an apple or an orange."},{"Start":"01:03.675 ","End":"01:05.355","Text":"Let\u0027s say an orange this time."},{"Start":"01:05.355 ","End":"01:07.125","Text":"We have 1 orange."},{"Start":"01:07.125 ","End":"01:08.430","Text":"We don\u0027t write the 1,"},{"Start":"01:08.430 ","End":"01:10.500","Text":"but there\u0027s a 1 there, of course."},{"Start":"01:10.500 ","End":"01:12.540","Text":"I have another 1 orange."},{"Start":"01:12.540 ","End":"01:14.420","Text":"All together we have 2 oranges,"},{"Start":"01:14.420 ","End":"01:16.950","Text":"which in our case is 2a^5."},{"Start":"01:17.180 ","End":"01:21.900","Text":"It\u0027s almost as easy as saying that 1 plus 1 equals 2."},{"Start":"01:21.900 ","End":"01:25.760","Text":"In this 1, we have apples and oranges."},{"Start":"01:25.760 ","End":"01:30.890","Text":"We have terms containing m and you know what?"},{"Start":"01:30.890 ","End":"01:33.080","Text":"I\u0027ll emphasize that using colors,"},{"Start":"01:33.080 ","End":"01:34.535","Text":"I\u0027ll say, okay,"},{"Start":"01:34.535 ","End":"01:39.890","Text":"we have terms containing m and we"},{"Start":"01:39.890 ","End":"01:45.770","Text":"have terms which are just numbers like this and this, apples and oranges."},{"Start":"01:45.770 ","End":"01:50.380","Text":"Let\u0027s first of all collect together the terms containing"},{"Start":"01:50.380 ","End":"01:55.170","Text":"m. What we have with m is 7m minus 9m."},{"Start":"01:55.170 ","End":"02:00.765","Text":"At the side I think to myself 7 minus 9 and that\u0027s minus 2."},{"Start":"02:00.765 ","End":"02:03.795","Text":"Back here, it\u0027s minus 2,"},{"Start":"02:03.795 ","End":"02:10.170","Text":"not apples but m. If I look at the 3 numbers,"},{"Start":"02:10.170 ","End":"02:15.660","Text":"we have 11 minus 2 which I don\u0027t have to do at the side but just for"},{"Start":"02:15.660 ","End":"02:22.680","Text":"consistency and so the answer is minus 2m plus 9."},{"Start":"02:22.680 ","End":"02:25.160","Text":"Really I should be highlighting my answers,"},{"Start":"02:25.160 ","End":"02:26.465","Text":"which is what I usually do."},{"Start":"02:26.465 ","End":"02:28.185","Text":"I\u0027ll catch up now."},{"Start":"02:28.185 ","End":"02:31.515","Text":"The last 1, here,"},{"Start":"02:31.515 ","End":"02:34.360","Text":"we have 3 kinds of fruit."},{"Start":"02:34.360 ","End":"02:38.300","Text":"We have things containing b,"},{"Start":"02:38.300 ","End":"02:42.540","Text":"we have just numbers and we have things containing b^2."},{"Start":"02:42.540 ","End":"02:47.460","Text":"It\u0027s important to note that b^2 and b will not be considered like terms."},{"Start":"02:47.460 ","End":"02:54.120","Text":"In other words, let\u0027s say that the ones containing b^2 are like the apples."},{"Start":"02:54.120 ","End":"02:56.130","Text":"That\u0027s this 1 and this 1."},{"Start":"02:56.130 ","End":"02:59.490","Text":"Ones containing b,"},{"Start":"02:59.490 ","End":"03:03.930","Text":"are like oranges and we only have 1 of those and 3 numbers will be,"},{"Start":"03:03.930 ","End":"03:07.065","Text":"let\u0027s go with lemons."},{"Start":"03:07.065 ","End":"03:10.275","Text":"We have a 1 and a minus 3."},{"Start":"03:10.275 ","End":"03:13.640","Text":"Just using colors to start off with items and we\u0027ll dispense"},{"Start":"03:13.640 ","End":"03:16.445","Text":"with that just an educational aid."},{"Start":"03:16.445 ","End":"03:19.925","Text":"Getting back to the problem,"},{"Start":"03:19.925 ","End":"03:23.200","Text":"first of all, let\u0027s take the b^2."},{"Start":"03:23.200 ","End":"03:25.955","Text":"It\u0027s usual to take the highest ones first."},{"Start":"03:25.955 ","End":"03:27.620","Text":"Though it doesn\u0027t really matter in what order."},{"Start":"03:27.620 ","End":"03:29.255","Text":"I\u0027ll take the b^2 first."},{"Start":"03:29.255 ","End":"03:31.670","Text":"I have plus 1, there is a 1 here,"},{"Start":"03:31.670 ","End":"03:34.385","Text":"though you don\u0027t see it, minus a 2."},{"Start":"03:34.385 ","End":"03:36.545","Text":"I have a 1 minus 2 for b^2,"},{"Start":"03:36.545 ","End":"03:37.880","Text":"which is minus 1."},{"Start":"03:37.880 ","End":"03:41.420","Text":"I start off with minus b^2."},{"Start":"03:41.420 ","End":"03:43.700","Text":"Let\u0027s go next to the orange. There is only 1 of them."},{"Start":"03:43.700 ","End":"03:44.720","Text":"There is nothing to collect."},{"Start":"03:44.720 ","End":"03:46.880","Text":"I just write it as minus 2b."},{"Start":"03:46.880 ","End":"03:50.360","Text":"I couldn\u0027t resist that 1."},{"Start":"03:50.360 ","End":"03:53.360","Text":"Then we have 3 numbers,"},{"Start":"03:53.360 ","End":"03:57.210","Text":"1 and minus 3."},{"Start":"03:57.710 ","End":"04:00.140","Text":"1 and minus 3, again,"},{"Start":"04:00.140 ","End":"04:02.780","Text":"we\u0027ll do it in our heads. It\u0027s minus 2."},{"Start":"04:02.780 ","End":"04:07.560","Text":"This is our answer and this time I\u0027ll remember to highlight it."},{"Start":"04:07.720 ","End":"04:11.250","Text":"We are done."}],"ID":4899},{"Watched":false,"Name":"Exercise 4","Duration":"9m 3s","ChapterTopicVideoID":4897,"CourseChapterTopicPlaylistID":3857,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.775","Text":"In this exercise, we have to combine like terms or collect like terms, some say."},{"Start":"00:05.775 ","End":"00:09.210","Text":"I prefer to work on a copy down here."},{"Start":"00:09.210 ","End":"00:11.265","Text":"Let\u0027s just scroll down a bit."},{"Start":"00:11.265 ","End":"00:13.540","Text":"We have more room now."},{"Start":"00:14.690 ","End":"00:20.055","Text":"Let\u0027s begin and I\u0027m going to continue using the fruit analogy here."},{"Start":"00:20.055 ","End":"00:22.440","Text":"Let\u0027s see what kind of fruits we have."},{"Start":"00:22.440 ","End":"00:28.875","Text":"We have ab and I\u0027ll cover those in green."},{"Start":"00:28.875 ","End":"00:33.255","Text":"I have a, b, and c. I have another one which is a, b."},{"Start":"00:33.255 ","End":"00:36.330","Text":"Then here I have a^2b,"},{"Start":"00:36.330 ","End":"00:38.040","Text":"but here I have b^2a,"},{"Start":"00:38.040 ","End":"00:40.020","Text":"and those are entirely different."},{"Start":"00:40.020 ","End":"00:45.410","Text":"One of them will be in one color and the other will be yet another color."},{"Start":"00:45.410 ","End":"00:54.095","Text":"Really all I can do here is collect the green ones together so 4ab minus 5ab,"},{"Start":"00:54.095 ","End":"00:59.325","Text":"so at the side I do 4 minus 5 is minus 1,"},{"Start":"00:59.325 ","End":"01:03.600","Text":"so minus 1ab it\u0027s just minus ab."},{"Start":"01:03.600 ","End":"01:11.430","Text":"Next I have what I can\u0027t collect just stays as is minus 3a^2b."},{"Start":"01:11.620 ","End":"01:15.440","Text":"This is getting in the way, excuse me,"},{"Start":"01:15.440 ","End":"01:19.400","Text":"there we are gone and the yellow one,"},{"Start":"01:19.400 ","End":"01:25.790","Text":"the lemon or banana, is 3b^2a."},{"Start":"01:25.930 ","End":"01:29.930","Text":"All we collected really are the first and last."},{"Start":"01:29.930 ","End":"01:32.435","Text":"Onto the next."},{"Start":"01:32.435 ","End":"01:35.000","Text":"Continuing with the fruit analogy,"},{"Start":"01:35.000 ","End":"01:39.380","Text":"let\u0027s see the first one we have is an x^2y and see if we have any more of those."},{"Start":"01:39.380 ","End":"01:41.550","Text":"Let\u0027s say those are apples."},{"Start":"01:42.770 ","End":"01:45.255","Text":"This is not."},{"Start":"01:45.255 ","End":"01:47.405","Text":"That\u0027s all on its own."},{"Start":"01:47.405 ","End":"01:53.224","Text":"Next one we see will be the x, y, oranges."},{"Start":"01:53.224 ","End":"01:56.165","Text":"Yes, I have another one of those here."},{"Start":"01:56.165 ","End":"02:01.655","Text":"Then we have two that look different what remains?"},{"Start":"02:01.655 ","End":"02:08.420","Text":"But here\u0027s a little trick they played on us because y^2x is the same as xy^2."},{"Start":"02:08.420 ","End":"02:12.830","Text":"After all, the order of multiplication doesn\u0027t make any difference."},{"Start":"02:12.830 ","End":"02:17.510","Text":"If I say y^2 times x or x times y^2 it\u0027s the same."},{"Start":"02:17.510 ","End":"02:19.685","Text":"So you\u0027ve got to watch out because these are tricky."},{"Start":"02:19.685 ","End":"02:20.810","Text":"They may look different,"},{"Start":"02:20.810 ","End":"02:22.970","Text":"but in actual fact they are the same."},{"Start":"02:22.970 ","End":"02:26.090","Text":"Now let\u0027s collect the like terms."},{"Start":"02:26.090 ","End":"02:28.160","Text":"The green one, the apples,"},{"Start":"02:28.160 ","End":"02:30.065","Text":"that\u0027s the x^2y,"},{"Start":"02:30.065 ","End":"02:34.760","Text":"then the two in orange, that\u0027s minus 1."},{"Start":"02:34.760 ","End":"02:37.820","Text":"If there\u0027s nothing here, that\u0027s 1xy minus 1,"},{"Start":"02:37.820 ","End":"02:39.585","Text":"and then plus 9."},{"Start":"02:39.585 ","End":"02:44.060","Text":"Altogether plus 8xy."},{"Start":"02:44.060 ","End":"02:48.060","Text":"Then the lemons."},{"Start":"02:48.100 ","End":"02:54.180","Text":"We have 3 of them minus 5 of them altogether minus 2"},{"Start":"02:54.180 ","End":"02:55.910","Text":"and you have to decide whether you want to"},{"Start":"02:55.910 ","End":"02:59.450","Text":"write xy^2 or y^2x because they\u0027re both the same,"},{"Start":"02:59.450 ","End":"03:01.190","Text":"but I like to put the extra for the y,"},{"Start":"03:01.190 ","End":"03:03.750","Text":"so I\u0027ll write it as xy^2."},{"Start":"03:04.570 ","End":"03:10.520","Text":"This exercise is a bit trickier because before we can collect like terms,"},{"Start":"03:10.520 ","End":"03:12.530","Text":"we have to do some expansion."},{"Start":"03:12.530 ","End":"03:16.010","Text":"There\u0027s two sets of brackets or parentheses."},{"Start":"03:16.010 ","End":"03:18.125","Text":"We have the outer one,"},{"Start":"03:18.125 ","End":"03:26.990","Text":"which I\u0027ll highlight in one color and then we also have an inner one,"},{"Start":"03:26.990 ","End":"03:28.640","Text":"which I\u0027ll do in a different color."},{"Start":"03:28.640 ","End":"03:32.060","Text":"That\u0027s the square ones, the inner ones."},{"Start":"03:32.060 ","End":"03:33.650","Text":"Now when you have brackets,"},{"Start":"03:33.650 ","End":"03:36.305","Text":"you should check on either side if there\u0027s a multiplier."},{"Start":"03:36.305 ","End":"03:38.435","Text":"For example, in this one,"},{"Start":"03:38.435 ","End":"03:43.100","Text":"there\u0027s a minus in front of it and is also a 5 after it."},{"Start":"03:43.100 ","End":"03:46.100","Text":"Whereas in the case of this color,"},{"Start":"03:46.100 ","End":"03:49.340","Text":"whatever it is, we just have a minus in front of it."},{"Start":"03:49.340 ","End":"03:52.730","Text":"Now we expand from the inside out."},{"Start":"03:52.730 ","End":"03:58.969","Text":"But what I\u0027d like to suggest is that because we have a minus in front and a 5 afterwards,"},{"Start":"03:58.969 ","End":"04:05.930","Text":"we should do a small rewrite and write this as a minus 5 times the whole thing."},{"Start":"04:05.930 ","End":"04:09.290","Text":"I won\u0027t waste your time. I\u0027ll just copy it like that."},{"Start":"04:09.290 ","End":"04:16.555","Text":"There are copied it with the only difference is that the minus 5."},{"Start":"04:16.555 ","End":"04:21.545","Text":"I brought it all in front and I still have these square brackets."},{"Start":"04:21.545 ","End":"04:24.330","Text":"Brackets should be opened from the inside out."},{"Start":"04:24.330 ","End":"04:29.535","Text":"The first thing we\u0027ll do is we\u0027ll do the minus 5 times this."},{"Start":"04:29.535 ","End":"04:34.635","Text":"Let\u0027s see, 10m^2n,"},{"Start":"04:34.635 ","End":"04:38.250","Text":"I\u0027ll put an equals here and an equals here."},{"Start":"04:38.290 ","End":"04:42.830","Text":"10m^2n minus."},{"Start":"04:42.830 ","End":"04:46.930","Text":"Everything stays the same except what I\u0027m going to expand and now here,"},{"Start":"04:46.930 ","End":"04:51.950","Text":"minus 5 is going to multiply both of them inside the brackets."},{"Start":"04:51.950 ","End":"04:59.070","Text":"I have minus 5 times m^2n and minus"},{"Start":"04:59.070 ","End":"05:06.795","Text":"5 times minus 2 times m. Now minus 5 times minus 2 is plus 10."},{"Start":"05:06.795 ","End":"05:10.450","Text":"So I have plus 10m."},{"Start":"05:11.740 ","End":"05:16.940","Text":"I\u0027ve expanded these, give me the last two terms here."},{"Start":"05:16.940 ","End":"05:21.260","Text":"Next thing we want to do is to deal with"},{"Start":"05:21.260 ","End":"05:29.780","Text":"the other curly braces they\u0027re called sometimes and the minus here."},{"Start":"05:29.780 ","End":"05:33.830","Text":"Now when we multiply a minus,"},{"Start":"05:33.830 ","End":"05:36.800","Text":"it just makes everything minus."},{"Start":"05:36.800 ","End":"05:46.925","Text":"So this is equal to 10m^2n minus 3mn^2."},{"Start":"05:46.925 ","End":"05:50.240","Text":"When you have a minus multiplying several things,"},{"Start":"05:50.240 ","End":"05:52.354","Text":"it reverses each of the signs."},{"Start":"05:52.354 ","End":"05:55.620","Text":"Minus times minus is plus."},{"Start":"05:55.750 ","End":"05:59.180","Text":"Minus times plus is minus."},{"Start":"05:59.180 ","End":"06:01.940","Text":"We end up getting this expression,"},{"Start":"06:01.940 ","End":"06:07.130","Text":"minus 3mn^2 at the end plus 5m^2n and minus 10m."},{"Start":"06:07.130 ","End":"06:12.510","Text":"Finally, we\u0027ve gotten rid of the brackets and we\u0027re ready to collect like terms."},{"Start":"06:12.860 ","End":"06:16.755","Text":"Let\u0027s see now, we have,"},{"Start":"06:16.755 ","End":"06:20.975","Text":"first of all, let\u0027s take the m^2n,"},{"Start":"06:20.975 ","End":"06:23.555","Text":"one kind of fruit or one kind of color."},{"Start":"06:23.555 ","End":"06:25.130","Text":"Let\u0027s see if we have any more of those."},{"Start":"06:25.130 ","End":"06:27.320","Text":"m^2n. Yes, I have another one here,"},{"Start":"06:27.320 ","End":"06:29.670","Text":"so that gets the same color."},{"Start":"06:29.840 ","End":"06:34.280","Text":"This is something else altogether."},{"Start":"06:34.280 ","End":"06:40.100","Text":"Mn^2 is not the same, that\u0027s a different color and m is still another color."},{"Start":"06:40.100 ","End":"06:42.860","Text":"When I\u0027ve collected everything."},{"Start":"06:42.860 ","End":"06:47.800","Text":"What I\u0027m left with now is,"},{"Start":"06:48.040 ","End":"06:51.095","Text":"let\u0027s see, the 10 and the 5,"},{"Start":"06:51.095 ","End":"06:52.775","Text":"10 plus 5 is 15."},{"Start":"06:52.775 ","End":"07:01.370","Text":"So I have, let me write it over here, 15m^2n."},{"Start":"07:01.370 ","End":"07:05.240","Text":"Then I have minus 3mn^2."},{"Start":"07:05.240 ","End":"07:14.190","Text":"I have to leave as is minus 3mn^2 and finally minus 10m."},{"Start":"07:14.320 ","End":"07:17.645","Text":"Just squeezing it in."},{"Start":"07:17.645 ","End":"07:20.255","Text":"That\u0027s the answer."},{"Start":"07:20.255 ","End":"07:22.805","Text":"I forgot to highlight."},{"Start":"07:22.805 ","End":"07:25.740","Text":"Never mind, not too late."},{"Start":"07:25.840 ","End":"07:28.580","Text":"Onto the fourth one,"},{"Start":"07:28.580 ","End":"07:33.110","Text":"Number D. Let\u0027s collect like terms."},{"Start":"07:33.110 ","End":"07:34.400","Text":"Start with the first one."},{"Start":"07:34.400 ","End":"07:35.990","Text":"This one is something a^2."},{"Start":"07:35.990 ","End":"07:37.565","Text":"Do I have any more a^2?"},{"Start":"07:37.565 ","End":"07:41.035","Text":"Yes. I have here and I have here."},{"Start":"07:41.035 ","End":"07:44.150","Text":"Next one is the apples and oranges,"},{"Start":"07:44.150 ","End":"07:46.715","Text":"so to speak, a is not the same as a^2."},{"Start":"07:46.715 ","End":"07:50.760","Text":"We have 10a and minus 11a. That goes together."},{"Start":"07:50.760 ","End":"07:52.790","Text":"We\u0027re going to get two kinds of terms,"},{"Start":"07:52.790 ","End":"07:54.485","Text":"just apples and oranges."},{"Start":"07:54.485 ","End":"07:55.850","Text":"Let\u0027s see for a^2,"},{"Start":"07:55.850 ","End":"07:59.485","Text":"I\u0027ve got 8 minus 5 plus 1."},{"Start":"07:59.485 ","End":"08:02.295","Text":"If you do 8 minus 5 plus 1, you get 4."},{"Start":"08:02.295 ","End":"08:05.750","Text":"It\u0027s 4a^2, 4 apples, if you like."},{"Start":"08:05.750 ","End":"08:07.385","Text":"Let\u0027s see how many oranges,"},{"Start":"08:07.385 ","End":"08:10.670","Text":"10 minus 11, which is minus 1."},{"Start":"08:10.670 ","End":"08:16.490","Text":"This is minus a and that is the answer to the fourth one."},{"Start":"08:16.490 ","End":"08:19.055","Text":"Now to the last question."},{"Start":"08:19.055 ","End":"08:23.030","Text":"Let\u0027s see what kind of fruits we have here."},{"Start":"08:23.030 ","End":"08:25.115","Text":"We have an a^2b,"},{"Start":"08:25.115 ","End":"08:27.030","Text":"and we have an a^2b."},{"Start":"08:27.030 ","End":"08:28.529","Text":"These will go together,"},{"Start":"08:28.529 ","End":"08:35.520","Text":"and ab^2 will go with ab^2."},{"Start":"08:35.520 ","End":"08:38.700","Text":"5 and 20 is 25 apples,"},{"Start":"08:38.700 ","End":"08:44.040","Text":"I mean 25 a^2b and oranges, let\u0027s see,"},{"Start":"08:44.040 ","End":"08:48.825","Text":"minus 8 minus 14,"},{"Start":"08:48.825 ","End":"08:50.900","Text":"two minuses, so they strengthen each other."},{"Start":"08:50.900 ","End":"08:53.060","Text":"8 and 14 is 22,"},{"Start":"08:53.060 ","End":"08:55.520","Text":"so it\u0027s minus 22 oranges."},{"Start":"08:55.520 ","End":"08:57.605","Text":"I mean a b^2."},{"Start":"08:57.605 ","End":"09:03.340","Text":"That\u0027s our answer for the last one and so we are done."}],"ID":4900},{"Watched":false,"Name":"Exercise 5","Duration":"2m 16s","ChapterTopicVideoID":4898,"CourseChapterTopicPlaylistID":3857,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.665","Text":"In this exercise, you\u0027re asked to expand the brackets."},{"Start":"00:04.665 ","End":"00:07.950","Text":"We use what is called the distributive law."},{"Start":"00:07.950 ","End":"00:11.535","Text":"I prefer to copy it down here."},{"Start":"00:11.535 ","End":"00:16.395","Text":"Let\u0027s start. The first one."},{"Start":"00:16.395 ","End":"00:21.540","Text":"What we do is we multiply this 2 by each of them,"},{"Start":"00:21.540 ","End":"00:23.925","Text":"by x, and by 4."},{"Start":"00:23.925 ","End":"00:27.240","Text":"What we get is 2 times x is 2x."},{"Start":"00:27.240 ","End":"00:33.120","Text":"There\u0027s no need to write a dot and 2 times 4 is a number,"},{"Start":"00:33.120 ","End":"00:36.825","Text":"it\u0027s 8, so the answer is 2x+8."},{"Start":"00:36.825 ","End":"00:40.245","Text":"That\u0027s all. Next one,"},{"Start":"00:40.245 ","End":"00:45.840","Text":"x(x+5), so it\u0027s x times x plus x times 5."},{"Start":"00:45.840 ","End":"00:48.975","Text":"X times x is x^2."},{"Start":"00:48.975 ","End":"00:52.140","Text":"Should remember that, x times x is x^2,"},{"Start":"00:52.140 ","End":"00:55.605","Text":"just like x times x times x is x^3,"},{"Start":"00:55.605 ","End":"00:58.875","Text":"and so on. Remember the exponents?"},{"Start":"00:58.875 ","End":"01:01.365","Text":"Back here, x times x is x^2."},{"Start":"01:01.365 ","End":"01:05.325","Text":"X times 5, you could say x5,"},{"Start":"01:05.325 ","End":"01:07.950","Text":"but we don\u0027t say x5, we say 5x."},{"Start":"01:07.950 ","End":"01:10.530","Text":"It\u0027s neater to write the number in front."},{"Start":"01:10.530 ","End":"01:13.630","Text":"That\u0027s the answer to that one."},{"Start":"01:14.000 ","End":"01:20.605","Text":"Next, 7(a-3), multiply here and multiply here,"},{"Start":"01:20.605 ","End":"01:23.470","Text":"7 times a is 7a,"},{"Start":"01:23.540 ","End":"01:29.440","Text":"minus 7 times 3, minus 21."},{"Start":"01:31.040 ","End":"01:37.054","Text":"Finally, we have a -2 in front."},{"Start":"01:37.054 ","End":"01:43.054","Text":"You could think of it as a -2 in brackets and there\u0027s no harm in writing it that way,"},{"Start":"01:43.054 ","End":"01:45.680","Text":"so multiply it by this and by this."},{"Start":"01:45.680 ","End":"01:51.260","Text":"-2 times b is -2b."},{"Start":"01:51.260 ","End":"01:53.195","Text":"We don\u0027t put the times,"},{"Start":"01:53.195 ","End":"01:55.010","Text":"remember when 2 is next to b,"},{"Start":"01:55.010 ","End":"01:56.600","Text":"it means times."},{"Start":"01:56.600 ","End":"01:59.875","Text":"-2 times -2x."},{"Start":"01:59.875 ","End":"02:04.665","Text":"Remember that -2 times -2 is +4,"},{"Start":"02:04.665 ","End":"02:08.830","Text":"so -2 times -2x is 4x."},{"Start":"02:09.920 ","End":"02:15.970","Text":"That\u0027s the last one and we are done."}],"ID":4901},{"Watched":false,"Name":"Exercise 6","Duration":"3m 48s","ChapterTopicVideoID":4899,"CourseChapterTopicPlaylistID":3857,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.350 ","End":"00:05.670","Text":"Here we have to expand the brackets using what is called the"},{"Start":"00:05.670 ","End":"00:11.310","Text":"distributive law and I prefer to work on a copy down here,"},{"Start":"00:11.310 ","End":"00:14.980","Text":"where we have more space here."},{"Start":"00:15.710 ","End":"00:19.590","Text":"The first one, the distributive law,"},{"Start":"00:19.590 ","End":"00:21.060","Text":"but you don\u0027t have to remember by name,"},{"Start":"00:21.060 ","End":"00:25.725","Text":"just means that we take this x and multiply it by each of them."},{"Start":"00:25.725 ","End":"00:28.080","Text":"This, this, and this."},{"Start":"00:28.080 ","End":"00:35.775","Text":"What we get is x times x^2 is x^3 and I\u0027ll explain why."},{"Start":"00:35.775 ","End":"00:39.120","Text":"Now x^2 is x times x."},{"Start":"00:39.120 ","End":"00:41.810","Text":"If we have, this is the x^2, this is the x,"},{"Start":"00:41.810 ","End":"00:46.020","Text":"altogether, we have x times x times x is x^3."},{"Start":"00:46.450 ","End":"00:50.200","Text":"Here we get x times 3x."},{"Start":"00:50.200 ","End":"00:51.965","Text":"We can do it in any order."},{"Start":"00:51.965 ","End":"00:54.500","Text":"We can, instead of saying x times 3 times x,"},{"Start":"00:54.500 ","End":"00:58.925","Text":"we can say 3 times x times x, and x times x is x^2."},{"Start":"00:58.925 ","End":"01:02.275","Text":"So this is 3x^2."},{"Start":"01:02.275 ","End":"01:06.695","Text":"Finally, minus x times 2 is the same as 2 times x."},{"Start":"01:06.695 ","End":"01:09.455","Text":"We write the number in front,"},{"Start":"01:09.455 ","End":"01:12.030","Text":"it\u0027s just customary to do it that way."},{"Start":"01:12.030 ","End":"01:15.700","Text":"Somehow neater. That\u0027s done with the first one."},{"Start":"01:15.700 ","End":"01:18.040","Text":"Now, the next one."},{"Start":"01:18.040 ","End":"01:19.825","Text":"This time we have a fraction,"},{"Start":"01:19.825 ","End":"01:21.160","Text":"but same rules apply."},{"Start":"01:21.160 ","End":"01:25.735","Text":"We have 2/3 times this and 2/3 times this."},{"Start":"01:25.735 ","End":"01:28.000","Text":"Now, what\u0027s 2/3 times 6x?"},{"Start":"01:28.000 ","End":"01:30.165","Text":"It\u0027s 2/3 times 6 times x."},{"Start":"01:30.165 ","End":"01:32.590","Text":"At the side, I do a little exercise in fractions and say,"},{"Start":"01:32.590 ","End":"01:34.795","Text":"what is 2/3 times 6?"},{"Start":"01:34.795 ","End":"01:37.045","Text":"I can cancel."},{"Start":"01:37.045 ","End":"01:44.530","Text":"Well, first of all, we write it as 2/3 times 6 over 1."},{"Start":"01:44.530 ","End":"01:49.295","Text":"Then we can cancel the 3 with the 6."},{"Start":"01:49.295 ","End":"01:51.900","Text":"This is twice."},{"Start":"01:51.900 ","End":"01:56.235","Text":"We get 2 times 2 is 4 over 1."},{"Start":"01:56.235 ","End":"01:58.290","Text":"I\u0027ll keep them with this color never mind,"},{"Start":"01:58.290 ","End":"02:02.205","Text":"4 over 1, which is 4."},{"Start":"02:02.205 ","End":"02:05.400","Text":"This is 4, but we still have an x there."},{"Start":"02:05.400 ","End":"02:10.470","Text":"So it\u0027s 4x, then minus 2/3 times 3."},{"Start":"02:10.470 ","End":"02:16.640","Text":"2/3 times 3 is 2/3 times 3 over 1."},{"Start":"02:16.640 ","End":"02:23.500","Text":"Once again, we can cancel this with this and that leaves us"},{"Start":"02:23.500 ","End":"02:31.625","Text":"with 2 over 1, which equals 2."},{"Start":"02:31.625 ","End":"02:40.670","Text":"Back here, it\u0027s 2/3 times 3 is minus 2y."},{"Start":"02:40.670 ","End":"02:42.950","Text":"Because we still have to keep the y."},{"Start":"02:42.950 ","End":"02:46.460","Text":"That\u0027s another one completed."},{"Start":"02:46.460 ","End":"02:48.155","Text":"Onto the next."},{"Start":"02:48.155 ","End":"02:50.270","Text":"This time we have a minus in front."},{"Start":"02:50.270 ","End":"02:52.070","Text":"A minus is like a minus 1."},{"Start":"02:52.070 ","End":"02:54.695","Text":"The minus also goes with each of them."},{"Start":"02:54.695 ","End":"02:57.755","Text":"But remember that minus times minus is plus."},{"Start":"02:57.755 ","End":"03:00.570","Text":"So what we get is minus 5y."},{"Start":"03:00.670 ","End":"03:03.950","Text":"Here\u0027s where people often slip up."},{"Start":"03:03.950 ","End":"03:05.690","Text":"Minus times minus is plus,"},{"Start":"03:05.690 ","End":"03:08.165","Text":"so it\u0027s plus 7,"},{"Start":"03:08.165 ","End":"03:11.075","Text":"and that one\u0027s done."},{"Start":"03:11.075 ","End":"03:17.480","Text":"Finally, here we have the 5 on the right."},{"Start":"03:17.480 ","End":"03:19.850","Text":"We\u0027re used to having it on the left."},{"Start":"03:19.850 ","End":"03:23.690","Text":"Since we used to it, why don\u0027t we write it on the left and write this"},{"Start":"03:23.690 ","End":"03:27.740","Text":"as 5 times 3x plus 2y,"},{"Start":"03:27.740 ","End":"03:29.180","Text":"which we\u0027re more familiar with."},{"Start":"03:29.180 ","End":"03:32.480","Text":"Once again, 5 multiplies by each of them."},{"Start":"03:32.480 ","End":"03:42.835","Text":"We get 5 times 3x is 5 times 3 times x is 15x plus 5 times 2 times y is 10y,"},{"Start":"03:42.835 ","End":"03:44.820","Text":"and that\u0027s that one done."},{"Start":"03:44.820 ","End":"03:48.250","Text":"That means that we\u0027re done here."}],"ID":4902},{"Watched":false,"Name":"Exercise 7","Duration":"4m 20s","ChapterTopicVideoID":4900,"CourseChapterTopicPlaylistID":3857,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.245","Text":"This is another exercise where we have to expand the brackets."},{"Start":"00:05.245 ","End":"00:10.095","Text":"Let\u0027s begin, but we\u0027ll work on a copy."},{"Start":"00:10.095 ","End":"00:12.090","Text":"Here we are,"},{"Start":"00:12.090 ","End":"00:14.320","Text":"we have more room now."},{"Start":"00:14.480 ","End":"00:21.585","Text":"Minus 3x times 2x minus y. I prefer to think of it as being in brackets"},{"Start":"00:21.585 ","End":"00:28.905","Text":"and this is multiplied by this and this minus 3x times 2x."},{"Start":"00:28.905 ","End":"00:30.780","Text":"Let\u0027s multiply the numbers first."},{"Start":"00:30.780 ","End":"00:35.670","Text":"Minus 3 times 2 is minus 6 and x times x,"},{"Start":"00:35.670 ","End":"00:40.680","Text":"we\u0027ve seen this already is x^2 any number times itself is that number squared."},{"Start":"00:40.680 ","End":"00:44.755","Text":"Then we have minus 3x times minus y,"},{"Start":"00:44.755 ","End":"00:47.855","Text":"minus times minus is plus."},{"Start":"00:47.855 ","End":"00:51.940","Text":"Note the plus people often slip up here."},{"Start":"00:51.940 ","End":"00:57.750","Text":"It\u0027s plus now minus 3 times the 1 here that\u0027s the minus 1"},{"Start":"00:57.750 ","End":"01:03.450","Text":"really is plus 3 and x times y is xy."},{"Start":"01:03.450 ","End":"01:07.020","Text":"That\u0027s the answer to this one."},{"Start":"01:07.020 ","End":"01:10.210","Text":"Onto the next."},{"Start":"01:10.990 ","End":"01:15.230","Text":"Notice that we have x plus 5,"},{"Start":"01:15.230 ","End":"01:18.160","Text":"but we need to do the multiplication before the addition,"},{"Start":"01:18.160 ","End":"01:20.435","Text":"there\u0027s a multiplication here."},{"Start":"01:20.435 ","End":"01:24.205","Text":"We want to do first of all this,"},{"Start":"01:24.205 ","End":"01:27.185","Text":"so we copy the x as is,"},{"Start":"01:27.185 ","End":"01:28.940","Text":"and then we do the multiplication,"},{"Start":"01:28.940 ","End":"01:32.255","Text":"5 times 2x is 10x,"},{"Start":"01:32.255 ","End":"01:37.280","Text":"5 times minus 1 is minus 5."},{"Start":"01:37.280 ","End":"01:39.785","Text":"We\u0027ve expanded the brackets,"},{"Start":"01:39.785 ","End":"01:44.960","Text":"but I think they expected you to collect,"},{"Start":"01:44.960 ","End":"01:47.195","Text":"combine like terms together."},{"Start":"01:47.195 ","End":"01:50.630","Text":"Here we have something with x,"},{"Start":"01:50.630 ","End":"01:52.580","Text":"and here we have something with x,"},{"Start":"01:52.580 ","End":"01:55.580","Text":"we have basically 1x plus 10x,"},{"Start":"01:55.580 ","End":"01:58.100","Text":"so that gives us 11x."},{"Start":"01:58.100 ","End":"02:05.520","Text":"A simplified answer is 11 x minus 5 and that\u0027s another one done."},{"Start":"02:05.520 ","End":"02:11.765","Text":"Next, this time we have 2 things times 2 things."},{"Start":"02:11.765 ","End":"02:15.730","Text":"What we do is each one in here is multiplied by each one in here."},{"Start":"02:15.730 ","End":"02:16.975","Text":"The order doesn\u0027t matter."},{"Start":"02:16.975 ","End":"02:19.510","Text":"This times this, this times this,"},{"Start":"02:19.510 ","End":"02:24.085","Text":"this times this, this times this, in any order."},{"Start":"02:24.085 ","End":"02:26.940","Text":"I\u0027ll just take it as this x times everything."},{"Start":"02:26.940 ","End":"02:30.640","Text":"x times x is x^2 as you know,"},{"Start":"02:30.640 ","End":"02:32.370","Text":"x times 5,"},{"Start":"02:32.370 ","End":"02:33.900","Text":"we don\u0027t say x5,"},{"Start":"02:33.900 ","End":"02:38.940","Text":"we say 5x, 4 times x is 4x,"},{"Start":"02:38.940 ","End":"02:42.695","Text":"4 times 5, 20."},{"Start":"02:42.695 ","End":"02:46.210","Text":"Once again, we collect like terms together because here we"},{"Start":"02:46.210 ","End":"02:50.340","Text":"have stuff with something times x,"},{"Start":"02:50.340 ","End":"02:51.495","Text":"and something times x."},{"Start":"02:51.495 ","End":"02:56.620","Text":"These two, we can collect like terms and we get x^2"},{"Start":"02:56.620 ","End":"03:02.760","Text":"and how many x altogether 5 of them and 4 of them altogether 9 of them plus 20."},{"Start":"03:02.760 ","End":"03:05.485","Text":"That\u0027s the answer to that one."},{"Start":"03:05.485 ","End":"03:07.850","Text":"Now the last one,"},{"Start":"03:07.850 ","End":"03:11.150","Text":"same thing as before, this times this,"},{"Start":"03:11.150 ","End":"03:13.760","Text":"this times this, this times this,"},{"Start":"03:13.760 ","End":"03:17.270","Text":"this times this, each one in here times each one in here."},{"Start":"03:17.270 ","End":"03:19.010","Text":"Let\u0027s see what we get."},{"Start":"03:19.010 ","End":"03:20.890","Text":"x times 5,"},{"Start":"03:20.890 ","End":"03:24.449","Text":"5x, x times x,"},{"Start":"03:24.449 ","End":"03:25.875","Text":"but it\u0027s a minus,"},{"Start":"03:25.875 ","End":"03:28.410","Text":"so minus x^2,"},{"Start":"03:28.410 ","End":"03:30.750","Text":"3 times 5,"},{"Start":"03:30.750 ","End":"03:37.105","Text":"15, 3 times minus x, minus 3x."},{"Start":"03:37.105 ","End":"03:39.860","Text":"Once again, collecting like terms."},{"Start":"03:39.860 ","End":"03:41.660","Text":"What is like? This is x^2,"},{"Start":"03:41.660 ","End":"03:43.700","Text":"x^2 and x are different."},{"Start":"03:43.700 ","End":"03:47.175","Text":"The 5x and then minus 3x,"},{"Start":"03:47.175 ","End":"03:49.365","Text":"they\u0027re all something times x."},{"Start":"03:49.365 ","End":"03:51.665","Text":"That\u0027s the only things we can combine."},{"Start":"03:51.665 ","End":"03:56.449","Text":"Besides combining, we also want to change the order just because it\u0027s customary,"},{"Start":"03:56.449 ","End":"03:59.495","Text":"seems neater to start off with x squareds\u0027,"},{"Start":"03:59.495 ","End":"04:01.565","Text":"then xs, and then just numbers."},{"Start":"04:01.565 ","End":"04:05.720","Text":"For x^2, we have minus x^2 nothing to combine with."},{"Start":"04:05.720 ","End":"04:09.450","Text":"5x minus 3x leaves only 2x,"},{"Start":"04:09.450 ","End":"04:12.936","Text":"5 of them minus 3 of them leaves 2 of them,"},{"Start":"04:12.936 ","End":"04:20.140","Text":"plus 15 and that\u0027s the answer to this one. We are done."}],"ID":4903},{"Watched":false,"Name":"Exercise 8","Duration":"6m 35s","ChapterTopicVideoID":4901,"CourseChapterTopicPlaylistID":3857,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.170 ","End":"00:04.710","Text":"Yet another exercise in expanding the brackets."},{"Start":"00:04.710 ","End":"00:08.845","Text":"As usual, I prefer to work on a copy,"},{"Start":"00:08.845 ","End":"00:12.540","Text":"and here we are with more room."},{"Start":"00:12.540 ","End":"00:13.920","Text":"Start with the first one."},{"Start":"00:13.920 ","End":"00:17.010","Text":"What\u0027s different about this is that this is a product of three things."},{"Start":"00:17.010 ","End":"00:20.100","Text":"Something times, something times something."},{"Start":"00:20.100 ","End":"00:23.565","Text":"Normally we multiply from left to right."},{"Start":"00:23.565 ","End":"00:28.155","Text":"But experience shows that it\u0027s better to multiply these two first,"},{"Start":"00:28.155 ","End":"00:31.875","Text":"and only then to multiply by 3."},{"Start":"00:31.875 ","End":"00:36.225","Text":"What we do, is we open a bracket,"},{"Start":"00:36.225 ","End":"00:39.305","Text":"this is 3, and then we\u0027re going to multiply these two."},{"Start":"00:39.305 ","End":"00:43.970","Text":"I\u0027m going to put them inside a brackets here."},{"Start":"00:43.970 ","End":"00:45.545","Text":"Now as usual,"},{"Start":"00:45.545 ","End":"00:48.620","Text":"with the product of this times this is,"},{"Start":"00:48.620 ","End":"00:49.955","Text":"2 things times 2 things,"},{"Start":"00:49.955 ","End":"00:51.230","Text":"each one with each one."},{"Start":"00:51.230 ","End":"00:52.625","Text":"This with this, this with this,"},{"Start":"00:52.625 ","End":"00:55.010","Text":"and then this with this and this with this."},{"Start":"00:55.010 ","End":"00:58.640","Text":"But we\u0027re going to put it inside the brackets because we still have the 3."},{"Start":"00:58.640 ","End":"01:01.760","Text":"You could use round brackets, it doesn\u0027t matter."},{"Start":"01:01.760 ","End":"01:05.375","Text":"X times x is x^2."},{"Start":"01:05.375 ","End":"01:08.585","Text":"X times 3 is 3x,"},{"Start":"01:08.585 ","End":"01:10.855","Text":"but it\u0027s minus 3x."},{"Start":"01:10.855 ","End":"01:14.685","Text":"Minus 1 times x is minus x,"},{"Start":"01:14.685 ","End":"01:16.860","Text":"minus 1 times minus 3,"},{"Start":"01:16.860 ","End":"01:19.040","Text":"negative times negative is positive,"},{"Start":"01:19.040 ","End":"01:21.670","Text":"so it\u0027s plus 3."},{"Start":"01:21.670 ","End":"01:24.164","Text":"Collect like terms."},{"Start":"01:24.164 ","End":"01:28.740","Text":"The 3x and the minus x are like terms."},{"Start":"01:28.740 ","End":"01:34.324","Text":"What we have is 3 times,"},{"Start":"01:34.324 ","End":"01:36.575","Text":"I\u0027ll go back to round brackets."},{"Start":"01:36.575 ","End":"01:42.470","Text":"X^2 minus 3 minus 1 together is minus 4,"},{"Start":"01:42.470 ","End":"01:48.630","Text":"so it\u0027s minus 4x plus 3 and finally,"},{"Start":"01:48.630 ","End":"01:50.625","Text":"I multiply by 3."},{"Start":"01:50.625 ","End":"01:52.610","Text":"This times this, this times this,"},{"Start":"01:52.610 ","End":"01:54.110","Text":"and this times this."},{"Start":"01:54.110 ","End":"01:57.565","Text":"We get 3x^2."},{"Start":"01:57.565 ","End":"01:59.475","Text":"3 times minus 4x,"},{"Start":"01:59.475 ","End":"02:05.895","Text":"3 times minus 4 is minus 12x and 3 times 3 is 9."},{"Start":"02:05.895 ","End":"02:10.620","Text":"That\u0027s this one add onto the next."},{"Start":"02:10.620 ","End":"02:12.945","Text":"Here, just 2 things times 2 things."},{"Start":"02:12.945 ","End":"02:14.660","Text":"This times this, this times this,"},{"Start":"02:14.660 ","End":"02:17.165","Text":"this times this, this times this."},{"Start":"02:17.165 ","End":"02:21.305","Text":"What we get is 2x times 2x."},{"Start":"02:21.305 ","End":"02:23.970","Text":"2 times 2 times x times x."},{"Start":"02:23.970 ","End":"02:25.560","Text":"2 times 2 is 4,"},{"Start":"02:25.560 ","End":"02:28.155","Text":"x times x is x^2."},{"Start":"02:28.155 ","End":"02:30.690","Text":"2x times 5, I can do it as 2,"},{"Start":"02:30.690 ","End":"02:35.630","Text":"times 5 times x, 10x."},{"Start":"02:35.630 ","End":"02:37.195","Text":"Minus 5 times 2x,"},{"Start":"02:37.195 ","End":"02:39.540","Text":"is minus 5 times 2 times x,"},{"Start":"02:39.540 ","End":"02:43.620","Text":"which is minus 10x minus 5,"},{"Start":"02:43.620 ","End":"02:46.500","Text":"times plus 5, minus times plus is minus."},{"Start":"02:46.500 ","End":"02:49.820","Text":"It\u0027s minus 25."},{"Start":"02:49.820 ","End":"02:53.060","Text":"Notice that when I collect like terms,"},{"Start":"02:53.060 ","End":"02:54.920","Text":"10 minus 10 is 0,"},{"Start":"02:54.920 ","End":"02:58.700","Text":"you can pretty much say that this and this cancel each other out."},{"Start":"02:58.700 ","End":"03:06.270","Text":"Which means that our answer is 4x^2 minus 25,"},{"Start":"03:06.270 ","End":"03:08.475","Text":"and there is no middle term."},{"Start":"03:08.475 ","End":"03:11.200","Text":"I mean no x term."},{"Start":"03:12.170 ","End":"03:15.120","Text":"Onto the next."},{"Start":"03:15.120 ","End":"03:20.650","Text":"Here we have an expansion here in an expansion here,"},{"Start":"03:20.650 ","End":"03:21.990","Text":"and the minus in-between."},{"Start":"03:21.990 ","End":"03:23.710","Text":"Let\u0027s do the first bit."},{"Start":"03:23.710 ","End":"03:26.840","Text":"The first bit gives us 4 times 3x,"},{"Start":"03:26.840 ","End":"03:32.540","Text":"is 12x, 4 times minus 2 is minus 8."},{"Start":"03:32.540 ","End":"03:38.350","Text":"Now here we do have it\u0027s like 3 things is like minus or minus 1 times these."},{"Start":"03:38.350 ","End":"03:41.700","Text":"I use a square brackets here,"},{"Start":"03:41.700 ","End":"03:47.200","Text":"and I expand this times this inside the square brackets to expand just without the minus."},{"Start":"03:47.200 ","End":"03:48.910","Text":"2x times 3x,"},{"Start":"03:48.910 ","End":"03:51.780","Text":"2x times 5 minus 1,"},{"Start":"03:51.780 ","End":"03:53.850","Text":"times 3x minus 1 times 5,"},{"Start":"03:53.850 ","End":"03:55.485","Text":"2x times 3x,"},{"Start":"03:55.485 ","End":"03:58.995","Text":"2 times 3 is 6, x times x is x^2."},{"Start":"03:58.995 ","End":"04:01.890","Text":"2x times 5,"},{"Start":"04:01.890 ","End":"04:05.625","Text":"10x minus 1 times 3x,"},{"Start":"04:05.625 ","End":"04:11.520","Text":"minus 3x, minus 1 times 5 minus 5."},{"Start":"04:11.520 ","End":"04:18.936","Text":"Like terms, the 10x and the 3x, are like terms."},{"Start":"04:18.936 ","End":"04:22.110","Text":"We\u0027ll continue over here."},{"Start":"04:22.110 ","End":"04:27.365","Text":"The 12x minus 8 stays minus."},{"Start":"04:27.365 ","End":"04:31.410","Text":"I\u0027ll just change the square brackets to round brackets."},{"Start":"04:31.900 ","End":"04:35.290","Text":"6x^2, nothing to combine,"},{"Start":"04:35.290 ","End":"04:37.400","Text":"10x, minus 3x,"},{"Start":"04:37.400 ","End":"04:42.315","Text":"leaves 7x, and then minus 5."},{"Start":"04:42.315 ","End":"04:46.015","Text":"Now we have this minus in front of the brackets."},{"Start":"04:46.015 ","End":"04:48.955","Text":"The minus reverses each of these."},{"Start":"04:48.955 ","End":"04:54.770","Text":"We get 12x minus 8 minus 6x^2,"},{"Start":"04:54.770 ","End":"04:57.830","Text":"minus times plus is minus 7x,"},{"Start":"04:57.830 ","End":"05:01.561","Text":"and minus times minus is plus 5."},{"Start":"05:01.561 ","End":"05:04.880","Text":"Once again, we\u0027d like to combine things."},{"Start":"05:04.880 ","End":"05:07.100","Text":"This time we can combine,"},{"Start":"05:07.100 ","End":"05:09.920","Text":"first of all, we can combine x\u0027s."},{"Start":"05:09.920 ","End":"05:13.070","Text":"I see there\u0027s a 12x and a minus 7x,"},{"Start":"05:13.070 ","End":"05:15.680","Text":"x^2, can\u0027t combine with anything."},{"Start":"05:15.680 ","End":"05:21.225","Text":"But minus 8 is a number and plus 5 is a number."},{"Start":"05:21.225 ","End":"05:24.095","Text":"Those things we can combine,"},{"Start":"05:24.095 ","End":"05:29.595","Text":"and what we get is, let\u0027s see."},{"Start":"05:29.595 ","End":"05:31.340","Text":"I\u0027ll take the x squareds\u0027 first."},{"Start":"05:31.340 ","End":"05:34.550","Text":"That\u0027s what we usually do in decreasing order of"},{"Start":"05:34.550 ","End":"05:39.585","Text":"the power x squareds\u0027 first, minus 6x^2,"},{"Start":"05:39.585 ","End":"05:43.080","Text":"then x\u0027s we have 12 minus 7 of them,"},{"Start":"05:43.080 ","End":"05:46.425","Text":"12 minus 7 is 5 x\u0027s,"},{"Start":"05:46.425 ","End":"05:51.315","Text":"and minus 8 plus 5 is minus 3."},{"Start":"05:51.315 ","End":"05:55.330","Text":"That\u0027s the answer to this one."},{"Start":"05:55.960 ","End":"05:58.370","Text":"Then the last one."},{"Start":"05:58.370 ","End":"06:01.850","Text":"Again, it\u0027s 1 thing times 3 things,"},{"Start":"06:01.850 ","End":"06:03.335","Text":"but it\u0027s all in letters."},{"Start":"06:03.335 ","End":"06:04.970","Text":"This it\u0027s still, this times this,"},{"Start":"06:04.970 ","End":"06:06.830","Text":"this times this, and this times this."},{"Start":"06:06.830 ","End":"06:09.575","Text":"A times a is a^2,"},{"Start":"06:09.575 ","End":"06:10.790","Text":"just like x times x,"},{"Start":"06:10.790 ","End":"06:15.840","Text":"is x^2, minus a times 2b."},{"Start":"06:15.840 ","End":"06:17.610","Text":"I don\u0027t write a2b."},{"Start":"06:17.610 ","End":"06:22.700","Text":"I write 2ab. You write the numbers in front and then the letters."},{"Start":"06:22.700 ","End":"06:26.650","Text":"a times c is ac."},{"Start":"06:26.650 ","End":"06:35.520","Text":"The answer to the last one is a ^2 minus 2ab plus ac. We are done."}],"ID":4904}],"Thumbnail":null,"ID":3857},{"Name":"Special Binomial Products","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Special Binomial Products","Duration":"5m 57s","ChapterTopicVideoID":4917,"CourseChapterTopicPlaylistID":3858,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.810","Text":"In this clip and in the next one in fact,"},{"Start":"00:03.810 ","End":"00:08.145","Text":"I\u0027ll be talking about something I call special binomial products."},{"Start":"00:08.145 ","End":"00:13.920","Text":"There are actually 3 formulas and I\u0027m going to do one in this lesson."},{"Start":"00:13.920 ","End":"00:18.000","Text":"This lesson will be devoted to this one formula which I\u0027ll give you right away,"},{"Start":"00:18.000 ","End":"00:20.655","Text":"and there it is,"},{"Start":"00:20.655 ","End":"00:21.960","Text":"I\u0027m just introducing it."},{"Start":"00:21.960 ","End":"00:28.365","Text":"I\u0027ll go into it just to have an idea what the monster looks like. It\u0027s not so bad."},{"Start":"00:28.365 ","End":"00:33.180","Text":"Let me give an example and then I\u0027ll tell you why it\u0027s"},{"Start":"00:33.180 ","End":"00:35.640","Text":"called the product even when I don\u0027t see"},{"Start":"00:35.640 ","End":"00:39.195","Text":"a product here and what it\u0027s good for and all that."},{"Start":"00:39.195 ","End":"00:41.240","Text":"Let\u0027s take an example."},{"Start":"00:41.240 ","End":"00:48.085","Text":"Suppose I asked you what is (5+1)^2?"},{"Start":"00:48.085 ","End":"00:54.405","Text":"Well, in arithmetic you would do if 5 plus 1 is 6 and 6^2 is 36."},{"Start":"00:54.405 ","End":"01:00.550","Text":"We know we\u0027re expecting the answer of 6^2 which is 36."},{"Start":"01:00.550 ","End":"01:04.070","Text":"But in algebra, you can\u0027t always do the addition,"},{"Start":"01:04.070 ","End":"01:07.115","Text":"because I don\u0027t know what a plus b is in general."},{"Start":"01:07.115 ","End":"01:10.640","Text":"Now, if I follow what the formula says,"},{"Start":"01:10.640 ","End":"01:14.510","Text":"let\u0027s just see what it says we should do in the case of numbers."},{"Start":"01:14.510 ","End":"01:17.135","Text":"A is going to be 5 and b is going to be 1."},{"Start":"01:17.135 ","End":"01:18.455","Text":"So we want a^2,"},{"Start":"01:18.455 ","End":"01:19.936","Text":"which is 5^2 plus,"},{"Start":"01:19.936 ","End":"01:24.755","Text":"2ab means twice times the first times the second,"},{"Start":"01:24.755 ","End":"01:29.645","Text":"2 times 5 times 1 and the last term,"},{"Start":"01:29.645 ","End":"01:32.150","Text":"b^2 is the last 1^2."},{"Start":"01:32.150 ","End":"01:39.105","Text":"Now, don\u0027t need this here, 5^2 is 25."},{"Start":"01:39.105 ","End":"01:42.270","Text":"Let me use the last term next,"},{"Start":"01:42.270 ","End":"01:48.380","Text":"and that\u0027s plus 1 and if I just did the 25 plus 1,"},{"Start":"01:48.380 ","End":"01:51.380","Text":"I would get 26, which would not be the answer."},{"Start":"01:51.380 ","End":"01:54.230","Text":"I\u0027m saying this because there is"},{"Start":"01:54.230 ","End":"01:58.310","Text":"a natural temptation that most people fall into is to say,"},{"Start":"01:58.310 ","End":"02:01.550","Text":"well 5^2 is 25 and 1^2 is 1,"},{"Start":"02:01.550 ","End":"02:05.050","Text":"so 25 and 1 is 26 and that is not true."},{"Start":"02:05.050 ","End":"02:10.570","Text":"It\u0027s the middle term that makes all the difference between a right and a wrong formula."},{"Start":"02:10.570 ","End":"02:12.370","Text":"Now if I do take the middle one,"},{"Start":"02:12.370 ","End":"02:14.860","Text":"2 times 5 times 1 is 10,"},{"Start":"02:14.860 ","End":"02:22.134","Text":"25 plus 10 plus 1 and now I really do get the 36 I was supposed to get."},{"Start":"02:22.134 ","End":"02:24.775","Text":"Because if I did from here to here,"},{"Start":"02:24.775 ","End":"02:27.050","Text":"once again, 6^2, it is 36."},{"Start":"02:28.080 ","End":"02:33.865","Text":"The answer is not a^2 plus b^2, not."},{"Start":"02:33.865 ","End":"02:35.965","Text":"There is the middle term here."},{"Start":"02:35.965 ","End":"02:38.560","Text":"The reason it\u0027s confusing is that with addition,"},{"Start":"02:38.560 ","End":"02:41.140","Text":"it doesn\u0027t work, but with multiplication, it does work."},{"Start":"02:41.140 ","End":"02:47.880","Text":"For example, if I have (a.b)^2 that is equal to a^2 times b^2."},{"Start":"02:47.880 ","End":"02:49.580","Text":"In the case of multiplication, it works,"},{"Start":"02:49.580 ","End":"02:51.665","Text":"but in the case of addition, it\u0027s not."},{"Start":"02:51.665 ","End":"02:54.530","Text":"We have this extra middle term."},{"Start":"02:54.530 ","End":"02:58.030","Text":"I feel I owe it to explain why this is so"},{"Start":"02:58.030 ","End":"03:01.855","Text":"and it also explains why I\u0027ve used the word product."},{"Start":"03:01.855 ","End":"03:07.845","Text":"See I can think of (a+b)^2 as a plus b times a plus b."},{"Start":"03:07.845 ","End":"03:11.135","Text":"After all, a number squared means it\u0027s times itself,"},{"Start":"03:11.135 ","End":"03:13.550","Text":"with or without a dot here."},{"Start":"03:13.550 ","End":"03:16.865","Text":"We already have the formula,"},{"Start":"03:16.865 ","End":"03:21.380","Text":"what we call the distributive law of how to multiply each thing with each thing."},{"Start":"03:21.380 ","End":"03:23.840","Text":"You remember with the arc,"},{"Start":"03:23.840 ","End":"03:25.115","Text":"so say this times this,"},{"Start":"03:25.115 ","End":"03:29.090","Text":"this times this, this times this, this times this."},{"Start":"03:29.090 ","End":"03:33.785","Text":"What we get if we expand is a times a is a^2,"},{"Start":"03:33.785 ","End":"03:36.365","Text":"a times b is ab,"},{"Start":"03:36.365 ","End":"03:43.600","Text":"b times a is ba and b times b is b^2."},{"Start":"03:43.600 ","End":"03:45.080","Text":"Now on the face of it,"},{"Start":"03:45.080 ","End":"03:47.510","Text":"if I now go to collect like terms,"},{"Start":"03:47.510 ","End":"03:50.045","Text":"I don\u0027t see any like terms,"},{"Start":"03:50.045 ","End":"03:52.055","Text":"but in actual fact,"},{"Start":"03:52.055 ","End":"03:53.720","Text":"there are 2 which are like,"},{"Start":"03:53.720 ","End":"03:55.955","Text":"they just don\u0027t look exactly the same,"},{"Start":"03:55.955 ","End":"03:58.070","Text":"ab and ba,"},{"Start":"03:58.070 ","End":"04:00.740","Text":"a times b, b times a."},{"Start":"04:00.740 ","End":"04:04.040","Text":"Order of multiplication doesn\u0027t make any difference."},{"Start":"04:04.040 ","End":"04:07.650","Text":"They could have been 3 times 7 and 7 times 3."},{"Start":"04:07.650 ","End":"04:13.680","Text":"So these are both the same and now I can rewrite it over"},{"Start":"04:13.680 ","End":"04:20.010","Text":"here as a^2 plus twice and I can write ab or I can write ba,"},{"Start":"04:20.010 ","End":"04:21.690","Text":"let\u0027s keep it in alphabetical order,"},{"Start":"04:21.690 ","End":"04:25.275","Text":"so it\u0027s twice ab plus b^2,"},{"Start":"04:25.275 ","End":"04:28.305","Text":"which is exactly what\u0027s written here."},{"Start":"04:28.305 ","End":"04:31.382","Text":"This is the proof of the important formula."},{"Start":"04:31.382 ","End":"04:34.204","Text":"There\u0027s also a geometric illustration,"},{"Start":"04:34.204 ","End":"04:36.725","Text":"just for those who are geometrically inclined."},{"Start":"04:36.725 ","End":"04:43.194","Text":"If I had, let\u0027s say a square like this,"},{"Start":"04:43.194 ","End":"04:46.610","Text":"let me just put the line here and the line here."},{"Start":"04:46.610 ","End":"04:52.250","Text":"Suppose this was a and this was b and this is a and this is b,"},{"Start":"04:52.250 ","End":"04:58.337","Text":"then altogether the length of side from here to here is a plus b,"},{"Start":"04:58.337 ","End":"05:00.755","Text":"and similarly from here to here it\u0027s"},{"Start":"05:00.755 ","End":"05:05.510","Text":"a plus b. I\u0027m going to compute the area in two ways,"},{"Start":"05:05.510 ","End":"05:08.660","Text":"one of them is to say the length times width,"},{"Start":"05:08.660 ","End":"05:10.770","Text":"which is a plus b times a plus b,"},{"Start":"05:10.770 ","End":"05:13.310","Text":"and that gives us a plus b^2."},{"Start":"05:13.310 ","End":"05:21.350","Text":"On the other hand, it\u0027s composed of 4 separate little squares or rectangles."},{"Start":"05:21.350 ","End":"05:24.230","Text":"These two are deliberately shaded in green like this."},{"Start":"05:24.230 ","End":"05:27.110","Text":"This one, I\u0027m going to underline in"},{"Start":"05:27.110 ","End":"05:30.139","Text":"orange and that corresponds to this and I\u0027ll explain,"},{"Start":"05:30.139 ","End":"05:33.667","Text":"and the last one, in this color."},{"Start":"05:33.667 ","End":"05:34.850","Text":"If you think about it,"},{"Start":"05:34.850 ","End":"05:39.410","Text":"what it geometrically means is that I have a^2,"},{"Start":"05:39.410 ","End":"05:41.690","Text":"which is this orange bit."},{"Start":"05:41.690 ","End":"05:45.346","Text":"The b^2 corresponds to b times b is this bit,"},{"Start":"05:45.346 ","End":"05:47.915","Text":"and the two green ones are this and this."},{"Start":"05:47.915 ","End":"05:50.450","Text":"Yes, they look different at first, but they are the same."},{"Start":"05:50.450 ","End":"05:52.190","Text":"Just like here, they look slightly different,"},{"Start":"05:52.190 ","End":"05:53.510","Text":"but they are the same in area,"},{"Start":"05:53.510 ","End":"05:55.460","Text":"and that\u0027s the 2ab."},{"Start":"05:55.460 ","End":"05:58.530","Text":"Let\u0027s get to the exercises."}],"ID":4905},{"Watched":false,"Name":"Factoring","Duration":"8m 58s","ChapterTopicVideoID":4913,"CourseChapterTopicPlaylistID":3858,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.260","Text":"In this clip, I\u0027m going talk about factorization,"},{"Start":"00:04.260 ","End":"00:08.020","Text":"also known as factoring."},{"Start":"00:08.090 ","End":"00:12.644","Text":"It\u0027s just the opposite of expansion or expanding the brackets."},{"Start":"00:12.644 ","End":"00:14.040","Text":"I\u0027ll show you what I mean."},{"Start":"00:14.040 ","End":"00:15.780","Text":"If I said to you,"},{"Start":"00:15.780 ","End":"00:24.165","Text":"what is 2 times x minus 3,"},{"Start":"00:24.165 ","End":"00:29.220","Text":"then you would immediately or almost immediately be able to tell me"},{"Start":"00:29.220 ","End":"00:34.770","Text":"you might have to do a little bit of redox,"},{"Start":"00:34.770 ","End":"00:37.590","Text":"but you would be able to tell me that"},{"Start":"00:37.590 ","End":"00:42.720","Text":"2 times x is 2x and 2 times 3 is 6, but it\u0027s minus."},{"Start":"00:42.720 ","End":"00:44.620","Text":"This direction here,"},{"Start":"00:44.620 ","End":"00:51.775","Text":"from here to here is what we call expansion or expanding the brackets."},{"Start":"00:51.775 ","End":"00:58.300","Text":"The other way is if I gave you this 2x minus 6 to write it like this,"},{"Start":"00:58.300 ","End":"01:01.930","Text":"that would be the factoring."},{"Start":"01:01.930 ","End":"01:07.330","Text":"I\u0027ll just write that expansion or opening the brackets is"},{"Start":"01:07.330 ","End":"01:13.015","Text":"this way and going back is factoring or factorization."},{"Start":"01:13.015 ","End":"01:16.280","Text":"I think it\u0027s a British-American or American thing."},{"Start":"01:16.280 ","End":"01:19.880","Text":"I think the Americans say more factoring and the English say"},{"Start":"01:19.880 ","End":"01:24.640","Text":"factorization and they even write it with an s. Anyway,"},{"Start":"01:24.640 ","End":"01:27.010","Text":"let\u0027s do some example."},{"Start":"01:27.010 ","End":"01:32.820","Text":"This time I\u0027ll give you what would be on the right-hand side, say for example,"},{"Start":"01:32.820 ","End":"01:34.890","Text":"2y minus 4,"},{"Start":"01:34.890 ","End":"01:40.655","Text":"and I have to say something and then a bracket something to equal this."},{"Start":"01:40.655 ","End":"01:44.690","Text":"What we do is we look for something that\u0027s in common to both of these."},{"Start":"01:44.690 ","End":"01:48.350","Text":"Some number or letter that goes into both of them."},{"Start":"01:48.350 ","End":"01:51.560","Text":"I can see that 2 will go into both of them."},{"Start":"01:51.560 ","End":"01:58.310","Text":"I pull this 2 in front of the brackets and then I have to say in reverse,"},{"Start":"01:58.310 ","End":"02:06.885","Text":"I can still use this and I can deduce that here it will be a minus."},{"Start":"02:06.885 ","End":"02:09.240","Text":"I\u0027m going to say 2 times what is 2y?"},{"Start":"02:09.240 ","End":"02:11.160","Text":"It\u0027s y clearly."},{"Start":"02:11.160 ","End":"02:13.605","Text":"Then 2 times what is 4?"},{"Start":"02:13.605 ","End":"02:15.385","Text":"That\u0027s going to be 2."},{"Start":"02:15.385 ","End":"02:18.365","Text":"You can always check yourself by multiplying out."},{"Start":"02:18.365 ","End":"02:20.300","Text":"Whenever you get an answer to a factoring,"},{"Start":"02:20.300 ","End":"02:24.110","Text":"you can always do an expanding because if I go here with the factoring,"},{"Start":"02:24.110 ","End":"02:25.385","Text":"I can always go back here,"},{"Start":"02:25.385 ","End":"02:27.050","Text":"2 times y is 2y,"},{"Start":"02:27.050 ","End":"02:30.107","Text":"2 times minus 2 is minus 4 so it\u0027s okay,"},{"Start":"02:30.107 ","End":"02:32.814","Text":"even able to check yourself."},{"Start":"02:32.814 ","End":"02:38.430","Text":"Another example, 60 minus 3m."},{"Start":"02:39.730 ","End":"02:42.480","Text":"What do they have in common?"},{"Start":"02:42.480 ","End":"02:44.290","Text":"A 3 clearly."},{"Start":"02:44.290 ","End":"02:46.554","Text":"Take 3 out of the brackets,"},{"Start":"02:46.554 ","End":"02:48.410","Text":"that\u0027s what we also called factoring sometimes,"},{"Start":"02:48.410 ","End":"02:50.690","Text":"taking something outside the brackets."},{"Start":"02:50.690 ","End":"02:53.445","Text":"I\u0027ll take 3 outside the brackets."},{"Start":"02:53.445 ","End":"02:57.890","Text":"3 times something has to give me 60, so that\u0027s 20."},{"Start":"02:57.890 ","End":"03:00.395","Text":"Then I have to get the minus 3m."},{"Start":"03:00.395 ","End":"03:03.270","Text":"I have to put minus m here."},{"Start":"03:03.580 ","End":"03:05.640","Text":"At the end I can check,"},{"Start":"03:05.640 ","End":"03:07.350","Text":"3 times 20 is 60."},{"Start":"03:07.350 ","End":"03:09.330","Text":"3 times minus m,"},{"Start":"03:09.330 ","End":"03:12.450","Text":"minus 3m. That\u0027s fine."},{"Start":"03:12.450 ","End":"03:15.170","Text":"Now, another one,"},{"Start":"03:15.170 ","End":"03:23.710","Text":"how about 81 plus 9a?"},{"Start":"03:24.130 ","End":"03:27.700","Text":"What could I take out the brackets?"},{"Start":"03:27.700 ","End":"03:30.900","Text":"Clearly a 9. What am I left with?"},{"Start":"03:30.900 ","End":"03:39.315","Text":"9 into 81 goes 9 times and 9 into 9a goes a times."},{"Start":"03:39.315 ","End":"03:41.520","Text":"Optionally at the end,"},{"Start":"03:41.520 ","End":"03:44.250","Text":"you can check by multiplying out 9 times 9,"},{"Start":"03:44.250 ","End":"03:48.180","Text":"81, 9 times a is 9a."},{"Start":"03:48.180 ","End":"03:50.180","Text":"These are very simple cases,"},{"Start":"03:50.180 ","End":"03:52.155","Text":"but just so you get the idea."},{"Start":"03:52.155 ","End":"03:57.680","Text":"We start with something that\u0027s expanded and sometimes this is called the simplified form,"},{"Start":"03:57.680 ","End":"03:59.855","Text":"but I\u0027m not sure which is simpler."},{"Start":"03:59.855 ","End":"04:03.200","Text":"Anyway, expansion and factoring, opposites."},{"Start":"04:03.200 ","End":"04:04.730","Text":"Sometimes we need to do one,"},{"Start":"04:04.730 ","End":"04:06.590","Text":"sometimes we need to do another."},{"Start":"04:06.590 ","End":"04:11.555","Text":"Let\u0027s move on to some more examples."},{"Start":"04:11.555 ","End":"04:16.405","Text":"These are so short. I think we\u0027ll do 4 more. Let\u0027s begin."},{"Start":"04:16.405 ","End":"04:22.220","Text":"x^3 plus 4x."},{"Start":"04:22.220 ","End":"04:24.920","Text":"This time we\u0027re not taking a number out,"},{"Start":"04:24.920 ","End":"04:27.065","Text":"we\u0027re going to take a variable out,"},{"Start":"04:27.065 ","End":"04:29.060","Text":"x is in common."},{"Start":"04:29.060 ","End":"04:30.860","Text":"How do I know x is in common?"},{"Start":"04:30.860 ","End":"04:32.240","Text":"Well, you can see x here,"},{"Start":"04:32.240 ","End":"04:35.410","Text":"but really it\u0027s x times x times x is x^3."},{"Start":"04:35.410 ","End":"04:37.355","Text":"It\u0027s x times something."},{"Start":"04:37.355 ","End":"04:42.710","Text":"Here, I put x and here we\u0027ve got brackets and I have a plus here."},{"Start":"04:42.710 ","End":"04:45.110","Text":"I\u0027ll do the easy part first."},{"Start":"04:45.110 ","End":"04:47.185","Text":"This part is going to be plus 4,"},{"Start":"04:47.185 ","End":"04:49.065","Text":"because x times 4 is 4x."},{"Start":"04:49.065 ","End":"04:51.480","Text":"But x times what is x^3?"},{"Start":"04:51.480 ","End":"04:53.970","Text":"Well, if you\u0027re not sure, you can do it at the side,"},{"Start":"04:53.970 ","End":"04:58.525","Text":"x^3, if you write it out is x times x times x."},{"Start":"04:58.525 ","End":"05:03.230","Text":"If I expand, I have to say this is equal to x times something."},{"Start":"05:03.230 ","End":"05:08.220","Text":"Well, that something is going to be x times x, which is x^2."},{"Start":"05:08.350 ","End":"05:12.500","Text":"That\u0027s the answer. Next one."},{"Start":"05:12.500 ","End":"05:16.940","Text":"Let\u0027s not use x. I\u0027ll go for y this time."},{"Start":"05:16.940 ","End":"05:23.880","Text":"How about y^3 minus 2y?"},{"Start":"05:24.220 ","End":"05:27.065","Text":"It\u0027s a bit similar to the one before,"},{"Start":"05:27.065 ","End":"05:29.420","Text":"y in common this time,"},{"Start":"05:29.420 ","End":"05:31.580","Text":"so I take y out."},{"Start":"05:31.580 ","End":"05:38.905","Text":"We see the y^3, the same trick will mean that it\u0027s y^2 here and here minus 2."},{"Start":"05:38.905 ","End":"05:41.570","Text":"As always, if you\u0027re not sure, you multiply out."},{"Start":"05:41.570 ","End":"05:42.620","Text":"Like in the first one,"},{"Start":"05:42.620 ","End":"05:44.240","Text":"x times x^2 is x^3,"},{"Start":"05:44.240 ","End":"05:45.965","Text":"x times 4 is 4x."},{"Start":"05:45.965 ","End":"05:48.860","Text":"Here, y times y^2 is y^3,"},{"Start":"05:48.860 ","End":"05:51.185","Text":"y times minus 2 is minus 2y."},{"Start":"05:51.185 ","End":"05:54.530","Text":"That\u0027s checked. The advantage of this exercise of"},{"Start":"05:54.530 ","End":"05:58.520","Text":"factorization is that you can always expand the brackets and get back,"},{"Start":"05:58.520 ","End":"06:00.625","Text":"and that\u0027s a check."},{"Start":"06:00.625 ","End":"06:04.260","Text":"Next one, I\u0027m going to use still a different letter."},{"Start":"06:04.260 ","End":"06:09.180","Text":"How about a^5"},{"Start":"06:09.180 ","End":"06:16.630","Text":"minus 3a^2?"},{"Start":"06:16.630 ","End":"06:18.960","Text":"Now, a is a common factor."},{"Start":"06:18.960 ","End":"06:20.750","Text":"If you\u0027re thinking of taking a out,"},{"Start":"06:20.750 ","End":"06:22.610","Text":"that\u0027s not so bad,"},{"Start":"06:22.610 ","End":"06:24.020","Text":"but it\u0027s not the best we can do."},{"Start":"06:24.020 ","End":"06:27.640","Text":"In actual fact, a^2 goes into both of these."},{"Start":"06:27.640 ","End":"06:31.110","Text":"I\u0027ll show you. Let\u0027s just write the a^2 first."},{"Start":"06:31.110 ","End":"06:33.110","Text":"I\u0027ll do the easy one first."},{"Start":"06:33.110 ","End":"06:36.445","Text":"The second half is minus 3,"},{"Start":"06:36.445 ","End":"06:39.325","Text":"because a^2 to the minus 3 is minus 3a^2."},{"Start":"06:39.325 ","End":"06:43.475","Text":"But now I want to know what to multiply this by. What is this?"},{"Start":"06:43.475 ","End":"06:50.730","Text":"I have to say that a^2 times something has got to be a^5."},{"Start":"06:50.730 ","End":"06:57.655","Text":"If you\u0027re not sure, just expand the exponents or wherever."},{"Start":"06:57.655 ","End":"07:02.780","Text":"a times a times something has got to equal a^5,"},{"Start":"07:02.780 ","End":"07:06.155","Text":"which is a times a times a times a times a."},{"Start":"07:06.155 ","End":"07:12.540","Text":"That something is going to be this, which means a^3."},{"Start":"07:12.800 ","End":"07:17.355","Text":"If we multiply out a^2 times a^3, well here it is,"},{"Start":"07:17.355 ","End":"07:19.215","Text":"a^2, a^3 is a^5,"},{"Start":"07:19.215 ","End":"07:21.960","Text":"and a^2 times 3 is 3a^2. It\u0027s a minus."},{"Start":"07:21.960 ","End":"07:26.685","Text":"Checked. Last one then."},{"Start":"07:26.685 ","End":"07:29.730","Text":"I\u0027m going to use the letter t. How"},{"Start":"07:29.730 ","End":"07:36.145","Text":"about 5t^3"},{"Start":"07:36.145 ","End":"07:44.250","Text":"plus 15t^2?"},{"Start":"07:44.250 ","End":"07:46.845","Text":"Once again, we could take t out."},{"Start":"07:46.845 ","End":"07:52.725","Text":"We can also take t^2 out. That\u0027s what we\u0027ll do."},{"Start":"07:52.725 ","End":"07:55.455","Text":"We\u0027ll take t^2 outside the brackets."},{"Start":"07:55.455 ","End":"07:57.915","Text":"Very similar to what we had before,"},{"Start":"07:57.915 ","End":"08:00.560","Text":"I have to say t^2 times something is 5t^3."},{"Start":"08:00.560 ","End":"08:02.370","Text":"Well, I need a 5,"},{"Start":"08:02.370 ","End":"08:04.365","Text":"but what else do I need?"},{"Start":"08:04.365 ","End":"08:07.990","Text":"t^2 into t^3, very similar to this."},{"Start":"08:07.990 ","End":"08:10.015","Text":"I won\u0027t draw these sketches again,"},{"Start":"08:10.015 ","End":"08:13.630","Text":"is t. If you multiply t^2 by t,"},{"Start":"08:13.630 ","End":"08:15.250","Text":"you will get t^3."},{"Start":"08:15.250 ","End":"08:20.290","Text":"That\u0027s so because this is t times t times t it\u0027ll be t^3."},{"Start":"08:20.290 ","End":"08:30.190","Text":"Then t^2"},{"Start":"08:30.190 ","End":"08:32.005","Text":"into 15t^2"},{"Start":"08:32.005 ","End":"08:34.330","Text":"goes 15 times,"},{"Start":"08:34.330 ","End":"08:37.565","Text":"t^2 times 15 is 15t^2."},{"Start":"08:37.565 ","End":"08:39.195","Text":"For checking the first one,"},{"Start":"08:39.195 ","End":"08:47.660","Text":"t^2 times 5t is 5t^2 times t, 5t^3, yes."},{"Start":"08:47.660 ","End":"08:49.940","Text":"That\u0027s it for factoring,"},{"Start":"08:49.940 ","End":"08:53.020","Text":"but we will continue this topic in the next clip."},{"Start":"08:53.020 ","End":"08:55.790","Text":"You\u0027re not done with factoring altogether just yet,"},{"Start":"08:55.790 ","End":"08:58.140","Text":"but we are for now."}],"ID":4908},{"Watched":false,"Name":"Factoring (continued)","Duration":"15m 41s","ChapterTopicVideoID":4914,"CourseChapterTopicPlaylistID":3858,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.260","Text":"In this clip, we\u0027re continuing with the factorization,"},{"Start":"00:04.260 ","End":"00:07.920","Text":"so I\u0027ll just write that this is continued."},{"Start":"00:07.920 ","End":"00:10.350","Text":"It\u0027s still the opposite of expansion,"},{"Start":"00:10.350 ","End":"00:14.220","Text":"but this particular clip will be about the opposite"},{"Start":"00:14.220 ","End":"00:19.035","Text":"of the special binomial expansion so a special binomial products."},{"Start":"00:19.035 ","End":"00:23.340","Text":"I\u0027ve written here three formulas which are new,"},{"Start":"00:23.340 ","End":"00:27.600","Text":"but then again they\u0027re not because we\u0027ve been reading them from"},{"Start":"00:27.600 ","End":"00:32.335","Text":"right to left and this direction is the expansion,"},{"Start":"00:32.335 ","End":"00:38.540","Text":"but we\u0027re going to be discussing the other direction which is factoring."},{"Start":"00:38.540 ","End":"00:42.470","Text":"This was the direction of the expansion,"},{"Start":"00:42.470 ","End":"00:45.050","Text":"which is what we call the special"},{"Start":"00:45.050 ","End":"00:49.670","Text":"binomial product and now we\u0027re going to do the opposite,"},{"Start":"00:49.670 ","End":"00:53.630","Text":"which is the factoring of the same formulas,"},{"Start":"00:53.630 ","End":"00:55.265","Text":"we\u0027re going to do them in reverse."},{"Start":"00:55.265 ","End":"00:59.550","Text":"Let\u0027s start with the first one and I\u0027ll"},{"Start":"00:59.550 ","End":"01:04.655","Text":"just highlight it and I\u0027ll give a few examples of the first one."},{"Start":"01:04.655 ","End":"01:11.550","Text":"So remember we had (a+b)^2 is a^2+2ab+b^2 now we\u0027re going to do the opposite."},{"Start":"01:11.950 ","End":"01:22.050","Text":"First example, x^2+8x+16,"},{"Start":"01:22.050 ","End":"01:24.105","Text":"and the way we do it is this,"},{"Start":"01:24.105 ","End":"01:29.375","Text":"for a moment we ignore the middle term and just focus on the first and last,"},{"Start":"01:29.375 ","End":"01:32.600","Text":"this has to be a^2 and this has to be b^2,"},{"Start":"01:32.600 ","End":"01:36.020","Text":"so if this is going to be something squared,"},{"Start":"01:36.020 ","End":"01:38.495","Text":"it\u0027s going to be something plus something squared"},{"Start":"01:38.495 ","End":"01:43.010","Text":"even then this thing is squared has to be this so this has to be"},{"Start":"01:43.010 ","End":"01:46.320","Text":"x and whatever is here squared has to be the"},{"Start":"01:46.320 ","End":"01:51.865","Text":"16 so this has to be 4 because 4^2 is 4 times 4 is 16."},{"Start":"01:51.865 ","End":"01:54.110","Text":"Now, the middle term,"},{"Start":"01:54.110 ","End":"01:56.180","Text":"we haven\u0027t used yet,"},{"Start":"01:56.180 ","End":"02:00.185","Text":"this basically just serves as a check."},{"Start":"02:00.185 ","End":"02:03.260","Text":"We expand this and if we get the 8x,"},{"Start":"02:03.260 ","End":"02:07.070","Text":"then all is well otherwise there\u0027s something wrong with the exercise."},{"Start":"02:07.070 ","End":"02:10.730","Text":"What I do is I expand it by using"},{"Start":"02:10.730 ","End":"02:15.245","Text":"the formula the other way and saying this plus this squared,"},{"Start":"02:15.245 ","End":"02:24.620","Text":"so it would be x^2 plus twice this times this and twice this times this."},{"Start":"02:24.620 ","End":"02:26.585","Text":"If you figure it is 8x,"},{"Start":"02:26.585 ","End":"02:29.360","Text":"and then of course, it\u0027s plus 16."},{"Start":"02:29.360 ","End":"02:34.160","Text":"But the middle term is the 1 that you really have to check that matches,"},{"Start":"02:34.160 ","End":"02:36.860","Text":"it does match so we\u0027re okay,"},{"Start":"02:36.860 ","End":"02:38.630","Text":"this is the answer."},{"Start":"02:38.630 ","End":"02:41.430","Text":"Okay, next one."},{"Start":"02:43.780 ","End":"02:53.150","Text":"4a^2+12a+9. Now we want to use this formula again,"},{"Start":"02:53.150 ","End":"02:56.570","Text":"so like I said, we just focus on the first and last terms."},{"Start":"02:56.570 ","End":"03:01.535","Text":"The first, and we want to make it as to be something plus something squared."},{"Start":"03:01.535 ","End":"03:05.250","Text":"Now what squared would give me 4a^2."},{"Start":"03:05.250 ","End":"03:13.820","Text":"4 is 2^2 so if I take 2a and I square that 2^2 is 4,"},{"Start":"03:13.820 ","End":"03:16.080","Text":"a^2 is a^2, so that\u0027s okay."},{"Start":"03:16.080 ","End":"03:18.980","Text":"For the last one, what squared gives me 9?"},{"Start":"03:18.980 ","End":"03:20.830","Text":"That will be 3,"},{"Start":"03:20.830 ","End":"03:24.125","Text":"and so if there is an answer at all,"},{"Start":"03:24.125 ","End":"03:26.960","Text":"if there\u0027s any justice here,"},{"Start":"03:26.960 ","End":"03:30.425","Text":"the answer will be (2a+3)^2,"},{"Start":"03:30.425 ","End":"03:32.795","Text":"but we haven\u0027t used the middle term at all."},{"Start":"03:32.795 ","End":"03:37.640","Text":"I mean, you don\u0027t always have to check the first and last, you can."},{"Start":"03:37.640 ","End":"03:42.540","Text":"The main thing we\u0027re going to do is check if the middle one is like the 2ab."},{"Start":"03:44.050 ","End":"03:48.980","Text":"2ab doesn\u0027t necessarily mean 2ab means twice this times this,"},{"Start":"03:48.980 ","End":"03:56.250","Text":"we have to multiply 2×2a×3 because instead of a we have 2a,"},{"Start":"03:56.250 ","End":"04:01.965","Text":"but it\u0027s 2 times twice this times this and if you figure it,"},{"Start":"04:01.965 ","End":"04:06.630","Text":"2×2×3 is 12, this is 12a so this is okay."},{"Start":"04:06.630 ","End":"04:10.960","Text":"The middle term checks out so this is the answer."},{"Start":"04:10.960 ","End":"04:13.820","Text":"Perhaps I should be highlighting the answers."},{"Start":"04:13.820 ","End":"04:20.100","Text":"Let\u0027s see here and here and now the next exercise."},{"Start":"04:28.350 ","End":"04:35.495","Text":"20x^2+100x+125, here\u0027s where we throw in a slight extra level of difficulty."},{"Start":"04:35.495 ","End":"04:40.400","Text":"For one thing, if you try to get this to be something squared,"},{"Start":"04:40.400 ","End":"04:42.050","Text":"it doesn\u0027t work out neatly,"},{"Start":"04:42.050 ","End":"04:45.650","Text":"in these exercises always use whole numbers,"},{"Start":"04:45.650 ","End":"04:47.329","Text":"same with the last one."},{"Start":"04:47.329 ","End":"04:50.630","Text":"Sometimes what you have to do is prepare it"},{"Start":"04:50.630 ","End":"04:53.855","Text":"to be in this form by taking out a common factor."},{"Start":"04:53.855 ","End":"04:55.580","Text":"Now if I look at all these,"},{"Start":"04:55.580 ","End":"05:05.175","Text":"I see that 5 is a common factor to everything so if I take 5 outside the brackets,"},{"Start":"05:05.175 ","End":"05:08.490","Text":"once again we\u0027re using factoring,"},{"Start":"05:08.490 ","End":"05:11.240","Text":"it\u0027s 5 times something plus something plus something."},{"Start":"05:11.240 ","End":"05:15.510","Text":"If I take 5 out of here, it\u0027s just 4x^2."},{"Start":"05:15.510 ","End":"05:17.560","Text":"If I take 5 out of here,"},{"Start":"05:17.560 ","End":"05:26.985","Text":"I get 20x and take 5 out of here I\u0027ve got 25 because 5×25 is 125."},{"Start":"05:26.985 ","End":"05:29.685","Text":"At this point and only at this point,"},{"Start":"05:29.685 ","End":"05:37.790","Text":"do I start to use this formula and what I say is that this has to equal to the 5,"},{"Start":"05:37.790 ","End":"05:41.690","Text":"I copy and it\u0027s going to be something squared."},{"Start":"05:41.690 ","End":"05:44.270","Text":"This time I have a better chance,"},{"Start":"05:44.270 ","End":"05:53.500","Text":"because 25 is 5^2 there\u0027s a plus in the middle and 4x^2 is,"},{"Start":"05:53.500 ","End":"05:59.250","Text":"you\u0027ve seen this enough times already to know that this is 2x^2 because 2^2 is"},{"Start":"05:59.250 ","End":"06:06.730","Text":"4 and x^2 is x^2 and so this is the answer except that we have to check the middle term."},{"Start":"06:06.730 ","End":"06:09.255","Text":"The middle term here in this expansion,"},{"Start":"06:09.255 ","End":"06:14.565","Text":"if I do 2 times this one times that one,"},{"Start":"06:14.565 ","End":"06:17.145","Text":"that\u0027s the 2ab thing that I\u0027m checking."},{"Start":"06:17.145 ","End":"06:21.495","Text":"Then it comes out as 2×2×5 is 4×5 is 20×x,"},{"Start":"06:21.495 ","End":"06:27.330","Text":"yes I\u0027m just checking the middle term that\u0027s always the difficult one,"},{"Start":"06:27.330 ","End":"06:31.440","Text":"this is the one to check here."},{"Start":"06:31.440 ","End":"06:39.005","Text":"You just work on the first and the last and hope that the middle one works out,"},{"Start":"06:39.005 ","End":"06:42.574","Text":"if it doesn\u0027t then you can\u0027t continue."},{"Start":"06:42.574 ","End":"06:48.760","Text":"But it does so this is the answer,"},{"Start":"06:48.760 ","End":"06:53.659","Text":"I\u0027ll highlight it and now we\u0027ll move on to the next formula."},{"Start":"06:53.659 ","End":"06:57.275","Text":"Here we are, we\u0027re going to practice the second formula."},{"Start":"06:57.275 ","End":"06:59.970","Text":"Start right away,"},{"Start":"07:00.140 ","End":"07:08.849","Text":"x^2-12x+36."},{"Start":"07:08.849 ","End":"07:12.370","Text":"I want to write it as something squared."},{"Start":"07:12.370 ","End":"07:15.295","Text":"But this time there\u0027s going to be a minus in the middle,"},{"Start":"07:15.295 ","End":"07:18.265","Text":"because it\u0027s something minus something plus something."},{"Start":"07:18.265 ","End":"07:21.355","Text":"As before, we look at the first and last terms"},{"Start":"07:21.355 ","End":"07:25.390","Text":"and then the middle term is just used a check to see that we\u0027re okay."},{"Start":"07:25.390 ","End":"07:27.731","Text":"Here, we put x."},{"Start":"07:27.731 ","End":"07:30.840","Text":"Here, something squared has to be 36,"},{"Start":"07:30.840 ","End":"07:32.310","Text":"so that\u0027s 6,"},{"Start":"07:32.310 ","End":"07:40.000","Text":"and then we just check for the middle term that we have twice x times 6,"},{"Start":"07:40.000 ","End":"07:44.050","Text":"that\u0027s the 2ab and it really is 12x,"},{"Start":"07:44.050 ","End":"07:45.640","Text":"so it\u0027s okay,"},{"Start":"07:45.640 ","End":"07:49.255","Text":"and this is the answer and I\u0027m highlighting it."},{"Start":"07:49.255 ","End":"07:59.575","Text":"Next example, a^2-16a+64."},{"Start":"07:59.575 ","End":"08:01.210","Text":"What will this equal?"},{"Start":"08:01.210 ","End":"08:03.100","Text":"Well, this is the formula we\u0027re using,"},{"Start":"08:03.100 ","End":"08:05.079","Text":"something something plus something."},{"Start":"08:05.079 ","End":"08:09.430","Text":"We want to write it as something minus something squared."},{"Start":"08:09.430 ","End":"08:13.000","Text":"We\u0027re looking for a and b from the first and the last terms."},{"Start":"08:13.000 ","End":"08:15.925","Text":"From the first term, I see that this has to be a,"},{"Start":"08:15.925 ","End":"08:17.740","Text":"because this squared has to be this,"},{"Start":"08:17.740 ","End":"08:21.865","Text":"and the last one has to be 8 because 8^2 is 64."},{"Start":"08:21.865 ","End":"08:27.540","Text":"All we have to do now is check that twice this times this is the middle term."},{"Start":"08:27.540 ","End":"08:32.470","Text":"In other words, the 16 has to be twice a times 8,"},{"Start":"08:32.470 ","End":"08:34.435","Text":"and clearly it is,"},{"Start":"08:34.435 ","End":"08:39.680","Text":"so this is the answer and onto the next."},{"Start":"08:41.130 ","End":"08:51.340","Text":"2x^2-28x+98."},{"Start":"08:51.340 ","End":"08:54.010","Text":"Remember what I said about factoring first,"},{"Start":"08:54.010 ","End":"08:56.140","Text":"taking a common factor outside the brackets,"},{"Start":"08:56.140 ","End":"08:57.640","Text":"this is one of those cases."},{"Start":"08:57.640 ","End":"08:58.840","Text":"They\u0027re all even numbers,"},{"Start":"08:58.840 ","End":"09:00.670","Text":"so I can take 2 out."},{"Start":"09:00.670 ","End":"09:02.920","Text":"That\u0027s the best I can do."},{"Start":"09:02.920 ","End":"09:04.690","Text":"If we take a 2 out,"},{"Start":"09:04.690 ","End":"09:09.430","Text":"we\u0027re left with x^2 here minus 14x here,"},{"Start":"09:09.430 ","End":"09:12.565","Text":"and 2 out of here, leaving me with a 49."},{"Start":"09:12.565 ","End":"09:17.140","Text":"That\u0027s just as well because 49 is a nice square number and so is 1."},{"Start":"09:17.140 ","End":"09:20.200","Text":"Now this equals the 2 stays."},{"Start":"09:20.200 ","End":"09:24.580","Text":"But what we\u0027re going to do is we\u0027re going to"},{"Start":"09:24.580 ","End":"09:29.680","Text":"replace with something minus something squared like in this formula."},{"Start":"09:29.680 ","End":"09:32.170","Text":"The first thing squared is x^2,"},{"Start":"09:32.170 ","End":"09:35.320","Text":"the last one is 7^2."},{"Start":"09:35.320 ","End":"09:37.750","Text":"All we have to do is check the middle term."},{"Start":"09:37.750 ","End":"09:41.170","Text":"I need to check that 2 times x times 7,"},{"Start":"09:41.170 ","End":"09:47.050","Text":"that\u0027s the 2ab is equal to 14x,"},{"Start":"09:47.050 ","End":"09:49.240","Text":"and it is because 2 times 7 is 14,"},{"Start":"09:49.240 ","End":"09:51.640","Text":"so we\u0027re all right here."},{"Start":"09:51.640 ","End":"09:54.415","Text":"I can highlight this,"},{"Start":"09:54.415 ","End":"09:57.610","Text":"and we\u0027re done with the third,"},{"Start":"09:57.610 ","End":"10:05.065","Text":"and so I can now clear the board and move the highlighting to the last formula,"},{"Start":"10:05.065 ","End":"10:07.164","Text":"the difference of squares formula."},{"Start":"10:07.164 ","End":"10:10.490","Text":"This is the one we\u0027re going to practice now."},{"Start":"10:10.800 ","End":"10:15.880","Text":"Let\u0027s start x^2 minus 9."},{"Start":"10:15.880 ","End":"10:17.800","Text":"Using this formula."},{"Start":"10:17.800 ","End":"10:21.265","Text":"It looks very much like something squared minus something squared."},{"Start":"10:21.265 ","End":"10:28.345","Text":"All I have to do is write it as x^2 minus 3^2 and it\u0027s very similar to this."},{"Start":"10:28.345 ","End":"10:32.425","Text":"Then we see that a can be x and b can be 3."},{"Start":"10:32.425 ","End":"10:35.380","Text":"I factorize it as the sum and the difference,"},{"Start":"10:35.380 ","End":"10:37.760","Text":"and it doesn\u0027t matter which order."},{"Start":"10:37.920 ","End":"10:42.970","Text":"I\u0027ll write the plus first x plus 3,"},{"Start":"10:42.970 ","End":"10:45.775","Text":"x minus 3,"},{"Start":"10:45.775 ","End":"10:55.735","Text":"and that\u0027s the answer, a^2-16."},{"Start":"10:55.735 ","End":"10:59.290","Text":"Once again, I want to write it as something squared minus something squared."},{"Start":"10:59.290 ","End":"11:01.465","Text":"The first one is already something squared."},{"Start":"11:01.465 ","End":"11:04.705","Text":"We\u0027ll have to do is write 16 is 4^2."},{"Start":"11:04.705 ","End":"11:07.285","Text":"Then I have a difference of squares,"},{"Start":"11:07.285 ","End":"11:11.320","Text":"and then I know it\u0027s the sum times the difference or the other way round."},{"Start":"11:11.320 ","End":"11:13.510","Text":"For example, here I might just,"},{"Start":"11:13.510 ","End":"11:16.180","Text":"to be different, write the minus first,"},{"Start":"11:16.180 ","End":"11:22.375","Text":"write it as (a-4)(a+4),"},{"Start":"11:22.375 ","End":"11:25.600","Text":"and just highlight that,"},{"Start":"11:25.600 ","End":"11:28.585","Text":"and onto the third."},{"Start":"11:28.585 ","End":"11:35.050","Text":"Take 121-x^2, this term,"},{"Start":"11:35.050 ","End":"11:37.240","Text":"the number\u0027s first and the x\u0027s second,"},{"Start":"11:37.240 ","End":"11:39.770","Text":"doesn\u0027t make any difference."},{"Start":"11:40.170 ","End":"11:43.600","Text":"I\u0027m going to write it as something squared minus something squared."},{"Start":"11:43.600 ","End":"11:45.460","Text":"Well, you have to know that 121 is 11^2,"},{"Start":"11:45.460 ","End":"11:48.535","Text":"11 times 11 is 121."},{"Start":"11:48.535 ","End":"11:51.530","Text":"It\u0027s 11^2 minus x^2."},{"Start":"11:51.530 ","End":"11:55.530","Text":"Then I write it as difference times the sum or the sometimes the difference,"},{"Start":"11:55.530 ","End":"11:57.765","Text":"we\u0027ll go with the plus first this time,"},{"Start":"11:57.765 ","End":"12:00.435","Text":"11 plus x,"},{"Start":"12:00.435 ","End":"12:06.610","Text":"11 minus x and highlight it."},{"Start":"12:06.610 ","End":"12:09.865","Text":"I\u0027d like to do a few extra examples."},{"Start":"12:09.865 ","End":"12:19.215","Text":"Let\u0027s start with 100y^2 minus 81."},{"Start":"12:19.215 ","End":"12:23.685","Text":"Write it as something squared minus something squared."},{"Start":"12:23.685 ","End":"12:28.340","Text":"Now, I might need a bracket here,"},{"Start":"12:28.590 ","End":"12:32.860","Text":"100y^2 is 10^2 times y^2."},{"Start":"12:32.860 ","End":"12:37.840","Text":"If I put here 10y and square it,"},{"Start":"12:37.840 ","End":"12:39.235","Text":"this will give me the first term,"},{"Start":"12:39.235 ","End":"12:41.935","Text":"10 times 10 times y times y is that."},{"Start":"12:41.935 ","End":"12:43.810","Text":"The second term 81,"},{"Start":"12:43.810 ","End":"12:48.565","Text":"I can write as 9^2 because everyone knows that 9 squared is 81."},{"Start":"12:48.565 ","End":"12:50.875","Text":"Now it really looks like this,"},{"Start":"12:50.875 ","End":"12:56.934","Text":"and so the answer will be the difference times the sum or the sometimes the difference,"},{"Start":"12:56.934 ","End":"13:00.175","Text":"10y plus 9,"},{"Start":"13:00.175 ","End":"13:04.360","Text":"10y minus 9,"},{"Start":"13:04.360 ","End":"13:07.510","Text":"and that\u0027s 1 of them done."},{"Start":"13:07.510 ","End":"13:17.086","Text":"81x minus x cubed,"},{"Start":"13:17.086 ","End":"13:21.310","Text":"this is one of those we have to do extra work of"},{"Start":"13:21.310 ","End":"13:26.665","Text":"pre-factoring because we already have some factor in common, which is x."},{"Start":"13:26.665 ","End":"13:28.285","Text":"If you don\u0027t take the x out,"},{"Start":"13:28.285 ","End":"13:29.605","Text":"it just won\u0027t work out,"},{"Start":"13:29.605 ","End":"13:32.620","Text":"when you see a common factor, take it out."},{"Start":"13:32.620 ","End":"13:36.130","Text":"This case x outside the brackets,"},{"Start":"13:36.130 ","End":"13:39.805","Text":"so we\u0027re left with 81, and x,"},{"Start":"13:39.805 ","End":"13:41.530","Text":"if we take it out of x^3,"},{"Start":"13:41.530 ","End":"13:44.455","Text":"we still have x^2 and other check,"},{"Start":"13:44.455 ","End":"13:50.800","Text":"x times x^2 is x^3 because it\u0027s x times x times x, and it\u0027s right."},{"Start":"13:50.800 ","End":"13:52.990","Text":"We leave the x here,"},{"Start":"13:52.990 ","End":"13:55.030","Text":"and now we try to write this."},{"Start":"13:55.030 ","End":"13:57.580","Text":"The left formula\u0027s the only one that"},{"Start":"13:57.580 ","End":"14:00.505","Text":"could fit as something squared minus something squared."},{"Start":"14:00.505 ","End":"14:04.225","Text":"We\u0027re okay here with something squared just here."},{"Start":"14:04.225 ","End":"14:08.080","Text":"But look in the previous exercise we had 9^2 is 81,"},{"Start":"14:08.080 ","End":"14:10.780","Text":"so this is 9^2."},{"Start":"14:10.780 ","End":"14:12.820","Text":"Now as the last step,"},{"Start":"14:12.820 ","End":"14:17.620","Text":"I write this as x, it stays."},{"Start":"14:17.620 ","End":"14:21.610","Text":"Then here we have 9 plus x,"},{"Start":"14:21.610 ","End":"14:24.400","Text":"9 minus x,"},{"Start":"14:24.400 ","End":"14:29.390","Text":"and this is the answer to this one."},{"Start":"14:29.390 ","End":"14:32.420","Text":"I\u0027ll do one last one."},{"Start":"14:32.420 ","End":"14:35.660","Text":"Let\u0027s see what letter haven\u0027t I used?"},{"Start":"14:35.660 ","End":"14:42.530","Text":"Let\u0027s go with c, c^3 minus c. Once again,"},{"Start":"14:42.530 ","End":"14:47.130","Text":"we have a common factor and we have to take it out first."},{"Start":"14:47.350 ","End":"14:53.625","Text":"First stage is to take c out and then I\u0027m left with c^2 squared minus 1."},{"Start":"14:53.625 ","End":"14:58.685","Text":"As a check, we multiply out c times c^2 is c times c times c is c cubed,"},{"Start":"14:58.685 ","End":"15:04.485","Text":"minus c. Now I just rewrite this,"},{"Start":"15:04.485 ","End":"15:08.450","Text":"c stays after write this as something squared minus something squared."},{"Start":"15:08.450 ","End":"15:13.985","Text":"This already is squared and 1 is 1 squared because 1 times 1 is 1."},{"Start":"15:13.985 ","End":"15:18.290","Text":"This last one, I can write, the c stays,"},{"Start":"15:18.290 ","End":"15:20.390","Text":"but this bit is the difference of squares,"},{"Start":"15:20.390 ","End":"15:23.975","Text":"so it\u0027s (c+1)(c-1),"},{"Start":"15:23.975 ","End":"15:25.070","Text":"and just to be different,"},{"Start":"15:25.070 ","End":"15:29.075","Text":"I\u0027ll do it as (c-1)(c+1),"},{"Start":"15:29.075 ","End":"15:34.740","Text":"and this is the answer."},{"Start":"15:34.740 ","End":"15:39.630","Text":"We\u0027re done with the factorization."}],"ID":4909},{"Watched":false,"Name":"Special Binomial Products (continued)","Duration":"10m 4s","ChapterTopicVideoID":4916,"CourseChapterTopicPlaylistID":3858,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.235","Text":"This clip is a continuation of the previous clip, special binomial products."},{"Start":"00:05.235 ","End":"00:07.887","Text":"There we talked about this formula."},{"Start":"00:07.887 ","End":"00:11.970","Text":"This time I\u0027m going to add another similar formula."},{"Start":"00:11.970 ","End":"00:15.885","Text":"Here\u0027s our second formula, very similar."},{"Start":"00:15.885 ","End":"00:21.810","Text":"Notice that the only difference is that here there\u0027s a minus and here there\u0027s a minus."},{"Start":"00:21.810 ","End":"00:24.150","Text":"The middle term changes sign,"},{"Start":"00:24.150 ","End":"00:26.980","Text":"but the last term is still a plus."},{"Start":"00:27.200 ","End":"00:30.000","Text":"This time I\u0027m not going to prove the formula."},{"Start":"00:30.000 ","End":"00:32.385","Text":"Let\u0027s get straight to the exercises."},{"Start":"00:32.385 ","End":"00:39.210","Text":"The first one is x minus 5^2."},{"Start":"00:39.210 ","End":"00:41.460","Text":"This is equal to,"},{"Start":"00:41.460 ","End":"00:44.389","Text":"as before, we see there\u0027s a minus."},{"Start":"00:44.389 ","End":"00:46.370","Text":"We identify before the minus,"},{"Start":"00:46.370 ","End":"00:48.644","Text":"it\u0027s an a after the minus it\u0027s b."},{"Start":"00:48.644 ","End":"00:52.470","Text":"Just like with this formula, we proceed;"},{"Start":"00:53.050 ","End":"00:56.145","Text":"a^2 to minus this time,"},{"Start":"00:56.145 ","End":"00:57.810","Text":"twice this times this,"},{"Start":"00:57.810 ","End":"01:02.895","Text":"2 times x times 5 plus the last 1 squared, that\u0027s the b^2."},{"Start":"01:02.895 ","End":"01:06.835","Text":"Then a small simplification, x^2,"},{"Start":"01:06.835 ","End":"01:09.410","Text":"2 times x times 5 is 10x,"},{"Start":"01:09.410 ","End":"01:13.050","Text":"and 5^2 squared is 25."},{"Start":"01:13.150 ","End":"01:19.020","Text":"Next one, 4 minus x^2."},{"Start":"01:19.580 ","End":"01:22.440","Text":"This time this is the a, this is the b,"},{"Start":"01:22.440 ","End":"01:25.487","Text":"and I\u0027m not going to label them."},{"Start":"01:25.487 ","End":"01:30.285","Text":"Let\u0027s write it as 4^2 minus twice,"},{"Start":"01:30.285 ","End":"01:35.295","Text":"4 times x plus x^2."},{"Start":"01:35.295 ","End":"01:40.140","Text":"Corrected. This is equal to 4^2 is 16,"},{"Start":"01:40.140 ","End":"01:45.045","Text":"2 times 4 is 8, x plus x^2."},{"Start":"01:45.045 ","End":"01:46.710","Text":"That\u0027s it, we\u0027re done."},{"Start":"01:46.710 ","End":"01:54.730","Text":"Next one, 4a minus 2^2."},{"Start":"01:54.920 ","End":"01:58.995","Text":"Once again, this formula,"},{"Start":"01:58.995 ","End":"02:03.010","Text":"it\u0027s a bit confusing because instead of a we have 4a,"},{"Start":"02:03.010 ","End":"02:08.270","Text":"but just imagine replacing a by 4a and b by 2."},{"Start":"02:08.270 ","End":"02:12.405","Text":"It\u0027s like the role of a is being played by 4a this time."},{"Start":"02:12.405 ","End":"02:14.895","Text":"If I look at this, instead of a^2,"},{"Start":"02:14.895 ","End":"02:17.130","Text":"I have 4a^2,"},{"Start":"02:17.130 ","End":"02:21.915","Text":"donated in brackets, minus twice 2ab."},{"Start":"02:21.915 ","End":"02:25.050","Text":"Minus twice, but instead of a, it\u0027s 4a."},{"Start":"02:25.050 ","End":"02:28.985","Text":"Sorry, I put brackets or put a little dot here,"},{"Start":"02:28.985 ","End":"02:35.560","Text":"times b, which is 2 also, and plus b^2."},{"Start":"02:35.960 ","End":"02:39.645","Text":"After simplification, what do we get?"},{"Start":"02:39.645 ","End":"02:43.335","Text":"4a^2 is 4^2 times a^2."},{"Start":"02:43.335 ","End":"02:46.760","Text":"That\u0027s what I said with addition or subtraction,"},{"Start":"02:46.760 ","End":"02:48.260","Text":"you can\u0027t just square each term,"},{"Start":"02:48.260 ","End":"02:50.390","Text":"but with multiplication, you can."},{"Start":"02:50.390 ","End":"02:56.435","Text":"4^2 is 16, a^2 is a^2."},{"Start":"02:56.435 ","End":"03:02.070","Text":"2 times 4 times 2 is 16a."},{"Start":"03:03.280 ","End":"03:07.700","Text":"Then 2 squared is 4 and that\u0027s it."},{"Start":"03:07.700 ","End":"03:14.940","Text":"One more example, 2c minus 4d^2."},{"Start":"03:15.850 ","End":"03:18.830","Text":"This time a is 2c,"},{"Start":"03:18.830 ","End":"03:25.405","Text":"so a^2 is (2c)^2 minus twice,"},{"Start":"03:25.405 ","End":"03:28.745","Text":"this time I\u0027ll use brackets instead of multiplication,"},{"Start":"03:28.745 ","End":"03:32.240","Text":"2c times 4d,"},{"Start":"03:32.240 ","End":"03:41.100","Text":"just for a change, and plus (4d)^2."},{"Start":"03:41.150 ","End":"03:44.670","Text":"Then the simplification,"},{"Start":"03:44.670 ","End":"03:48.210","Text":"(2c)^2 is 2^2 times c^2."},{"Start":"03:48.210 ","End":"03:51.360","Text":"4c^2 squared minus,"},{"Start":"03:51.360 ","End":"03:53.055","Text":"this is still multiplication,"},{"Start":"03:53.055 ","End":"03:57.300","Text":"brackets can go out."},{"Start":"03:57.300 ","End":"04:01.449","Text":"2 times 2 times 4 is 16."},{"Start":"04:01.449 ","End":"04:04.045","Text":"Remember I can do multiplication in any order,"},{"Start":"04:04.045 ","End":"04:08.550","Text":"and c times d is cd,"},{"Start":"04:08.550 ","End":"04:15.800","Text":"and then 4d, all this squared is 4^2 times d^2, which is 16d^2."},{"Start":"04:17.370 ","End":"04:20.290","Text":"That\u0027s enough examples for this."},{"Start":"04:20.290 ","End":"04:22.330","Text":"Let\u0027s go on to the third formula."},{"Start":"04:22.330 ","End":"04:23.680","Text":"Yes, there\u0027s a third."},{"Start":"04:23.680 ","End":"04:29.380","Text":"Now, actually you could guess what the third formula is or at least part of it."},{"Start":"04:29.380 ","End":"04:33.730","Text":"Because look, we said binomial products and we said this is"},{"Start":"04:33.730 ","End":"04:39.445","Text":"a product because this is a plus b times a plus b."},{"Start":"04:39.445 ","End":"04:47.135","Text":"Similarly, a minus b^2 is a minus b times a minus b."},{"Start":"04:47.135 ","End":"04:49.385","Text":"Let\u0027s play mix and match."},{"Start":"04:49.385 ","End":"04:51.500","Text":"Here we match now let\u0027s mix,"},{"Start":"04:51.500 ","End":"04:53.935","Text":"let\u0027s take one of them to be a plus,"},{"Start":"04:53.935 ","End":"04:56.265","Text":"and one of them to be a minus."},{"Start":"04:56.265 ","End":"05:00.560","Text":"That\u0027s exactly what\u0027s going to be the left-hand side of"},{"Start":"05:00.560 ","End":"05:06.460","Text":"the next formula or the other way round and it doesn\u0027t really matter. Here it goes."},{"Start":"05:06.460 ","End":"05:13.430","Text":"Here it is, a plus b times a minus b is equal to a^2 minus b^2."},{"Start":"05:13.430 ","End":"05:16.580","Text":"Actually, this formula has a name, well, they all do,"},{"Start":"05:16.580 ","End":"05:19.545","Text":"but this one is called the difference of squares."},{"Start":"05:19.545 ","End":"05:21.860","Text":"It\u0027s called the difference of squares formula because this"},{"Start":"05:21.860 ","End":"05:24.815","Text":"is a square and this is a square and we subtract them."},{"Start":"05:24.815 ","End":"05:27.680","Text":"I\u0027ll give you an arithmetical example."},{"Start":"05:27.680 ","End":"05:32.120","Text":"I actually use this formula sometimes when I multiply in my head,"},{"Start":"05:32.120 ","End":"05:34.535","Text":"sometimes I\u0027ll get an exercise,"},{"Start":"05:34.535 ","End":"05:40.445","Text":"like could be 28 times 32."},{"Start":"05:40.445 ","End":"05:43.020","Text":"I should have written it the other way around."},{"Start":"05:43.510 ","End":"05:45.830","Text":"I\u0027m thinking to myself,"},{"Start":"05:45.830 ","End":"05:55.715","Text":"what I have here is I have 30 plus 2 and I also have 30 minus 2 and I\u0027m multiplying them."},{"Start":"05:55.715 ","End":"05:58.535","Text":"Then I remember the difference of squares formulas."},{"Start":"05:58.535 ","End":"06:01.045","Text":"This is a and this is b."},{"Start":"06:01.045 ","End":"06:04.710","Text":"A plus b, a minus b."},{"Start":"06:04.710 ","End":"06:08.570","Text":"I\u0027m not doing all of this in my head because I already know this formula well,"},{"Start":"06:08.570 ","End":"06:11.390","Text":"but then I say, it\u0027s the first one squared,"},{"Start":"06:11.390 ","End":"06:17.475","Text":"the a^2, 30^2 minus 2^2."},{"Start":"06:17.475 ","End":"06:19.170","Text":"This I say to myself,"},{"Start":"06:19.170 ","End":"06:22.155","Text":"this is 30^2 is 900, 2^2 is 4."},{"Start":"06:22.155 ","End":"06:29.470","Text":"I can do 900 minus 4 in my head and I can get the answer as 896."},{"Start":"06:29.470 ","End":"06:35.135","Text":"In fact, you can multiply these on your calculator to check that it is 896."},{"Start":"06:35.135 ","End":"06:36.875","Text":"Sometimes it happens."},{"Start":"06:36.875 ","End":"06:39.440","Text":"I\u0027ll give you on your own and another one to do."},{"Start":"06:39.440 ","End":"06:44.210","Text":"How about trying to multiply 19 times 21,"},{"Start":"06:44.210 ","End":"06:45.800","Text":"you may have to reverse the order."},{"Start":"06:45.800 ","End":"06:47.960","Text":"I\u0027m not doing anymore of this."},{"Start":"06:47.960 ","End":"06:51.265","Text":"I\u0027m going to erase what\u0027s not necessary."},{"Start":"06:51.265 ","End":"06:57.135","Text":"That\u0027s got rid of that. I should highlight this one instead of this. There we are."},{"Start":"06:57.135 ","End":"07:02.610","Text":"I want to remind you or to point out that it doesn\u0027t matter which order,"},{"Start":"07:02.610 ","End":"07:07.880","Text":"obviously you can change the order of a product."},{"Start":"07:07.880 ","End":"07:10.760","Text":"It could have been, the first one is a minus and the second one is"},{"Start":"07:10.760 ","End":"07:13.895","Text":"a plus and you\u0027ll still get the same thing on the right,"},{"Start":"07:13.895 ","End":"07:17.540","Text":"just changing the order. Let\u0027s get started."},{"Start":"07:17.540 ","End":"07:22.600","Text":"I\u0027ll just give you three exercises this time because we already did a numerical one."},{"Start":"07:22.600 ","End":"07:30.945","Text":"The first one will be x minus 6, x plus 6."},{"Start":"07:30.945 ","End":"07:33.140","Text":"One is a plus, one is a minus."},{"Start":"07:33.140 ","End":"07:34.955","Text":"It doesn\u0027t matter in which order."},{"Start":"07:34.955 ","End":"07:37.500","Text":"The first one would be a,"},{"Start":"07:37.500 ","End":"07:39.000","Text":"the second one, b."},{"Start":"07:39.000 ","End":"07:43.235","Text":"It\u0027s the square root of the first minus the square root of the second,"},{"Start":"07:43.235 ","End":"07:46.810","Text":"so it\u0027s x^2 minus 6^2."},{"Start":"07:46.810 ","End":"07:52.050","Text":"Then I simplify it as x^2 minus 36."},{"Start":"07:52.050 ","End":"08:01.570","Text":"The next one, let\u0027s take 8 minus y, 8 plus y."},{"Start":"08:01.570 ","End":"08:05.585","Text":"Soon as you see the same thing with one to the plus 1 to the minus,"},{"Start":"08:05.585 ","End":"08:08.930","Text":"you know you\u0027re in a difference of squares so you take the first one"},{"Start":"08:08.930 ","End":"08:12.560","Text":"squared and then you subtract the second one squared."},{"Start":"08:12.560 ","End":"08:19.245","Text":"Difference tells you you should subtract squares per good name for this formula."},{"Start":"08:19.245 ","End":"08:25.700","Text":"Then again, we simplify at 64 minus y^2."},{"Start":"08:25.700 ","End":"08:33.270","Text":"Last one, 3t plus 4, 3t minus 4."},{"Start":"08:33.270 ","End":"08:37.395","Text":"You see those in example of the pluses first and the minuses second, it doesn\u0027t matter."},{"Start":"08:37.395 ","End":"08:40.309","Text":"Difference of squares, the first one squared,"},{"Start":"08:40.309 ","End":"08:46.280","Text":"but put it in brackets before you put the squared, minus 4^2."},{"Start":"08:46.280 ","End":"08:49.530","Text":"This becomes, we square each 1,"},{"Start":"08:49.530 ","End":"08:53.230","Text":"3^2 is 9, t^2 is t^2,"},{"Start":"08:53.230 ","End":"08:56.415","Text":"and 4^2 is 16."},{"Start":"08:56.415 ","End":"08:59.825","Text":"That\u0027s it. Now I\u0027m kind of done,"},{"Start":"08:59.825 ","End":"09:04.295","Text":"but for those who want to stay and see the proof, it\u0027s really quick."},{"Start":"09:04.295 ","End":"09:07.135","Text":"At least don\u0027t say I didn\u0027t offer."},{"Start":"09:07.135 ","End":"09:10.880","Text":"Remember the red arcs this times this,"},{"Start":"09:10.880 ","End":"09:13.685","Text":"this times this, this times this, this times this."},{"Start":"09:13.685 ","End":"09:15.345","Text":"We\u0027re going to do that here."},{"Start":"09:15.345 ","End":"09:17.910","Text":"A times a,"},{"Start":"09:18.080 ","End":"09:22.610","Text":"a^2, a times minus b,"},{"Start":"09:22.610 ","End":"09:29.655","Text":"minus ab, plus b times a is plus ba,"},{"Start":"09:29.655 ","End":"09:34.260","Text":"minus b times b minus b^2."},{"Start":"09:34.260 ","End":"09:40.984","Text":"Then all that\u0027s necessary is for you to notice that these are like terms,"},{"Start":"09:40.984 ","End":"09:45.525","Text":"because after all, ab is equal to ba."},{"Start":"09:45.525 ","End":"09:47.630","Text":"The order of multiplication doesn\u0027t make any difference,"},{"Start":"09:47.630 ","End":"09:53.045","Text":"so we can cross these two out and all we\u0027re left with is a^2 minus b^2,"},{"Start":"09:53.045 ","End":"09:54.815","Text":"which is the right-hand side."},{"Start":"09:54.815 ","End":"09:57.260","Text":"When you prove something at the end you write something in"},{"Start":"09:57.260 ","End":"10:00.995","Text":"Latin, quod erat demonstrandum."},{"Start":"10:00.995 ","End":"10:04.560","Text":"That\u0027s what we needed to show. We\u0027re done."}],"ID":4907},{"Watched":false,"Name":"Special Binomial Products - Examples","Duration":"9m 2s","ChapterTopicVideoID":4915,"CourseChapterTopicPlaylistID":3858,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.260 ","End":"00:04.080","Text":"I erased all this theoretical stuff but keeping the formula."},{"Start":"00:04.080 ","End":"00:05.910","Text":"Now, let\u0027s go and do some exercises."},{"Start":"00:05.910 ","End":"00:08.040","Text":"In fact, I\u0027ll do quite a few and then you\u0027ll"},{"Start":"00:08.040 ","End":"00:11.325","Text":"see that it\u0027s not so bad. Let\u0027s take the first one."},{"Start":"00:11.325 ","End":"00:15.355","Text":"Let\u0027s take x plus 3^2."},{"Start":"00:15.355 ","End":"00:17.312","Text":"This is an algebraic expression,"},{"Start":"00:17.312 ","End":"00:19.900","Text":"it\u0027s got numbers, it\u0027s got letters."},{"Start":"00:19.900 ","End":"00:25.920","Text":"What it is, is we look at this formula and we decide usually just mentally,"},{"Start":"00:25.920 ","End":"00:27.600","Text":"what is a and what is b."},{"Start":"00:27.600 ","End":"00:30.150","Text":"Usually, it\u0027s on either side of the plus sign so x"},{"Start":"00:30.150 ","End":"00:33.720","Text":"is going to be instead of a and 3 instead of b."},{"Start":"00:33.720 ","End":"00:39.300","Text":"So we get a^2 will be x^2 plus 2,"},{"Start":"00:39.300 ","End":"00:45.045","Text":"a we said was x and b is 3."},{"Start":"00:45.045 ","End":"00:47.610","Text":"2x times 3 plus and then b^2, 3^2."},{"Start":"00:47.610 ","End":"00:54.290","Text":"Then we just do a bit of simplification."},{"Start":"00:54.290 ","End":"01:02.200","Text":"This is x^2 plus 2 times 3 is 6, 6x plus 9."},{"Start":"01:02.200 ","End":"01:04.145","Text":"That\u0027s the first one done."},{"Start":"01:04.145 ","End":"01:05.795","Text":"Now another one,"},{"Start":"01:05.795 ","End":"01:13.650","Text":"let\u0027s take a plus 2^2."},{"Start":"01:13.880 ","End":"01:17.925","Text":"Well, the a is right, it just fits the a."},{"Start":"01:17.925 ","End":"01:20.565","Text":"So a plays itself here,"},{"Start":"01:20.565 ","End":"01:23.175","Text":"and b is 2."},{"Start":"01:23.175 ","End":"01:28.245","Text":"I look here and say a^2 is a squared,"},{"Start":"01:28.245 ","End":"01:38.520","Text":"plus 2a and b is 2 and then b^2 is 2^2."},{"Start":"01:38.520 ","End":"01:43.425","Text":"Then finally, just a little simplification,"},{"Start":"01:43.425 ","End":"01:46.515","Text":"a^2 plus 2 times a times 2 is,"},{"Start":"01:46.515 ","End":"01:48.840","Text":"I can do the 2 times 2 first is 4,"},{"Start":"01:48.840 ","End":"01:54.075","Text":"times a, 2^2 is 2 times 2 is 4."},{"Start":"01:54.075 ","End":"02:04.185","Text":"Next one, let\u0027s take y plus 1 squared."},{"Start":"02:04.185 ","End":"02:06.175","Text":"This is a and this is b."},{"Start":"02:06.175 ","End":"02:08.810","Text":"If you want, you can always just write it for yourself."},{"Start":"02:08.810 ","End":"02:12.420","Text":"I\u0027m just saying you could if you really want"},{"Start":"02:12.420 ","End":"02:17.810","Text":"to make sure you\u0027re doing it right so a is y, b is 1."},{"Start":"02:17.810 ","End":"02:24.200","Text":"We get a^2 so that means y^2,"},{"Start":"02:24.200 ","End":"02:27.630","Text":"2, a is y, b is 1."},{"Start":"02:27.630 ","End":"02:37.190","Text":"Then b^2, 1^2 and so this is y^2 plus 2y plus 1."},{"Start":"02:37.190 ","End":"02:40.400","Text":"Now yet another one,"},{"Start":"02:40.400 ","End":"02:50.220","Text":"let us take c plus 1/3 squared."},{"Start":"02:50.220 ","End":"02:54.070","Text":"It\u0027s always going to be something plus something squared."},{"Start":"02:54.290 ","End":"03:02.300","Text":"Now a is cast as c and b is cast as 1/3."},{"Start":"03:02.300 ","End":"03:05.235","Text":"So a^2 becomes c^2,"},{"Start":"03:05.235 ","End":"03:09.675","Text":"2ab is 2 times c times 1/3."},{"Start":"03:09.675 ","End":"03:15.995","Text":"B squared is 1/3 squared and then just a bit of simplification."},{"Start":"03:15.995 ","End":"03:18.755","Text":"C squared, 2 times c times 1/3,"},{"Start":"03:18.755 ","End":"03:22.508","Text":"2 with the 1/3 is 2/3."},{"Start":"03:22.508 ","End":"03:24.275","Text":"If you remember your fractions."},{"Start":"03:24.275 ","End":"03:28.790","Text":"1/3 squared, remember a fraction raised to the power of 2 is the top"},{"Start":"03:28.790 ","End":"03:33.150","Text":"squared over the bottom squared,1^2 over 3^2,"},{"Start":"03:33.150 ","End":"03:35.550","Text":"which I\u0027m going straightaway write as 1/9."},{"Start":"03:35.550 ","End":"03:39.130","Text":"Because that\u0027s 1^2 and 3^2 is 9."},{"Start":"03:40.250 ","End":"03:43.990","Text":"How about a few more?"},{"Start":"03:44.200 ","End":"03:47.120","Text":"I erase those. We\u0027re going to continue with,"},{"Start":"03:47.120 ","End":"03:49.100","Text":"say 3 more examples."},{"Start":"03:49.100 ","End":"03:57.440","Text":"How about if I take 2n plus 3 squared?"},{"Start":"03:57.440 ","End":"04:00.600","Text":"This time, a is all of 2n,"},{"Start":"04:00.600 ","End":"04:05.060","Text":"and there\u0027s b so this equals"},{"Start":"04:05.060 ","End":"04:11.300","Text":"a^2 is 2n but it\u0027s important that you put it in brackets squared."},{"Start":"04:11.300 ","End":"04:13.490","Text":"Otherwise, it\u0027s not the whole thing squared."},{"Start":"04:13.490 ","End":"04:15.890","Text":"If you just put 2n squared without brackets,"},{"Start":"04:15.890 ","End":"04:18.360","Text":"only the n is squared."},{"Start":"04:18.790 ","End":"04:22.900","Text":"Then twice 2n."},{"Start":"04:22.900 ","End":"04:24.960","Text":"You don\u0027t need brackets."},{"Start":"04:24.960 ","End":"04:29.625","Text":"I could put brackets but you don\u0027t need to, times 3."},{"Start":"04:29.625 ","End":"04:34.130","Text":"Then b^2 is 3^2."},{"Start":"04:34.130 ","End":"04:35.690","Text":"So this is equal to."},{"Start":"04:35.690 ","End":"04:38.840","Text":"Now remember when you have a product squared,"},{"Start":"04:38.840 ","End":"04:41.825","Text":"you do take each one separately squared."},{"Start":"04:41.825 ","End":"04:44.810","Text":"The whole point of this exercise is to say that yes,"},{"Start":"04:44.810 ","End":"04:48.770","Text":"it works for multiplication but it doesn\u0027t work for addition."},{"Start":"04:48.770 ","End":"04:54.440","Text":"With multiplication, I say 2^2 separately, that\u0027s 4."},{"Start":"04:54.440 ","End":"05:00.500","Text":"Then n^2 separately is n^2."},{"Start":"05:00.500 ","End":"05:06.163","Text":"Then 2 times 2 times 3 is 4 times 3 is 12n,"},{"Start":"05:06.163 ","End":"05:08.990","Text":"and 3^2 is 9."},{"Start":"05:08.990 ","End":"05:13.160","Text":"There are no like terms to collect because n^2 and n are different."},{"Start":"05:13.160 ","End":"05:14.764","Text":"It\u0027s apples and oranges."},{"Start":"05:14.764 ","End":"05:17.120","Text":"But this is the answer."},{"Start":"05:17.120 ","End":"05:20.550","Text":"Let\u0027s go for another one."},{"Start":"05:21.940 ","End":"05:25.023","Text":"Let\u0027s take some other letters we haven\u0027t used,"},{"Start":"05:25.023 ","End":"05:33.095","Text":"let\u0027s say u and v. 5u plus 4v, all this squared."},{"Start":"05:33.095 ","End":"05:35.240","Text":"This is equal to."},{"Start":"05:35.240 ","End":"05:37.655","Text":"Just reminding this is a and this is b."},{"Start":"05:37.655 ","End":"05:40.160","Text":"A squared is 5u,"},{"Start":"05:40.160 ","End":"05:45.020","Text":"but squared I need to put it in brackets to make sure the whole thing is squared."},{"Start":"05:45.020 ","End":"05:51.080","Text":"Plus 2, and here optionally brackets but I think it\u0027s clearer."},{"Start":"05:51.080 ","End":"05:55.040","Text":"Just for clarity, I\u0027ll put brackets, 5u, 4v."},{"Start":"05:55.040 ","End":"05:58.250","Text":"This is the a, b Part 2."},{"Start":"05:58.250 ","End":"06:01.430","Text":"This is the a and this is the b in brackets."},{"Start":"06:01.430 ","End":"06:03.710","Text":"Then the b squared again,"},{"Start":"06:03.710 ","End":"06:05.450","Text":"here I do need brackets."},{"Start":"06:05.450 ","End":"06:07.280","Text":"Here it\u0027s important."},{"Start":"06:07.280 ","End":"06:09.410","Text":"Opening the brackets."},{"Start":"06:09.410 ","End":"06:13.220","Text":"(5u)^2 is 5^2 times u^2 squared."},{"Start":"06:13.220 ","End":"06:15.230","Text":"5^2 is 25, of course."},{"Start":"06:15.230 ","End":"06:19.790","Text":"Then 2 times 5 times u times 4 times v,"},{"Start":"06:19.790 ","End":"06:21.530","Text":"they\u0027re all multiplied. I can do it in any order."},{"Start":"06:21.530 ","End":"06:24.169","Text":"I\u0027ll do the 2 times 5 times 4 first,"},{"Start":"06:24.169 ","End":"06:27.155","Text":"and that comes out to be 40."},{"Start":"06:27.155 ","End":"06:33.425","Text":"Then I have a u, and then I have a v. Multiplication can go in any order."},{"Start":"06:33.425 ","End":"06:36.570","Text":"Then the last one is 4v,"},{"Start":"06:36.570 ","End":"06:42.135","Text":"all squared is 4^2 is 16 and v^2 is just v^2."},{"Start":"06:42.135 ","End":"06:44.315","Text":"That\u0027s the answer to that."},{"Start":"06:44.315 ","End":"06:50.180","Text":"Now let\u0027s take something a bit more difficult."},{"Start":"06:50.180 ","End":"06:52.490","Text":"Let\u0027s suppose, I have"},{"Start":"06:52.490 ","End":"06:58.490","Text":"xy^2 plus"},{"Start":"06:58.490 ","End":"07:04.010","Text":"9 all this squared."},{"Start":"07:04.010 ","End":"07:06.615","Text":"Now xy^2 is my a,"},{"Start":"07:06.615 ","End":"07:09.360","Text":"9 is my b."},{"Start":"07:09.360 ","End":"07:11.114","Text":"So a squared,"},{"Start":"07:11.114 ","End":"07:14.080","Text":"you can always put brackets, it never hurts."},{"Start":"07:14.080 ","End":"07:17.935","Text":"If you\u0027re not sure, put brackets."},{"Start":"07:17.935 ","End":"07:22.150","Text":"The first one, a^2 plus 2,"},{"Start":"07:22.150 ","End":"07:26.760","Text":"xy^2 times 9, that\u0027s the 2ab."},{"Start":"07:26.760 ","End":"07:28.590","Text":"Then the last one,"},{"Start":"07:28.590 ","End":"07:34.790","Text":"the b^2 is this and the rest of it is just simplification."},{"Start":"07:35.040 ","End":"07:38.440","Text":"This time it isn\u0027t immediately clear in one step,"},{"Start":"07:38.440 ","End":"07:39.640","Text":"I\u0027ll do that the other side,"},{"Start":"07:39.640 ","End":"07:46.400","Text":"xy^2 all squared is equal to,"},{"Start":"07:46.600 ","End":"07:49.355","Text":"if I square something,"},{"Start":"07:49.355 ","End":"07:51.200","Text":"this is the product here."},{"Start":"07:51.200 ","End":"07:54.095","Text":"I square the first and square the second."},{"Start":"07:54.095 ","End":"08:00.470","Text":"So it\u0027s x^2 and then y^2 is also squared."},{"Start":"08:00.470 ","End":"08:04.798","Text":"This is equal to x^2 times,"},{"Start":"08:04.798 ","End":"08:07.010","Text":"now what is y^2 squared?"},{"Start":"08:07.010 ","End":"08:11.350","Text":"If you\u0027re not sure, you can always say y^2 is y times y."},{"Start":"08:11.350 ","End":"08:12.920","Text":"If I square it,"},{"Start":"08:12.920 ","End":"08:17.210","Text":"it means I take another y times y."},{"Start":"08:17.210 ","End":"08:20.360","Text":"This is just y, y, y,"},{"Start":"08:20.360 ","End":"08:24.390","Text":"y and so this is y^4."},{"Start":"08:24.820 ","End":"08:27.605","Text":"That\u0027s just this bit."},{"Start":"08:27.605 ","End":"08:32.495","Text":"Now, getting back here."},{"Start":"08:32.495 ","End":"08:35.300","Text":"The first one is x^2y^4,"},{"Start":"08:35.300 ","End":"08:39.137","Text":"just copied it from there,"},{"Start":"08:39.137 ","End":"08:44.550","Text":"plus 2 times 9 is 18."},{"Start":"08:44.550 ","End":"08:53.260","Text":"So it\u0027s 18 times xy^2 and lastly, 9^2 is 81."},{"Start":"08:53.530 ","End":"08:55.580","Text":"This is the answer."},{"Start":"08:55.580 ","End":"08:56.840","Text":"There\u0027s nothing to collect."},{"Start":"08:56.840 ","End":"08:58.790","Text":"It\u0027s already simplified as it can be,"},{"Start":"08:58.790 ","End":"09:01.620","Text":"and we are done."}],"ID":4906},{"Watched":false,"Name":"Exercise 1","Duration":"4m 9s","ChapterTopicVideoID":4905,"CourseChapterTopicPlaylistID":3858,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.120","Text":"In this exercise, we have to expand"},{"Start":"00:03.120 ","End":"00:06.195","Text":"the following square binomials"},{"Start":"00:06.195 ","End":"00:10.845","Text":"and I\u0027m going to copy them over here and give you a formula."},{"Start":"00:10.845 ","End":"00:13.845","Text":"This is the formula we\u0027ll be using."},{"Start":"00:13.845 ","End":"00:17.670","Text":"Let\u0027s begin."},{"Start":"00:17.670 ","End":"00:20.625","Text":"In the first exercise,"},{"Start":"00:20.625 ","End":"00:23.190","Text":"we look at this and we look at this."},{"Start":"00:23.190 ","End":"00:25.920","Text":"We see what is a and what is b?"},{"Start":"00:25.920 ","End":"00:29.265","Text":"What\u0027s before the plus is the a,"},{"Start":"00:29.265 ","End":"00:34.545","Text":"and after the plus is what we would call b to use this formula,"},{"Start":"00:34.545 ","End":"00:37.290","Text":"and then we have exactly a+b^2."},{"Start":"00:37.290 ","End":"00:38.520","Text":"We have the left-hand side,"},{"Start":"00:38.520 ","End":"00:41.355","Text":"and now we just simply have to write it as the right-hand side."},{"Start":"00:41.355 ","End":"00:43.110","Text":"We look a^2,"},{"Start":"00:43.110 ","End":"00:46.350","Text":"but a is x, so that\u0027s x^2,"},{"Start":"00:46.350 ","End":"00:49.230","Text":"then plus 2,"},{"Start":"00:49.230 ","End":"00:55.545","Text":"a is x, b is 2."},{"Start":"00:55.545 ","End":"00:58.785","Text":"I\u0027m writing dots here to show the product,"},{"Start":"00:58.785 ","End":"01:04.035","Text":"and then +b^2, b^2 is 2^2."},{"Start":"01:04.035 ","End":"01:08.310","Text":"I\u0027m now just simplifying a little bit, x^2 stays x^2,"},{"Start":"01:08.310 ","End":"01:11.250","Text":"2 times x times 2,"},{"Start":"01:11.250 ","End":"01:13.185","Text":"2 times 2 is 4 times x."},{"Start":"01:13.185 ","End":"01:17.895","Text":"This is 4x and 2^2 is 4."},{"Start":"01:17.895 ","End":"01:20.650","Text":"This is the answer."},{"Start":"01:24.230 ","End":"01:27.270","Text":"Now, the next one."},{"Start":"01:27.270 ","End":"01:29.955","Text":"Here, well a is a,"},{"Start":"01:29.955 ","End":"01:35.400","Text":"but it just happens to be the same letter."},{"Start":"01:35.400 ","End":"01:37.275","Text":"Then 3 would be b,"},{"Start":"01:37.275 ","End":"01:40.275","Text":"this is my a, this is my b."},{"Start":"01:40.275 ","End":"01:44.115","Text":"We once again use the formula,"},{"Start":"01:44.115 ","End":"01:50.610","Text":"a^2 is a^2 plus 2. a times b,"},{"Start":"01:50.610 ","End":"01:53.509","Text":"which is 3 plus 3^2,"},{"Start":"01:53.509 ","End":"01:57.045","Text":"that\u0027s b^2 and 3^2."},{"Start":"01:57.045 ","End":"01:59.340","Text":"Simplifying, it\u0027s a^2,"},{"Start":"01:59.340 ","End":"02:02.520","Text":"2 times 3 is 6 so it\u0027s 6a,"},{"Start":"02:02.520 ","End":"02:07.740","Text":"3^2 is 9,"},{"Start":"02:07.740 ","End":"02:10.060","Text":"and so that\u0027s the next one."},{"Start":"02:10.390 ","End":"02:18.380","Text":"Now, the third one is a little bit confusing because actually in the role of a here,"},{"Start":"02:18.380 ","End":"02:22.880","Text":"we have our b and in the role of b, we have 1."},{"Start":"02:22.880 ","End":"02:26.460","Text":"That shouldn\u0027t be confusing because in this formula,"},{"Start":"02:26.460 ","End":"02:27.700","Text":"a just means the first one,"},{"Start":"02:27.700 ","End":"02:33.380","Text":"so a^2 really means b^2 and 2ab,"},{"Start":"02:33.380 ","End":"02:36.925","Text":"really means 2 times b times 1,"},{"Start":"02:36.925 ","End":"02:42.930","Text":"and b^2 here is this b is 1^2."},{"Start":"02:42.930 ","End":"02:45.615","Text":"This becomes b^2,"},{"Start":"02:45.615 ","End":"02:52.275","Text":"2 times b times 1 is just 2b and 1^2 is 1,"},{"Start":"02:52.275 ","End":"02:55.460","Text":"and that\u0027s the answer."},{"Start":"02:55.460 ","End":"02:58.900","Text":"The last one contains a fraction,"},{"Start":"02:58.900 ","End":"03:01.984","Text":"but not to worry, the same principle applies."},{"Start":"03:01.984 ","End":"03:04.995","Text":"The c will be in the role of a,"},{"Start":"03:04.995 ","End":"03:07.785","Text":"1/4 is cast in the role of b."},{"Start":"03:07.785 ","End":"03:12.553","Text":"Again, a^2 is c^2, 2 ab,"},{"Start":"03:12.553 ","End":"03:14.970","Text":"2 this times this,"},{"Start":"03:14.970 ","End":"03:21.540","Text":"times 1/4, then 1/4^2."},{"Start":"03:21.540 ","End":"03:23.215","Text":"Now, when you square a fraction,"},{"Start":"03:23.215 ","End":"03:28.450","Text":"you square the numerator and you square the denominator."},{"Start":"03:28.450 ","End":"03:31.600","Text":"Let\u0027s do a bit of simplification."},{"Start":"03:31.600 ","End":"03:35.595","Text":"Here, we can cancel 2 with a 1/4."},{"Start":"03:35.595 ","End":"03:39.230","Text":"It\u0027s 2/1, but we don\u0027t need that crutch anymore."},{"Start":"03:39.230 ","End":"03:43.534","Text":"Just 2 into 4 goes twice."},{"Start":"03:43.534 ","End":"03:51.990","Text":"What we get is c^2 plus c times 1/2."},{"Start":"03:51.990 ","End":"03:57.270","Text":"I\u0027ll write it as 1/2c and 1^2 is 1,"},{"Start":"03:57.270 ","End":"04:04.670","Text":"4^2 is 16, so it\u0027s 1/16."},{"Start":"04:04.670 ","End":"04:06.110","Text":"That\u0027s the answer to the last one,"},{"Start":"04:06.110 ","End":"04:08.670","Text":"and we are done."}],"ID":4910},{"Watched":false,"Name":"Exercise 2","Duration":"4m 13s","ChapterTopicVideoID":4906,"CourseChapterTopicPlaylistID":3858,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.395","Text":"In this exercise, we have to expand the following square binomials."},{"Start":"00:04.395 ","End":"00:07.200","Text":"I need to give you a formula."},{"Start":"00:07.200 ","End":"00:11.710","Text":"Here it is, just scroll down a bit."},{"Start":"00:16.100 ","End":"00:20.135","Text":"In this case, we\u0027ve used this before so I\u0027m just proceeding more quickly."},{"Start":"00:20.135 ","End":"00:23.105","Text":"In this case, 2m is the role of a,"},{"Start":"00:23.105 ","End":"00:25.735","Text":"and 5 is in the role of b,"},{"Start":"00:25.735 ","End":"00:32.580","Text":"and so what we get if we apply a squared is 2m squared,"},{"Start":"00:32.580 ","End":"00:35.570","Text":"I put it in brackets because the whole thing is squared,"},{"Start":"00:35.570 ","End":"00:46.470","Text":"plus 2 a is 2m and b is 5."},{"Start":"00:47.200 ","End":"00:55.955","Text":"Then we have plus the b^2, which is 5^2."},{"Start":"00:55.955 ","End":"00:59.210","Text":"All that remains is to simplify a bit,"},{"Start":"00:59.210 ","End":"01:04.445","Text":"2m squared is 2^2 times m^2, which is 4m^2."},{"Start":"01:04.445 ","End":"01:06.515","Text":"You square each one separately,"},{"Start":"01:06.515 ","End":"01:09.485","Text":"2^2 is 4, m^2 is m^2."},{"Start":"01:09.485 ","End":"01:14.375","Text":"Then, here 2 times 2 times 5 is 4 times 5 is 20m,"},{"Start":"01:14.375 ","End":"01:19.460","Text":"and 5^2 is 25."},{"Start":"01:19.460 ","End":"01:21.440","Text":"That\u0027s the first one."},{"Start":"01:21.440 ","End":"01:25.475","Text":"Then the second one, 5y is a,"},{"Start":"01:25.475 ","End":"01:28.395","Text":"4t is b,"},{"Start":"01:28.395 ","End":"01:31.530","Text":"and so what we get is (5y)^2,"},{"Start":"01:31.530 ","End":"01:34.020","Text":"but I need brackets,"},{"Start":"01:34.020 ","End":"01:39.720","Text":"plus 2 times a times b 5y"},{"Start":"01:39.720 ","End":"01:45.870","Text":"times 4t plus the last one squared."},{"Start":"01:45.870 ","End":"01:48.680","Text":"Again, I\u0027ll need brackets because if I don\u0027t put the brackets,"},{"Start":"01:48.680 ","End":"01:49.970","Text":"it\u0027ll just be the t squared,"},{"Start":"01:49.970 ","End":"01:51.755","Text":"I want the whole thing squared."},{"Start":"01:51.755 ","End":"02:00.050","Text":"What we get if we multiply out here is 5y^2 is 5^2, y^2 is 25y^2."},{"Start":"02:00.050 ","End":"02:02.450","Text":"Here, 2 times 5 times 4,"},{"Start":"02:02.450 ","End":"02:07.210","Text":"all the numbers, is 10 times 4 is 40."},{"Start":"02:07.210 ","End":"02:09.290","Text":"40 as opposed to 4t."},{"Start":"02:09.290 ","End":"02:14.030","Text":"40yt"},{"Start":"02:15.200 ","End":"02:20.510","Text":"plus 4t,"},{"Start":"02:20.510 ","End":"02:24.095","Text":"as opposed to 40, all squared."},{"Start":"02:24.095 ","End":"02:26.530","Text":"I meant to do that in 1 step."},{"Start":"02:26.530 ","End":"02:30.700","Text":"I\u0027m going to write instead of this 16t^2."},{"Start":"02:30.700 ","End":"02:35.520","Text":"Let me erase that and we write straightaway,"},{"Start":"02:35.520 ","End":"02:38.940","Text":"4^2 is 16t^2 squared."},{"Start":"02:38.940 ","End":"02:42.855","Text":"Finally, just to highlight it, that\u0027s another one."},{"Start":"02:42.855 ","End":"02:45.435","Text":"Now we have a third one,"},{"Start":"02:45.435 ","End":"02:53.115","Text":"in this case, all of the x^2 y is a and 11 is b."},{"Start":"02:53.115 ","End":"02:53.580","Text":"We"},{"Start":"02:53.580 ","End":"02:55.000","Text":"get"},{"Start":"02:58.250 ","End":"03:04.750","Text":"x^2y all"},{"Start":"03:04.750 ","End":"03:10.700","Text":"squared plus 2x^2y times 11 plus 11^2."},{"Start":"03:10.700 ","End":"03:12.905","Text":"The question is, what is this thing here?"},{"Start":"03:12.905 ","End":"03:14.795","Text":"Well, x^2y all squared,"},{"Start":"03:14.795 ","End":"03:16.100","Text":"I\u0027ll just do this at the side,"},{"Start":"03:16.100 ","End":"03:19.885","Text":"is x^2y times x^2y."},{"Start":"03:19.885 ","End":"03:22.955","Text":"Square means take 2 of them, this times this."},{"Start":"03:22.955 ","End":"03:26.390","Text":"Now I can write the multiplication in any order I want."},{"Start":"03:26.390 ","End":"03:36.250","Text":"It\u0027s x^2 times x^2 times y times y. x^2 times x^2,"},{"Start":"03:36.250 ","End":"03:38.075","Text":"I can again expand this."},{"Start":"03:38.075 ","End":"03:40.160","Text":"This is x times x."},{"Start":"03:40.160 ","End":"03:43.460","Text":"The other x^2 is also x times x."},{"Start":"03:43.460 ","End":"03:47.680","Text":"What I get is x^4y^2."},{"Start":"03:47.680 ","End":"03:54.585","Text":"That\u0027s what I put in the first term here, x^4y^2 plus,"},{"Start":"03:54.585 ","End":"03:57.450","Text":"2 times 11 is 22,"},{"Start":"03:57.450 ","End":"04:02.640","Text":"times x^2 y, and 11^2."},{"Start":"04:02.640 ","End":"04:06.030","Text":"11 times 11 is 121."},{"Start":"04:06.030 ","End":"04:13.480","Text":"That is the third of the exercises and the last and so we are done."}],"ID":4911},{"Watched":false,"Name":"Exercise 3","Duration":"1m 48s","ChapterTopicVideoID":4907,"CourseChapterTopicPlaylistID":3858,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.230 ","End":"00:05.010","Text":"In this exercise, we have to expand each of these as a difference"},{"Start":"00:05.010 ","End":"00:09.870","Text":"of squares and now bring the formula in a second."},{"Start":"00:09.870 ","End":"00:13.230","Text":"Here it is, I\u0027ll just scroll down a bit."},{"Start":"00:13.230 ","End":"00:17.175","Text":"I made a copy of the exercises as well."},{"Start":"00:17.175 ","End":"00:18.855","Text":"This is the formula,"},{"Start":"00:18.855 ","End":"00:21.255","Text":"although it\u0027s sometimes written the other way around."},{"Start":"00:21.255 ","End":"00:25.290","Text":"You know what, I look at here and most of them are a minus before the plus."},{"Start":"00:25.290 ","End":"00:27.974","Text":"Well, write it the other way around."},{"Start":"00:27.974 ","End":"00:30.165","Text":"I switched the order around,"},{"Start":"00:30.165 ","End":"00:31.890","Text":"here\u0027s minus, here is plus."},{"Start":"00:31.890 ","End":"00:34.194","Text":"The first one,"},{"Start":"00:34.194 ","End":"00:38.040","Text":"x is in the role of a and 7 is in the role of b."},{"Start":"00:38.040 ","End":"00:40.775","Text":"We do have a minus ba plus b."},{"Start":"00:40.775 ","End":"00:44.090","Text":"What this equals is a squared,"},{"Start":"00:44.090 ","End":"00:49.040","Text":"in our case, x^2 minus b^2, which is 7^2."},{"Start":"00:49.040 ","End":"00:54.140","Text":"This is just x^2 minus 49."},{"Start":"00:54.140 ","End":"00:57.845","Text":"This is a very useful formula."},{"Start":"00:57.845 ","End":"01:00.800","Text":"Similarly here, if you see something"},{"Start":"01:00.800 ","End":"01:04.055","Text":"minus something and then something plus the same thing."},{"Start":"01:04.055 ","End":"01:08.705","Text":"This is just 9^2 minus x^2,"},{"Start":"01:08.705 ","End":"01:13.470","Text":"which is 81 minus x^2."},{"Start":"01:13.580 ","End":"01:18.260","Text":"The last one, 3x minus 4 3x plus 4."},{"Start":"01:18.260 ","End":"01:19.925","Text":"We have a difference in a sum."},{"Start":"01:19.925 ","End":"01:25.730","Text":"We write it as the first one squared minus the second one squared."},{"Start":"01:25.730 ","End":"01:28.175","Text":"Notice the parenthesis,"},{"Start":"01:28.175 ","End":"01:30.605","Text":"brackets from the 3x because the whole thing,"},{"Start":"01:30.605 ","End":"01:32.294","Text":"all of this is a,"},{"Start":"01:32.294 ","End":"01:35.090","Text":"so a^2 is all of this."},{"Start":"01:35.090 ","End":"01:38.750","Text":"This is equal to 3 times x times 3 times x,"},{"Start":"01:38.750 ","End":"01:44.025","Text":"9x^2, 4^2 is 16."},{"Start":"01:44.025 ","End":"01:48.160","Text":"That\u0027s this one. Then we\u0027ve done them all we\u0027re done."}],"ID":4912},{"Watched":false,"Name":"Exercise 4","Duration":"2m 17s","ChapterTopicVideoID":4908,"CourseChapterTopicPlaylistID":3858,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.550","Text":"In this exercise, we have to simplify it by taking out a common factor,"},{"Start":"00:05.550 ","End":"00:09.210","Text":"which is the opposite process of expanding brackets."},{"Start":"00:09.210 ","End":"00:12.640","Text":"I prefer to work on a copy here."},{"Start":"00:13.280 ","End":"00:18.835","Text":"Let\u0027s see, and you will get to the first one, 2x minus 4."},{"Start":"00:18.835 ","End":"00:20.310","Text":"What do they have in common?"},{"Start":"00:20.310 ","End":"00:21.870","Text":"What can I divide both by?"},{"Start":"00:21.870 ","End":"00:23.865","Text":"Well, clearly 2."},{"Start":"00:23.865 ","End":"00:28.905","Text":"If I take 2 outside the brackets, it\u0027s 2."},{"Start":"00:28.905 ","End":"00:31.880","Text":"Now if I take the 2(2x), it\u0027s x."},{"Start":"00:31.880 ","End":"00:36.275","Text":"If I take the 2 out of minus 4 it\u0027s minus 2."},{"Start":"00:36.275 ","End":"00:39.500","Text":"You can always check by expanding the brackets."},{"Start":"00:39.500 ","End":"00:41.720","Text":"If I multiply this by this and this,"},{"Start":"00:41.720 ","End":"00:45.575","Text":"I get 2x minus 2 times 2, which is 4."},{"Start":"00:45.575 ","End":"00:49.700","Text":"In fact, I recommend checking this taking outside"},{"Start":"00:49.700 ","End":"00:54.275","Text":"the brackets by expanding afterwards and seeing if we get back to the original."},{"Start":"00:54.275 ","End":"00:56.825","Text":"That\u0027s the first one."},{"Start":"00:56.825 ","End":"00:58.720","Text":"Now onto the second one."},{"Start":"00:58.720 ","End":"01:03.250","Text":"In this one, I think you can see that the common factor is 3."},{"Start":"01:03.250 ","End":"01:07.215","Text":"If we take 3 outside the brackets,"},{"Start":"01:07.215 ","End":"01:10.230","Text":"then we are safe to say 3x."},{"Start":"01:10.230 ","End":"01:11.400","Text":"If we take 3 out,"},{"Start":"01:11.400 ","End":"01:17.300","Text":"we\u0027re left with x and minus 6 is minus 3 times 2,"},{"Start":"01:17.300 ","End":"01:20.090","Text":"so this is just minus 2."},{"Start":"01:20.090 ","End":"01:25.285","Text":"As I said, you can always check by expanding 3 times x is 3x,"},{"Start":"01:25.285 ","End":"01:26.710","Text":"3 times 2 is 6,"},{"Start":"01:26.710 ","End":"01:29.155","Text":"and there\u0027s a minus here. This is fine."},{"Start":"01:29.155 ","End":"01:31.395","Text":"That\u0027s the next one done."},{"Start":"01:31.395 ","End":"01:35.300","Text":"Then, here we have an 80 and a 4x."},{"Start":"01:35.300 ","End":"01:40.545","Text":"The only thing they have in common is 4, which is something."},{"Start":"01:40.545 ","End":"01:44.175","Text":"Taking 4 out, we take 4 out of 80,"},{"Start":"01:44.175 ","End":"01:46.665","Text":"so 80 over 4, it\u0027s 20."},{"Start":"01:46.665 ","End":"01:48.690","Text":"Take the 4 out of minus 4x,"},{"Start":"01:48.690 ","End":"01:50.895","Text":"we\u0027re left with the minus x."},{"Start":"01:50.895 ","End":"01:52.950","Text":"Again, 4 times 20 is 80,"},{"Start":"01:52.950 ","End":"01:54.795","Text":"4 times minus 6 minus 4x."},{"Start":"01:54.795 ","End":"01:58.340","Text":"We check by expansion and it works."},{"Start":"01:58.340 ","End":"02:01.970","Text":"The last one, common factor is 8."},{"Start":"02:01.970 ","End":"02:05.360","Text":"Take 8 out, we take it out of 64,"},{"Start":"02:05.360 ","End":"02:08.765","Text":"we\u0027re left with 8 and indeed 8 times 8 to 64."},{"Start":"02:08.765 ","End":"02:11.885","Text":"Here we have plus a."},{"Start":"02:11.885 ","End":"02:16.410","Text":"That\u0027s the last one done. That\u0027s it."}],"ID":4913},{"Watched":false,"Name":"Exercise 5","Duration":"5m 34s","ChapterTopicVideoID":4909,"CourseChapterTopicPlaylistID":3858,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.865","Text":"This is a continuation in the exercises on taking out a common factor."},{"Start":"00:05.865 ","End":"00:10.210","Text":"As usual, I\u0027ll make a copy over here."},{"Start":"00:10.550 ","End":"00:16.245","Text":"Let\u0027s start with the first one which is this."},{"Start":"00:16.245 ","End":"00:19.814","Text":"Now what can we take out as a common factor?"},{"Start":"00:19.814 ","End":"00:22.875","Text":"Here we have x^2 and here we have x."},{"Start":"00:22.875 ","End":"00:24.765","Text":"Now if you remember,"},{"Start":"00:24.765 ","End":"00:28.440","Text":"of course that x^2 equals x times x."},{"Start":"00:28.440 ","End":"00:32.895","Text":"So in a way, this contains x which could be removed."},{"Start":"00:32.895 ","End":"00:41.335","Text":"In fact, let me just write it as x times x plus 3 times x."},{"Start":"00:41.335 ","End":"00:47.090","Text":"Now you more clearly see that the x is the common factor and we can take"},{"Start":"00:47.090 ","End":"00:53.730","Text":"that x outside the brackets and write it as x,"},{"Start":"00:53.730 ","End":"00:55.805","Text":"now here is x times x,"},{"Start":"00:55.805 ","End":"00:58.280","Text":"and here it\u0027s x times 3."},{"Start":"00:58.280 ","End":"01:02.245","Text":"If I write this as x times x plus 3,"},{"Start":"01:02.245 ","End":"01:07.845","Text":"this is in the way, then that will do it."},{"Start":"01:07.845 ","End":"01:14.480","Text":"As a check, you multiply out and you see x times x is x^2 or x times x,"},{"Start":"01:14.480 ","End":"01:17.210","Text":"x times 3 is 3x,"},{"Start":"01:17.210 ","End":"01:19.880","Text":"it works so this is our answer."},{"Start":"01:19.880 ","End":"01:26.340","Text":"We took a variable outside the brackets. Next one."},{"Start":"01:26.340 ","End":"01:31.100","Text":"Here, we also have a power of x is time x^3."},{"Start":"01:31.100 ","End":"01:37.460","Text":"x^3 is x times x times x."},{"Start":"01:37.460 ","End":"01:39.875","Text":"If I take x out,"},{"Start":"01:39.875 ","End":"01:44.540","Text":"this is like x times x times x, which is x^2."},{"Start":"01:44.540 ","End":"01:49.380","Text":"In other words, x times x^2 gives me x^3."},{"Start":"01:49.660 ","End":"01:58.710","Text":"What I can do is write this as x^2 times x minus x."},{"Start":"01:58.710 ","End":"02:00.975","Text":"To help us with factoring,"},{"Start":"02:00.975 ","End":"02:03.350","Text":"taking a factor out,"},{"Start":"02:03.350 ","End":"02:06.190","Text":"we\u0027ll write it as 1 times x."},{"Start":"02:06.190 ","End":"02:10.280","Text":"Altogether, we see that we have x in common."},{"Start":"02:10.280 ","End":"02:15.785","Text":"We take the x out and what we\u0027re left with here is the x^2 and the minus 1."},{"Start":"02:15.785 ","End":"02:19.860","Text":"Just to check, we multiply out, expand,"},{"Start":"02:19.860 ","End":"02:22.440","Text":"x times x^2 is x times x times x,"},{"Start":"02:22.440 ","End":"02:25.455","Text":"which is x^3, and minus x."},{"Start":"02:25.455 ","End":"02:33.360","Text":"We\u0027re all okay. We just need to highlight the answer and move on to the next."},{"Start":"02:33.590 ","End":"02:37.639","Text":"Here, we have same idea,"},{"Start":"02:37.639 ","End":"02:39.260","Text":"but higher powers of x."},{"Start":"02:39.260 ","End":"02:42.305","Text":"Here I have x^5 and x^2."},{"Start":"02:42.305 ","End":"02:46.700","Text":"The practical rule is that the least power of the x\u0027s,"},{"Start":"02:46.700 ","End":"02:50.495","Text":"in this case, the x^2 is what you can take outside the brackets."},{"Start":"02:50.495 ","End":"02:51.935","Text":"To show you this,"},{"Start":"02:51.935 ","End":"02:53.870","Text":"let\u0027s look at x^5."},{"Start":"02:53.870 ","End":"02:58.835","Text":"If I write it as x times x times x times x times x,"},{"Start":"02:58.835 ","End":"03:09.050","Text":"what I can see is that I can take x^2 out and say that this is x^2 times x^3."},{"Start":"03:09.050 ","End":"03:16.550","Text":"I\u0027ll rewrite the x^5 as x^3 times x^2 or x^2 times x^3,"},{"Start":"03:16.550 ","End":"03:20.330","Text":"doesn\u0027t matter, minus 2x^2."},{"Start":"03:20.330 ","End":"03:25.035","Text":"This way I see that I have the x^2 in common here and here,"},{"Start":"03:25.035 ","End":"03:33.040","Text":"so I can take that x^2 outside the brackets and I\u0027m left with x^3 minus 2."},{"Start":"03:33.040 ","End":"03:36.060","Text":"Again, I always like to check,"},{"Start":"03:36.060 ","End":"03:40.670","Text":"x^2 times x^3 is x^5 because it\u0027s x times x times x times x times x,"},{"Start":"03:40.670 ","End":"03:47.305","Text":"as we see here and x^2 times minus 2 is minus 2x^2, that looks right."},{"Start":"03:47.305 ","End":"03:49.800","Text":"Onto the last one."},{"Start":"03:49.800 ","End":"03:53.810","Text":"The last one is a little bit more difficult because you have"},{"Start":"03:53.810 ","End":"03:57.620","Text":"to take something outside the x\u0027s and the numbers."},{"Start":"03:57.620 ","End":"03:59.830","Text":"We have two factoring."},{"Start":"03:59.830 ","End":"04:02.100","Text":"For x\u0027s, like I said,"},{"Start":"04:02.100 ","End":"04:07.665","Text":"the smallest power will do."},{"Start":"04:07.665 ","End":"04:11.525","Text":"We can take x^2 outside the brackets."},{"Start":"04:11.525 ","End":"04:16.295","Text":"Just to show you, the x-squared already is x^2."},{"Start":"04:16.295 ","End":"04:18.470","Text":"In fact, we saw this above,"},{"Start":"04:18.470 ","End":"04:23.090","Text":"is x times x^2."},{"Start":"04:23.250 ","End":"04:25.760","Text":"Now as far as the numbers go,"},{"Start":"04:25.760 ","End":"04:27.710","Text":"we can also take factors out."},{"Start":"04:27.710 ","End":"04:31.070","Text":"4 actually goes into 12 three times."},{"Start":"04:31.070 ","End":"04:37.080","Text":"The 12 is equal to 4 times 3."},{"Start":"04:37.080 ","End":"04:44.735","Text":"What I\u0027m suggesting is we take the x^2 out of the x\u0027s and the 4 for the numbers."},{"Start":"04:44.735 ","End":"04:47.120","Text":"Let\u0027s see what we\u0027re left with."},{"Start":"04:47.120 ","End":"04:51.770","Text":"We\u0027re going to multiply 4x^2 by something to get 4x^3."},{"Start":"04:51.770 ","End":"04:53.420","Text":"The number we\u0027re okay,"},{"Start":"04:53.420 ","End":"04:58.030","Text":"but x-squared still has to be multiplied by x to get x^3,"},{"Start":"04:58.030 ","End":"05:03.785","Text":"x^2 times x is x^3."},{"Start":"05:03.785 ","End":"05:11.670","Text":"Then 4 taken out of the 12 is 3 and the x^2 is already outside,"},{"Start":"05:11.670 ","End":"05:14.010","Text":"so this is what we get."},{"Start":"05:14.010 ","End":"05:17.617","Text":"Just to check. Again,"},{"Start":"05:17.617 ","End":"05:20.675","Text":"multiply out 4 x^2 times x is 4x^3."},{"Start":"05:20.675 ","End":"05:22.760","Text":"4x^2 times 3 is,"},{"Start":"05:22.760 ","End":"05:26.540","Text":"3 times 4 is 12x^2."},{"Start":"05:26.540 ","End":"05:33.810","Text":"This is what it is. Looks like we\u0027re done."}],"ID":4914},{"Watched":false,"Name":"Exercise 6","Duration":"7m 29s","ChapterTopicVideoID":4910,"CourseChapterTopicPlaylistID":3858,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.710","Text":"In this exercise, we have to factorize the following as square binomials."},{"Start":"00:04.710 ","End":"00:09.390","Text":"I\u0027ll bring you the formula. Here we are."},{"Start":"00:09.390 ","End":"00:13.950","Text":"I also copied the exercises,"},{"Start":"00:13.950 ","End":"00:15.930","Text":"so we have more room."},{"Start":"00:15.930 ","End":"00:19.350","Text":"This is essentially the reverse of what we\u0027ve been doing."},{"Start":"00:19.350 ","End":"00:23.685","Text":"We\u0027ve been expanding using this equals this,"},{"Start":"00:23.685 ","End":"00:28.830","Text":"written the equality the other way and now we\u0027re going to be factoring."},{"Start":"00:28.830 ","End":"00:32.340","Text":"Let\u0027s see, let\u0027s look at the first one and see what could"},{"Start":"00:32.340 ","End":"00:37.635","Text":"a be and what could b be. Just see what I mean."},{"Start":"00:37.635 ","End":"00:42.650","Text":"In this case, if we look at the left-hand side, first thing is an a^2,"},{"Start":"00:42.650 ","End":"00:47.835","Text":"a is going to be x."},{"Start":"00:47.835 ","End":"00:49.935","Text":"If we look at the left term,"},{"Start":"00:49.935 ","End":"00:57.935","Text":"9 is going to be b^2 so b is 3 in our case."},{"Start":"00:57.935 ","End":"01:03.300","Text":"What we have to check is that the middle term now really is 2ab."},{"Start":"01:04.060 ","End":"01:07.740","Text":"If we check what is 2ab,"},{"Start":"01:08.320 ","End":"01:13.060","Text":"it\u0027s 2 times x times 3,"},{"Start":"01:13.060 ","End":"01:17.730","Text":"which is 6x,"},{"Start":"01:17.730 ","End":"01:19.275","Text":"which is what we have."},{"Start":"01:19.275 ","End":"01:21.430","Text":"This is really 2ab,"},{"Start":"01:21.430 ","End":"01:31.265","Text":"this is really a^2 and this is really b^2 so the factorization is just a plus b squared,"},{"Start":"01:31.265 ","End":"01:36.300","Text":"x plus 3 squared."},{"Start":"01:37.970 ","End":"01:41.810","Text":"I should have stressed that of course the middle term is plus,"},{"Start":"01:41.810 ","End":"01:45.010","Text":"so we have to use the a plus b formula."},{"Start":"01:45.010 ","End":"01:47.625","Text":"That\u0027s what we\u0027re going to use in this one, in this one,"},{"Start":"01:47.625 ","End":"01:50.685","Text":"and in this one but in this one it\u0027s going to be a minus,"},{"Start":"01:50.685 ","End":"01:54.035","Text":"that\u0027s so important that I\u0027m even going to highlight that."},{"Start":"01:54.035 ","End":"01:59.825","Text":"That this is the thing that decides which of the 2 we\u0027re going to use."},{"Start":"01:59.825 ","End":"02:04.320","Text":"In the last case, we\u0027re going to use the second formula."},{"Start":"02:05.050 ","End":"02:09.485","Text":"Back to here, the second one."},{"Start":"02:09.485 ","End":"02:14.525","Text":"Once again, I have to look and I see it\u0027s this formula here."},{"Start":"02:14.525 ","End":"02:17.435","Text":"What am I going to replace a and b by?"},{"Start":"02:17.435 ","End":"02:22.169","Text":"I have to say what a is and what b is in this formula."},{"Start":"02:24.200 ","End":"02:29.855","Text":"This is funny because a^2 is got to be 9a^2."},{"Start":"02:29.855 ","End":"02:34.730","Text":"What I mean is if we replace a by 3a,"},{"Start":"02:35.200 ","End":"02:38.990","Text":"like a different a, then if we square this,"},{"Start":"02:38.990 ","End":"02:43.670","Text":"the a^2 here corresponds to 3a^2 which is 9a^2."},{"Start":"02:43.670 ","End":"02:46.445","Text":"The common letter a can be confusing."},{"Start":"02:46.445 ","End":"02:54.305","Text":"I think you have to do an example of this where the role of a is served by 3a and b,"},{"Start":"02:54.305 ","End":"02:56.930","Text":"we have to look b^2 is going to be 4,"},{"Start":"02:56.930 ","End":"03:00.625","Text":"so b is 2 here."},{"Start":"03:00.625 ","End":"03:04.740","Text":"Let\u0027s see. If it works,"},{"Start":"03:04.740 ","End":"03:06.540","Text":"it\u0027s going to be 3a plus 2,"},{"Start":"03:06.540 ","End":"03:10.980","Text":"which is a plus b, all squared."},{"Start":"03:10.980 ","End":"03:14.955","Text":"Let\u0027s check it out, 3a^2 is 9a^2."},{"Start":"03:14.955 ","End":"03:18.000","Text":"That\u0027s how we got to it. It is 3 times 3 times a times a."},{"Start":"03:18.000 ","End":"03:20.965","Text":"2^2 is 4, that\u0027s all right."},{"Start":"03:20.965 ","End":"03:22.460","Text":"What about the middle term?"},{"Start":"03:22.460 ","End":"03:25.742","Text":"The middle term we check what is called 2ab,"},{"Start":"03:25.742 ","End":"03:32.670","Text":"in our case is 2 times 3a times 2 that\u0027s the equivalent of ab."},{"Start":"03:32.670 ","End":"03:39.780","Text":"This is equal 2 times 3 times 2 is 12a so that\u0027s okay."},{"Start":"03:39.780 ","End":"03:44.055","Text":"This means that this is okay."},{"Start":"03:44.055 ","End":"03:46.840","Text":"Wrong color."},{"Start":"03:50.420 ","End":"03:52.695","Text":"Let\u0027s just highlight it."},{"Start":"03:52.695 ","End":"03:54.290","Text":"If you\u0027re in doubt,"},{"Start":"03:54.290 ","End":"03:57.815","Text":"you can always check by multiplying out"},{"Start":"03:57.815 ","End":"04:02.965","Text":"using the formula the other way that this plus this squared is this thing squared,"},{"Start":"04:02.965 ","End":"04:07.065","Text":"9a^2 twice this times this 12a,"},{"Start":"04:07.065 ","End":"04:09.880","Text":"and this thing squared is 4."},{"Start":"04:10.800 ","End":"04:13.840","Text":"The next one. Once again,"},{"Start":"04:13.840 ","End":"04:16.400","Text":"we\u0027re using the plus formula."},{"Start":"04:17.510 ","End":"04:22.270","Text":"This time, it doesn\u0027t quite work according to this."},{"Start":"04:22.270 ","End":"04:26.230","Text":"This isn\u0027t anything squared but easily notice"},{"Start":"04:26.230 ","End":"04:31.225","Text":"this and that\u0027s because they made the exercise a bit more difficult."},{"Start":"04:31.225 ","End":"04:36.490","Text":"We have to actually take out a factor first so this is a new thing."},{"Start":"04:36.490 ","End":"04:37.750","Text":"We went to two steps."},{"Start":"04:37.750 ","End":"04:39.520","Text":"First of all, take out the common factor."},{"Start":"04:39.520 ","End":"04:45.125","Text":"Let\u0027s see they\u0027re all divisible by 3 but we can\u0027t take anything more than 3 out."},{"Start":"04:45.125 ","End":"04:48.565","Text":"This will be 3."},{"Start":"04:48.565 ","End":"04:53.630","Text":"Now if I take 3 out of 12, it\u0027s 4x^2."},{"Start":"04:53.630 ","End":"04:56.795","Text":"I take 3 out of 60, it\u0027s 20x,"},{"Start":"04:56.795 ","End":"05:01.235","Text":"take 3 out of here, it\u0027s 25."},{"Start":"05:01.235 ","End":"05:08.345","Text":"Now, we do have a chance with this because if we ignore the 3 for a moment, the 4x^2,"},{"Start":"05:08.345 ","End":"05:11.940","Text":"if we take a to be 2x,"},{"Start":"05:13.120 ","End":"05:19.235","Text":"then you can see that a^2 is 2 times 2 times x times x is 4x^2."},{"Start":"05:19.235 ","End":"05:23.605","Text":"If we take b as 5,"},{"Start":"05:23.605 ","End":"05:26.015","Text":"then this thing is b^2."},{"Start":"05:26.015 ","End":"05:31.160","Text":"What we\u0027re going to guess is that this is 3 as it was and then it\u0027s"},{"Start":"05:31.160 ","End":"05:37.705","Text":"going to be the 2x plus 5 squared."},{"Start":"05:37.705 ","End":"05:41.435","Text":"We\u0027ve really checked the first and last term,"},{"Start":"05:41.435 ","End":"05:45.110","Text":"this thing squared and the last one squared,"},{"Start":"05:45.110 ","End":"05:47.315","Text":"what we have to check is the middle term."},{"Start":"05:47.315 ","End":"05:50.067","Text":"I really only have to check the 2ab gives us this."},{"Start":"05:50.067 ","End":"05:57.120","Text":"Let\u0027s see, 2ab is 2 times 2x times 5,"},{"Start":"05:57.120 ","End":"06:03.440","Text":"2 times 2 times 5 is 20 times x so this is being checked."},{"Start":"06:03.440 ","End":"06:05.420","Text":"This is indeed correct."},{"Start":"06:05.420 ","End":"06:07.580","Text":"As I said, you can always expand"},{"Start":"06:07.580 ","End":"06:11.045","Text":"and see that you get back to what you had in the first place."},{"Start":"06:11.045 ","End":"06:14.810","Text":"This was a bit different is that we couldn\u0027t use the formula right away."},{"Start":"06:14.810 ","End":"06:19.710","Text":"We had to take out a factor first."},{"Start":"06:20.830 ","End":"06:24.170","Text":"Now we come to the last one."},{"Start":"06:24.170 ","End":"06:27.560","Text":"The last one is just straight forward."},{"Start":"06:27.560 ","End":"06:31.715","Text":"We take a as x and then we have a^2 here."},{"Start":"06:31.715 ","End":"06:33.770","Text":"What squared is 64?"},{"Start":"06:33.770 ","End":"06:37.035","Text":"It\u0027s 8 so I take b as 8."},{"Start":"06:37.035 ","End":"06:40.730","Text":"Now here, I certainly have a^2 and here I have b^2."},{"Start":"06:40.730 ","End":"06:46.310","Text":"I would like this to be x minus 8^2."},{"Start":"06:46.310 ","End":"06:48.920","Text":"Notice the minus, and I\u0027ll"},{"Start":"06:48.920 ","End":"06:54.240","Text":"just highlight that to emphasize that this time we have a minus."},{"Start":"06:55.400 ","End":"07:03.015","Text":"What we need to check is that twice ab is 16x. Let\u0027s do that."},{"Start":"07:03.015 ","End":"07:09.015","Text":"2ab is 2 times x times"},{"Start":"07:09.015 ","End":"07:15.180","Text":"8 and that is 16x because 2 times 8 is 16 so that\u0027s okay."},{"Start":"07:15.180 ","End":"07:22.785","Text":"This is our answer and I will just highlight it."},{"Start":"07:22.785 ","End":"07:29.140","Text":"Got double highlighting, fine. We\u0027re done."}],"ID":4915},{"Watched":false,"Name":"Exercise 7","Duration":"4m 42s","ChapterTopicVideoID":4911,"CourseChapterTopicPlaylistID":3858,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.830","Text":"This exercise, we have to factorize and this is mix and match."},{"Start":"00:04.830 ","End":"00:09.060","Text":"Some of them will be difference of squares and some of them will be square binomials."},{"Start":"00:09.060 ","End":"00:11.940","Text":"Let me bring you all the formulas."},{"Start":"00:11.940 ","End":"00:16.605","Text":"Here we are, these are all three formulas."},{"Start":"00:16.605 ","End":"00:18.497","Text":"We\u0027ve used all three of them,"},{"Start":"00:18.497 ","End":"00:20.970","Text":"I\u0027m just collecting them in one place."},{"Start":"00:20.970 ","End":"00:26.100","Text":"The first two are the difference of squares."},{"Start":"00:26.100 ","End":"00:28.200","Text":"Sorry, the other way round."},{"Start":"00:28.200 ","End":"00:34.320","Text":"The last one is the difference of squares and this one here is the square binomial."},{"Start":"00:34.320 ","End":"00:40.050","Text":"We\u0027ll use one of the three formulas in each of these cases and scroll down here."},{"Start":"00:40.050 ","End":"00:43.250","Text":"In the first one, what does it look like?"},{"Start":"00:43.250 ","End":"00:44.990","Text":"Well first of all, there\u0027s three terms."},{"Start":"00:44.990 ","End":"00:48.395","Text":"It\u0027s what we call a trinomial so that\u0027s going to be one of the first two."},{"Start":"00:48.395 ","End":"00:50.870","Text":"Because it\u0027s a minus here,"},{"Start":"00:50.870 ","End":"00:54.410","Text":"we\u0027re going to go with this middle formula."},{"Start":"00:54.410 ","End":"00:56.600","Text":"If it\u0027s the middle formula,"},{"Start":"00:56.600 ","End":"01:01.590","Text":"we can see that this will work if we take a just"},{"Start":"01:01.590 ","End":"01:07.040","Text":"as is in the role of itself as a and we\u0027ll take b as 5."},{"Start":"01:07.040 ","End":"01:08.355","Text":"I\u0027ll just write that down,"},{"Start":"01:08.355 ","End":"01:12.260","Text":"a is just a and b is 5."},{"Start":"01:12.260 ","End":"01:17.503","Text":"Well, a is a,"},{"Start":"01:17.503 ","End":"01:19.655","Text":"and we wanted to need a minus in the middle,"},{"Start":"01:19.655 ","End":"01:21.755","Text":"and the b is 5."},{"Start":"01:21.755 ","End":"01:23.305","Text":"It will have to be this,"},{"Start":"01:23.305 ","End":"01:25.730","Text":"but we need to check that the middle term works."},{"Start":"01:25.730 ","End":"01:30.050","Text":"Because we\u0027ve checked that a^2 is this and 5^2 is 25,"},{"Start":"01:30.050 ","End":"01:32.780","Text":"we just need to check that 2ab works,"},{"Start":"01:32.780 ","End":"01:36.500","Text":"and 2ab is 2 times a times 5,"},{"Start":"01:36.500 ","End":"01:42.465","Text":"which is 10a so we\u0027re all checked."},{"Start":"01:42.465 ","End":"01:44.580","Text":"That means that this is correct."},{"Start":"01:44.580 ","End":"01:46.160","Text":"This is the factorization."},{"Start":"01:46.160 ","End":"01:50.315","Text":"Of course, you can multiply out and see if you get back to where you started."},{"Start":"01:50.315 ","End":"01:52.160","Text":"In the next one."},{"Start":"01:52.160 ","End":"01:57.560","Text":"Once again, there\u0027s three terms so it\u0027s going to be one of these."},{"Start":"01:57.560 ","End":"02:00.605","Text":"It has a minus so it\u0027s going to be this, again,"},{"Start":"02:00.605 ","End":"02:05.600","Text":"but we have to take out a common factor first,"},{"Start":"02:05.600 ","End":"02:07.835","Text":"which we sometimes have to do."},{"Start":"02:07.835 ","End":"02:11.030","Text":"You can tell because usually in these kind of exercises,"},{"Start":"02:11.030 ","End":"02:13.190","Text":"this is going to be a square number and it isn\u0027t."},{"Start":"02:13.190 ","End":"02:16.850","Text":"You notice they\u0027re all divisible by 2 so the first thing we do is"},{"Start":"02:16.850 ","End":"02:21.210","Text":"we take 2 outside the brackets."},{"Start":"02:21.210 ","End":"02:27.985","Text":"What we\u0027re left with is x^2 minus 36 is 18 times 2,"},{"Start":"02:27.985 ","End":"02:31.770","Text":"and 162 is 81 times 2,"},{"Start":"02:31.770 ","End":"02:33.435","Text":"so that fits in with 2."},{"Start":"02:33.435 ","End":"02:39.570","Text":"Now, this really looks like the middle one because we want to take a,"},{"Start":"02:39.570 ","End":"02:41.540","Text":"we want to take as x,"},{"Start":"02:41.540 ","End":"02:44.110","Text":"and b we want to take as, let\u0027s see,"},{"Start":"02:44.110 ","End":"02:45.730","Text":"b^2 is going to be 81,"},{"Start":"02:45.730 ","End":"02:47.020","Text":"so that would be 9,"},{"Start":"02:47.020 ","End":"02:49.485","Text":"because 81 is 9 times 9."},{"Start":"02:49.485 ","End":"02:55.850","Text":"Now hopefully, this will be a minus b all squared,"},{"Start":"02:55.850 ","End":"02:59.030","Text":"which is x minus 9 all squared."},{"Start":"02:59.030 ","End":"03:01.940","Text":"But we just have to check the middle term here,"},{"Start":"03:01.940 ","End":"03:06.883","Text":"that this is minus 18 or that 2ab is 18. Let us see that."},{"Start":"03:06.883 ","End":"03:10.870","Text":"It should be 18x, I mean. Let\u0027s see."},{"Start":"03:10.870 ","End":"03:13.545","Text":"It\u0027s 2 times,"},{"Start":"03:13.545 ","End":"03:15.300","Text":"a is x, and here,"},{"Start":"03:15.300 ","End":"03:20.370","Text":"b is 9, and 2 x 9 really is 18x,"},{"Start":"03:20.370 ","End":"03:22.695","Text":"because 2 times 9 is 18."},{"Start":"03:22.695 ","End":"03:25.170","Text":"So I know that this is okay."},{"Start":"03:25.170 ","End":"03:29.925","Text":"This is the answer to part b."},{"Start":"03:29.925 ","End":"03:35.300","Text":"In this one, has only two terms and it\u0027s a difference,"},{"Start":"03:35.300 ","End":"03:37.790","Text":"so I\u0027m betting on the last one."},{"Start":"03:37.790 ","End":"03:40.340","Text":"Let\u0027s try, a^2 already is a^2,"},{"Start":"03:40.340 ","End":"03:43.505","Text":"and if I take b is 3,"},{"Start":"03:43.505 ","End":"03:49.065","Text":"like jump says, a is itself and b is 3,"},{"Start":"03:49.065 ","End":"03:50.784","Text":"and 3^2 is 9."},{"Start":"03:50.784 ","End":"03:55.070","Text":"This is going to be just a plus b,"},{"Start":"03:55.070 ","End":"03:58.085","Text":"a minus b, or the other way round sometimes."},{"Start":"03:58.085 ","End":"03:59.705","Text":"We\u0027ll write it the way it is there,"},{"Start":"03:59.705 ","End":"04:03.020","Text":"a plus 3, a minus 3 of."},{"Start":"04:03.020 ","End":"04:05.400","Text":"Although some books write it as a minus 3,"},{"Start":"04:05.400 ","End":"04:07.635","Text":"a plus 3, it really doesn\u0027t matter."},{"Start":"04:07.635 ","End":"04:12.904","Text":"We\u0027re done with another one and onto the last."},{"Start":"04:12.904 ","End":"04:16.580","Text":"Once again, it looks like it\u0027s a difference of squares."},{"Start":"04:16.580 ","End":"04:21.275","Text":"As you can see, a is going to be x and b is going to be,"},{"Start":"04:21.275 ","End":"04:24.680","Text":"well, 16 is 4^2 so we want 4 here."},{"Start":"04:24.680 ","End":"04:28.160","Text":"Now, we want to write a plus b, a minus b."},{"Start":"04:28.160 ","End":"04:31.130","Text":"This is going to be x plus 4,"},{"Start":"04:31.130 ","End":"04:32.720","Text":"which is the a plus b,"},{"Start":"04:32.720 ","End":"04:36.545","Text":"and a minus b, which is x minus 4."},{"Start":"04:36.545 ","End":"04:42.540","Text":"That\u0027s this one and we\u0027re done."}],"ID":4916},{"Watched":false,"Name":"Exercise 8","Duration":"5m 47s","ChapterTopicVideoID":4912,"CourseChapterTopicPlaylistID":3858,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.170","Text":"This is a continuation of the previous exercise."},{"Start":"00:04.170 ","End":"00:08.430","Text":"We still have to do some simplification or"},{"Start":"00:08.430 ","End":"00:13.365","Text":"factorization using either the difference of squares or squared binomials."},{"Start":"00:13.365 ","End":"00:17.080","Text":"I\u0027ll bring the formulas for you."},{"Start":"00:17.140 ","End":"00:21.270","Text":"Here we are with the formulas."},{"Start":"00:22.360 ","End":"00:26.750","Text":"We have 5 to do here."},{"Start":"00:26.750 ","End":"00:30.410","Text":"Let me just say that besides these formulas and some of"},{"Start":"00:30.410 ","End":"00:33.305","Text":"them you might have to also factor,"},{"Start":"00:33.305 ","End":"00:34.955","Text":"so just bear that in mind,"},{"Start":"00:34.955 ","End":"00:36.680","Text":"take a factor outside the brackets."},{"Start":"00:36.680 ","End":"00:43.685","Text":"In fact, I can already see that\u0027s going to be the case in c and d. Let\u0027s just start."},{"Start":"00:43.685 ","End":"00:48.360","Text":"In this one, there\u0027s a difference of two things,"},{"Start":"00:48.360 ","End":"00:51.015","Text":"so it\u0027s got to be the last formula."},{"Start":"00:51.015 ","End":"00:54.990","Text":"I can see that if I want it to be a^2 minus b^2,"},{"Start":"00:54.990 ","End":"00:58.995","Text":"it will be if I take a as 9 because 9^2 is 81,"},{"Start":"00:58.995 ","End":"01:01.680","Text":"and I\u0027ll take b as x,"},{"Start":"01:01.680 ","End":"01:04.050","Text":"so b^2 will be x^2."},{"Start":"01:04.050 ","End":"01:05.669","Text":"Now, using this formula,"},{"Start":"01:05.669 ","End":"01:07.860","Text":"it\u0027s simply a plus b, a minus b,"},{"Start":"01:07.860 ","End":"01:11.400","Text":"which is 9 plus x,"},{"Start":"01:11.400 ","End":"01:17.260","Text":"9 minus x, that\u0027s this one."},{"Start":"01:18.080 ","End":"01:20.540","Text":"Let\u0027s go onto the next one."},{"Start":"01:20.540 ","End":"01:22.930","Text":"Once again, there\u0027s only two terms and it\u0027s a difference,"},{"Start":"01:22.930 ","End":"01:24.970","Text":"so it\u0027s got to be the last formula."},{"Start":"01:24.970 ","End":"01:27.665","Text":"The question is what squared gives me this?"},{"Start":"01:27.665 ","End":"01:31.940","Text":"Well, if you think about it, if I want something squared to be 100x^2 I look at"},{"Start":"01:31.940 ","End":"01:37.130","Text":"the number part separately and it\u0027s going to be 10^2 and here it\u0027s x^2."},{"Start":"01:37.130 ","End":"01:38.330","Text":"If I take a as 10x,"},{"Start":"01:38.330 ","End":"01:43.810","Text":"then a^2 will be 10x times 10x, which is 100x^2."},{"Start":"01:43.850 ","End":"01:47.220","Text":"For b, since 49 is 7^2,"},{"Start":"01:47.220 ","End":"01:49.390","Text":"I\u0027ll take b is 7."},{"Start":"01:49.390 ","End":"01:55.250","Text":"Now I can be guaranteed that this is a plus b, a minus b."},{"Start":"01:55.250 ","End":"02:01.690","Text":"In other words, 10x plus 7 times 10x minus 7."},{"Start":"02:01.690 ","End":"02:06.080","Text":"There\u0027s nothing to check like in the first two formulas, we had to check the 2ab."},{"Start":"02:06.080 ","End":"02:07.580","Text":"Here, there\u0027s nothing more to check."},{"Start":"02:07.580 ","End":"02:09.055","Text":"We know that this is right,"},{"Start":"02:09.055 ","End":"02:12.430","Text":"so it deserves a highlight."},{"Start":"02:15.160 ","End":"02:19.225","Text":"In fact, they\u0027re all using difference of squares."},{"Start":"02:19.225 ","End":"02:23.540","Text":"I guess, I mislabeled the question as using this or this,"},{"Start":"02:23.540 ","End":"02:25.925","Text":"they\u0027re all difference of squares."},{"Start":"02:25.925 ","End":"02:28.535","Text":"Clear that matter up."},{"Start":"02:28.535 ","End":"02:30.590","Text":"Let\u0027s go onto the next one."},{"Start":"02:30.590 ","End":"02:33.590","Text":"Next one, is also"},{"Start":"02:33.590 ","End":"02:38.534","Text":"a difference but there\u0027s a common factor that needs to be taken out first."},{"Start":"02:38.534 ","End":"02:45.500","Text":"We\u0027ve seen this before that x^3 is x^2 times x,"},{"Start":"02:45.500 ","End":"02:47.165","Text":"because this is x times x times x,"},{"Start":"02:47.165 ","End":"02:54.570","Text":"this is x times x times x. I\u0027m going to take x outside the brackets."},{"Start":"02:54.570 ","End":"02:57.620","Text":"Well, just to make it really clear,"},{"Start":"02:57.620 ","End":"03:01.754","Text":"this is 49x minus x^2,"},{"Start":"03:01.754 ","End":"03:05.265","Text":"x. I have x in common to both."},{"Start":"03:05.265 ","End":"03:09.135","Text":"I can take x outside the brackets and I get"},{"Start":"03:09.135 ","End":"03:15.990","Text":"x times 49 minus x^2."},{"Start":"03:15.990 ","End":"03:19.490","Text":"Now, the x stays and this bit is a difference of squares,"},{"Start":"03:19.490 ","End":"03:25.695","Text":"because I can take a as 7 and that\u0027s 7^2 and b,"},{"Start":"03:25.695 ","End":"03:28.480","Text":"I can take as x, and that\u0027s b^2."},{"Start":"03:28.480 ","End":"03:35.940","Text":"What I get from here is this is equal to x times the difference of squares a minus b,"},{"Start":"03:35.940 ","End":"03:38.200","Text":"a plus b,"},{"Start":"03:38.930 ","End":"03:42.485","Text":"or is it the other way around, it doesn\u0027t matter actually."},{"Start":"03:42.485 ","End":"03:44.960","Text":"We\u0027ll just stick to the order here,"},{"Start":"03:44.960 ","End":"03:51.540","Text":"7 plus x, 7 minus x and that\u0027s it."},{"Start":"03:52.800 ","End":"03:56.420","Text":"In the next one, we also have to take out a factor."},{"Start":"03:56.420 ","End":"04:01.665","Text":"Now since we\u0027ve already seen how to take out x^2, I\u0027ll rewrite this."},{"Start":"04:01.665 ","End":"04:12.150","Text":"This is x^2 times x and this is 1 times x. I meant just x outside the brackets."},{"Start":"04:12.150 ","End":"04:18.300","Text":"What I\u0027m left here is with x^2 and if I take x out of here,"},{"Start":"04:18.300 ","End":"04:22.350","Text":"I\u0027ve got just minus 1 because this is really minus 1x."},{"Start":"04:22.350 ","End":"04:27.805","Text":"Now, I can look at this and see that, just this bit,"},{"Start":"04:27.805 ","End":"04:31.690","Text":"if I take a as x and b as,"},{"Start":"04:31.690 ","End":"04:33.175","Text":"well, what am I going to take b as?"},{"Start":"04:33.175 ","End":"04:34.630","Text":"Something squared has to be 1,"},{"Start":"04:34.630 ","End":"04:36.220","Text":"so b has to be 1."},{"Start":"04:36.220 ","End":"04:39.830","Text":"Now I really do have a^2 minus b^2."},{"Start":"04:39.830 ","End":"04:41.840","Text":"Now, the x stays,"},{"Start":"04:41.840 ","End":"04:45.320","Text":"and here I use this difference of squares and I get a plus b,"},{"Start":"04:45.320 ","End":"04:47.210","Text":"which is x plus 1,"},{"Start":"04:47.210 ","End":"04:48.965","Text":"and a minus b,"},{"Start":"04:48.965 ","End":"04:51.395","Text":"which is x minus 1."},{"Start":"04:51.395 ","End":"04:53.230","Text":"That\u0027s this one."},{"Start":"04:53.230 ","End":"05:01.100","Text":"Last, m^2 minus 9."},{"Start":"05:01.100 ","End":"05:14.520","Text":"What we\u0027ll do is we\u0027ll take,"},{"Start":"05:14.520 ","End":"05:19.725","Text":"a as m and b as 3."},{"Start":"05:19.725 ","End":"05:25.515","Text":"Then a^2 is just m^2 and b^2 is 9,"},{"Start":"05:25.515 ","End":"05:27.790","Text":"so this fits this."},{"Start":"05:27.790 ","End":"05:34.620","Text":"We can write that this will equal, a plus b,"},{"Start":"05:34.620 ","End":"05:42.750","Text":"a minus b, so it\u0027s m plus 3 times m minus 3."},{"Start":"05:42.750 ","End":"05:45.120","Text":"That\u0027s the last one done,"},{"Start":"05:45.120 ","End":"05:47.860","Text":"so we\u0027re done altogether."}],"ID":4917}],"Thumbnail":null,"ID":3858},{"Name":"Rational Expressions","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Reduction","Duration":"15m 1s","ChapterTopicVideoID":5046,"CourseChapterTopicPlaylistID":3859,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.725","Text":"Here we are beginning a new concept, rational expressions."},{"Start":"00:04.725 ","End":"00:09.750","Text":"This usually comes right after polynomials and they\u0027re related."},{"Start":"00:09.750 ","End":"00:15.840","Text":"A rational expression is"},{"Start":"00:15.840 ","End":"00:22.860","Text":"just a polynomial over a polynomial."},{"Start":"00:22.860 ","End":"00:24.660","Text":"I believe the word"},{"Start":"00:24.660 ","End":"00:31.150","Text":"rational comes from an analogy that a rational number is an integer of an integer,"},{"Start":"00:31.370 ","End":"00:35.460","Text":"and this extends the concept from integer to polynomial."},{"Start":"00:35.460 ","End":"00:37.554","Text":"Anyway, it\u0027s just a name;"},{"Start":"00:37.554 ","End":"00:39.435","Text":"that\u0027s what it is."},{"Start":"00:39.435 ","End":"00:44.614","Text":"I\u0027ll start with giving a few examples. Let\u0027s see."},{"Start":"00:44.614 ","End":"00:49.570","Text":"We\u0027ll take examples in different variables, not just x."},{"Start":"00:49.570 ","End":"00:52.110","Text":"Here there\u0027s four examples."},{"Start":"00:52.110 ","End":"00:57.275","Text":"This one is an x and y and m and t. Of course,"},{"Start":"00:57.275 ","End":"01:01.370","Text":"we have to take the same variable in the top and in the bottom."},{"Start":"01:01.370 ","End":"01:03.780","Text":"That\u0027s just the way it is."},{"Start":"01:04.190 ","End":"01:07.910","Text":"Remember that a constant,"},{"Start":"01:07.910 ","End":"01:10.955","Text":"like five, is also considered a polynomial."},{"Start":"01:10.955 ","End":"01:13.740","Text":"This is a rational expression in x,"},{"Start":"01:13.740 ","End":"01:17.300","Text":"this one is the rational expression."},{"Start":"01:17.300 ","End":"01:20.985","Text":"The last one is a bit different."},{"Start":"01:20.985 ","End":"01:25.015","Text":"Notice that I have just a 1 in the denominator."},{"Start":"01:25.015 ","End":"01:27.580","Text":"Like we said, a constant is a polynomial."},{"Start":"01:27.580 ","End":"01:34.040","Text":"But when we have 1 in the denominator, it\u0027s unnecessary."},{"Start":"01:34.130 ","End":"01:37.060","Text":"I could erase the over 1 part."},{"Start":"01:37.060 ","End":"01:38.665","Text":"I just made it faint."},{"Start":"01:38.665 ","End":"01:41.290","Text":"What I\u0027m trying to say here is that a polynomial"},{"Start":"01:41.290 ","End":"01:44.095","Text":"is also considered as a rational expression."},{"Start":"01:44.095 ","End":"01:48.325","Text":"A rational expression is an extension of the concept of a polynomial."},{"Start":"01:48.325 ","End":"01:51.260","Text":"Polynomials are also rational."},{"Start":"01:51.720 ","End":"01:58.520","Text":"I\u0027ll just point out one example of something that is not a rational expression."},{"Start":"01:58.520 ","End":"02:00.840","Text":"That would, for example,"},{"Start":"02:00.840 ","End":"02:10.390","Text":"be 3x plus x^2 over 0."},{"Start":"02:10.390 ","End":"02:15.830","Text":"The constant 0 is not the same as the polynomial 0."},{"Start":"02:15.830 ","End":"02:18.275","Text":"Here we had the polynomial 5,"},{"Start":"02:18.275 ","End":"02:21.110","Text":"the polynomial 1, the polynomial 0,"},{"Start":"02:21.110 ","End":"02:26.720","Text":"and that\u0027s the polynomial that is excluded from being in the denominator here."},{"Start":"02:26.720 ","End":"02:33.390","Text":"I\u0027ll put it in red that this is not a rational expression."},{"Start":"02:34.600 ","End":"02:39.500","Text":"I\u0027ll also point out that polynomials and"},{"Start":"02:39.500 ","End":"02:45.583","Text":"rational expressions are entities in their own right but also,"},{"Start":"02:45.583 ","End":"02:49.560","Text":"they\u0027re used for substitution sometimes."},{"Start":"02:49.560 ","End":"02:52.040","Text":"We don\u0027t always go and substitute values in them."},{"Start":"02:52.040 ","End":"02:54.898","Text":"If we do substitute values, of course,"},{"Start":"02:54.898 ","End":"03:00.695","Text":"we have to be careful that we don\u0027t get the number 0 in the denominator."},{"Start":"03:00.695 ","End":"03:09.253","Text":"For example, I couldn\u0027t let x=2/3 here if I was substituting a value,"},{"Start":"03:09.253 ","End":"03:11.840","Text":"because that would make the denominator 0."},{"Start":"03:11.840 ","End":"03:14.730","Text":"I just mentioned that."},{"Start":"03:15.100 ","End":"03:22.580","Text":"Now let\u0027s continue into the first topic in rational expressions and that"},{"Start":"03:22.580 ","End":"03:30.210","Text":"is the topic of reduction or reducing a rational expression."},{"Start":"03:30.460 ","End":"03:39.605","Text":"The name the reducing or reduction comes also by analogy from integers."},{"Start":"03:39.605 ","End":"03:43.055","Text":"For example, if I had a fraction,"},{"Start":"03:43.055 ","End":"03:47.975","Text":"a rational number 18 over 30,"},{"Start":"03:47.975 ","End":"03:51.140","Text":"then I could reduce it."},{"Start":"03:51.140 ","End":"03:58.955","Text":"I could divide top and bottom by 2 and get 9 over 15."},{"Start":"03:58.955 ","End":"04:02.960","Text":"But this is not reduced to the lowest terms because I could keep"},{"Start":"04:02.960 ","End":"04:09.120","Text":"going and divide also by 3 and get 3 over 5."},{"Start":"04:09.120 ","End":"04:11.510","Text":"Sometimes or usually,"},{"Start":"04:11.510 ","End":"04:14.810","Text":"we want to reduce to the lowest terms."},{"Start":"04:14.810 ","End":"04:26.460","Text":"I\u0027ll just put that in brackets to lowest terms."},{"Start":"04:27.160 ","End":"04:30.470","Text":"I\u0027ll give an example of that."},{"Start":"04:30.470 ","End":"04:37.210","Text":"Let\u0027s take 2x^2 plus"},{"Start":"04:37.210 ","End":"04:47.210","Text":"8x over 6x^2 minus 10x."},{"Start":"04:47.210 ","End":"04:54.470","Text":"The way to do it is to just complete the factor as far as possible,"},{"Start":"04:54.470 ","End":"04:58.360","Text":"both the numerator and the denominator separately."},{"Start":"04:58.360 ","End":"05:01.370","Text":"In the case of the numerator, certainly,"},{"Start":"05:01.370 ","End":"05:03.140","Text":"we can take a 2 out,"},{"Start":"05:03.140 ","End":"05:04.730","Text":"but that\u0027s not the best we can do."},{"Start":"05:04.730 ","End":"05:06.480","Text":"We can also take an x,"},{"Start":"05:06.480 ","End":"05:14.010","Text":"we can take 2x outside the brackets and what we\u0027re left with is x plus 4."},{"Start":"05:14.010 ","End":"05:17.805","Text":"On the denominator,"},{"Start":"05:17.805 ","End":"05:21.480","Text":"we can also take 2x,"},{"Start":"05:21.480 ","End":"05:24.885","Text":"and that\u0027s the best we can do outside the brackets."},{"Start":"05:24.885 ","End":"05:27.195","Text":"Again, 2x."},{"Start":"05:27.195 ","End":"05:32.760","Text":"Here we\u0027re left with 3x minus 5."},{"Start":"05:32.760 ","End":"05:37.830","Text":"Then we just cancel the 2xs,"},{"Start":"05:37.830 ","End":"05:41.007","Text":"like with numbers,"},{"Start":"05:41.007 ","End":"05:50.020","Text":"and so the answer is x plus 4 over 3x minus 5."},{"Start":"05:50.020 ","End":"05:52.400","Text":"That\u0027s the lowest terms,"},{"Start":"05:52.400 ","End":"05:54.890","Text":"we can\u0027t reduce it anymore."},{"Start":"05:54.890 ","End":"06:00.365","Text":"Another example, and maybe I\u0027ll number them."},{"Start":"06:00.365 ","End":"06:05.525","Text":"Let\u0027s say this is example 1 and this is example 2."},{"Start":"06:05.525 ","End":"06:09.050","Text":"Here we\u0027ll take different letters,"},{"Start":"06:09.050 ","End":"06:13.265","Text":"say m minus 5"},{"Start":"06:13.265 ","End":"06:20.950","Text":"over m^2 minus 4m minus 5."},{"Start":"06:20.950 ","End":"06:27.140","Text":"As before, we factor the numerator and denominator."},{"Start":"06:27.140 ","End":"06:30.560","Text":"There\u0027s nothing to do in the numerator,"},{"Start":"06:30.560 ","End":"06:32.490","Text":"but the denominator,"},{"Start":"06:32.490 ","End":"06:35.925","Text":"and I won\u0027t give the details,"},{"Start":"06:35.925 ","End":"06:43.510","Text":"factors into m plus 1 times m minus 5."},{"Start":"06:43.510 ","End":"06:45.530","Text":"I\u0027ll let you do that one."},{"Start":"06:45.530 ","End":"06:47.735","Text":"But we can check at least,"},{"Start":"06:47.735 ","End":"06:55.215","Text":"m times m is m^2 plus m minus 5m minus 4m and 1/3 minus 5 is this."},{"Start":"06:55.215 ","End":"06:59.000","Text":"Now I see that I have here an m minus 5."},{"Start":"06:59.000 ","End":"07:02.210","Text":"I mean, I could put it in brackets just for emphasis."},{"Start":"07:02.210 ","End":"07:05.480","Text":"I can see that this is exactly like this."},{"Start":"07:05.480 ","End":"07:08.480","Text":"We can cancel that common factor,"},{"Start":"07:08.480 ","End":"07:09.800","Text":"but we don\u0027t just leave a blank."},{"Start":"07:09.800 ","End":"07:11.940","Text":"We put 1 here."},{"Start":"07:12.540 ","End":"07:20.185","Text":"That = 1 over m +1."},{"Start":"07:20.185 ","End":"07:23.020","Text":"I\u0027ll give 1 more example,"},{"Start":"07:23.020 ","End":"07:25.600","Text":"that would be number 3."},{"Start":"07:25.600 ","End":"07:29.920","Text":"Let\u0027s take it this time in variable t,"},{"Start":"07:29.920 ","End":"07:33.682","Text":"let\u0027s take"},{"Start":"07:33.682 ","End":"07:43.540","Text":"3t-t^2 over t^2 minus 9."},{"Start":"07:43.540 ","End":"07:47.875","Text":"We factorize numerator and denominator separately."},{"Start":"07:47.875 ","End":"07:53.005","Text":"In the numerator, the best we can do is to take the t out."},{"Start":"07:53.005 ","End":"08:01.707","Text":"Then we\u0027re left with 3 minus t. On the denominator we have a difference of squares,"},{"Start":"08:01.707 ","End":"08:10.410","Text":"so we can write that as t minus 3 times t plus 3,"},{"Start":"08:10.410 ","End":"08:13.525","Text":"because of course 9 is 3^2."},{"Start":"08:13.525 ","End":"08:20.177","Text":"Now, we don\u0027t see something identical in the top and in the bottom,"},{"Start":"08:20.177 ","End":"08:23.950","Text":"but 3 minus t and t minus 3 are almost the same,"},{"Start":"08:23.950 ","End":"08:26.690","Text":"they\u0027re just in the opposite order."},{"Start":"08:26.820 ","End":"08:37.630","Text":"What we can do is rewrite 3 minus t as minus of t minus 3."},{"Start":"08:37.630 ","End":"08:43.945","Text":"Sometimes this is called taking the minus out of the brackets or factoring out the minus."},{"Start":"08:43.945 ","End":"08:49.880","Text":"Basically, you can reverse the order of a difference as long as you put a minus sign."},{"Start":"08:50.310 ","End":"08:57.250","Text":"Back here, we can continue and rewrite this with the minus,"},{"Start":"08:57.250 ","End":"09:02.800","Text":"which we\u0027ll put in front as minus t times t minus 3"},{"Start":"09:02.800 ","End":"09:09.205","Text":"over t minus 3, t plus 3."},{"Start":"09:09.205 ","End":"09:14.390","Text":"Now we can cancel the common factor."},{"Start":"09:14.700 ","End":"09:23.290","Text":"The answer is just minus t over t plus 3."},{"Start":"09:23.290 ","End":"09:25.270","Text":"If you\u0027re not sure about this,"},{"Start":"09:25.270 ","End":"09:28.855","Text":"we could take a numerical example."},{"Start":"09:28.855 ","End":"09:36.145","Text":"First, let me write it in general that a minus b is minus b minus a."},{"Start":"09:36.145 ","End":"09:42.835","Text":"If we take an example of 3 minus 7,"},{"Start":"09:42.835 ","End":"09:47.845","Text":"what you can do is subtract them in the reverse order,"},{"Start":"09:47.845 ","End":"09:50.395","Text":"7 minus 3,"},{"Start":"09:50.395 ","End":"09:52.975","Text":"and then stick a minus in front of it."},{"Start":"09:52.975 ","End":"09:54.910","Text":"That\u0027s what I\u0027ll be doing in my head anyway,"},{"Start":"09:54.910 ","End":"09:57.664","Text":"I\u0027d be doing 7 minus 3 is 4,"},{"Start":"09:57.664 ","End":"10:03.640","Text":"and then the answer would be minus 4 because I\u0027m taking the smaller minus the bigger."},{"Start":"10:03.640 ","End":"10:09.220","Text":"Anyway, going to do a little bit more of this reducing of a rational expression."},{"Start":"10:09.220 ","End":"10:11.994","Text":"Let me just erase what I don\u0027t need."},{"Start":"10:11.994 ","End":"10:14.350","Text":"Let me just move this up here."},{"Start":"10:14.350 ","End":"10:16.480","Text":"I\u0027ll mention that it\u0027s continued."},{"Start":"10:16.480 ","End":"10:20.162","Text":"What we\u0027ll be doing is reducing,"},{"Start":"10:20.162 ","End":"10:24.160","Text":"but after we do a multiplication or division problem."},{"Start":"10:24.160 ","End":"10:25.720","Text":"Best to explain with an example."},{"Start":"10:25.720 ","End":"10:31.045","Text":"We\u0027ll take an example of the multiplication and then I\u0027ll take an example with division."},{"Start":"10:31.045 ","End":"10:38.020","Text":"The exercise is x^2 plus 3x minus"},{"Start":"10:38.020 ","End":"10:44.605","Text":"4 over x^2 plus x minus 12,"},{"Start":"10:44.605 ","End":"10:49.405","Text":"then multiplied by another 1 of these."},{"Start":"10:49.405 ","End":"10:55.720","Text":"This time x^2 plus 4x minus 21 over"},{"Start":"10:55.720 ","End":"11:03.595","Text":"x^2 plus 4x minus 5."},{"Start":"11:03.595 ","End":"11:05.980","Text":"What we do is this, first of all,"},{"Start":"11:05.980 ","End":"11:08.650","Text":"we factorize each of the polynomials,"},{"Start":"11:08.650 ","End":"11:11.600","Text":"they are all quadratics in this case."},{"Start":"11:13.410 ","End":"11:18.040","Text":"I\u0027m not going to do each of the factorizations,"},{"Start":"11:18.040 ","End":"11:20.095","Text":"I\u0027ll just give you the answers."},{"Start":"11:20.095 ","End":"11:27.970","Text":"This 1 comes out to x plus 4 x minus 1."},{"Start":"11:27.970 ","End":"11:30.955","Text":"We can check x times x is x^2,"},{"Start":"11:30.955 ","End":"11:34.885","Text":"plus 4x minus x is 3x, and minus 4."},{"Start":"11:34.885 ","End":"11:38.050","Text":"You know what? I\u0027ll just write the rest out for you."},{"Start":"11:38.050 ","End":"11:41.305","Text":"This is what we get."},{"Start":"11:41.305 ","End":"11:47.215","Text":"When you cancel, it\u0027s okay to cancel across a multiplication."},{"Start":"11:47.215 ","End":"11:49.675","Text":"You couldn\u0027t do this if there was a division here,"},{"Start":"11:49.675 ","End":"11:51.220","Text":"but when it\u0027s a multiplication,"},{"Start":"11:51.220 ","End":"11:57.415","Text":"I can say that this x minus 1 cancels with this x minus 1."},{"Start":"11:57.415 ","End":"11:59.215","Text":"Let\u0027s see what else."},{"Start":"11:59.215 ","End":"12:03.880","Text":"Here I have an x minus 3 and an x minus 3."},{"Start":"12:03.880 ","End":"12:10.630","Text":"Within the same quotient of course you can cancel,"},{"Start":"12:10.630 ","End":"12:13.915","Text":"so this x plus 4 with this x plus 4."},{"Start":"12:13.915 ","End":"12:24.265","Text":"All we\u0027re left with is x plus 7 over x plus 5."},{"Start":"12:24.265 ","End":"12:29.500","Text":"That\u0027s it. I\u0027m just giving you 1 example of multiplication."},{"Start":"12:29.500 ","End":"12:32.990","Text":"I\u0027ll now give an example with a division."},{"Start":"12:33.840 ","End":"12:38.230","Text":"Let\u0027s take for our division,"},{"Start":"12:38.230 ","End":"12:45.130","Text":"5 over x^2 minus 4 divided by,"},{"Start":"12:45.130 ","End":"12:48.685","Text":"so I write divided by, sometimes it\u0027s written like this,"},{"Start":"12:48.685 ","End":"12:54.410","Text":"divided by x plus"},{"Start":"12:54.410 ","End":"13:01.660","Text":"6 over x^2 plus 11x plus 18."},{"Start":"13:01.660 ","End":"13:06.220","Text":"What we\u0027ll do first is convert the division to a multiplication,"},{"Start":"13:06.220 ","End":"13:09.940","Text":"just like with the fractions with numbers."},{"Start":"13:09.940 ","End":"13:14.110","Text":"We take 5 over x^2 minus 4."},{"Start":"13:14.110 ","End":"13:15.280","Text":"Instead of dividing,"},{"Start":"13:15.280 ","End":"13:17.500","Text":"we multiply by the reciprocal,"},{"Start":"13:17.500 ","End":"13:19.585","Text":"by the opposite fraction."},{"Start":"13:19.585 ","End":"13:24.805","Text":"On the numerator, x^2 plus 11x plus 18,"},{"Start":"13:24.805 ","End":"13:26.755","Text":"and on the denominator,"},{"Start":"13:26.755 ","End":"13:29.605","Text":"the x plus 6."},{"Start":"13:29.605 ","End":"13:32.980","Text":"Now, just like a multiplication problem,"},{"Start":"13:32.980 ","End":"13:36.745","Text":"factorize, 5 is just 5,"},{"Start":"13:36.745 ","End":"13:39.610","Text":"x^2 minus 4 is a difference of squares,"},{"Start":"13:39.610 ","End":"13:41.035","Text":"4 is 2^2,"},{"Start":"13:41.035 ","End":"13:46.160","Text":"so it\u0027s x minus 2, x plus 2."},{"Start":"13:49.410 ","End":"13:52.660","Text":"Well, I\u0027ll just tell you the factorization."},{"Start":"13:52.660 ","End":"13:54.460","Text":"We won\u0027t spend time doing it."},{"Start":"13:54.460 ","End":"14:01.990","Text":"This turns out to be x plus 2 times x plus 9."},{"Start":"14:01.990 ","End":"14:07.150","Text":"But we\u0027ll check it, x times x is x^2 plus 2x plus 9x is 11x,"},{"Start":"14:07.150 ","End":"14:10.090","Text":"2 times 9 is 18. So that\u0027s fine."},{"Start":"14:10.090 ","End":"14:13.615","Text":"Over x plus 6."},{"Start":"14:13.615 ","End":"14:16.150","Text":"Now let\u0027s see what we can cancel."},{"Start":"14:16.150 ","End":"14:20.360","Text":"Looks like the only thing is the x plus 2."},{"Start":"14:20.370 ","End":"14:24.415","Text":"That\u0027s fine, even cancel 1 thing is good."},{"Start":"14:24.415 ","End":"14:28.960","Text":"Now we just rearrange this and write it as"},{"Start":"14:28.960 ","End":"14:39.130","Text":"5 times x plus 9 over x minus 2."},{"Start":"14:39.130 ","End":"14:41.860","Text":"Then x plus 6."},{"Start":"14:41.860 ","End":"14:43.190","Text":"That\u0027s the best we can do,"},{"Start":"14:43.190 ","End":"14:47.880","Text":"and that\u0027s it for divisions."},{"Start":"14:47.880 ","End":"14:50.870","Text":"There are example questions,"},{"Start":"14:50.870 ","End":"14:54.500","Text":"solved examples besides the tutorial."},{"Start":"14:54.500 ","End":"15:00.420","Text":"So I\u0027m done with this topic of reducing. That\u0027s it."}],"ID":5045},{"Watched":false,"Name":"Addition and Subtraction","Duration":"15m 56s","ChapterTopicVideoID":5045,"CourseChapterTopicPlaylistID":3859,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.395","Text":"Continuing the previous clip on rational expressions,"},{"Start":"00:04.395 ","End":"00:06.720","Text":"where we learned about reducing,"},{"Start":"00:06.720 ","End":"00:15.615","Text":"but in particular, we learned how to multiply and how to divide rational expressions."},{"Start":"00:15.615 ","End":"00:18.540","Text":"Multiplication, multiply the tops,"},{"Start":"00:18.540 ","End":"00:21.930","Text":"multiply the bottoms, and then reduce."},{"Start":"00:21.930 ","End":"00:24.539","Text":"With division, you invert,"},{"Start":"00:24.539 ","End":"00:27.870","Text":"multiply, and then again you reduce."},{"Start":"00:27.870 ","End":"00:32.055","Text":"It is like simplify."},{"Start":"00:32.055 ","End":"00:36.470","Text":"Now, that we\u0027ve got the multiplication and division done,"},{"Start":"00:36.470 ","End":"00:40.320","Text":"let\u0027s not talk about addition and subtraction."},{"Start":"00:40.420 ","End":"00:46.970","Text":"I think we can explain addition and subtraction just with an example."},{"Start":"00:46.970 ","End":"00:51.505","Text":"Let\u0027s take as our first example,"},{"Start":"00:51.505 ","End":"00:59.490","Text":"5/6x^2"},{"Start":"01:02.390 ","End":"01:11.100","Text":"-"},{"Start":"01:14.660 ","End":"01:19.890","Text":"4/5x^6 + 3/8x^3."},{"Start":"01:24.260 ","End":"01:26.470","Text":"10x^3."},{"Start":"01:26.470 ","End":"01:29.080","Text":"When we do addition and subtraction,"},{"Start":"01:29.080 ","End":"01:33.640","Text":"the most important concept is that of a common denominator,"},{"Start":"01:33.640 ","End":"01:39.655","Text":"or even a least common denominator."},{"Start":"01:39.655 ","End":"01:43.480","Text":"Actually the least is not that important."},{"Start":"01:43.480 ","End":"01:49.090","Text":"By the way, this has an abbreviation LCD, least common denominator."},{"Start":"01:49.090 ","End":"01:50.980","Text":"But if it\u0027s least,"},{"Start":"01:50.980 ","End":"01:52.210","Text":"it just works better."},{"Start":"01:52.210 ","End":"01:53.950","Text":"We get smaller numbers."},{"Start":"01:53.950 ","End":"01:59.660","Text":"What we do is we see which denominator."},{"Start":"01:59.660 ","End":"02:01.790","Text":"All of these will divide into,"},{"Start":"02:01.790 ","End":"02:03.785","Text":"look at the numbers separately,"},{"Start":"02:03.785 ","End":"02:07.820","Text":"6, 5, and 10 and after some thought,"},{"Start":"02:07.820 ","End":"02:12.170","Text":"we realize that 30 is going to be"},{"Start":"02:12.170 ","End":"02:17.488","Text":"divisible by all of them and we can\u0027t do better than 30,"},{"Start":"02:17.488 ","End":"02:20.550","Text":"so we make a fraction."},{"Start":"02:23.240 ","End":"02:28.190","Text":"I\u0027m going to write that 30 here that takes care of the number part."},{"Start":"02:28.190 ","End":"02:29.480","Text":"Now, for the xs,"},{"Start":"02:29.480 ","End":"02:34.070","Text":"I have x^2, x^6, and x^3."},{"Start":"02:34.070 ","End":"02:40.370","Text":"They all go into a look at the highest power x^6."},{"Start":"02:40.370 ","End":"02:46.220","Text":"This is actually the least I mean they all go into x^7, x^20 and so on."},{"Start":"02:46.220 ","End":"02:51.365","Text":"But the least that they all go into is x^6."},{"Start":"02:51.365 ","End":"02:59.630","Text":"Now, we look at each denominator and see what factor it\u0027s missing."},{"Start":"02:59.630 ","End":"03:04.885","Text":"To make it 30x^6."},{"Start":"03:04.885 ","End":"03:11.625","Text":"I will just do some examples and you can learn through examples."},{"Start":"03:11.625 ","End":"03:13.260","Text":"For the first example,"},{"Start":"03:13.260 ","End":"03:23.040","Text":"I\u0027ll take what I wrote here, 5/6x^2-4/5x^6+3/10x^3."},{"Start":"03:23.040 ","End":"03:27.740","Text":"The key concept in addition and subtraction of fractions is"},{"Start":"03:27.740 ","End":"03:34.310","Text":"common denominator or better still, least common denominator."},{"Start":"03:34.310 ","End":"03:37.010","Text":"I wrote the term, like I said,"},{"Start":"03:37.010 ","End":"03:39.500","Text":"least is not essential."},{"Start":"03:39.500 ","End":"03:43.135","Text":"It just helps to work with smaller numbers."},{"Start":"03:43.135 ","End":"03:47.420","Text":"Main thing is to find a common denominator,"},{"Start":"03:47.420 ","End":"03:52.115","Text":"which is something that each of these separate denominators divides into."},{"Start":"03:52.115 ","End":"03:55.800","Text":"I\u0027m, I try to make it as small as possible."},{"Start":"03:56.030 ","End":"04:03.770","Text":"6,5 and 10 I just look at the number parts first and say what do they all go into?"},{"Start":"04:03.770 ","End":"04:05.930","Text":"They all go into 60,"},{"Start":"04:05.930 ","End":"04:11.075","Text":"but I can do better 30 would be good account do any less than that."},{"Start":"04:11.075 ","End":"04:14.420","Text":"30 is the common denominator of 6,"},{"Start":"04:14.420 ","End":"04:17.570","Text":"5, and 10, but it also have powers of x here,"},{"Start":"04:17.570 ","End":"04:19.310","Text":"so I look at x^2,"},{"Start":"04:19.310 ","End":"04:25.300","Text":"x^6 and x^3 and the largest of these is x^6."},{"Start":"04:25.300 ","End":"04:28.110","Text":"All of them will go into x^6,"},{"Start":"04:28.110 ","End":"04:33.890","Text":"and so 30x^6 will be something that all my denominators will go into."},{"Start":"04:33.890 ","End":"04:37.400","Text":"Then I put each of these over a common denominator."},{"Start":"04:37.400 ","End":"04:42.620","Text":"Remember, fractions with the same denominator can be added"},{"Start":"04:42.620 ","End":"04:48.335","Text":"just by adding the numerators and keeping that same common denominator,"},{"Start":"04:48.335 ","End":"04:51.430","Text":"just like with number fractions."},{"Start":"04:53.690 ","End":"05:00.185","Text":"The way I find convenient is to take each one and say as follows 6x^2,"},{"Start":"05:00.185 ","End":"05:05.600","Text":"goes into 30x^6 how many times or what factors are missing?"},{"Start":"05:05.600 ","End":"05:10.525","Text":"Well, I\u0027m missing a 5 and I\u0027m missing a factor of x^4."},{"Start":"05:10.525 ","End":"05:12.720","Text":"I write that here,"},{"Start":"05:12.720 ","End":"05:19.260","Text":"that 5x^4 is what I have to multiply this by to get this."},{"Start":"05:19.260 ","End":"05:25.360","Text":"Then the next one, 5x^6 into 30x^6."},{"Start":"05:25.360 ","End":"05:29.520","Text":"Well x^6th goes in just once into x to the 6th."},{"Start":"05:29.520 ","End":"05:31.170","Text":"It\u0027s only the numbers that count."},{"Start":"05:31.170 ","End":"05:34.410","Text":"5 into 30 goes 6 times,"},{"Start":"05:34.410 ","End":"05:36.570","Text":"I write 6 here."},{"Start":"05:36.570 ","End":"05:46.905","Text":"10x^3 into 30x^6 the number goes in 3 times and x^3 into x^6 goes x^3 times."},{"Start":"05:46.905 ","End":"05:49.995","Text":"This is 3x^3."},{"Start":"05:49.995 ","End":"05:54.040","Text":"This 3 is just this 6-3."},{"Start":"05:54.850 ","End":"06:00.750","Text":"Then what we do is we just multiply in pairs."},{"Start":"06:01.120 ","End":"06:05.160","Text":"This times, this is"},{"Start":"06:08.720 ","End":"06:12.950","Text":"25x^4 and 6x4 is 24,"},{"Start":"06:12.950 ","End":"06:14.990","Text":"but I\u0027ve got to keep the sign also,"},{"Start":"06:14.990 ","End":"06:18.275","Text":"so it\u0027s minus 24."},{"Start":"06:18.275 ","End":"06:22.950","Text":"Then this times this is 9x^3."},{"Start":"06:26.510 ","End":"06:28.640","Text":"That\u0027s the answer."},{"Start":"06:28.640 ","End":"06:30.350","Text":"We could tidy up a bit,"},{"Start":"06:30.350 ","End":"06:32.899","Text":"rearranged the polynomial in order."},{"Start":"06:32.899 ","End":"06:37.830","Text":"But basically, this is the answer."},{"Start":"06:37.940 ","End":"06:43.310","Text":"You should have been familiar already with this simplification."},{"Start":"06:43.310 ","End":"06:47.150","Text":"That\u0027s why I cut out the middle step where I tried 3 separate fractions,"},{"Start":"06:47.150 ","End":"06:50.165","Text":"this over this, minus this over this,"},{"Start":"06:50.165 ","End":"06:51.330","Text":"plus this over this."},{"Start":"06:51.330 ","End":"06:54.305","Text":"Then I would say fractions with the same denominator."},{"Start":"06:54.305 ","End":"06:56.195","Text":"We just combine the denominator."},{"Start":"06:56.195 ","End":"06:57.350","Text":"But you\u0027ve seen all this,"},{"Start":"06:57.350 ","End":"07:00.875","Text":"so I just did it briefly."},{"Start":"07:00.875 ","End":"07:03.335","Text":"It\u0027s more of a reminder."},{"Start":"07:03.335 ","End":"07:06.565","Text":"Let\u0027s take another example."},{"Start":"07:06.565 ","End":"07:09.375","Text":"Number 2."},{"Start":"07:09.375 ","End":"07:12.370","Text":"Here\u0027s example 2,"},{"Start":"07:12.370 ","End":"07:16.510","Text":"this time in a different letters time Y."},{"Start":"07:16.510 ","End":"07:21.170","Text":"It\u0027s addition and subtraction combined just like the other one was."},{"Start":"07:21.170 ","End":"07:25.730","Text":"There\u0027s really not very much difference between addition and subtraction in this context."},{"Start":"07:25.730 ","End":"07:28.700","Text":"Anyway, same principle."},{"Start":"07:28.700 ","End":"07:33.889","Text":"We have to find the common denominator and it\u0027s better if we make it the least."},{"Start":"07:33.889 ","End":"07:37.160","Text":"I mean this is definitely optional."},{"Start":"07:37.160 ","End":"07:44.510","Text":"Also wanted to say that you can sometimes see LCD in the books."},{"Start":"07:44.510 ","End":"07:49.385","Text":"The abbreviation for least common denominator,"},{"Start":"07:49.385 ","End":"07:53.970","Text":"just like GCF was greatest common factor."},{"Start":"07:54.600 ","End":"08:00.153","Text":"Before we decide on the common denominator,"},{"Start":"08:00.153 ","End":"08:05.425","Text":"it\u0027s often usually even helpful to factorize,"},{"Start":"08:05.425 ","End":"08:08.500","Text":"so this y^2 minus 25,"},{"Start":"08:08.500 ","End":"08:11.593","Text":"identified as a difference of squares,"},{"Start":"08:11.593 ","End":"08:18.550","Text":"and I can write this as y minus 5 times y plus 5."},{"Start":"08:18.550 ","End":"08:22.165","Text":"That\u0027s just the denominator part."},{"Start":"08:22.165 ","End":"08:25.360","Text":"I won\u0027t copy the whole thing again I\u0027ll just say,"},{"Start":"08:25.360 ","End":"08:28.580","Text":"instead of this we could write this."},{"Start":"08:28.770 ","End":"08:34.240","Text":"Now, we\u0027ll get to the part with the common denominator."},{"Start":"08:34.240 ","End":"08:37.795","Text":"We have a y minus 5, a y plus 5,"},{"Start":"08:37.795 ","End":"08:39.490","Text":"and then again a y plus 5,"},{"Start":"08:39.490 ","End":"08:42.340","Text":"and again a y minus 5."},{"Start":"08:42.340 ","End":"08:48.740","Text":"As long as I take 1 of each of these and I\u0027m covered,"},{"Start":"08:48.740 ","End":"08:50.320","Text":"y minus 5,"},{"Start":"08:50.320 ","End":"08:52.345","Text":"y plus 5 covers everything."},{"Start":"08:52.345 ","End":"08:55.870","Text":"It is this and it\u0027s this time something,"},{"Start":"08:55.870 ","End":"08:59.373","Text":"and it\u0027s this time something."},{"Start":"08:59.373 ","End":"09:03.100","Text":"As above, I\u0027ll write the missing factors,"},{"Start":"09:03.100 ","End":"09:05.830","Text":"like to put a little separator on each and"},{"Start":"09:05.830 ","End":"09:10.315","Text":"say how many times does this denominator go into this?"},{"Start":"09:10.315 ","End":"09:11.480","Text":"Well, just once,"},{"Start":"09:11.480 ","End":"09:13.015","Text":"it is the same thing."},{"Start":"09:13.015 ","End":"09:18.205","Text":"How many times is y minus 5 goes into y minus 5, y plus 5?"},{"Start":"09:18.205 ","End":"09:21.410","Text":"It\u0027s just the y plus 5."},{"Start":"09:21.930 ","End":"09:25.030","Text":"How many times does this go into this?"},{"Start":"09:25.030 ","End":"09:29.284","Text":"It\u0027s y minus 5."},{"Start":"09:29.284 ","End":"09:31.680","Text":"Now we do as before,"},{"Start":"09:31.680 ","End":"09:36.985","Text":"we multiply in pairs and add and subtract as appropriate."},{"Start":"09:36.985 ","End":"09:41.800","Text":"We get 1 times 2y is 2y,"},{"Start":"09:41.800 ","End":"09:48.340","Text":"and then minus 2 times y plus 5."},{"Start":"09:48.340 ","End":"09:51.380","Text":"This is considered to be in a bracket."},{"Start":"09:53.010 ","End":"09:59.380","Text":"Then minus 1 times y minus 5."},{"Start":"09:59.380 ","End":"10:00.850","Text":"I\u0027ll keep the 1 there."},{"Start":"10:00.850 ","End":"10:02.110","Text":"Didn\u0027t have to write it,"},{"Start":"10:02.110 ","End":"10:05.553","Text":"but I\u0027ll write it,"},{"Start":"10:05.553 ","End":"10:10.090","Text":"and that\u0027s already an answer."},{"Start":"10:10.090 ","End":"10:14.028","Text":"But we should really simplify it first,"},{"Start":"10:14.028 ","End":"10:20.200","Text":"because I prefer to have just a polynomials that have little bits and pieces."},{"Start":"10:20.200 ","End":"10:22.413","Text":"Let\u0027s keep going,"},{"Start":"10:22.413 ","End":"10:31.120","Text":"and say that this is equal to 2y minus 2y minus"},{"Start":"10:31.120 ","End":"10:41.020","Text":"10 from here minus y and then plus 5 over the same thing."},{"Start":"10:41.020 ","End":"10:44.650","Text":"Here again, y minus 5,"},{"Start":"10:44.650 ","End":"10:48.168","Text":"y plus 5. Let\u0027s see."},{"Start":"10:48.168 ","End":"10:52.300","Text":"Combine like terms."},{"Start":"10:52.300 ","End":"10:54.175","Text":"What do I have?"},{"Start":"10:54.175 ","End":"10:58.270","Text":"2y minus 2y cancels."},{"Start":"10:58.270 ","End":"11:07.015","Text":"We get minus y and minus 10 plus 5"},{"Start":"11:07.015 ","End":"11:16.390","Text":"is minus 5 over y minus 5,"},{"Start":"11:16.390 ","End":"11:19.180","Text":"y plus 5,"},{"Start":"11:19.180 ","End":"11:22.765","Text":"and you could say that this is the end."},{"Start":"11:22.765 ","End":"11:26.695","Text":"But if you have a sharp eye,"},{"Start":"11:26.695 ","End":"11:33.130","Text":"you will notice that we have 2 things very similar minus y minus 5 and y plus 5."},{"Start":"11:33.130 ","End":"11:36.520","Text":"If I just factor out the minus,"},{"Start":"11:36.520 ","End":"11:39.567","Text":"then I\u0027ll have a y plus 5 in both,"},{"Start":"11:39.567 ","End":"11:41.710","Text":"so let\u0027s do that."},{"Start":"11:41.710 ","End":"11:44.470","Text":"I\u0027ll continue it over here."},{"Start":"11:44.470 ","End":"11:51.580","Text":"I have minus y plus 5 over."},{"Start":"11:51.580 ","End":"11:56.005","Text":"Just copied it the denominator from here to here."},{"Start":"11:56.005 ","End":"12:03.760","Text":"Now, we can say y minus 5 and y minus 5 cancels."},{"Start":"12:03.760 ","End":"12:06.880","Text":"But you don\u0027t just leave it like a blank,"},{"Start":"12:06.880 ","End":"12:09.460","Text":"we just put a 1 in after it\u0027s canceled."},{"Start":"12:09.460 ","End":"12:12.170","Text":"This goes into this 1 time."},{"Start":"12:12.540 ","End":"12:22.465","Text":"The final answer is minus 1 over y minus 5,"},{"Start":"12:22.465 ","End":"12:25.119","Text":"and this can\u0027t be simplified anymore."},{"Start":"12:25.119 ","End":"12:28.810","Text":"But this was a surprise and you might not"},{"Start":"12:28.810 ","End":"12:32.469","Text":"have noticed it that sometimes after we do an addition and subtraction,"},{"Start":"12:32.469 ","End":"12:40.673","Text":"there\u0027s an additional cancellation that occurs,"},{"Start":"12:40.673 ","End":"12:44.140","Text":"and this is an example of it."},{"Start":"12:44.700 ","End":"12:47.140","Text":"Here\u0027s the 3rd example."},{"Start":"12:47.140 ","End":"12:52.270","Text":"I\u0027ll make it the last bit more space down."},{"Start":"12:53.870 ","End":"13:03.120","Text":"Using the least common denominator method dividing"},{"Start":"13:03.120 ","End":"13:13.015","Text":"line I notice that the 2x plus 1 appears here and here, but different exponents."},{"Start":"13:13.015 ","End":"13:15.610","Text":"For a common denominator,"},{"Start":"13:15.610 ","End":"13:18.820","Text":"I need to take the higher 1."},{"Start":"13:18.820 ","End":"13:27.250","Text":"That would be the (2x+1)^3 and that\u0027s the best I can do."},{"Start":"13:27.250 ","End":"13:34.370","Text":"Can\u0027t take anything less because then 2x plus 1^3 wouldn\u0027t go into it. This is the LCD."},{"Start":"13:35.550 ","End":"13:46.345","Text":"Then I proceed by seeing what\u0027s missing in each denominator to get it to be the LCD."},{"Start":"13:46.345 ","End":"13:48.310","Text":"Start with the easier one."},{"Start":"13:48.310 ","End":"13:51.400","Text":"This goes into this once."},{"Start":"13:51.400 ","End":"13:54.514","Text":"There\u0027s nothing missing, so we write that as a 1."},{"Start":"13:54.514 ","End":"13:58.750","Text":"This into this, or something squared into something cubed,"},{"Start":"13:58.750 ","End":"14:05.480","Text":"it just goes in 2x plus 1 times."},{"Start":"14:05.610 ","End":"14:11.995","Text":"If I say how many times does a squared go into a cubed?"},{"Start":"14:11.995 ","End":"14:16.870","Text":"It goes in a times because I times I squared is a cubed."},{"Start":"14:16.870 ","End":"14:25.450","Text":"I don\u0027t need that. Getting a bit cramped. That\u0027s a bit better."},{"Start":"14:25.450 ","End":"14:29.725","Text":"Now, we do the multiplications for the numerators."},{"Start":"14:29.725 ","End":"14:35.620","Text":"Here\u0027s 2x plus 1 times x minus 3 here,"},{"Start":"14:35.620 ","End":"14:37.705","Text":"1 times x,"},{"Start":"14:37.705 ","End":"14:43.969","Text":"which is just x. I can just copy the denominator,"},{"Start":"14:43.969 ","End":"14:47.830","Text":"and then just multiply out at 2x with x is (2x)^2,"},{"Start":"14:47.830 ","End":"14:52.330","Text":"2x with minus 3 is minus 6x."},{"Start":"14:52.330 ","End":"14:58.120","Text":"Now the plus 1 gives me plus x minus 3 and then we"},{"Start":"14:58.120 ","End":"15:04.915","Text":"have a plus x from here and it\u0027ll continue on the same line,"},{"Start":"15:04.915 ","End":"15:08.395","Text":"same denominator just copied it."},{"Start":"15:08.395 ","End":"15:13.750","Text":"Here, collecting like terms (2x)^2."},{"Start":"15:13.750 ","End":"15:19.939","Text":"Have to collect the x are 3 of the minus 6 plus 1 and plus 1,"},{"Start":"15:19.939 ","End":"15:23.140","Text":"so that\u0027s minus 4X,"},{"Start":"15:23.140 ","End":"15:28.150","Text":"and constant term just 1 minus 3"},{"Start":"15:28.150 ","End":"15:35.030","Text":"and that\u0027s the answer."},{"Start":"15:35.070 ","End":"15:40.915","Text":"Remember there are also examples exercises which are solved"},{"Start":"15:40.915 ","End":"15:48.170","Text":"besides the tutorial and you can go and look at those if you need more practice."},{"Start":"15:49.060 ","End":"15:55.320","Text":"This concludes the subject of rational expressions."}],"ID":5046},{"Watched":false,"Name":"Exercise 1","Duration":"9m 57s","ChapterTopicVideoID":5047,"CourseChapterTopicPlaylistID":3859,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.080","Text":"In this exercise, there are three parts and in each one we have"},{"Start":"00:04.080 ","End":"00:08.265","Text":"to reduce the expression to lowest terms."},{"Start":"00:08.265 ","End":"00:13.785","Text":"Remember that rational expression means polynomial over polynomial,"},{"Start":"00:13.785 ","End":"00:16.140","Text":"which is what we have in these three cases."},{"Start":"00:16.140 ","End":"00:20.550","Text":"Also, I\u0027m going assume you know how to factorize polynomials."},{"Start":"00:20.550 ","End":"00:23.010","Text":"I\u0027m not going to waste time on that."},{"Start":"00:23.010 ","End":"00:28.725","Text":"Let\u0027s scroll down where I have a bit more room."},{"Start":"00:28.725 ","End":"00:31.380","Text":"Start with the first one."},{"Start":"00:31.380 ","End":"00:36.090","Text":"The first thing to do in factorization or in reducing is"},{"Start":"00:36.090 ","End":"00:41.480","Text":"to factorize the numerator and the denominator."},{"Start":"00:41.480 ","End":"00:46.408","Text":"In the numerator the best we can do is to take out x,"},{"Start":"00:46.408 ","End":"00:52.020","Text":"and so we get x and what we\u0027re left with is x plus 4."},{"Start":"00:52.400 ","End":"00:57.185","Text":"On the denominator 2 and 5 have nothing in common,"},{"Start":"00:57.185 ","End":"00:59.900","Text":"but I can take the x out."},{"Start":"00:59.900 ","End":"01:06.425","Text":"I\u0027ve got x times 2x minus 5."},{"Start":"01:06.425 ","End":"01:12.200","Text":"At this point, I can see that the common part is x."},{"Start":"01:12.200 ","End":"01:18.690","Text":"Now I\u0027ll just indicate that I\u0027m canceling or putting a line through."},{"Start":"01:18.690 ","End":"01:25.990","Text":"Our answer is x plus 4 over 2x minus 5."},{"Start":"01:25.990 ","End":"01:28.565","Text":"Just want to point out that in these exercises,"},{"Start":"01:28.565 ","End":"01:34.415","Text":"we\u0027re not going to worry too much or at all about domain questions like,"},{"Start":"01:34.415 ","End":"01:37.339","Text":"what happens if x is 0?"},{"Start":"01:37.339 ","End":"01:40.520","Text":"Certainly we can substitute x=0 here,"},{"Start":"01:40.520 ","End":"01:43.610","Text":"but we couldn\u0027t substitute x=0 here."},{"Start":"01:43.610 ","End":"01:49.685","Text":"We\u0027ll just assume that we\u0027re factorizing on the domain or where possible."},{"Start":"01:49.685 ","End":"01:53.090","Text":"Anyway, I don\u0027t want to get bogged down with that question."},{"Start":"01:53.090 ","End":"01:54.950","Text":"We\u0027ll ignore that for the moment,"},{"Start":"01:54.950 ","End":"01:57.040","Text":"I mean, in these exercises."},{"Start":"01:57.040 ","End":"01:58.730","Text":"This will be the answer."},{"Start":"01:58.730 ","End":"02:00.650","Text":"Nothing further to reduce."},{"Start":"02:00.650 ","End":"02:02.300","Text":"Let\u0027s go onto the next one."},{"Start":"02:02.300 ","End":"02:02.301","Text":"In this one, the numerator, there\u0027s nothing to be done there;"},{"Start":"02:02.301 ","End":"02:03.302","Text":"it\u0027s already factorized as much as it would go."},{"Start":"02:13.600 ","End":"02:17.225","Text":"In the denominator, we have a quadratic."},{"Start":"02:17.225 ","End":"02:19.700","Text":"Like I said, I\u0027m not going spend time doing that,"},{"Start":"02:19.700 ","End":"02:23.210","Text":"you can do that as a side exercise on your own."},{"Start":"02:23.210 ","End":"02:31.390","Text":"What this comes out to is x minus 1 times x plus 3."},{"Start":"02:31.390 ","End":"02:34.580","Text":"Certainly we can mentally check that it works out."},{"Start":"02:34.580 ","End":"02:37.010","Text":"X times x is x^2,"},{"Start":"02:37.010 ","End":"02:39.418","Text":"minus 1 with 3 is minus 3,"},{"Start":"02:39.418 ","End":"02:44.690","Text":"and the middle term comes from minus x plus 3x. We can see it works."},{"Start":"02:44.690 ","End":"02:49.675","Text":"The common bit is the x minus 1."},{"Start":"02:49.675 ","End":"02:51.410","Text":"But we can\u0027t just leave nothing,"},{"Start":"02:51.410 ","End":"03:00.390","Text":"we\u0027ll put a 1 there and so the answer is 1 over x plus 3."},{"Start":"03:00.390 ","End":"03:03.520","Text":"Now, the last one."},{"Start":"03:04.810 ","End":"03:14.980","Text":"Let\u0027s see, the numerator is a quadratic and I\u0027ll tell you what the factorization is."},{"Start":"03:14.980 ","End":"03:20.950","Text":"It is x minus 4 times x plus 2."},{"Start":"03:20.950 ","End":"03:24.336","Text":"We\u0027ll just quickly do a mental check; x^2, fine,"},{"Start":"03:24.336 ","End":"03:26.975","Text":"minus 4 times 2 is minus 8,"},{"Start":"03:26.975 ","End":"03:31.689","Text":"minus 4x plus 2x is that, so that\u0027s fine."},{"Start":"03:31.689 ","End":"03:36.470","Text":"The denominator is one of those special a^2 minus b^2,"},{"Start":"03:36.470 ","End":"03:41.895","Text":"so it factorizes as x plus 2,"},{"Start":"03:41.895 ","End":"03:45.910","Text":"x minus 2, 4 is 2^2."},{"Start":"03:46.280 ","End":"03:52.320","Text":"Here what we can cancel is the x plus 2 here and the x plus 2"},{"Start":"03:52.320 ","End":"04:00.090","Text":"here and all we\u0027re left with is x minus 4 over x minus 2,"},{"Start":"04:00.090 ","End":"04:02.835","Text":"and we don\u0027t even need the brackets."},{"Start":"04:02.835 ","End":"04:05.310","Text":"Done for this one."},{"Start":"04:05.310 ","End":"04:12.230","Text":"Well, I\u0027ve reconsidered and for those of you who are not so proficient at factorization,"},{"Start":"04:12.230 ","End":"04:14.720","Text":"I\u0027m going to go back and show you how I got these."},{"Start":"04:14.720 ","End":"04:17.015","Text":"I just produced them out of thin air."},{"Start":"04:17.015 ","End":"04:23.210","Text":"Let me go back even to the previous one and show you how I did the factorizations."},{"Start":"04:23.210 ","End":"04:26.120","Text":"Let\u0027s take this denominator."},{"Start":"04:26.120 ","End":"04:29.450","Text":"I will pretend I don\u0027t know what the answer is."},{"Start":"04:29.450 ","End":"04:32.330","Text":"What we do is first of all identify a,"},{"Start":"04:32.330 ","End":"04:34.190","Text":"b, and c as in the formula."},{"Start":"04:34.190 ","End":"04:37.190","Text":"I have a=1,"},{"Start":"04:37.190 ","End":"04:41.475","Text":"b=2, c=minus 3."},{"Start":"04:41.475 ","End":"04:42.810","Text":"I\u0027m going to use this technique,"},{"Start":"04:42.810 ","End":"04:44.960","Text":"I called it trinomial decomposition,"},{"Start":"04:44.960 ","End":"04:47.150","Text":"but it doesn\u0027t have an official name."},{"Start":"04:47.150 ","End":"04:50.120","Text":"What we do is we look for two numbers,"},{"Start":"04:50.120 ","End":"04:53.765","Text":"I\u0027ll call them square and triangle,"},{"Start":"04:53.765 ","End":"04:58.765","Text":"such that their sum is b,"},{"Start":"04:58.765 ","End":"05:02.610","Text":"which is 2, it\u0027s always b here,"},{"Start":"05:02.610 ","End":"05:06.225","Text":"and the product is going to be ac."},{"Start":"05:06.225 ","End":"05:12.170","Text":"The same number times this number is going to be ac,"},{"Start":"05:12.170 ","End":"05:15.700","Text":"which is minus 3."},{"Start":"05:16.120 ","End":"05:22.040","Text":"I usually start with the product because there\u0027s less combinations of whole numbers."},{"Start":"05:22.040 ","End":"05:26.059","Text":"I know because of the minus sign,"},{"Start":"05:26.059 ","End":"05:29.850","Text":"that one\u0027s going to be plus and one\u0027s is going be minus."},{"Start":"05:30.950 ","End":"05:33.484","Text":"If I ignore a sign,"},{"Start":"05:33.484 ","End":"05:40.060","Text":"it has to be 1 times 3 or 3 times 1 and one of them is going to be negative."},{"Start":"05:40.060 ","End":"05:43.365","Text":"Now, since the sum has to be 2,"},{"Start":"05:43.365 ","End":"05:48.155","Text":"it\u0027s pretty clear that I have to take the 3 as positive and the 1 as negative."},{"Start":"05:48.155 ","End":"05:57.323","Text":"What I get is that here it\u0027s 3,"},{"Start":"05:57.323 ","End":"06:03.825","Text":"and minus 1 gives me 2 and 3 times minus 1 gives me minus 3."},{"Start":"06:03.825 ","End":"06:09.830","Text":"Now, these two numbers give me the key of how to break up the middle term."},{"Start":"06:09.830 ","End":"06:14.070","Text":"I take this x^2 plus"},{"Start":"06:14.070 ","End":"06:19.805","Text":"2x minus 3 and I rewrite it by breaking up the two according to this,"},{"Start":"06:19.805 ","End":"06:28.830","Text":"so it\u0027s x^2 plus 3x minus x and then minus 3."},{"Start":"06:28.830 ","End":"06:33.615","Text":"This 2 is what I broke up as 3 and minus 1."},{"Start":"06:33.615 ","End":"06:38.270","Text":"Now we use group factorization or factoring in groups."},{"Start":"06:38.270 ","End":"06:40.535","Text":"The first two terms,"},{"Start":"06:40.535 ","End":"06:43.865","Text":"I can take x out and I get x plus 3."},{"Start":"06:43.865 ","End":"06:46.745","Text":"In the next two terms, I take the minus out,"},{"Start":"06:46.745 ","End":"06:48.930","Text":"if you like minus 1,"},{"Start":"06:48.930 ","End":"06:51.450","Text":"and it\u0027s also x plus 3."},{"Start":"06:51.450 ","End":"06:55.775","Text":"Finally, I say I have x of these minus 1 of these,"},{"Start":"06:55.775 ","End":"06:58.735","Text":"I have x minus 1 of these."},{"Start":"06:58.735 ","End":"07:01.480","Text":"These are x plus 3."},{"Start":"07:01.480 ","End":"07:04.775","Text":"If we look over here, that\u0027s exactly what we got."},{"Start":"07:04.775 ","End":"07:11.855","Text":"Now let\u0027s try the same technique on this example on this numerator."},{"Start":"07:11.855 ","End":"07:18.445","Text":"Let\u0027s see. We have a=1,"},{"Start":"07:18.445 ","End":"07:25.935","Text":"b= minus 2, c= minus 8."},{"Start":"07:25.935 ","End":"07:28.460","Text":"We can skip writing this out."},{"Start":"07:28.460 ","End":"07:29.853","Text":"I mean, you can straight away say,"},{"Start":"07:29.853 ","End":"07:35.600","Text":"what I need is two numbers whose sum is b and whose product is ac."},{"Start":"07:35.600 ","End":"07:37.655","Text":"I\u0027ll just write it again to remind you."},{"Start":"07:37.655 ","End":"07:41.250","Text":"The sum and the product."},{"Start":"07:41.260 ","End":"07:50.805","Text":"What we need are something plus something to equal negative 2."},{"Start":"07:50.805 ","End":"07:52.370","Text":"As far as the product goes,"},{"Start":"07:52.370 ","End":"07:54.260","Text":"the product ac is minus 8,"},{"Start":"07:54.260 ","End":"07:59.975","Text":"so I need this thing times this thing to be minus 8."},{"Start":"07:59.975 ","End":"08:05.000","Text":"Now once again, we know that they have opposite signs."},{"Start":"08:05.000 ","End":"08:10.700","Text":"If I ignore the sign it\u0027s going to be either 1 times 8 or 2 times 4,"},{"Start":"08:10.700 ","End":"08:13.945","Text":"but I need the sum to be negative 2."},{"Start":"08:13.945 ","End":"08:19.625","Text":"Since 1 is plus or minus, it\u0027s a difference."},{"Start":"08:19.625 ","End":"08:23.610","Text":"You can see that what I need is 2 and 4,"},{"Start":"08:23.610 ","End":"08:25.920","Text":"but the 4 has to be the negative 1,"},{"Start":"08:25.920 ","End":"08:28.185","Text":"so I get 2 minus 4."},{"Start":"08:28.185 ","End":"08:37.155","Text":"What we get is 2 times minus 4 is minus 8 and 2 plus minus 4 is minus 2."},{"Start":"08:37.155 ","End":"08:39.970","Text":"Once I have the 2 and the minus 4,"},{"Start":"08:39.970 ","End":"08:45.090","Text":"then I can rewrite the quadratic, the one here,"},{"Start":"08:45.090 ","End":"08:49.350","Text":"as x^2, instead of the minus 2,"},{"Start":"08:49.350 ","End":"08:51.960","Text":"I\u0027ll write it as 2 and minus 4,"},{"Start":"08:51.960 ","End":"08:54.385","Text":"so it\u0027s plus 2x minus 4x."},{"Start":"08:54.385 ","End":"08:57.650","Text":"The middle term is what always gets split."},{"Start":"08:58.490 ","End":"09:02.410","Text":"Then we still have the minus 8."},{"Start":"09:02.410 ","End":"09:07.010","Text":"Now we can use group factoring."},{"Start":"09:07.010 ","End":"09:11.720","Text":"This is equal to x times x plus 2 from"},{"Start":"09:11.720 ","End":"09:17.290","Text":"the first two terms and from the second two terms I can take out the minus 4,"},{"Start":"09:17.290 ","End":"09:20.930","Text":"and if I do that I also get x plus 2."},{"Start":"09:20.930 ","End":"09:25.065","Text":"Then finally I can take the, doesn\u0027t matter,"},{"Start":"09:25.065 ","End":"09:34.015","Text":"x apples minus 4 apples is x minus 4 apples and the apples happen to be x plus 2."},{"Start":"09:34.015 ","End":"09:38.255","Text":"This is exactly what we have up here,"},{"Start":"09:38.255 ","End":"09:40.880","Text":"and so this method works."},{"Start":"09:40.880 ","End":"09:47.735","Text":"This was extra practice for those of you who like this technique."},{"Start":"09:47.735 ","End":"09:50.120","Text":"Of course, there\u0027s always the other technique using roots,"},{"Start":"09:50.120 ","End":"09:52.620","Text":"but that\u0027s more straightforward."},{"Start":"09:54.060 ","End":"09:57.530","Text":"Now, really I\u0027m done."}],"ID":5047},{"Watched":false,"Name":"Exercise 2","Duration":"36m 7s","ChapterTopicVideoID":5048,"CourseChapterTopicPlaylistID":3859,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.170 ","End":"00:05.345","Text":"This exercise has 4 part."},{"Start":"00:05.345 ","End":"00:09.190","Text":"In each case we have to do the indicated operation."},{"Start":"00:09.190 ","End":"00:10.750","Text":"Let me explain what that means."},{"Start":"00:10.750 ","End":"00:12.370","Text":"There is an operation like here,"},{"Start":"00:12.370 ","End":"00:16.869","Text":"it\u0027s multiplication, here, it\u0027s division."},{"Start":"00:16.869 ","End":"00:21.155","Text":"Using this colon symbol for division here we have a division."},{"Start":"00:21.155 ","End":"00:25.365","Text":"I guess here we have a division also which is this."},{"Start":"00:25.365 ","End":"00:29.490","Text":"We do the operation and afterwards we need to reduce the answer,"},{"Start":"00:29.490 ","End":"00:31.510","Text":"just like in a previous exercise,"},{"Start":"00:31.510 ","End":"00:34.115","Text":"we reduced to lowest terms."},{"Start":"00:34.115 ","End":"00:36.780","Text":"I have more room below."},{"Start":"00:36.780 ","End":"00:39.910","Text":"It start with number 1."},{"Start":"00:39.950 ","End":"00:45.049","Text":"When we multiply, just like with numerical fractions,"},{"Start":"00:45.049 ","End":"00:47.870","Text":"you multiply the tops and multiply the bottoms."},{"Start":"00:47.870 ","End":"00:53.105","Text":"What we have is just basically the same thing, slightly rewritten."},{"Start":"00:53.105 ","End":"01:01.440","Text":"This times x^2 plus 2x minus 15 and"},{"Start":"01:01.440 ","End":"01:05.709","Text":"all this one big dividing line over"},{"Start":"01:05.709 ","End":"01:10.799","Text":"x^2 minus x minus"},{"Start":"01:10.799 ","End":"01:18.295","Text":"6 times x^2 plus 3x minus 4."},{"Start":"01:18.295 ","End":"01:20.750","Text":"Now that\u0027s the operation done,"},{"Start":"01:20.750 ","End":"01:23.645","Text":"but now we have to reduce."},{"Start":"01:23.645 ","End":"01:26.540","Text":"I\u0027m not going to multiply out."},{"Start":"01:26.540 ","End":"01:29.675","Text":"I mean we\u0027re reducing this is already,"},{"Start":"01:29.675 ","End":"01:33.679","Text":"we\u0027re going to get smaller, break up into smaller pieces not to expand."},{"Start":"01:33.679 ","End":"01:37.069","Text":"Let\u0027s see if each one of them can be reduced."},{"Start":"01:37.069 ","End":"01:39.754","Text":"Each one of them is a quadratic."},{"Start":"01:39.754 ","End":"01:45.930","Text":"In principle they could each be reduced by factorizing."},{"Start":"01:45.930 ","End":"01:51.635","Text":"Let\u0027s see, I\u0027ll factor these quadratics for you,"},{"Start":"01:51.635 ","End":"01:58.995","Text":"the first one is x plus 2, x minus 1."},{"Start":"01:58.995 ","End":"02:09.990","Text":"Then the next one is x plus 5, x minus 3."},{"Start":"02:09.990 ","End":"02:17.890","Text":"This one here is x minus 3, x plus 2."},{"Start":"02:18.650 ","End":"02:24.940","Text":"This one is x"},{"Start":"02:26.150 ","End":"02:33.325","Text":"plus 4, x minus 1."},{"Start":"02:33.325 ","End":"02:36.410","Text":"We have 4 factors here and 4 factors here."},{"Start":"02:36.410 ","End":"02:38.970","Text":"Let\u0027s see what cancels,"},{"Start":"02:39.220 ","End":"02:45.625","Text":"this x plus 2 with this x plus 2."},{"Start":"02:45.625 ","End":"02:53.295","Text":"This x minus 1 with this x minus 1. Anything else?"},{"Start":"02:53.295 ","End":"02:57.525","Text":"Yes. X minus 3."},{"Start":"02:57.525 ","End":"03:01.770","Text":"With x minus 3."},{"Start":"03:01.770 ","End":"03:04.409","Text":"Not much left."},{"Start":"03:04.409 ","End":"03:12.600","Text":"Just x plus 5/x plus 4."},{"Start":"03:12.600 ","End":"03:14.754","Text":"That\u0027s that one."},{"Start":"03:14.754 ","End":"03:17.179","Text":"Onto the next one."},{"Start":"03:17.179 ","End":"03:19.900","Text":"Here we have a division."},{"Start":"03:19.900 ","End":"03:22.340","Text":"When you divide fractions,"},{"Start":"03:22.340 ","End":"03:24.605","Text":"you multiply by the reciprocal."},{"Start":"03:24.605 ","End":"03:29.070","Text":"The first thing is x^2 minus 9/x^2 plus"},{"Start":"03:30.320 ","End":"03:41.709","Text":"4x minus 21 times the inverse of this."},{"Start":"03:42.680 ","End":"03:47.699","Text":"That\u0027s x^2 plus 9 x"},{"Start":"03:47.699 ","End":"03:55.120","Text":"plus 14/x^2 plus 2x minus 3."},{"Start":"03:56.470 ","End":"04:01.640","Text":"I\u0027m not going to bother with the stage of writing it out"},{"Start":"04:01.640 ","End":"04:06.079","Text":"as a product because between this form and this form is not much difference."},{"Start":"04:06.079 ","End":"04:10.470","Text":"What I\u0027ll do is just factorize each of the quadratics,"},{"Start":"04:10.470 ","End":"04:12.735","Text":"so all 4 of them are quadratics."},{"Start":"04:12.735 ","End":"04:18.515","Text":"The first one, I\u0027m going to use a difference of squares because it\u0027s x^2 minus 3^2."},{"Start":"04:18.515 ","End":"04:23.825","Text":"It\u0027s x plus 3, x minus 3."},{"Start":"04:23.825 ","End":"04:28.769","Text":"Then this one x^2 plus 9x plus"},{"Start":"04:28.769 ","End":"04:36.690","Text":"14 factorizes to x plus 7, x plus 2."},{"Start":"04:36.690 ","End":"04:42.210","Text":"We can combine them all on one dividing line."},{"Start":"04:42.460 ","End":"04:45.109","Text":"Close this bracket."},{"Start":"04:45.109 ","End":"04:47.210","Text":"Let\u0027s see the denominator."},{"Start":"04:47.210 ","End":"04:53.949","Text":"This one, x^2 plus 4x minus 21 is x"},{"Start":"04:53.949 ","End":"05:02.215","Text":"plus 7 times x minus 3."},{"Start":"05:02.215 ","End":"05:05.110","Text":"This one x^2 plus 2,"},{"Start":"05:05.110 ","End":"05:10.000","Text":"x minus 3 is"},{"Start":"05:10.000 ","End":"05:18.800","Text":"x plus 3 times x minus 1."},{"Start":"05:18.830 ","End":"05:26.014","Text":"If you\u0027re not sure, just at least may be multiplier out in your head just to look,"},{"Start":"05:26.014 ","End":"05:28.400","Text":"for example, the last one let\u0027s just check,"},{"Start":"05:28.400 ","End":"05:30.530","Text":"x times x is x^2,"},{"Start":"05:30.530 ","End":"05:33.095","Text":"3 times minus 1 is minus 3."},{"Start":"05:33.095 ","End":"05:37.160","Text":"The middle term. Is plus 3x minus x is 2x,"},{"Start":"05:37.160 ","End":"05:39.634","Text":"so it seems all fine."},{"Start":"05:39.634 ","End":"05:43.115","Text":"Now let\u0027s do some cancellation."},{"Start":"05:43.115 ","End":"05:49.745","Text":"Canceling x plus 3 with x plus 3,"},{"Start":"05:49.745 ","End":"05:55.000","Text":"x minus 3 with x minus 3,"},{"Start":"05:55.000 ","End":"05:59.130","Text":"x plus 7 with x plus 7."},{"Start":"05:59.130 ","End":"06:02.554","Text":"Once again, we\u0027re just left with very little,"},{"Start":"06:02.554 ","End":"06:10.030","Text":"which is just x plus 2/x minus 1."},{"Start":"06:10.030 ","End":"06:12.944","Text":"That\u0027s the answer to the second one."},{"Start":"06:12.944 ","End":"06:16.630","Text":"Onto the third one."},{"Start":"06:17.820 ","End":"06:21.039","Text":"The third one, once again,"},{"Start":"06:21.039 ","End":"06:23.320","Text":"it\u0027s a division here,"},{"Start":"06:23.320 ","End":"06:27.025","Text":"so we multiply by the inverse fraction."},{"Start":"06:27.025 ","End":"06:30.654","Text":"It\u0027s x^2 minus 4x plus"},{"Start":"06:30.654 ","End":"06:38.610","Text":"3/x^2 plus 4x minus"},{"Start":"06:38.610 ","End":"06:42.104","Text":"5 is just copying, sorry, it\u0027s boring."},{"Start":"06:42.104 ","End":"06:48.369","Text":"Times than the inverse x^2"},{"Start":"06:48.369 ","End":"06:57.240","Text":"plus 45/x^2 plus 6x minus 27."},{"Start":"06:57.240 ","End":"07:04.744","Text":"Then as before, we have 4 quadratics factorize each one."},{"Start":"07:04.744 ","End":"07:07.730","Text":"General, not every quadratic factorizes,"},{"Start":"07:07.730 ","End":"07:11.284","Text":"but in this exercise they do."},{"Start":"07:11.284 ","End":"07:15.049","Text":"Remember sometimes a quadratic cannot be factorized if"},{"Start":"07:15.049 ","End":"07:19.084","Text":"the b^2 minus 4ac comes out negative or something."},{"Start":"07:19.084 ","End":"07:23.630","Text":"Anyway, here they do work out and as I say,"},{"Start":"07:23.630 ","End":"07:24.799","Text":"I\u0027m not going to do them here."},{"Start":"07:24.799 ","End":"07:29.044","Text":"It\u0027s going to give you the answers to the side."},{"Start":"07:29.044 ","End":"07:37.930","Text":"This one comes out to be x minus 1, x minus 3."},{"Start":"07:38.660 ","End":"07:45.650","Text":"We\u0027ll use one big dividing line."},{"Start":"07:45.650 ","End":"07:51.690","Text":"Next one is x"},{"Start":"07:51.690 ","End":"07:56.970","Text":"plus 9, x plus 5."},{"Start":"07:56.970 ","End":"08:05.910","Text":"This one is x plus 5, x minus 1."},{"Start":"08:05.910 ","End":"08:16.605","Text":"This one is x plus 9, x minus 3."},{"Start":"08:16.605 ","End":"08:21.900","Text":"Now let\u0027s see x minus 1 here with x minus 1"},{"Start":"08:21.900 ","End":"08:27.400","Text":"here that cancels x minus 3 here with x minus 3,"},{"Start":"08:28.550 ","End":"08:30.900","Text":"x minus 5,"},{"Start":"08:30.900 ","End":"08:33.180","Text":"with x minus 5,"},{"Start":"08:33.180 ","End":"08:37.230","Text":"and x plus 9 with x plus 9."},{"Start":"08:37.230 ","End":"08:40.465","Text":"Now there\u0027s nothing left we put a 1."},{"Start":"08:40.465 ","End":"08:45.345","Text":"The answer to this is just 1."},{"Start":"08:45.345 ","End":"08:47.950","Text":"Everything canceled."},{"Start":"08:48.960 ","End":"08:54.230","Text":"Fair enough. I suppose we could check it."},{"Start":"08:54.630 ","End":"08:58.105","Text":"I mean, one thing I could do just as a quick check."},{"Start":"08:58.105 ","End":"09:00.850","Text":"Suppose I put x=0 in this."},{"Start":"09:00.850 ","End":"09:03.490","Text":"Let\u0027s see if it\u0027s true for all x,"},{"Start":"09:03.490 ","End":"09:05.335","Text":"wherever x is, we\u0027ve got 1."},{"Start":"09:05.335 ","End":"09:07.495","Text":"Let\u0027s try x=0."},{"Start":"09:07.495 ","End":"09:14.530","Text":"We just got 3 over minus 5 I just do to the side here."},{"Start":"09:14.530 ","End":"09:17.890","Text":"Let\u0027s put x=0 just as a check."},{"Start":"09:17.890 ","End":"09:23.229","Text":"What we have is 3 over minus 5 divided"},{"Start":"09:23.229 ","End":"09:30.720","Text":"by minus 27 over 45,"},{"Start":"09:30.720 ","End":"09:34.845","Text":"which means that we have 3."},{"Start":"09:34.845 ","End":"09:38.189","Text":"Well the minus with minus cancels, Let\u0027s see."},{"Start":"09:38.189 ","End":"09:47.349","Text":"It\u0027s 3 times 45 over 5 times 27."},{"Start":"09:47.349 ","End":"09:51.380","Text":"I\u0027ve written this 1 upside down and the minuses,"},{"Start":"09:51.720 ","End":"09:54.819","Text":"well the minuses would be here and here,"},{"Start":"09:54.819 ","End":"09:57.115","Text":"but 2 minuses give me a plus."},{"Start":"09:57.115 ","End":"10:05.395","Text":"Let\u0027s see 3 times 45 is 135,"},{"Start":"10:05.395 ","End":"10:09.685","Text":"5 times 27 is 135, it\u0027s 1."},{"Start":"10:09.685 ","End":"10:12.530","Text":"At least for x=0 it works."},{"Start":"10:12.750 ","End":"10:18.979","Text":"Plausible could be that everything cancels out, then it\u0027s a constant."},{"Start":"10:19.380 ","End":"10:22.840","Text":"Onto the last one,"},{"Start":"10:22.840 ","End":"10:24.880","Text":"which is Number 4."},{"Start":"10:24.880 ","End":"10:27.205","Text":"Here we have a division,"},{"Start":"10:27.205 ","End":"10:31.220","Text":"just like with the division that was written this way."},{"Start":"10:31.440 ","End":"10:39.580","Text":"We can write it as 3 over x^2 minus 4 divided by,"},{"Start":"10:39.580 ","End":"10:43.104","Text":"we multiply by the reciprocal by the inverse,"},{"Start":"10:43.104 ","End":"10:47.320","Text":"which is x^2 plus 8x minus"},{"Start":"10:47.320 ","End":"10:54.040","Text":"20 over x plus 5."},{"Start":"10:54.040 ","End":"11:00.190","Text":"Now we can start doing some factorizing."},{"Start":"11:00.190 ","End":"11:02.890","Text":"Here we have 3,"},{"Start":"11:02.890 ","End":"11:05.155","Text":"can\u0027t do anything with that."},{"Start":"11:05.155 ","End":"11:14.395","Text":"Here we have this thing comes out to be x plus 10,"},{"Start":"11:14.395 ","End":"11:16.990","Text":"x minus 2,"},{"Start":"11:16.990 ","End":"11:19.495","Text":"w are multiplying it all together."},{"Start":"11:19.495 ","End":"11:22.420","Text":"We just get 1 dividing line."},{"Start":"11:22.420 ","End":"11:27.625","Text":"Here, we have a difference of squares, x^2 minus 2^2."},{"Start":"11:27.625 ","End":"11:29.860","Text":"It\u0027s x plus 2,"},{"Start":"11:29.860 ","End":"11:31.870","Text":"x minus 2,"},{"Start":"11:31.870 ","End":"11:34.210","Text":"and here we have another x plus 5,"},{"Start":"11:34.210 ","End":"11:36.440","Text":"put it in brackets."},{"Start":"11:36.930 ","End":"11:41.470","Text":"The only thing that cancels as far as I can see is"},{"Start":"11:41.470 ","End":"11:47.140","Text":"an x minus 2 with an x minus 2, and so,"},{"Start":"11:47.140 ","End":"11:53.124","Text":"leave the answer as 3 times x plus 10"},{"Start":"11:53.124 ","End":"12:01.060","Text":"over x plus 2, x plus 5."},{"Start":"12:01.060 ","End":"12:05.060","Text":"That would be the answer for 4 and we\u0027re done."},{"Start":"12:05.220 ","End":"12:09.429","Text":"I\u0027ve just reconsidered and I skipped over"},{"Start":"12:09.429 ","End":"12:13.450","Text":"the step of vectorization thinking that you\u0027re proficient at it."},{"Start":"12:13.450 ","End":"12:15.699","Text":"But for those who need the extra practice,"},{"Start":"12:15.699 ","End":"12:21.655","Text":"I\u0027ll now show you how I do these decompositions and I\u0027ll start with the first one."},{"Start":"12:21.655 ","End":"12:24.039","Text":"Let me just write down the general idea."},{"Start":"12:24.039 ","End":"12:26.739","Text":"When you have a quadratic,"},{"Start":"12:26.739 ","End":"12:29.169","Text":"we have a, b, and c as the coefficients,"},{"Start":"12:29.169 ","End":"12:35.109","Text":"we need to find 2 numbers whose sum is equal to b and"},{"Start":"12:35.109 ","End":"12:45.070","Text":"whose product is equal to c. Let\u0027s take the first one,"},{"Start":"12:45.070 ","End":"12:48.955","Text":"and if it\u0027s x^2 plus x minus 2,"},{"Start":"12:48.955 ","End":"12:51.594","Text":"then I want the sum."},{"Start":"12:51.594 ","End":"12:53.859","Text":"I\u0027ll write it as a sum of 2 numbers,"},{"Start":"12:53.859 ","End":"13:00.505","Text":"square plus triangle, let\u0027s say is equal to b, which is 1."},{"Start":"13:00.505 ","End":"13:08.965","Text":"Then square times triangle is equal to ac,"},{"Start":"13:08.965 ","End":"13:11.245","Text":"which is 1 times minus 2,"},{"Start":"13:11.245 ","End":"13:14.095","Text":"which is minus 2."},{"Start":"13:14.095 ","End":"13:20.304","Text":"Then it\u0027s usually best to start with the product."},{"Start":"13:20.304 ","End":"13:22.900","Text":"It has to be 1 times 2,"},{"Start":"13:22.900 ","End":"13:25.960","Text":"except that 1 of them is going to be negative to get a negative."},{"Start":"13:25.960 ","End":"13:29.800","Text":"It\u0027s either 1 and minus 2 or minus 1 and 2."},{"Start":"13:29.800 ","End":"13:31.870","Text":"Because the sum has to be 1,"},{"Start":"13:31.870 ","End":"13:41.280","Text":"it has to be equal to plus 2 and minus 1 or minus 1 and plus 2."},{"Start":"13:41.280 ","End":"13:48.250","Text":"I\u0027ll take it as, let\u0027s say 2 here and minus 1 here."},{"Start":"13:48.250 ","End":"13:52.990","Text":"Then the sum is 1 and 2 times minus 1 is minus 2."},{"Start":"13:52.990 ","End":"13:55.240","Text":"Once I have these 2 numbers,"},{"Start":"13:55.240 ","End":"13:57.805","Text":"this tells me from here,"},{"Start":"13:57.805 ","End":"14:00.474","Text":"how to decompose the middle term."},{"Start":"14:00.474 ","End":"14:06.790","Text":"I rewrite this as x^2 and Instead of plus 1x,"},{"Start":"14:06.790 ","End":"14:13.195","Text":"I write it as plus 2x minus x,"},{"Start":"14:13.195 ","End":"14:17.845","Text":"breaking up the 2 minus 1 and 2 minus 2."},{"Start":"14:17.845 ","End":"14:20.739","Text":"Then we use group decomposition every pair,"},{"Start":"14:20.739 ","End":"14:26.335","Text":"the first pair gives me x times x plus 2,"},{"Start":"14:26.335 ","End":"14:34.390","Text":"and the second gives me minus or minus 1 if you prefer times x plus 2."},{"Start":"14:34.390 ","End":"14:42.655","Text":"Altogether, what we get is we have x apples minus 1 apple,"},{"Start":"14:42.655 ","End":"14:46.600","Text":"we have x minus 1 apples."},{"Start":"14:46.600 ","End":"14:51.625","Text":"Could\u0027ve been the other way round or you could take the x plus 2 out front either way,"},{"Start":"14:51.625 ","End":"14:55.239","Text":"what we have here is what we have here."},{"Start":"14:55.239 ","End":"14:57.400","Text":"This 1 is okay,"},{"Start":"14:57.400 ","End":"15:02.510","Text":"and now let\u0027s do the next 1."},{"Start":"15:03.000 ","End":"15:14.380","Text":"The next 1 is here or here and then we have that b is equal to minus 1."},{"Start":"15:14.380 ","End":"15:24.445","Text":"AC is 1 times minus 6 is minus 6."},{"Start":"15:24.445 ","End":"15:28.975","Text":"I need the sum, something plus something."},{"Start":"15:28.975 ","End":"15:31.570","Text":"It has got to equal minus 1."},{"Start":"15:31.570 ","End":"15:34.570","Text":"The same things when multiplied,"},{"Start":"15:34.570 ","End":"15:37.405","Text":"I\u0027ve got to give me minus 6."},{"Start":"15:37.405 ","End":"15:45.249","Text":"Now, minus means that the 2 have the ignoring signs."},{"Start":"15:45.249 ","End":"15:48.205","Text":"It\u0027s going to be the 1 times 6 or 2 times 3."},{"Start":"15:48.205 ","End":"15:50.769","Text":"Obviously we want the 2 times 3 because we want"},{"Start":"15:50.769 ","End":"15:54.980","Text":"the sum or difference in this case to be 1."},{"Start":"15:55.860 ","End":"16:00.370","Text":"What we do is, we have to have the larger 1 being the negative."},{"Start":"16:00.370 ","End":"16:03.115","Text":"The 3 has to be minus and the 2 has to be plus."},{"Start":"16:03.115 ","End":"16:07.105","Text":"I can put a 2 here and a minus 3 here or the other way round."},{"Start":"16:07.105 ","End":"16:13.735","Text":"Then the sum is minus 1 and then the product does give me minus 6."},{"Start":"16:13.735 ","End":"16:15.264","Text":"Once I know this,"},{"Start":"16:15.264 ","End":"16:23.184","Text":"then this decomposition here tells me how to rewrite this minus 1 here."},{"Start":"16:23.184 ","End":"16:25.165","Text":"I\u0027m working on this 1."},{"Start":"16:25.165 ","End":"16:29.995","Text":"It\u0027s going to be instead of x^2 minus 1x."},{"Start":"16:29.995 ","End":"16:34.600","Text":"I\u0027m going to write it as plus 2x minus 3x."},{"Start":"16:34.600 ","End":"16:37.855","Text":"Then the left term to say minus 6."},{"Start":"16:37.855 ","End":"16:48.009","Text":"This will equal, using group factorization will be x times x plus 2 from the first."},{"Start":"16:48.009 ","End":"16:49.659","Text":"In the second pair,"},{"Start":"16:49.659 ","End":"16:54.295","Text":"I can take out minus 3 x plus 2,"},{"Start":"16:54.295 ","End":"16:56.740","Text":"and then I can combine these."},{"Start":"16:56.740 ","End":"16:59.260","Text":"I have x to these minus 3 of these."},{"Start":"16:59.260 ","End":"17:01.645","Text":"I have x minus 3 of these,"},{"Start":"17:01.645 ","End":"17:04.375","Text":"these being x plus 2."},{"Start":"17:04.375 ","End":"17:07.674","Text":"It looks like it\u0027s exactly what we got here."},{"Start":"17:07.674 ","End":"17:10.130","Text":"I can check mark."},{"Start":"17:10.470 ","End":"17:13.179","Text":"The third 1,"},{"Start":"17:13.179 ","End":"17:17.110","Text":"now we come to the third 1 will speed things up a bit."},{"Start":"17:17.110 ","End":"17:23.425","Text":"I always start off with this scheme plus and times,"},{"Start":"17:23.425 ","End":"17:26.410","Text":"and here I write b, in this case,"},{"Start":"17:26.410 ","End":"17:27.939","Text":"working on this polynomial,"},{"Start":"17:27.939 ","End":"17:30.400","Text":"this quadratic, it\u0027s going to be 2."},{"Start":"17:30.400 ","End":"17:38.034","Text":"The ac is going to be 1 times negative 15 is minus 15."},{"Start":"17:38.034 ","End":"17:41.360","Text":"For some reason, mostly a is always equal to 1 here,"},{"Start":"17:41.360 ","End":"17:43.120","Text":"but that\u0027s not necessarily the case,"},{"Start":"17:43.120 ","End":"17:46.019","Text":"it\u0027s just in this exercise."},{"Start":"17:46.019 ","End":"17:51.860","Text":"We start with the product because there\u0027s not so many combinations to the 15 times 1 or"},{"Start":"17:51.860 ","End":"17:57.040","Text":"5 times 3 and 1 has to be positive,"},{"Start":"17:57.040 ","End":"17:58.764","Text":"1 has to be negative."},{"Start":"17:58.764 ","End":"18:00.925","Text":"It\u0027s clearly 5 and 3."},{"Start":"18:00.925 ","End":"18:07.020","Text":"We want the positive to be the larger 1 plus 5 and minus 3."},{"Start":"18:07.020 ","End":"18:15.245","Text":"Indeed 5 and minus 3 gives 2 and 5 times minus 3 is minus 15."},{"Start":"18:15.245 ","End":"18:18.454","Text":"Now we take the middle term,"},{"Start":"18:18.454 ","End":"18:21.445","Text":"the 2 here and break it up."},{"Start":"18:21.445 ","End":"18:25.450","Text":"We get from here x^2 instead of 2x,"},{"Start":"18:25.450 ","End":"18:29.800","Text":"I write 5x minus 3x."},{"Start":"18:29.800 ","End":"18:34.250","Text":"It\u0027s the middle term that I break up according to this."},{"Start":"18:34.250 ","End":"18:38.434","Text":"Then minus 15."},{"Start":"18:38.434 ","End":"18:41.884","Text":"Then the decomposition in groups or in pairs,"},{"Start":"18:41.884 ","End":"18:49.040","Text":"x times x plus 5 minus 3 times x plus 5."},{"Start":"18:49.040 ","End":"18:57.800","Text":"Then finally, we take x minus 3 and with x plus 5 or the other way round,"},{"Start":"18:57.800 ","End":"19:00.160","Text":"whichever way you prefer."},{"Start":"19:00.160 ","End":"19:05.635","Text":"That verifies this."},{"Start":"19:05.635 ","End":"19:10.179","Text":"This one here. 3 down,"},{"Start":"19:10.179 ","End":"19:13.069","Text":"1 to go erase this."},{"Start":"19:13.170 ","End":"19:19.659","Text":"For this one, I want 2 numbers whose sum is b,"},{"Start":"19:19.659 ","End":"19:21.235","Text":"which is 3,"},{"Start":"19:21.235 ","End":"19:25.810","Text":"and the product is ac,"},{"Start":"19:25.810 ","End":"19:28.060","Text":"which is minus 4."},{"Start":"19:28.060 ","End":"19:30.250","Text":"The only things that make 4 are,"},{"Start":"19:30.250 ","End":"19:32.995","Text":"4 times 1 or 2 times 2."},{"Start":"19:32.995 ","End":"19:35.260","Text":"I want 4 and 1,"},{"Start":"19:35.260 ","End":"19:37.134","Text":"and one of them has to be negative."},{"Start":"19:37.134 ","End":"19:38.655","Text":"In order to get 3,"},{"Start":"19:38.655 ","End":"19:41.960","Text":"I need the positive 4 and a negative 1."},{"Start":"19:41.960 ","End":"19:46.600","Text":"Then 4 times minus 1 is minus 4 and 4 plus 1 is 3."},{"Start":"19:46.600 ","End":"19:51.400","Text":"So this one here, becomes x^2."},{"Start":"19:51.400 ","End":"19:52.690","Text":"Instead of 3,"},{"Start":"19:52.690 ","End":"19:56.080","Text":"I look here and say 3 is 4 minus 1,"},{"Start":"19:56.080 ","End":"20:02.380","Text":"so it\u0027s 4x minus x and then minus 4."},{"Start":"20:02.380 ","End":"20:09.739","Text":"Then in pairs, x times x plus 4 minus,"},{"Start":"20:09.739 ","End":"20:16.165","Text":"just minus so I can put a 1 in front of x plus 4 and this equals to take the x minus 1,"},{"Start":"20:16.165 ","End":"20:24.699","Text":"x minus 1 thing is x minus 1 thing is not the same order,"},{"Start":"20:24.699 ","End":"20:26.679","Text":"but it\u0027s the same factorization."},{"Start":"20:26.679 ","End":"20:28.360","Text":"So that\u0027s the fourth one."},{"Start":"20:28.360 ","End":"20:31.180","Text":"That ends the number 1,"},{"Start":"20:31.180 ","End":"20:36.075","Text":"but we still have a few more to go."},{"Start":"20:36.075 ","End":"20:38.259","Text":"So let\u0027s continue."},{"Start":"20:38.259 ","End":"20:40.930","Text":"Continuing with number 2."},{"Start":"20:40.930 ","End":"20:45.309","Text":"We have 4 polynomials here, 4 quadratics."},{"Start":"20:45.309 ","End":"20:47.980","Text":"The first one, difference of squares,"},{"Start":"20:47.980 ","End":"20:50.934","Text":"so we don\u0027t need to use the trinomial decomposition."},{"Start":"20:50.934 ","End":"20:53.304","Text":"Let\u0027s start with this one."},{"Start":"20:53.304 ","End":"20:56.515","Text":"As before, and what\u0027s written here,"},{"Start":"20:56.515 ","End":"21:00.820","Text":"we\u0027re looking for 2 numbers whose sum is b and its product is ac."},{"Start":"21:00.820 ","End":"21:06.189","Text":"So what we have is that we have something,"},{"Start":"21:06.189 ","End":"21:09.190","Text":"let\u0027s call it square plus something else,"},{"Start":"21:09.190 ","End":"21:13.975","Text":"call it triangle is equal to b, which is 4."},{"Start":"21:13.975 ","End":"21:21.220","Text":"And the product times this is equal to ac,"},{"Start":"21:21.220 ","End":"21:23.990","Text":"which is minus 21."},{"Start":"21:24.950 ","End":"21:27.419","Text":"In most of these, as you see,"},{"Start":"21:27.419 ","End":"21:28.740","Text":"a is equal to 1,"},{"Start":"21:28.740 ","End":"21:34.830","Text":"and that\u0027s just very common coincidence or it\u0027s just that it\u0027s very common."},{"Start":"21:34.830 ","End":"21:40.389","Text":"Now, once again, we have a product which is negative."},{"Start":"21:40.389 ","End":"21:43.869","Text":"So we need one positive and one negative number."},{"Start":"21:43.869 ","End":"21:49.195","Text":"The only way to get 21 is either 1 times 21 or 3 times 7."},{"Start":"21:49.195 ","End":"21:53.545","Text":"Obviously we need a 3 times 7 and we can already think the difference is 4."},{"Start":"21:53.545 ","End":"21:58.285","Text":"So what we want is a plus 7 and a minus 3,"},{"Start":"21:58.285 ","End":"22:05.290","Text":"then the sum will be 4 and 7 times minus 3 will be minus 21."},{"Start":"22:05.290 ","End":"22:10.599","Text":"This is the key to decomposing the middle term, the full here."},{"Start":"22:10.599 ","End":"22:16.929","Text":"So we rewrite that as x^2 instead of plus"},{"Start":"22:16.929 ","End":"22:25.210","Text":"4x plus 7x minus 3x minus 21."},{"Start":"22:25.210 ","End":"22:31.555","Text":"Then this we do by group decomposition in pairs."},{"Start":"22:31.555 ","End":"22:39.190","Text":"Group factorization of whatever x times x plus 7 and then minus 3,"},{"Start":"22:39.190 ","End":"22:42.560","Text":"also x plus 7."},{"Start":"22:42.630 ","End":"22:47.934","Text":"Finally, we just collect together the like terms."},{"Start":"22:47.934 ","End":"22:50.620","Text":"Doesn\u0027t really matter which order."},{"Start":"22:50.620 ","End":"22:54.079","Text":"Let\u0027s take the x plus 7 out."},{"Start":"22:54.390 ","End":"23:02.245","Text":"Then what we\u0027re left with is x and minus 3 and that is exactly what we have here."},{"Start":"23:02.245 ","End":"23:06.790","Text":"That\u0027s one down, 2 to go next one."},{"Start":"23:06.790 ","End":"23:08.589","Text":"That\u0027s this one here."},{"Start":"23:08.589 ","End":"23:10.359","Text":"So we want the sum."},{"Start":"23:10.359 ","End":"23:14.259","Text":"This time instead of a square and a triangle,"},{"Start":"23:14.259 ","End":"23:16.930","Text":"I\u0027ll just put underlines."},{"Start":"23:16.930 ","End":"23:19.300","Text":"So the sum is equal to b,"},{"Start":"23:19.300 ","End":"23:25.060","Text":"which is 2 and the product this times this is going to equal ac,"},{"Start":"23:25.060 ","End":"23:27.685","Text":"which is minus 3."},{"Start":"23:27.685 ","End":"23:32.880","Text":"The only way to get 2 numbers whose product is 3 is 1 and 3,"},{"Start":"23:32.880 ","End":"23:35.429","Text":"and one is positive, one is negative."},{"Start":"23:35.429 ","End":"23:40.180","Text":"So the 3 is going to be positive and the minus 1,"},{"Start":"23:40.180 ","End":"23:45.460","Text":"it\u0027ll be negative in order to get the 2 and 3 times minus 1 is indeed minus 3."},{"Start":"23:45.460 ","End":"23:46.780","Text":"So that this thing,"},{"Start":"23:46.780 ","End":"23:52.375","Text":"we decompose the middle term according to 3 minus 1."},{"Start":"23:52.375 ","End":"23:55.465","Text":"So we get x^2,"},{"Start":"23:55.465 ","End":"23:56.950","Text":"instead of the 2x,"},{"Start":"23:56.950 ","End":"24:03.130","Text":"3x minus x or minus 1x minus 3."},{"Start":"24:03.130 ","End":"24:06.174","Text":"From here we take out the x from the first pair,"},{"Start":"24:06.174 ","End":"24:07.765","Text":"that\u0027s x plus 3,"},{"Start":"24:07.765 ","End":"24:10.120","Text":"and here we just take out the minus,"},{"Start":"24:10.120 ","End":"24:12.815","Text":"it\u0027s x plus 3."},{"Start":"24:12.815 ","End":"24:14.879","Text":"Just to help us see things,"},{"Start":"24:14.879 ","End":"24:16.425","Text":"let\u0027s put a 1 here."},{"Start":"24:16.425 ","End":"24:18.750","Text":"So we get x plus 3,"},{"Start":"24:18.750 ","End":"24:22.395","Text":"x minus 1 or the other way round."},{"Start":"24:22.395 ","End":"24:26.204","Text":"I\u0027ll take the x plus 3 in front this"},{"Start":"24:26.204 ","End":"24:33.015","Text":"time and we\u0027re left with from here x from here minus 1."},{"Start":"24:33.015 ","End":"24:42.340","Text":"That is indeed what we have here."},{"Start":"24:42.340 ","End":"24:44.260","Text":"I got confused for a moment because it\u0027s now"},{"Start":"24:44.260 ","End":"24:46.525","Text":"in the denominator because this was a division."},{"Start":"24:46.525 ","End":"24:51.430","Text":"So this here became this 2 here, and that\u0027s that."},{"Start":"24:51.430 ","End":"24:55.914","Text":"The last one is this one which comes out over here."},{"Start":"24:55.914 ","End":"24:59.200","Text":"It\u0027s the 9 that we want."},{"Start":"24:59.200 ","End":"25:01.120","Text":"Let me just clear the board."},{"Start":"25:01.120 ","End":"25:03.564","Text":"Now we want this one,"},{"Start":"25:03.564 ","End":"25:06.740","Text":"which is the one over here."},{"Start":"25:06.900 ","End":"25:09.385","Text":"Just for a change,"},{"Start":"25:09.385 ","End":"25:13.270","Text":"let me do it using another technique for factorization."},{"Start":"25:13.270 ","End":"25:15.430","Text":"Just so you don\u0027t think this is the only way."},{"Start":"25:15.430 ","End":"25:16.645","Text":"There are several ways."},{"Start":"25:16.645 ","End":"25:20.919","Text":"The other main one is solving the equation and using the roots."},{"Start":"25:20.919 ","End":"25:29.739","Text":"So the idea for the other technique is if you have ax^2 plus bx plus c,"},{"Start":"25:29.739 ","End":"25:37.930","Text":"and you solve the equation when it\u0027s equal to 0 and you get 2 roots, x1 and x2."},{"Start":"25:37.930 ","End":"25:44.080","Text":"Let\u0027s call them. Then this factorizes as ax minus x1,"},{"Start":"25:44.080 ","End":"25:48.850","Text":"x minus x2 and a is usually 1,"},{"Start":"25:48.850 ","End":"25:51.610","Text":"or at least in our case it is, though not necessarily."},{"Start":"25:51.610 ","End":"25:54.130","Text":"Let\u0027s do it just for a change this way."},{"Start":"25:54.130 ","End":"26:00.369","Text":"We solve the quadratic equation and let this thing equal 0."},{"Start":"26:00.369 ","End":"26:07.150","Text":"So we get x^2 plus 9x plus 14 equals 0."},{"Start":"26:07.150 ","End":"26:10.149","Text":"Then we\u0027ll use the formula to solve it."},{"Start":"26:10.149 ","End":"26:18.160","Text":"So we get x is equal to minus b plus or minus the square root of b^2,"},{"Start":"26:18.160 ","End":"26:25.704","Text":"which is 81 minus 4 times 1 times c, which is 14."},{"Start":"26:25.704 ","End":"26:30.430","Text":"All this over 2a, which is 2."},{"Start":"26:30.430 ","End":"26:33.640","Text":"4 times 14 is 56,"},{"Start":"26:33.640 ","End":"26:36.654","Text":"56 from 81 is 25."},{"Start":"26:36.654 ","End":"26:40.119","Text":"So what we have is 9 plus or minus root 25,"},{"Start":"26:40.119 ","End":"26:43.660","Text":"which is 5 over 2."},{"Start":"26:43.660 ","End":"26:47.500","Text":"Then this gives us 2 solutions."},{"Start":"26:47.500 ","End":"26:52.255","Text":"Minus 9 plus 5 over 2 is minus 4 over 2,"},{"Start":"26:52.255 ","End":"26:55.765","Text":"which is minus 2."},{"Start":"26:55.765 ","End":"26:57.550","Text":"If we take the minus,"},{"Start":"26:57.550 ","End":"26:59.710","Text":"we\u0027ve got minus 14 over 2,"},{"Start":"26:59.710 ","End":"27:01.405","Text":"which is minus 7."},{"Start":"27:01.405 ","End":"27:08.604","Text":"One of them will be x1 and the other one will be x2, doesn\u0027t really matter."},{"Start":"27:08.604 ","End":"27:12.430","Text":"Ultimately what we want to do to factorize this is to"},{"Start":"27:12.430 ","End":"27:19.180","Text":"write a which is 1 times x minus x1,"},{"Start":"27:19.180 ","End":"27:24.220","Text":"x minus, minus 2 is x plus 2."},{"Start":"27:24.220 ","End":"27:26.500","Text":"Same in the second x minus,"},{"Start":"27:26.500 ","End":"27:30.445","Text":"minus 7 is x plus 7."},{"Start":"27:30.445 ","End":"27:33.114","Text":"We can just get rid of the one."},{"Start":"27:33.114 ","End":"27:35.560","Text":"This is what we have here, well,"},{"Start":"27:35.560 ","End":"27:36.729","Text":"in the reverse order,"},{"Start":"27:36.729 ","End":"27:39.139","Text":"so third one checked."},{"Start":"27:39.330 ","End":"27:43.555","Text":"Next we\u0027ll move on to number 3."},{"Start":"27:43.555 ","End":"27:47.649","Text":"I\u0027ll just scroll down so we can see something."},{"Start":"27:47.649 ","End":"27:51.950","Text":"What\u0027s going on here?"},{"Start":"27:52.830 ","End":"27:57.115","Text":"Now this one is next in line."},{"Start":"27:57.115 ","End":"28:02.290","Text":"I\u0027ll just write mnemonic that we want 2 numbers"},{"Start":"28:02.290 ","End":"28:09.490","Text":"whose sum is b and whose product is ac."},{"Start":"28:09.490 ","End":"28:13.299","Text":"That tells us how to decompose the middle term,"},{"Start":"28:13.299 ","End":"28:15.385","Text":"this minus 4 here."},{"Start":"28:15.385 ","End":"28:18.219","Text":"We want something, sum means something plus"},{"Start":"28:18.219 ","End":"28:23.020","Text":"something that has to equal b which is minus 4."},{"Start":"28:23.020 ","End":"28:26.410","Text":"The product this times this,"},{"Start":"28:26.410 ","End":"28:34.940","Text":"it\u0027s got to equal ac which is 1 times 3 which is 3."},{"Start":"28:36.390 ","End":"28:41.469","Text":"Product is 3 means they\u0027re both positive or both negative."},{"Start":"28:41.469 ","End":"28:45.999","Text":"In this case we want both negative in order to get a negative sum,"},{"Start":"28:45.999 ","End":"28:49.090","Text":"so it\u0027s clearly minus 3 and minus 1,"},{"Start":"28:49.090 ","End":"28:50.949","Text":"and it doesn\u0027t matter in what order."},{"Start":"28:50.949 ","End":"28:55.555","Text":"We can say that minus 3 times minus 1 is 3,"},{"Start":"28:55.555 ","End":"28:59.335","Text":"and minus 3 plus minus 1 is minus 4."},{"Start":"28:59.335 ","End":"29:02.320","Text":"We\u0027re all right, so this tells us how to decompose this."},{"Start":"29:02.320 ","End":"29:06.130","Text":"This becomes now x^2 instead of minus 4x,"},{"Start":"29:06.130 ","End":"29:14.025","Text":"I\u0027ll write minus 3x minus x according to these numbers and it\u0027s still plus 3."},{"Start":"29:14.025 ","End":"29:18.014","Text":"Then group factoring."},{"Start":"29:18.014 ","End":"29:27.970","Text":"So x(x-3) minus x minus 3."},{"Start":"29:27.970 ","End":"29:29.049","Text":"Then this is equal to,"},{"Start":"29:29.049 ","End":"29:31.930","Text":"we stick a 1 here to make it clearer."},{"Start":"29:31.930 ","End":"29:36.090","Text":"It\u0027s equal to x-1, x-3,"},{"Start":"29:36.090 ","End":"29:38.770","Text":"or x-3, x-1,"},{"Start":"29:38.770 ","End":"29:41.604","Text":"depending on how you want to factor it, doesn\u0027t matter."},{"Start":"29:41.604 ","End":"29:45.939","Text":"That verifies this one here."},{"Start":"29:45.939 ","End":"29:53.620","Text":"Next one will be this one here."},{"Start":"29:53.620 ","End":"29:55.660","Text":"Let me just clear this."},{"Start":"29:55.660 ","End":"29:58.060","Text":"Decomposing this one now."},{"Start":"29:58.060 ","End":"30:03.850","Text":"Sum b, the b is 4 something plus something is 4,"},{"Start":"30:03.850 ","End":"30:09.530","Text":"and the something times something is ac which is minus 5."},{"Start":"30:09.540 ","End":"30:12.699","Text":"Look at the product first, it\u0027s negative,"},{"Start":"30:12.699 ","End":"30:17.245","Text":"so we have a plus and a minus and it has to add together to give 4."},{"Start":"30:17.245 ","End":"30:22.000","Text":"The only way that\u0027s going to happen is if we have a plus 5 and a minus 1,"},{"Start":"30:22.000 ","End":"30:25.330","Text":"and then the product is minus 5,"},{"Start":"30:25.330 ","End":"30:27.010","Text":"and the sum is 4."},{"Start":"30:27.010 ","End":"30:31.225","Text":"This tells us how to decompose this 4."},{"Start":"30:31.225 ","End":"30:40.405","Text":"We can rewrite it now as x^2 plus 5x minus x minus 5,"},{"Start":"30:40.405 ","End":"30:44.215","Text":"and that equals, take out x from the first pair."},{"Start":"30:44.215 ","End":"30:49.764","Text":"Take out the minus from the second pair,"},{"Start":"30:49.764 ","End":"30:53.229","Text":"writing the one so it\u0027s easier to see."},{"Start":"30:53.229 ","End":"30:55.180","Text":"Then we have x-1,"},{"Start":"30:55.180 ","End":"31:00.340","Text":"x+5, whichever way you want to do it."},{"Start":"31:00.340 ","End":"31:03.680","Text":"Let\u0027s say we have x of these minus 1 of these,"},{"Start":"31:03.680 ","End":"31:06.354","Text":"we have x minus 1of these."},{"Start":"31:06.354 ","End":"31:09.400","Text":"If you wanted it in the other order,"},{"Start":"31:09.400 ","End":"31:10.780","Text":"then you would say, okay,"},{"Start":"31:10.780 ","End":"31:14.470","Text":"take x+5 out front and then we\u0027d get it exactly as here,"},{"Start":"31:14.470 ","End":"31:16.720","Text":"but the order doesn\u0027t matter,"},{"Start":"31:16.720 ","End":"31:18.505","Text":"so we\u0027ve done with the second one."},{"Start":"31:18.505 ","End":"31:20.740","Text":"Now onto this one."},{"Start":"31:20.740 ","End":"31:22.810","Text":"Next is this one,"},{"Start":"31:22.810 ","End":"31:26.004","Text":"which is actually the one down here."},{"Start":"31:26.004 ","End":"31:30.370","Text":"We need a sum of b,"},{"Start":"31:30.370 ","End":"31:31.630","Text":"which is 6,"},{"Start":"31:31.630 ","End":"31:37.584","Text":"and we need a product of ac which is minus 27."},{"Start":"31:37.584 ","End":"31:45.010","Text":"We need something positive and something negative and the sum has to be 6."},{"Start":"31:45.010 ","End":"31:48.669","Text":"It\u0027s a difference because it\u0027s a plus and a minus."},{"Start":"31:48.669 ","End":"31:50.785","Text":"Well, we can see that it\u0027s 9 and 3."},{"Start":"31:50.785 ","End":"31:54.085","Text":"The only other possibility is 27 times 1 is not going to help."},{"Start":"31:54.085 ","End":"31:57.295","Text":"A 9 and a 3 and just have to arrange the signs right."},{"Start":"31:57.295 ","End":"32:03.265","Text":"Let\u0027s take it as 9 times minus 3 is minus 27,"},{"Start":"32:03.265 ","End":"32:06.700","Text":"9 plus minus 3 is 6."},{"Start":"32:06.700 ","End":"32:13.840","Text":"This tells us how to decompose this and so we get x^2 instead of plus 6x,"},{"Start":"32:13.840 ","End":"32:19.285","Text":"we write it as 9x minus 3x, minus 27 stays."},{"Start":"32:19.285 ","End":"32:24.325","Text":"Then x out of the first pair x+9."},{"Start":"32:24.325 ","End":"32:28.465","Text":"Then minus 3 comes out of the second pair,"},{"Start":"32:28.465 ","End":"32:31.630","Text":"so it\u0027s x+9,"},{"Start":"32:31.630 ","End":"32:36.745","Text":"and then we just combine, sometimes just keep the order as the x-3."},{"Start":"32:36.745 ","End":"32:41.154","Text":"The second one as is x+9, and let\u0027s see."},{"Start":"32:41.154 ","End":"32:45.280","Text":"Yeah, the opposite order but that\u0027s what we got."},{"Start":"32:45.280 ","End":"32:49.450","Text":"One more to go is this one, where this one."},{"Start":"32:49.450 ","End":"32:55.000","Text":"This time we want something plus something to be 14,"},{"Start":"32:55.000 ","End":"33:04.315","Text":"the b and then the same something times the other something is going to equal 45."},{"Start":"33:04.315 ","End":"33:07.790","Text":"We need 2 positive numbers."},{"Start":"33:09.980 ","End":"33:13.770","Text":"There\u0027s a few ways to break up 45,"},{"Start":"33:13.770 ","End":"33:17.654","Text":"1 times 45 or 3 times 15,"},{"Start":"33:17.654 ","End":"33:21.430","Text":"or 5 times 9,"},{"Start":"33:21.430 ","End":"33:27.700","Text":"and the one that\u0027s going to do the trick is 5 times 9 because 5 plus 9 is 14."},{"Start":"33:27.700 ","End":"33:29.980","Text":"Let\u0027s make it 5 and 9."},{"Start":"33:29.980 ","End":"33:40.450","Text":"Then this thing here decomposes x^2+5x plus 9x plus 45,"},{"Start":"33:40.450 ","End":"33:45.115","Text":"and this is equal to x times x+5."},{"Start":"33:45.115 ","End":"33:49.644","Text":"Second pair, 9 times x+5, and finally,"},{"Start":"33:49.644 ","End":"33:52.105","Text":"in either order you can take the x+5 first,"},{"Start":"33:52.105 ","End":"33:54.475","Text":"and we see the other factor is x+9."},{"Start":"33:54.475 ","End":"33:58.435","Text":"I\u0027ll write the x+9 first and x+5."},{"Start":"33:58.435 ","End":"34:00.040","Text":"This is what we have here,"},{"Start":"34:00.040 ","End":"34:01.090","Text":"even in the same order,"},{"Start":"34:01.090 ","End":"34:03.969","Text":"so check both of them."},{"Start":"34:03.969 ","End":"34:06.190","Text":"Okay. That\u0027s this one."},{"Start":"34:06.190 ","End":"34:09.860","Text":"Then we come to number 4 next."},{"Start":"34:09.960 ","End":"34:12.535","Text":"In number 4,"},{"Start":"34:12.535 ","End":"34:15.969","Text":"when we look here there was only one decomposition."},{"Start":"34:15.969 ","End":"34:19.090","Text":"There was this one too but that was the difference of squares,"},{"Start":"34:19.090 ","End":"34:20.590","Text":"which just this on,"},{"Start":"34:20.590 ","End":"34:25.749","Text":"and once in awhile for a change I do it with the other method with finding the roots."},{"Start":"34:25.749 ","End":"34:30.745","Text":"Which basically means we solve the equation that this is equal to 0,"},{"Start":"34:30.745 ","End":"34:34.375","Text":"the quadratic, and then we solve x1 and x2."},{"Start":"34:34.375 ","End":"34:41.829","Text":"We get that x is minus b plus or"},{"Start":"34:41.829 ","End":"34:49.795","Text":"minus the square root of b^2 is 64 minus 4ac,"},{"Start":"34:49.795 ","End":"34:55.659","Text":"minus 4 times 1 times minus 20."},{"Start":"34:55.659 ","End":"35:00.640","Text":"All this over 2a which is 2."},{"Start":"35:00.640 ","End":"35:03.859","Text":"Let\u0027s compute what\u0027s under the square root sign,"},{"Start":"35:03.859 ","End":"35:07.756","Text":"4 times 20 is 80 and it\u0027s plus 80,"},{"Start":"35:07.756 ","End":"35:10.359","Text":"80 and 64 is 144."},{"Start":"35:10.359 ","End":"35:12.939","Text":"We get minus 8 plus or minus."},{"Start":"35:12.939 ","End":"35:18.505","Text":"The square root of a 144 is 12 over 2,"},{"Start":"35:18.505 ","End":"35:20.140","Text":"and that gives us 2 possibilities."},{"Start":"35:20.140 ","End":"35:25.075","Text":"If I take the plus it\u0027s 4 over 2, which is 2."},{"Start":"35:25.075 ","End":"35:26.590","Text":"If I take the minus,"},{"Start":"35:26.590 ","End":"35:29.830","Text":"I\u0027ve got minus 20 over 2,"},{"Start":"35:29.830 ","End":"35:33.700","Text":"which is minus 10."},{"Start":"35:33.700 ","End":"35:44.515","Text":"Then the factorization remember is a which is 1 times X minus, officially it\u0027s 1."},{"Start":"35:44.515 ","End":"35:46.914","Text":"Then x-1 one other roots,"},{"Start":"35:46.914 ","End":"35:49.345","Text":"which is minus 2,"},{"Start":"35:49.345 ","End":"35:51.640","Text":"and x minus the other root,"},{"Start":"35:51.640 ","End":"35:55.885","Text":"minus 10 makes it plus 10."},{"Start":"35:55.885 ","End":"35:57.700","Text":"You can ignore the 1,"},{"Start":"35:57.700 ","End":"36:02.965","Text":"and this is what we have here in the other order, check."},{"Start":"36:02.965 ","End":"36:06.679","Text":"Done with this exercise."}],"ID":5048}],"Thumbnail":null,"ID":3859},{"Name":"Complex Numbers","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Introduction","Duration":"7m 51s","ChapterTopicVideoID":4925,"CourseChapterTopicPlaylistID":3860,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.960","Text":"We\u0027re going to start a new subject called complex numbers."},{"Start":"00:03.960 ","End":"00:09.360","Text":"We\u0027re going to extend the whole concept of number as we know it up till now."},{"Start":"00:09.360 ","End":"00:13.710","Text":"Up to now, the only kind of numbers we came across were"},{"Start":"00:13.710 ","End":"00:18.360","Text":"called real numbers and they are numbers on the number line,"},{"Start":"00:18.360 ","End":"00:23.785","Text":"which include whole numbers, negative numbers, fractions,"},{"Start":"00:23.785 ","End":"00:31.010","Text":"and even irrational numbers like Pi and the square root of 2 and e and so on."},{"Start":"00:31.010 ","End":"00:33.365","Text":"But they\u0027re all on the number line."},{"Start":"00:33.365 ","End":"00:36.400","Text":"Now we\u0027re going to go outside the box,"},{"Start":"00:36.400 ","End":"00:38.990","Text":"so beyond the number line."},{"Start":"00:38.990 ","End":"00:42.680","Text":"The problem begins that we\u0027re limited when we"},{"Start":"00:42.680 ","End":"00:46.265","Text":"don\u0027t know what the square root of a negative number is."},{"Start":"00:46.265 ","End":"00:49.415","Text":"For example, what is the square root of minus 1?"},{"Start":"00:49.415 ","End":"00:52.430","Text":"Up till now, we said,"},{"Start":"00:52.430 ","End":"00:55.415","Text":"\u0027\u0027Well, we can\u0027t have a square root of a negative number.\u0027\u0027"},{"Start":"00:55.415 ","End":"01:04.085","Text":"But supposing that we invent a new entity and we give it a name,"},{"Start":"01:04.085 ","End":"01:06.800","Text":"call it the letter i, and say,"},{"Start":"01:06.800 ","End":"01:10.820","Text":"the square root of minus 1 is i. i is a new number."},{"Start":"01:10.820 ","End":"01:12.545","Text":"It\u0027s not on the number line."},{"Start":"01:12.545 ","End":"01:13.759","Text":"It\u0027s not a variable,"},{"Start":"01:13.759 ","End":"01:15.920","Text":"even though it is a letter,"},{"Start":"01:15.920 ","End":"01:17.501","Text":"it\u0027s a number,"},{"Start":"01:17.501 ","End":"01:19.820","Text":"and we say it\u0027s the square root of minus 1."},{"Start":"01:19.820 ","End":"01:22.475","Text":"If we continue with this,"},{"Start":"01:22.475 ","End":"01:28.610","Text":"we can develop a whole mathematics which is consistent and which is actually useful,"},{"Start":"01:28.610 ","End":"01:30.815","Text":"but that the use comes later on."},{"Start":"01:30.815 ","End":"01:33.890","Text":"Meanwhile, let\u0027s just take this as an axiom of"},{"Start":"01:33.890 ","End":"01:37.415","Text":"faith and say that there is a square root of minus 1."},{"Start":"01:37.415 ","End":"01:42.695","Text":"It\u0027s a new number and it\u0027s called i. i stands by the way, for imaginary."},{"Start":"01:42.695 ","End":"01:45.170","Text":"We\u0027re actually going to develop, first of all,"},{"Start":"01:45.170 ","End":"01:50.660","Text":"imaginary numbers, and then we\u0027ll go onto complex numbers."},{"Start":"01:50.660 ","End":"01:57.320","Text":"Imaginary numbers are something times i."},{"Start":"01:57.320 ","End":"02:03.815","Text":"3i would be an imaginary number,"},{"Start":"02:03.815 ","End":"02:09.755","Text":"minus 8i, 1.74i."},{"Start":"02:09.755 ","End":"02:13.670","Text":"Even, say square root of 2 times i,"},{"Start":"02:13.670 ","End":"02:18.110","Text":"where the number here is a real number."},{"Start":"02:18.110 ","End":"02:23.970","Text":"The numbers that we know up to now and all these are called imaginary numbers."},{"Start":"02:24.220 ","End":"02:26.795","Text":"They can be drawn."},{"Start":"02:26.795 ","End":"02:28.130","Text":"There is a diagram,"},{"Start":"02:28.130 ","End":"02:32.720","Text":"but I don\u0027t feel it\u0027s necessary to do that."},{"Start":"02:32.720 ","End":"02:36.650","Text":"They actually can go from the number line to the number plane,"},{"Start":"02:36.650 ","End":"02:40.096","Text":"and we might do that later on but meanwhile not."},{"Start":"02:40.096 ","End":"02:45.590","Text":"One of the things that this gives us is that we can now solve"},{"Start":"02:45.590 ","End":"02:52.845","Text":"the equations like x squared equals minus 1."},{"Start":"02:52.845 ","End":"02:59.510","Text":"If x squared is minus 1 then just like we would do with real numbers, we say,"},{"Start":"02:59.510 ","End":"03:05.737","Text":"the next is plus or minus the square root of minus 1,"},{"Start":"03:05.737 ","End":"03:11.855","Text":"and then we can say that x is plus or minus I. Yeah."},{"Start":"03:11.855 ","End":"03:16.200","Text":"Minus i is also an imaginary number, it\u0027s minus 1i."},{"Start":"03:17.410 ","End":"03:23.825","Text":"For example, if I wanted to know x squared is minus 9,"},{"Start":"03:23.825 ","End":"03:31.160","Text":"then I could say that x is plus or minus the square root of minus 9."},{"Start":"03:31.160 ","End":"03:34.040","Text":"If the usual rules are still going to hold,"},{"Start":"03:34.040 ","End":"03:36.709","Text":"this is going to be plus or minus."},{"Start":"03:36.709 ","End":"03:41.330","Text":"Since minus 9 is 9 times i will"},{"Start":"03:41.330 ","End":"03:46.280","Text":"get plus or minus the square root of 9 times the square root of minus 1,"},{"Start":"03:46.280 ","End":"03:49.750","Text":"which is plus or minus 3i."},{"Start":"03:49.750 ","End":"03:52.520","Text":"So much for imaginary numbers."},{"Start":"03:52.520 ","End":"03:56.645","Text":"Now, let\u0027s introduce complex numbers."},{"Start":"03:56.645 ","End":"04:06.395","Text":"Now, complex numbers are just a combination of a real number plus an imaginary number."},{"Start":"04:06.395 ","End":"04:12.740","Text":"For example, 3 plus 4i is a complex number,"},{"Start":"04:12.740 ","End":"04:14.435","Text":"a typical complex number."},{"Start":"04:14.435 ","End":"04:17.464","Text":"In general, a complex number,"},{"Start":"04:17.464 ","End":"04:22.850","Text":"we could define as a plus bi is"},{"Start":"04:22.850 ","End":"04:29.180","Text":"the general complex number where a and b are real numbers,"},{"Start":"04:29.180 ","End":"04:32.160","Text":"the numbers that we know up to now,"},{"Start":"04:32.450 ","End":"04:42.200","Text":"and i has the property that i equals the square root of minus 1,"},{"Start":"04:42.200 ","End":"04:44.150","Text":"and that\u0027s complex numbers."},{"Start":"04:44.150 ","End":"04:47.360","Text":"Now, let\u0027s learn how to work with them."},{"Start":"04:47.360 ","End":"04:49.695","Text":"How to add, multiply, subtract,"},{"Start":"04:49.695 ","End":"04:53.195","Text":"divide, take exponents of, and so on."},{"Start":"04:53.195 ","End":"04:58.020","Text":"Let\u0027s do that one operation at a time."},{"Start":"04:58.090 ","End":"05:03.830","Text":"Before that, this thing is so important that I feel I\u0027ve got a highlight it."},{"Start":"05:03.830 ","End":"05:07.190","Text":"I\u0027m also going to highlight the fact that,"},{"Start":"05:07.190 ","End":"05:12.965","Text":"in other words, this says that i squared is minus 1."},{"Start":"05:12.965 ","End":"05:16.044","Text":"Let\u0027s also highlight that."},{"Start":"05:16.044 ","End":"05:17.660","Text":"That\u0027s the basis for everything."},{"Start":"05:17.660 ","End":"05:21.275","Text":"A complex number is a plus bi,"},{"Start":"05:21.275 ","End":"05:28.400","Text":"where i is this new mathematical entity and let\u0027s just go along with it."},{"Start":"05:28.400 ","End":"05:32.480","Text":"People used to say that negative numbers don\u0027t exist."},{"Start":"05:32.480 ","End":"05:36.170","Text":"They were almost imaginary but we got used to"},{"Start":"05:36.170 ","End":"05:40.790","Text":"them and negative numbers seem to you just as valid as any other."},{"Start":"05:40.790 ","End":"05:45.590","Text":"After a while, we\u0027ll get used to the fact that there\u0027s a new kind of number called i."},{"Start":"05:45.590 ","End":"05:49.969","Text":"The name imaginary stuck,"},{"Start":"05:49.969 ","End":"05:54.785","Text":"but it doesn\u0027t mean that they are really imaginary."},{"Start":"05:54.785 ","End":"05:56.330","Text":"It depends on your perspective."},{"Start":"05:56.330 ","End":"06:00.620","Text":"Anyway, let\u0027s start with addition and subtraction."},{"Start":"06:00.620 ","End":"06:05.840","Text":"Addition and subtraction work in the obvious way that you might expect."},{"Start":"06:05.840 ","End":"06:07.895","Text":"Let\u0027s just make up an example."},{"Start":"06:07.895 ","End":"06:17.565","Text":"Suppose I have one complex number is 3 plus 4i just like I took it from here."},{"Start":"06:17.565 ","End":"06:20.265","Text":"I\u0027ll just put it in brackets for emphasis."},{"Start":"06:20.265 ","End":"06:22.550","Text":"I want to add another complex number."},{"Start":"06:22.550 ","End":"06:27.655","Text":"Let\u0027s say 7 minus 2i."},{"Start":"06:27.655 ","End":"06:31.695","Text":"You do what you would expect to do where you would just"},{"Start":"06:31.695 ","End":"06:36.870","Text":"add the 3 to the 7 and say that\u0027s 10."},{"Start":"06:36.870 ","End":"06:40.005","Text":"Then 4i minus 2i,"},{"Start":"06:40.005 ","End":"06:41.895","Text":"4 minus 2 is 2,"},{"Start":"06:41.895 ","End":"06:45.645","Text":"so it\u0027s plus 2i."},{"Start":"06:45.645 ","End":"06:49.240","Text":"Same thing works for subtraction."},{"Start":"06:49.240 ","End":"06:52.205","Text":"Just make an example up."},{"Start":"06:52.205 ","End":"06:56.225","Text":"Minus 5, I don\u0027t know plus 2i."},{"Start":"06:56.225 ","End":"07:01.685","Text":"Let say less minus"},{"Start":"07:01.685 ","End":"07:11.340","Text":"8 minus 7i."},{"Start":"07:11.340 ","End":"07:12.720","Text":"Again, just the obvious."},{"Start":"07:12.720 ","End":"07:19.170","Text":"Minus 5 take away minus 8 is minus 5 plus 8 is 3,"},{"Start":"07:19.170 ","End":"07:24.915","Text":"and 2i minus minus 7i is 2i plus 7i."},{"Start":"07:24.915 ","End":"07:28.595","Text":"This is the answer, just the obvious."},{"Start":"07:28.595 ","End":"07:33.515","Text":"I\u0027m not going to write a general rule for a plus bi plus c plus di."},{"Start":"07:33.515 ","End":"07:38.120","Text":"I could, but it\u0027s just the obvious thing to do."},{"Start":"07:38.120 ","End":"07:44.060","Text":"Let\u0027s take a break now and after the break,"},{"Start":"07:44.060 ","End":"07:51.120","Text":"we\u0027ll continue with the next topic which is multiplication of complex numbers."}],"ID":4918},{"Watched":false,"Name":"Multiplication","Duration":"7m 22s","ChapterTopicVideoID":4926,"CourseChapterTopicPlaylistID":3860,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.620","Text":"Here we are after the break."},{"Start":"00:01.620 ","End":"00:03.990","Text":"We are continuing with complex numbers."},{"Start":"00:03.990 ","End":"00:07.590","Text":"We learned what they are and we learned about addition and"},{"Start":"00:07.590 ","End":"00:11.520","Text":"subtraction and we\u0027re about to do multiplication."},{"Start":"00:11.520 ","End":"00:17.080","Text":"Is just something I forgot to mention earlier that when we write a complex number,"},{"Start":"00:17.080 ","End":"00:19.240","Text":"a plus bi,"},{"Start":"00:19.240 ","End":"00:21.455","Text":"then there\u0027s a name."},{"Start":"00:21.455 ","End":"00:31.530","Text":"This part here is called the real part and the bi part is called the imaginary part."},{"Start":"00:31.930 ","End":"00:37.115","Text":"Another thing is that when we work with real numbers,"},{"Start":"00:37.115 ","End":"00:40.910","Text":"the typical variable is the letter x."},{"Start":"00:40.910 ","End":"00:43.279","Text":"But when we work with complex numbers,"},{"Start":"00:43.279 ","End":"00:46.100","Text":"then we would use the letter z."},{"Start":"00:46.100 ","End":"00:52.615","Text":"You might say that z is equal to a plus bi."},{"Start":"00:52.615 ","End":"00:57.215","Text":"Z posed to x gives us a hint that we\u0027re talking about complex numbers."},{"Start":"00:57.215 ","End":"01:00.580","Text":"Now about that multiplication. Let\u0027s see."},{"Start":"01:00.580 ","End":"01:05.125","Text":"Let\u0027s start off with something easy."},{"Start":"01:05.125 ","End":"01:15.130","Text":"Let\u0027s say we have 4 times 3 plus 2i."},{"Start":"01:15.130 ","End":"01:17.530","Text":"This is a real number,"},{"Start":"01:17.530 ","End":"01:19.315","Text":"which is also a complex number,"},{"Start":"01:19.315 ","End":"01:21.085","Text":"times another complex number."},{"Start":"01:21.085 ","End":"01:23.430","Text":"In this case, it\u0027s just straightforward,"},{"Start":"01:23.430 ","End":"01:32.160","Text":"4 times 3 is 12 and 4 times 2i is 8i so the answer is 12 plus 8i."},{"Start":"01:32.160 ","End":"01:33.550","Text":"Let\u0027s make it a little bit more difficult."},{"Start":"01:33.550 ","End":"01:36.985","Text":"Let\u0027s take an imaginary times a general complex number."},{"Start":"01:36.985 ","End":"01:38.910","Text":"I\u0027ll take the same 3 plus 2i,"},{"Start":"01:38.910 ","End":"01:45.980","Text":"but suppose I had 4i here times 3 plus 2i,"},{"Start":"01:45.980 ","End":"01:47.785","Text":"what would it be?"},{"Start":"01:47.785 ","End":"01:53.290","Text":"As usual, the distributive law applies and we bracket so meaning, this times this,"},{"Start":"01:53.290 ","End":"02:00.255","Text":"this times this, so 4i times 3 is 4 times 3 times i is 12i."},{"Start":"02:00.255 ","End":"02:04.035","Text":"Then 4i times 2i,"},{"Start":"02:04.035 ","End":"02:05.940","Text":"4 times 2 is 8,"},{"Start":"02:05.940 ","End":"02:11.620","Text":"so it\u0027s plus 8i^2."},{"Start":"02:11.620 ","End":"02:16.675","Text":"But it doesn\u0027t end here because remember the most important thing about"},{"Start":"02:16.675 ","End":"02:23.465","Text":"complex numbers is that i^2 is minus 1 or that i is the square root of minus 1."},{"Start":"02:23.465 ","End":"02:27.155","Text":"I replace i^2 by minus 1,"},{"Start":"02:27.155 ","End":"02:29.910","Text":"not 1 minus 1."},{"Start":"02:29.910 ","End":"02:36.700","Text":"This is equal to 12i minus 8."},{"Start":"02:36.700 ","End":"02:39.290","Text":"But when we write complex numbers,"},{"Start":"02:39.290 ","End":"02:42.140","Text":"it\u0027s customary to write the real part first,"},{"Start":"02:42.140 ","End":"02:47.700","Text":"so I write the answer as minus 8 plus 12i."},{"Start":"02:48.440 ","End":"02:51.680","Text":"Now let\u0027s go to the most general case,"},{"Start":"02:51.680 ","End":"02:53.615","Text":"a complex times a complex."},{"Start":"02:53.615 ","End":"02:56.975","Text":"Let\u0027s take, I don\u0027t know,"},{"Start":"02:56.975 ","End":"03:07.570","Text":"minus 5 plus 4i and let\u0027s keep the same 3 plus 2i and see what we get this time."},{"Start":"03:07.570 ","End":"03:11.360","Text":"Remember when we have a sum of two things times the sum of two things,"},{"Start":"03:11.360 ","End":"03:13.460","Text":"we just multiply everything in"},{"Start":"03:13.460 ","End":"03:18.260","Text":"this bracket with everything in this bracket and sometimes we put little arcs."},{"Start":"03:18.260 ","End":"03:21.260","Text":"This with this, this with this, this with this,"},{"Start":"03:21.260 ","End":"03:25.630","Text":"this with this, the order doesn\u0027t matter. But let\u0027s see."},{"Start":"03:25.630 ","End":"03:31.590","Text":"Minus 5 times 3 is minus 15,"},{"Start":"03:31.590 ","End":"03:38.430","Text":"minus 5 with 2i is minus 10i."},{"Start":"03:38.430 ","End":"03:43.605","Text":"Then 4i times 3 is 12i."},{"Start":"03:43.605 ","End":"03:50.800","Text":"Finally 4i times 2i is 8i^2."},{"Start":"03:50.800 ","End":"03:53.110","Text":"Now once you get into the hang of it,"},{"Start":"03:53.110 ","End":"03:56.665","Text":"and you\u0027ll remember that i^2 is minus 1,"},{"Start":"03:56.665 ","End":"03:59.410","Text":"which I\u0027ll highlight again,"},{"Start":"03:59.410 ","End":"04:04.250","Text":"then you would have written this straightaway as minus 8."},{"Start":"04:04.250 ","End":"04:10.475","Text":"In fact, I\u0027ll just erase this and say this is minus 8 because i^2 is minus 1."},{"Start":"04:10.475 ","End":"04:12.610","Text":"Then we just collect stuff together."},{"Start":"04:12.610 ","End":"04:17.910","Text":"Minus 15 and minus 8 gives me minus 23,"},{"Start":"04:17.910 ","End":"04:26.360","Text":"and minus 10i plus 12i is plus 2i and that\u0027s the answer to the multiplication."},{"Start":"04:26.540 ","End":"04:30.910","Text":"I\u0027d like to do one more example with multiplication."},{"Start":"04:30.910 ","End":"04:32.880","Text":"Let\u0027s take the following."},{"Start":"04:32.880 ","End":"04:43.060","Text":"Let\u0027s take 7 plus 5i times 7 minus 5i."},{"Start":"04:44.440 ","End":"04:46.925","Text":"Let\u0027s see what we get."},{"Start":"04:46.925 ","End":"04:50.360","Text":"Of course we could do it the same as before or we could"},{"Start":"04:50.360 ","End":"04:54.800","Text":"notice that this is the same thing just with a plus and with a minus."},{"Start":"04:54.800 ","End":"05:00.185","Text":"Then you might recall a certain formula called the difference of squares formula,"},{"Start":"05:00.185 ","End":"05:04.010","Text":"that a plus b times a minus b."},{"Start":"05:04.010 ","End":"05:05.660","Text":"If you have two things ones with a plus,"},{"Start":"05:05.660 ","End":"05:08.929","Text":"and ones with a minus, the answer is the difference of the squares."},{"Start":"05:08.929 ","End":"05:12.575","Text":"It\u0027s the first one squared less the second one squared."},{"Start":"05:12.575 ","End":"05:15.095","Text":"If I apply that rule here,"},{"Start":"05:15.095 ","End":"05:18.535","Text":"then I get that this is 7^2,"},{"Start":"05:18.535 ","End":"05:23.890","Text":"I\u0027m putting the brackets just for emphasis, minus 5i^2."},{"Start":"05:23.890 ","End":"05:26.120","Text":"Here I do need the brackets,"},{"Start":"05:26.120 ","End":"05:28.925","Text":"otherwise it would just be the i^2."},{"Start":"05:28.925 ","End":"05:31.295","Text":"This is equal to,"},{"Start":"05:31.295 ","End":"05:35.840","Text":"7^2 squared is 49 and initially we would"},{"Start":"05:35.840 ","End":"05:41.940","Text":"write minus 25i^2 because 5i times 5i,"},{"Start":"05:41.940 ","End":"05:49.049","Text":"but remember that i^2 is minus 1 so really this is equal to"},{"Start":"05:49.049 ","End":"05:58.065","Text":"49 plus 25 and"},{"Start":"05:58.065 ","End":"06:00.180","Text":"that is equal to,"},{"Start":"06:00.180 ","End":"06:07.090","Text":"let\u0027s see, 74, I make it."},{"Start":"06:07.090 ","End":"06:11.615","Text":"Turns out the answer is a real number without the imaginary part."},{"Start":"06:11.615 ","End":"06:18.470","Text":"Now, I actually knew this was going to happen because whenever you have a pair like this,"},{"Start":"06:18.470 ","End":"06:20.630","Text":"ones with a plus and ones with a minus,"},{"Start":"06:20.630 ","End":"06:23.855","Text":"it comes out real and there is actually a name."},{"Start":"06:23.855 ","End":"06:26.120","Text":"If a complex number."},{"Start":"06:26.120 ","End":"06:32.925","Text":"Let\u0027s call it z is 7 plus 5i in this case,"},{"Start":"06:32.925 ","End":"06:39.830","Text":"then the same thing with a negative on the imaginary part,"},{"Start":"06:39.830 ","End":"06:44.090","Text":"the opposite, is called the complex conjugate."},{"Start":"06:44.090 ","End":"06:47.930","Text":"Sometimes we call that Z bar."},{"Start":"06:47.930 ","End":"06:51.355","Text":"This is the, I\u0027ll write the word,"},{"Start":"06:51.355 ","End":"06:53.750","Text":"two words, complex conjugate."},{"Start":"06:53.750 ","End":"06:56.480","Text":"Sometimes in the context of complex numbers,"},{"Start":"06:56.480 ","End":"06:59.720","Text":"I just say this is the conjugate of this and the other way round,"},{"Start":"06:59.720 ","End":"07:01.310","Text":"and this is the conjugate of this."},{"Start":"07:01.310 ","End":"07:05.190","Text":"In general, when we have a plus bi,"},{"Start":"07:05.190 ","End":"07:10.380","Text":"then the conjugate is a minus bi."},{"Start":"07:10.380 ","End":"07:16.430","Text":"These two are conjugates and this will be very useful later on for example,"},{"Start":"07:16.430 ","End":"07:19.385","Text":"when we do the section on division."},{"Start":"07:19.385 ","End":"07:22.170","Text":"Let\u0027s take a break now."}],"ID":4919},{"Watched":false,"Name":"Exponents","Duration":"10m 29s","ChapterTopicVideoID":4927,"CourseChapterTopicPlaylistID":3860,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.640","Text":"Here we are back with complex numbers."},{"Start":"00:02.640 ","End":"00:09.270","Text":"We\u0027ve just done addition and subtraction and multiplication."},{"Start":"00:09.270 ","End":"00:14.340","Text":"Let\u0027s continue now with exponents or if you like,"},{"Start":"00:14.340 ","End":"00:19.170","Text":"powers, and after that we\u0027ll get to division."},{"Start":"00:19.170 ","End":"00:22.785","Text":"Let\u0027s start with powers of i."},{"Start":"00:22.785 ","End":"00:28.365","Text":"Now, the main starting point is that i^2 is minus 1."},{"Start":"00:28.365 ","End":"00:32.625","Text":"We\u0027ll keep highlighting it again and again, it\u0027s so important."},{"Start":"00:32.625 ","End":"00:34.890","Text":"But I\u0027d like to look at some other powers."},{"Start":"00:34.890 ","End":"00:39.780","Text":"In fact, let\u0027s look at natural numbers as 0, 1, 2, 3,"},{"Start":"00:39.780 ","End":"00:47.010","Text":"4. i^0, and anything to the 0 is 1 and i^1 is just i,"},{"Start":"00:47.010 ","End":"00:49.625","Text":"I\u0027m just writing that for completeness."},{"Start":"00:49.625 ","End":"00:51.455","Text":"We\u0027ve got 0, 1, 2."},{"Start":"00:51.455 ","End":"00:55.910","Text":"What about i^3?"},{"Start":"00:55.910 ","End":"01:04.935","Text":"Well, I can write i^3 as i^2 times i^1 and i^2 is minus 1,"},{"Start":"01:04.935 ","End":"01:07.620","Text":"so it\u0027s minus 1 times i,"},{"Start":"01:07.620 ","End":"01:11.535","Text":"so the answer is minus i."},{"Start":"01:11.535 ","End":"01:14.655","Text":"Now let\u0027s go to the next one, i^4."},{"Start":"01:14.655 ","End":"01:21.000","Text":"i^4, I could write it as"},{"Start":"01:21.000 ","End":"01:28.170","Text":"( i^2)^2 using the laws of exponents."},{"Start":"01:28.170 ","End":"01:30.510","Text":"Since i^2 is minus 1,"},{"Start":"01:30.510 ","End":"01:32.715","Text":"it\u0027s minus 1^2,"},{"Start":"01:32.715 ","End":"01:36.570","Text":"which is 1, so the answer is 1."},{"Start":"01:36.570 ","End":"01:39.120","Text":"Let\u0027s do one more."},{"Start":"01:39.120 ","End":"01:42.390","Text":"i^5, let\u0027s see what that equals."},{"Start":"01:42.390 ","End":"01:51.585","Text":"i^5= I can either start from the beginning or I could use i^4 times i."},{"Start":"01:51.585 ","End":"01:55.140","Text":"Let\u0027s say I use it\u0027s i^4 times i,"},{"Start":"01:55.140 ","End":"01:57.180","Text":"i^4 is 1,"},{"Start":"01:57.180 ","End":"01:59.265","Text":"1 times i,"},{"Start":"01:59.265 ","End":"02:02.170","Text":"so it\u0027s equal to i. i^6."},{"Start":"02:04.160 ","End":"02:08.790","Text":"I can either say i^5 times i or we can start a fresh and say"},{"Start":"02:08.790 ","End":"02:13.970","Text":"it\u0027s (i^2)^3 when it\u0027s an even number, I can do that."},{"Start":"02:13.970 ","End":"02:17.210","Text":"Then i^2 is (minus 1)^3,"},{"Start":"02:17.210 ","End":"02:19.650","Text":"which is minus 1."},{"Start":"02:20.890 ","End":"02:24.750","Text":"Just do as many as I have room for."},{"Start":"02:26.120 ","End":"02:33.290","Text":"i^7, I\u0027ll do it as if from scratch."},{"Start":"02:33.290 ","End":"02:35.030","Text":"What we can do is we always say,"},{"Start":"02:35.030 ","End":"02:42.343","Text":"if it\u0027s an odd number I can say that it\u0027s i^6 times i"},{"Start":"02:42.343 ","End":"02:50.570","Text":"and then (i^2 )^3s in laws of exponents times i,"},{"Start":"02:50.570 ","End":"02:55.920","Text":"i^2 is (minus 1)^3 times i."},{"Start":"02:55.920 ","End":"02:58.020","Text":"(Minus 1)^3 is minus 1,"},{"Start":"02:58.020 ","End":"03:01.695","Text":"so it\u0027s minus i. i^8,"},{"Start":"03:01.695 ","End":"03:05.340","Text":"it\u0027s the last one we\u0027ll do here, i^8,"},{"Start":"03:05.340 ","End":"03:10.125","Text":"you can say it\u0027s (i^2)^4,"},{"Start":"03:10.125 ","End":"03:13.710","Text":"which is (minus 10)^4,"},{"Start":"03:13.710 ","End":"03:16.555","Text":"which is plus 1."},{"Start":"03:16.555 ","End":"03:19.250","Text":"Notice that it\u0027s actually periodic."},{"Start":"03:19.250 ","End":"03:26.115","Text":"It goes,1i minus i. I said that all wrong. I\u0027ll start again."},{"Start":"03:26.115 ","End":"03:29.190","Text":"1i minus 1 minus i,"},{"Start":"03:29.190 ","End":"03:32.765","Text":"then 1i minus 1 minus i, and so on."},{"Start":"03:32.765 ","End":"03:34.190","Text":"It has a period of 4;"},{"Start":"03:34.190 ","End":"03:36.875","Text":"every 4 it repeats itself."},{"Start":"03:36.875 ","End":"03:39.755","Text":"The reason for that is that i^4 is 1,"},{"Start":"03:39.755 ","End":"03:42.080","Text":"so i^8 is 1,"},{"Start":"03:42.080 ","End":"03:43.260","Text":"i^12 is 1,"},{"Start":"03:43.260 ","End":"03:45.690","Text":"every 4 we start over again."},{"Start":"03:45.690 ","End":"03:48.705","Text":"That\u0027s as far as exponents of just i."},{"Start":"03:48.705 ","End":"03:51.980","Text":"Now let\u0027s make it a bit more complicated."},{"Start":"03:51.980 ","End":"03:54.830","Text":"Let me first clean the board."},{"Start":"03:54.830 ","End":"03:59.580","Text":"Let\u0027s try something like minus i^11."},{"Start":"04:01.870 ","End":"04:06.530","Text":"What I would do here is, first of all,"},{"Start":"04:06.530 ","End":"04:17.460","Text":"I see minus i is minus 1 times i. I can say it\u0027s minus 1^11 times i^11."},{"Start":"04:18.400 ","End":"04:20.870","Text":"Let\u0027s do each piece separately."},{"Start":"04:20.870 ","End":"04:24.740","Text":"This piece here is minus"},{"Start":"04:24.740 ","End":"04:31.175","Text":"1^11 because it\u0027s an odd power of minus 1, it\u0027s minus 1."},{"Start":"04:31.175 ","End":"04:37.940","Text":"But i^11, we use a usual trick of saying that"},{"Start":"04:37.940 ","End":"04:45.240","Text":"(i^2)^5 is i^10 times i. I skipped a step,"},{"Start":"04:45.240 ","End":"04:51.390","Text":"it\u0027s i^10 times i and i^10 is (i^2)^5 so on."},{"Start":"04:51.390 ","End":"04:56.325","Text":"i^2 is minus 1^5 times i,"},{"Start":"04:56.325 ","End":"04:58.320","Text":"which is minus i."},{"Start":"04:58.320 ","End":"05:04.470","Text":"Finally, I combine this minus 1 with this minus i,"},{"Start":"05:04.470 ","End":"05:10.770","Text":"so actually, the answer comes out to be just 1 plus 1."},{"Start":"05:10.900 ","End":"05:18.445","Text":"Another example, 3 minus (4i)^2."},{"Start":"05:18.445 ","End":"05:21.440","Text":"Power of 2 we can use the formula."},{"Start":"05:21.440 ","End":"05:29.565","Text":"I\u0027m talking about a minus b squared is a^2 minus 2ab plus b^2,"},{"Start":"05:29.565 ","End":"05:31.220","Text":"same if it was a plus here,"},{"Start":"05:31.220 ","End":"05:33.245","Text":"it would be a plus here."},{"Start":"05:33.245 ","End":"05:39.195","Text":"We get 3^2 is 9 minus 2ab,"},{"Start":"05:39.195 ","End":"05:42.525","Text":"2 times 3 times 4 is 24,"},{"Start":"05:42.525 ","End":"05:44.535","Text":"so minus 24i,"},{"Start":"05:44.535 ","End":"05:51.940","Text":"and b^2 is (4i)^2."},{"Start":"05:51.940 ","End":"05:59.175","Text":"Now, (4i)^2 is 4^2 times i^2,"},{"Start":"05:59.175 ","End":"06:01.275","Text":"which is (16i)^2,"},{"Start":"06:01.275 ","End":"06:06.540","Text":"actually minus 16 because i^2 is minus 1,"},{"Start":"06:06.540 ","End":"06:15.400","Text":"and so the answer is 9 minus 16 is minus 7 minus 24i."},{"Start":"06:16.820 ","End":"06:20.894","Text":"Now, let\u0027s take a higher power."},{"Start":"06:20.894 ","End":"06:23.785","Text":"Now let\u0027s stick to the power of 2."},{"Start":"06:23.785 ","End":"06:26.060","Text":"Let\u0027s make it with a plus in the middle."},{"Start":"06:26.060 ","End":"06:37.080","Text":"But let\u0027s say I have here something like i^9 plus i^15 squared."},{"Start":"06:37.310 ","End":"06:40.560","Text":"Using the corresponding formula,"},{"Start":"06:40.560 ","End":"06:42.520","Text":"well, I\u0027ll just make it 2 in 1."},{"Start":"06:42.520 ","End":"06:44.345","Text":"If it\u0027s plus here,"},{"Start":"06:44.345 ","End":"06:46.715","Text":"then it\u0027s also a plus here."},{"Start":"06:46.715 ","End":"06:49.800","Text":"We use a plus b^2."},{"Start":"06:50.020 ","End":"06:52.940","Text":"This by the way, is one way of solving it"},{"Start":"06:52.940 ","End":"06:55.055","Text":"because at the end I\u0027m going to show you an easier way."},{"Start":"06:55.055 ","End":"06:57.935","Text":"But let\u0027s do it this way with the formula."},{"Start":"06:57.935 ","End":"07:01.510","Text":"We get (i^9)^2."},{"Start":"07:01.510 ","End":"07:06.065","Text":"We can straight away see that that\u0027s i^18 with the rules of exponents."},{"Start":"07:06.065 ","End":"07:12.610","Text":"Then plus twice i^9 times i^15."},{"Start":"07:12.610 ","End":"07:14.370","Text":"Again, the rules of exponents,"},{"Start":"07:14.370 ","End":"07:18.735","Text":"I add 9 plus 15 is 24,"},{"Start":"07:18.735 ","End":"07:26.820","Text":"and then i^15 all squared is i^30."},{"Start":"07:26.820 ","End":"07:28.275","Text":"This=."},{"Start":"07:28.275 ","End":"07:32.740","Text":"Now, at this point you should already be able to"},{"Start":"07:32.740 ","End":"07:37.825","Text":"without doing all these expansions with i^2,"},{"Start":"07:37.825 ","End":"07:41.635","Text":"you could straight away see that i^16,"},{"Start":"07:41.635 ","End":"07:43.475","Text":"anything that divides by 4,"},{"Start":"07:43.475 ","End":"07:45.525","Text":"i to its power is 1."},{"Start":"07:45.525 ","End":"07:48.930","Text":"This is i^16 times i^2,"},{"Start":"07:48.930 ","End":"07:51.670","Text":"so that\u0027s minus 1."},{"Start":"07:51.890 ","End":"07:54.660","Text":"Then here, i^24,"},{"Start":"07:54.660 ","End":"07:56.850","Text":"if it\u0027s divisible by 4,"},{"Start":"07:56.850 ","End":"07:58.710","Text":"it\u0027s going to be 1."},{"Start":"07:58.710 ","End":"08:00.888","Text":"See, divisible by 4 is 1,"},{"Start":"08:00.888 ","End":"08:03.156","Text":"divisible by 4 is 1,"},{"Start":"08:03.156 ","End":"08:04.800","Text":"divisible by 4 is 1."},{"Start":"08:04.800 ","End":"08:06.030","Text":"All the 0,"},{"Start":"08:06.030 ","End":"08:08.190","Text":"4, 8, 12, 16,"},{"Start":"08:08.190 ","End":"08:10.200","Text":"all those powers of 1,"},{"Start":"08:10.200 ","End":"08:11.875","Text":"that\u0027s a shortcut way of doing it,"},{"Start":"08:11.875 ","End":"08:13.324","Text":"and you see I have this leftover,"},{"Start":"08:13.324 ","End":"08:15.485","Text":"leftover is 2, so it\u0027s i^2."},{"Start":"08:15.485 ","End":"08:18.215","Text":"Here, it\u0027s an even, so it\u0027s 1."},{"Start":"08:18.215 ","End":"08:22.535","Text":"This is just 2 times 1. i^30,"},{"Start":"08:22.535 ","End":"08:27.620","Text":"well, 28 divides by 4."},{"Start":"08:27.620 ","End":"08:31.275","Text":"1^28 is 1. Then so it\u0027s again just i^2,"},{"Start":"08:31.275 ","End":"08:33.480","Text":"which is minus 1,"},{"Start":"08:33.480 ","End":"08:34.530","Text":"plus minus 1,"},{"Start":"08:34.530 ","End":"08:36.375","Text":"which is minus 1."},{"Start":"08:36.375 ","End":"08:41.160","Text":"What we get is minus 1 plus 2 minus 1, 0."},{"Start":"08:41.160 ","End":"08:42.810","Text":"Okay, that\u0027s the answer."},{"Start":"08:42.810 ","End":"08:45.200","Text":"I said I\u0027d show you another way of doing it."},{"Start":"08:45.200 ","End":"08:51.440","Text":"The other way of doing it is right from the beginning to say i^9 is i^8 times"},{"Start":"08:51.440 ","End":"08:58.680","Text":"i. i^8 is 1 times i. I could say this is=i."},{"Start":"08:58.680 ","End":"09:03.945","Text":"Then again, i^15 is,"},{"Start":"09:03.945 ","End":"09:06.460","Text":"I\u0027m going to show you this one on the side,"},{"Start":"09:06.460 ","End":"09:12.190","Text":"i^15 is i^12 times i^3."},{"Start":"09:12.190 ","End":"09:15.710","Text":"You take out the multiple of 4. This is 1."},{"Start":"09:15.710 ","End":"09:19.910","Text":"Once again, it\u0027s because (i^4)^3 is 1,"},{"Start":"09:19.910 ","End":"09:23.480","Text":"you can either say it\u0027s (i^4)^3,"},{"Start":"09:23.480 ","End":"09:26.990","Text":"which is 1^3, which is 1."},{"Start":"09:26.990 ","End":"09:28.790","Text":"If you didn\u0027t remember that,"},{"Start":"09:28.790 ","End":"09:32.765","Text":"you could always go back to (i^2)^6,"},{"Start":"09:32.765 ","End":"09:37.380","Text":"which is (minus 1)^6,"},{"Start":"09:37.380 ","End":"09:39.460","Text":"minus 1 to an even power is 1."},{"Start":"09:39.460 ","End":"09:42.010","Text":"I\u0027m showing you alternative ways of doing all things."},{"Start":"09:42.010 ","End":"09:44.420","Text":"Sure, this is not just one way of doing it."},{"Start":"09:44.420 ","End":"09:48.218","Text":"Either way, the i^12 is 1 and i^3,"},{"Start":"09:48.218 ","End":"09:49.800","Text":"well, let\u0027s not repeat our work,"},{"Start":"09:49.800 ","End":"09:52.155","Text":"let\u0027s just look it up, it\u0027s minus i."},{"Start":"09:52.155 ","End":"09:54.552","Text":"What we get is here it\u0027s 1,"},{"Start":"09:54.552 ","End":"09:56.798","Text":"here it\u0027s minus i,"},{"Start":"09:56.798 ","End":"10:02.670","Text":"so altogether, i^15 is minus i."},{"Start":"10:02.670 ","End":"10:05.100","Text":"If i^15 is minus i,"},{"Start":"10:05.100 ","End":"10:08.835","Text":"we have i plus minus i squared,"},{"Start":"10:08.835 ","End":"10:11.400","Text":"but I can do i plus minus i, it\u0027s just 0."},{"Start":"10:11.400 ","End":"10:12.726","Text":"It\u0027s i minus i,"},{"Start":"10:12.726 ","End":"10:15.030","Text":"0^2, which is 0,"},{"Start":"10:15.030 ","End":"10:19.115","Text":"and it\u0027s a good job we got the same answer if we did it with 2 different ways,"},{"Start":"10:19.115 ","End":"10:23.160","Text":"0, 0, so that obviously is the answer."},{"Start":"10:23.160 ","End":"10:26.355","Text":"That\u0027s enough for exponents."},{"Start":"10:26.355 ","End":"10:28.690","Text":"Let\u0027s take a break here."}],"ID":4920},{"Watched":false,"Name":"Division","Duration":"6m 51s","ChapterTopicVideoID":4918,"CourseChapterTopicPlaylistID":3860,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.519","Text":"Continuing with complex numbers,"},{"Start":"00:02.519 ","End":"00:07.244","Text":"this time we are going talk about division of complex numbers."},{"Start":"00:07.244 ","End":"00:10.635","Text":"But to start with, I would like to remind you"},{"Start":"00:10.635 ","End":"00:14.040","Text":"of something we did in the section on multiplication."},{"Start":"00:14.040 ","End":"00:17.520","Text":"In fact, I\u0027m just going to copy and paste the example."},{"Start":"00:17.520 ","End":"00:22.320","Text":"This is the example we had in the chapter on multiplication I just"},{"Start":"00:22.320 ","End":"00:27.300","Text":"copied it and we had the product of 2 numbers,"},{"Start":"00:27.300 ","End":"00:31.730","Text":"which were not just any 2 complex numbers that were conjugates of each other,"},{"Start":"00:31.730 ","End":"00:36.269","Text":"remember if we have the imaginary part once with a plus and once with a minus,"},{"Start":"00:36.269 ","End":"00:39.443","Text":"then these 22 complex numbers are conjugates."},{"Start":"00:39.443 ","End":"00:41.325","Text":"When we multiply them together,"},{"Start":"00:41.325 ","End":"00:44.180","Text":"we got a positive real number."},{"Start":"00:44.180 ","End":"00:45.995","Text":"In fact, if you look at it,"},{"Start":"00:45.995 ","End":"00:48.155","Text":"what this 74 is,"},{"Start":"00:48.155 ","End":"00:54.225","Text":"it\u0027s actually 7^2 plus 5^2 it came out to be 49 plus 25,"},{"Start":"00:54.225 ","End":"00:59.030","Text":"and this is actually true in general as a rule that if we take a plus"},{"Start":"00:59.030 ","End":"01:04.625","Text":"bi as one complex number and then take its conjugate a minus bi,"},{"Start":"01:04.625 ","End":"01:07.700","Text":"the answer is always going to be positive real,"},{"Start":"01:07.700 ","End":"01:11.585","Text":"and it\u0027s equal to a^2 plus b^2."},{"Start":"01:11.585 ","End":"01:15.949","Text":"This is going to be very useful to us when we talk about division."},{"Start":"01:15.949 ","End":"01:18.319","Text":"Let\u0027s start straight away with an example."},{"Start":"01:18.319 ","End":"01:26.400","Text":"I want to compute 11 plus 2i divided by 2 minus i."},{"Start":"01:26.400 ","End":"01:29.585","Text":"The trick is this."},{"Start":"01:29.585 ","End":"01:34.160","Text":"We multiply apply this fraction top and bottom by the same thing,"},{"Start":"01:34.160 ","End":"01:38.765","Text":"and we multiply it by the conjugate of the denominator."},{"Start":"01:38.765 ","End":"01:41.330","Text":"To remind you again, the word conjugate,"},{"Start":"01:41.330 ","End":"01:44.060","Text":"these 2 numbers are conjugates of each other."},{"Start":"01:44.060 ","End":"01:47.660","Text":"Complex conjugates and these 2 are complex conjugates."},{"Start":"01:47.660 ","End":"01:49.700","Text":"It\u0027s the same real part,"},{"Start":"01:49.700 ","End":"01:52.460","Text":"but the imaginary part with the opposite sign."},{"Start":"01:52.460 ","End":"01:58.960","Text":"In this case, the conjugate of 2 minus i is going to be 2 plus i,"},{"Start":"01:58.960 ","End":"02:02.465","Text":"and if I multiply both top and bottom,"},{"Start":"02:02.465 ","End":"02:06.185","Text":"really I\u0027m just multiplying by 1 because this fraction is 1."},{"Start":"02:06.185 ","End":"02:09.635","Text":"This is going to help us, as you will see."},{"Start":"02:09.635 ","End":"02:17.875","Text":"What we get, is let\u0027s do the denominator first."},{"Start":"02:17.875 ","End":"02:20.430","Text":"That\u0027s going to be the easier part,"},{"Start":"02:20.430 ","End":"02:25.130","Text":"2 minus i times 1 plus i we already discussed from"},{"Start":"02:25.130 ","End":"02:30.430","Text":"this formula that this is 2^2 plus 1^2."},{"Start":"02:30.430 ","End":"02:36.540","Text":"Because this is 1i, this is 2 plus 1i that will be 5,"},{"Start":"02:36.540 ","End":"02:37.890","Text":"of course in the moment."},{"Start":"02:37.890 ","End":"02:41.250","Text":"Then on the numerator we just multiply,"},{"Start":"02:41.250 ","End":"02:49.175","Text":"just like we would multiply 2 complex numbers or just write it."},{"Start":"02:49.175 ","End":"02:50.960","Text":"We\u0027re doing the product here,"},{"Start":"02:50.960 ","End":"02:53.610","Text":"and then we get,"},{"Start":"02:53.720 ","End":"02:57.940","Text":"let\u0027s see, 11 times 2."},{"Start":"02:57.940 ","End":"02:59.675","Text":"Let\u0027s try and do it all in 1 go,"},{"Start":"02:59.675 ","End":"03:04.565","Text":"11 times 2 is 22,"},{"Start":"03:04.565 ","End":"03:07.095","Text":"I\u0027m just going to make a note."},{"Start":"03:07.095 ","End":"03:14.595","Text":"That\u0027s 22 and 2i times I is 2i^2,"},{"Start":"03:14.595 ","End":"03:16.290","Text":"so that\u0027s minus 2,"},{"Start":"03:16.290 ","End":"03:22.520","Text":"so I get 20 for the real part and then on the imaginary parts I\u0027ve got"},{"Start":"03:22.520 ","End":"03:28.780","Text":"2 times 2 is 4i and 11 times 1 is 11i,"},{"Start":"03:28.780 ","End":"03:33.520","Text":"so I have got 4 plus 11, it\u0027s plus 15i."},{"Start":"03:35.300 ","End":"03:39.010","Text":"I\u0027m going to erase the scratch work."},{"Start":"03:39.230 ","End":"03:44.865","Text":"Over the dividing line and 2^2 plus 1^2 is 5,"},{"Start":"03:44.865 ","End":"03:49.545","Text":"and finally, we can do the division,"},{"Start":"03:49.545 ","End":"03:51.380","Text":"when we divide by a number,"},{"Start":"03:51.380 ","End":"04:01.320","Text":"we just divide each bit separately 20 divided by 5 is 4 and 15 over 5 is 3,"},{"Start":"04:01.320 ","End":"04:04.995","Text":"so we get the answer as 4 plus 3i,"},{"Start":"04:04.995 ","End":"04:11.040","Text":"and that\u0027s all there is to division of complex numbers."},{"Start":"04:14.480 ","End":"04:17.630","Text":"When we divide by a complex number,"},{"Start":"04:17.630 ","End":"04:22.560","Text":"what we do is we then multiply top and bottom by the conjugate."},{"Start":"04:23.390 ","End":"04:26.865","Text":"Let\u0027s just do 1 more for practice,"},{"Start":"04:26.865 ","End":"04:34.480","Text":"3 plus 7i divided by 2 minus 5i,"},{"Start":"04:34.760 ","End":"04:41.270","Text":"so what we do is we multiply top and bottom by a fraction,"},{"Start":"04:41.270 ","End":"04:44.600","Text":"which is the conjugate of the denominator."},{"Start":"04:44.600 ","End":"04:48.350","Text":"The conjugate of 2 minus 5i is 2 plus 5i,"},{"Start":"04:48.350 ","End":"04:51.440","Text":"but we have to do the same thing to the numerator so as not"},{"Start":"04:51.440 ","End":"04:55.155","Text":"to change the sum the answer,"},{"Start":"04:55.155 ","End":"05:00.655","Text":"and then what we get is on the denominator,"},{"Start":"05:00.655 ","End":"05:04.300","Text":"we always get from this formula,"},{"Start":"05:04.300 ","End":"05:07.370","Text":"2^2 plus 5^2,"},{"Start":"05:08.030 ","End":"05:10.750","Text":"and on the numerator,"},{"Start":"05:10.750 ","End":"05:12.100","Text":"we just multiply it out."},{"Start":"05:12.100 ","End":"05:15.240","Text":"Let\u0027s see, I will do it slightly differently than I did before,"},{"Start":"05:15.240 ","End":"05:25.270","Text":"3 times 2 is 6 plus 7i times 2i is 14i."},{"Start":"05:27.290 ","End":"05:31.630","Text":"Then these 2 give me plus"},{"Start":"05:32.150 ","End":"05:39.800","Text":"15i and 7i times 5i is 35i^2 minus 35."},{"Start":"05:39.800 ","End":"05:42.670","Text":"Remember, i^2 is minus 1."},{"Start":"05:42.670 ","End":"05:46.845","Text":"Again, i^2 is minus 1."},{"Start":"05:46.845 ","End":"05:51.510","Text":"Then what we get is, let\u0027s see,"},{"Start":"05:51.510 ","End":"06:00.435","Text":"6 minus 35 is minus 29,"},{"Start":"06:00.435 ","End":"06:09.270","Text":"14 plus 15 is 29i and on the denominator,"},{"Start":"06:09.270 ","End":"06:13.350","Text":"4 plus 25 is 29."},{"Start":"06:13.350 ","End":"06:16.175","Text":"That works out nicely."},{"Start":"06:16.175 ","End":"06:22.155","Text":"That comes out to be minus 1 plus 1i,"},{"Start":"06:22.155 ","End":"06:25.025","Text":"which is just i, and that\u0027s the answer."},{"Start":"06:25.025 ","End":"06:29.435","Text":"Of course often you will get fractions in the answer just,"},{"Start":"06:29.435 ","End":"06:33.654","Text":"again examples where it came out neatly, where everything divided."},{"Start":"06:33.654 ","End":"06:37.095","Text":"That\u0027s enough for division,"},{"Start":"06:37.095 ","End":"06:42.650","Text":"and of course there are more solved example exercises besides the tutorial,"},{"Start":"06:42.650 ","End":"06:45.420","Text":"go and look at them and do them."},{"Start":"06:46.300 ","End":"06:48.530","Text":"Let\u0027s take a break now,"},{"Start":"06:48.530 ","End":"06:51.390","Text":"and after that the next topic."}],"ID":4921},{"Watched":false,"Name":"Quadratic Equations","Duration":"9m 59s","ChapterTopicVideoID":4928,"CourseChapterTopicPlaylistID":3860,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.700","Text":"Continuing with complex numbers is"},{"Start":"00:02.700 ","End":"00:08.220","Text":"one more topic which is related to quadratic equations."},{"Start":"00:08.220 ","End":"00:09.615","Text":"You might say, \"Oh yeah,"},{"Start":"00:09.615 ","End":"00:12.210","Text":"we\u0027ve already learned quadratic equations.\""},{"Start":"00:12.210 ","End":"00:19.065","Text":"But I want to relate to those quadratic equations which we used to say have no solution,"},{"Start":"00:19.065 ","End":"00:22.090","Text":"maybe now they do somehow."},{"Start":"00:22.340 ","End":"00:24.930","Text":"A note on notation,"},{"Start":"00:24.930 ","End":"00:30.060","Text":"we used to usually use x as the variable,"},{"Start":"00:30.060 ","End":"00:33.495","Text":"but when we talk about complex numbers,"},{"Start":"00:33.495 ","End":"00:38.835","Text":"we usually use the letter z and I put a line through it to distinguish it from a 2."},{"Start":"00:38.835 ","End":"00:48.740","Text":"So our first example is z^2 plus 9=0."},{"Start":"00:48.740 ","End":"00:50.000","Text":"We don\u0027t need to crank out"},{"Start":"00:50.000 ","End":"00:53.900","Text":"the whole quadratic formula because there\u0027s a missing middle term,"},{"Start":"00:53.900 ","End":"01:00.470","Text":"so what I can do is say z^2 is minus 9,"},{"Start":"01:00.470 ","End":"01:03.870","Text":"and then I take the square root and then z is going to"},{"Start":"01:03.870 ","End":"01:09.125","Text":"be plus or minus the square root of minus 9,"},{"Start":"01:09.125 ","End":"01:13.550","Text":"and we can simplify this because we did this actually in"},{"Start":"01:13.550 ","End":"01:18.410","Text":"the very first chapter on complex numbers, we\u0027ll do it again."},{"Start":"01:18.410 ","End":"01:25.570","Text":"This is equal to plus or minus the square root of 9 times the square root of minus 1,"},{"Start":"01:25.570 ","End":"01:27.560","Text":"because the properties of the square roots,"},{"Start":"01:27.560 ","End":"01:30.365","Text":"square root of the product is the product of the square roots."},{"Start":"01:30.365 ","End":"01:34.640","Text":"Then we finally get the answer plus or minus 3,"},{"Start":"01:34.640 ","End":"01:37.860","Text":"and the square root of minus 1 is i."},{"Start":"01:38.750 ","End":"01:42.240","Text":"This used to have no solution,"},{"Start":"01:42.240 ","End":"01:44.930","Text":"when we were talking about the numbers,"},{"Start":"01:44.930 ","End":"01:47.225","Text":"we used to be familiar with the real numbers,"},{"Start":"01:47.225 ","End":"01:49.385","Text":"but in terms of complex numbers,"},{"Start":"01:49.385 ","End":"01:52.190","Text":"it now has two solutions."},{"Start":"01:52.190 ","End":"01:56.570","Text":"Let\u0027s try another one that used to be impossible."},{"Start":"01:56.570 ","End":"02:01.610","Text":"Let\u0027s take now z^2"},{"Start":"02:01.610 ","End":"02:10.255","Text":"plus 4z plus 5=0."},{"Start":"02:10.255 ","End":"02:17.120","Text":"This time we will use the quadratic formula and say that z is"},{"Start":"02:17.120 ","End":"02:24.480","Text":"equal to minus b plus or minus the square root of b^2,"},{"Start":"02:24.480 ","End":"02:27.120","Text":"which is 16 minus 4ac,"},{"Start":"02:27.120 ","End":"02:30.780","Text":"4 times 1 times 5 is 20,"},{"Start":"02:30.780 ","End":"02:36.045","Text":"and all this over 2a which is 2 times 1 is 2."},{"Start":"02:36.045 ","End":"02:42.990","Text":"So what we get is equal to minus 4 plus or minus."},{"Start":"02:42.990 ","End":"02:46.503","Text":"Now this is the square root of minus 4."},{"Start":"02:46.503 ","End":"02:49.045","Text":"Let\u0027s just do that at the side."},{"Start":"02:49.045 ","End":"02:58.010","Text":"The square root of minus 4 is the square root of 4 times the square root of minus 1,"},{"Start":"02:58.010 ","End":"03:03.060","Text":"because minus 4 is 4 times minus 1 and this is equal to 2,"},{"Start":"03:03.060 ","End":"03:04.755","Text":"and this is equal to i."},{"Start":"03:04.755 ","End":"03:12.675","Text":"So we get minus 4 plus or minus 2i over 2,"},{"Start":"03:12.675 ","End":"03:15.865","Text":"and that gives us two answers."},{"Start":"03:15.865 ","End":"03:17.815","Text":"If I take the plus,"},{"Start":"03:17.815 ","End":"03:21.290","Text":"I\u0027ll get minus 4 plus 2i over 2,"},{"Start":"03:21.290 ","End":"03:23.010","Text":"and then we just cancel by 2,"},{"Start":"03:23.010 ","End":"03:25.725","Text":"so we get minus 2 plus i,"},{"Start":"03:25.725 ","End":"03:27.380","Text":"and if we take the minus,"},{"Start":"03:27.380 ","End":"03:30.365","Text":"we get minus 2 minus i."},{"Start":"03:30.365 ","End":"03:35.250","Text":"So these are the two solutions to this quadratic equation."},{"Start":"03:35.480 ","End":"03:39.240","Text":"Let\u0027s do one more,"},{"Start":"03:39.240 ","End":"03:42.045","Text":"but to point something out,"},{"Start":"03:42.045 ","End":"03:46.610","Text":"notice that the two solutions here are complex conjugates of each other,"},{"Start":"03:46.610 ","End":"03:49.655","Text":"minus 2 and it\u0027s plus or minus i."},{"Start":"03:49.655 ","End":"03:51.430","Text":"The same thing here,"},{"Start":"03:51.430 ","End":"03:52.900","Text":"the real part is zero."},{"Start":"03:52.900 ","End":"03:57.230","Text":"I could have written 0 plus or minus 3i."},{"Start":"03:57.230 ","End":"04:03.065","Text":"0 plus 3i is the conjugate of 0 minus 3i,"},{"Start":"04:03.065 ","End":"04:06.350","Text":"the opposite imaginary part, same as here."},{"Start":"04:06.350 ","End":"04:16.340","Text":"It happens when the coefficients of the quadratic equation, the real numbers."},{"Start":"04:16.340 ","End":"04:19.970","Text":"But actually even the coefficients could be complex numbers."},{"Start":"04:19.970 ","End":"04:23.197","Text":"Let\u0027s take another last example."},{"Start":"04:23.197 ","End":"04:26.325","Text":"Here it is I wrote it out when you weren\u0027t looking,"},{"Start":"04:26.325 ","End":"04:29.180","Text":"and now even the coefficients are complex,"},{"Start":"04:29.180 ","End":"04:30.650","Text":"but I still have an a, b,"},{"Start":"04:30.650 ","End":"04:33.400","Text":"and a c. This bit is a,"},{"Start":"04:33.400 ","End":"04:37.230","Text":"this bit is b and this bit is c,"},{"Start":"04:37.230 ","End":"04:40.299","Text":"and I can still use the quadratic equation."},{"Start":"04:40.299 ","End":"04:45.620","Text":"So what I get is that z is equal to minus"},{"Start":"04:45.620 ","End":"04:51.290","Text":"b plus or minus the square root of b^2,"},{"Start":"04:51.290 ","End":"04:54.680","Text":"which is 4 minus 4ac,"},{"Start":"04:54.680 ","End":"04:56.975","Text":"so it\u0027s 4 times a,"},{"Start":"04:56.975 ","End":"05:00.655","Text":"which is 1 plus i times c,"},{"Start":"05:00.655 ","End":"05:03.970","Text":"which is 1 minus i,"},{"Start":"05:03.970 ","End":"05:08.990","Text":"and all this is over 2a,"},{"Start":"05:08.990 ","End":"05:13.350","Text":"which is twice 1 plus i."},{"Start":"05:13.350 ","End":"05:15.190","Text":"There\u0027s a bit more work in"},{"Start":"05:15.190 ","End":"05:17.950","Text":"this exercise than the previous ones because we\u0027re going to have to do"},{"Start":"05:17.950 ","End":"05:22.000","Text":"some complex number multiplications and then we\u0027re going to have a division,"},{"Start":"05:22.000 ","End":"05:24.400","Text":"but we\u0027ll get there."},{"Start":"05:24.400 ","End":"05:27.720","Text":"Okay, so let\u0027s continue,"},{"Start":"05:27.720 ","End":"05:32.785","Text":"and what I suggest is to compute what\u0027s under the square root sign."},{"Start":"05:32.785 ","End":"05:35.555","Text":"Maybe we can do that at the side."},{"Start":"05:35.555 ","End":"05:39.810","Text":"Let\u0027s see, first of all, I\u0027ll do the 1 plus i times"},{"Start":"05:39.810 ","End":"05:43.975","Text":"1 minus i this bit and see what we get here."},{"Start":"05:43.975 ","End":"05:47.920","Text":"Well, this is a number times its conjugate and remember"},{"Start":"05:47.920 ","End":"05:52.990","Text":"the formula that when you multiply a complex number by its conjugate,"},{"Start":"05:52.990 ","End":"05:56.645","Text":"we get a^2 plus b^2."},{"Start":"05:56.645 ","End":"05:59.550","Text":"I\u0027m going to use that formula here,"},{"Start":"05:59.550 ","End":"06:03.810","Text":"and it\u0027s going to be 1^2 plus 1^2,"},{"Start":"06:03.810 ","End":"06:05.745","Text":"other words, 2."},{"Start":"06:05.745 ","End":"06:07.380","Text":"This part is 2,"},{"Start":"06:07.380 ","End":"06:09.180","Text":"and then under the square root sign,"},{"Start":"06:09.180 ","End":"06:15.660","Text":"I get the square root now 4 minus 4 times 2 because this part was 2 from here,"},{"Start":"06:15.660 ","End":"06:20.340","Text":"and that\u0027s 4 minus 8 is minus 4 so the square root of minus 4."},{"Start":"06:20.340 ","End":"06:27.800","Text":"Back here, we get minus 2 plus or minus"},{"Start":"06:27.800 ","End":"06:37.560","Text":"the square root of minus 4 over twice 1 plus i."},{"Start":"06:38.660 ","End":"06:44.445","Text":"Now back here, the square root of minus 4 is the square root of 4 times minus 1,"},{"Start":"06:44.445 ","End":"06:49.330","Text":"so its square root of 4 is 2 times square root of minus 1 is i, that\u0027s 2i."},{"Start":"06:50.030 ","End":"06:56.355","Text":"What I get, two solutions,"},{"Start":"06:56.355 ","End":"07:01.200","Text":"minus 2 plus 2i"},{"Start":"07:01.200 ","End":"07:09.030","Text":"over twice 1 plus i and minus 2,"},{"Start":"07:09.030 ","End":"07:17.340","Text":"this time, minus 2i over twice 1 plus i."},{"Start":"07:17.340 ","End":"07:23.165","Text":"Certainly, I can divide top and bottom by 2 in each case."},{"Start":"07:23.165 ","End":"07:25.460","Text":"In this case, if I cancel the 2,"},{"Start":"07:25.460 ","End":"07:32.490","Text":"I\u0027ve got minus 1 plus i over 1 plus i,"},{"Start":"07:32.490 ","End":"07:35.190","Text":"and in the other case, if I cancel by 2,"},{"Start":"07:35.190 ","End":"07:37.755","Text":"I have minus 1,"},{"Start":"07:37.755 ","End":"07:42.675","Text":"minus i over 1 plus i,"},{"Start":"07:42.675 ","End":"07:49.865","Text":"and you can see here that if I take the minus outside the brackets,"},{"Start":"07:49.865 ","End":"07:56.395","Text":"this is equal to minus 1 plus i over 1 plus i,"},{"Start":"07:56.395 ","End":"07:59.730","Text":"and that\u0027s going to equal minus 1."},{"Start":"07:59.730 ","End":"08:01.965","Text":"That\u0027s one of the solutions."},{"Start":"08:01.965 ","End":"08:03.810","Text":"As to this,"},{"Start":"08:03.810 ","End":"08:09.675","Text":"let\u0027s just continue this over here,"},{"Start":"08:09.675 ","End":"08:11.880","Text":"I\u0027m going to rewrite it. What was it?"},{"Start":"08:11.880 ","End":"08:16.590","Text":"Minus 1 plus i over 1 plus i,"},{"Start":"08:16.590 ","End":"08:20.165","Text":"and remember when we have a division,"},{"Start":"08:20.165 ","End":"08:25.110","Text":"we multiply top and bottom by the conjugate of the denominator,"},{"Start":"08:25.110 ","End":"08:31.020","Text":"so it\u0027s 1 minus i and 1 minus i."},{"Start":"08:31.020 ","End":"08:33.885","Text":"Let\u0027s see what we get."},{"Start":"08:33.885 ","End":"08:35.670","Text":"On the denominator,"},{"Start":"08:35.670 ","End":"08:37.070","Text":"it\u0027s always a positive number,"},{"Start":"08:37.070 ","End":"08:40.105","Text":"it\u0027s 1^2 plus 1^2,"},{"Start":"08:40.105 ","End":"08:42.420","Text":"pretty much like we did here."},{"Start":"08:42.420 ","End":"08:45.045","Text":"So that comes out to be 2."},{"Start":"08:45.045 ","End":"08:46.770","Text":"On the numerator, let\u0027s see,"},{"Start":"08:46.770 ","End":"08:54.150","Text":"we have minus 1 times 1 is minus 1,1 times i is"},{"Start":"08:54.150 ","End":"09:04.050","Text":"i minus 1 times minus i is also plus i and i times minus i is minus i^2,"},{"Start":"09:04.050 ","End":"09:09.360","Text":"but minus i^2 is plus 1,"},{"Start":"09:09.360 ","End":"09:11.830","Text":"because i^2 is minus 1."},{"Start":"09:11.830 ","End":"09:16.960","Text":"What we get is 1 is 1 with 1 cancels to get 2i over 2,"},{"Start":"09:16.960 ","End":"09:20.050","Text":"which is just equal to i."},{"Start":"09:20.050 ","End":"09:22.429","Text":"So collecting the two together,"},{"Start":"09:22.429 ","End":"09:32.585","Text":"we\u0027ve got that the solution is the Z is either equal to minus 1 or i two solutions."},{"Start":"09:32.585 ","End":"09:35.150","Text":"So from now on,"},{"Start":"09:35.150 ","End":"09:36.770","Text":"one word in complex numbers,"},{"Start":"09:36.770 ","End":"09:39.785","Text":"all quadratic equations will have two solutions."},{"Start":"09:39.785 ","End":"09:42.260","Text":"Well, except the case that there\u0027s only one solution,"},{"Start":"09:42.260 ","End":"09:45.410","Text":"but that\u0027s really two solutions that just happened to be the same."},{"Start":"09:45.410 ","End":"09:48.880","Text":"In any events, we don\u0027t get the situation where there\u0027s no solution,"},{"Start":"09:48.880 ","End":"09:51.495","Text":"because the square root can be of anything,"},{"Start":"09:51.495 ","End":"09:53.755","Text":"and that\u0027s what was impeding us."},{"Start":"09:53.755 ","End":"09:55.910","Text":"That\u0027s it. That\u0027s the last chapter."},{"Start":"09:55.910 ","End":"09:58.890","Text":"Done with complex numbers."}],"ID":4922},{"Watched":false,"Name":"Exercise 1","Duration":"4m 20s","ChapterTopicVideoID":4919,"CourseChapterTopicPlaylistID":3860,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.070","Text":"In this exercise,"},{"Start":"00:02.070 ","End":"00:05.700","Text":"a little bit of practice in computations"},{"Start":"00:05.700 ","End":"00:09.720","Text":"with complex numbers and addition or subtraction,"},{"Start":"00:09.720 ","End":"00:17.955","Text":"and then multiplied by a real number or an imaginary number or a mixture."},{"Start":"00:17.955 ","End":"00:19.725","Text":"Anyway, let\u0027s get started with"},{"Start":"00:19.725 ","End":"00:26.070","Text":"the first one which is a simple addition of two complex numbers."},{"Start":"00:26.070 ","End":"00:30.495","Text":"What we do is we add the real part;"},{"Start":"00:30.495 ","End":"00:32.055","Text":"4 plus 3,"},{"Start":"00:32.055 ","End":"00:34.960","Text":"and that would be 7."},{"Start":"00:34.960 ","End":"00:37.415","Text":"Then the imaginary part,"},{"Start":"00:37.415 ","End":"00:39.545","Text":"i and minus 2i,"},{"Start":"00:39.545 ","End":"00:44.545","Text":"so that\u0027s minus i and that\u0027s all there is to it."},{"Start":"00:44.545 ","End":"00:49.980","Text":"Here we have a subtraction almost the same as addition."},{"Start":"00:49.980 ","End":"00:52.965","Text":"We do the real part separately;"},{"Start":"00:52.965 ","End":"00:58.710","Text":"minus 2, minus 4 would be minus 6,"},{"Start":"00:58.710 ","End":"01:05.565","Text":"and then 3i takeaway minus 5i is 3i plus 5i,"},{"Start":"01:05.565 ","End":"01:09.180","Text":"so the answer is minus 6 plus 8i."},{"Start":"01:09.180 ","End":"01:13.935","Text":"Very straightforward. The next one,"},{"Start":"01:13.935 ","End":"01:21.785","Text":"we have to multiply first each bit by a real number."},{"Start":"01:21.785 ","End":"01:24.920","Text":"Multiplying a complex number by a real number is very"},{"Start":"01:24.920 ","End":"01:28.700","Text":"straightforward also because you just multiply each part."},{"Start":"01:28.700 ","End":"01:34.830","Text":"In the first one 3 times 2 is 6 and 3 times 1 is 3,"},{"Start":"01:34.830 ","End":"01:37.200","Text":"so we get 6 plus 3i."},{"Start":"01:37.200 ","End":"01:38.670","Text":"That\u0027s the first bit."},{"Start":"01:38.670 ","End":"01:42.315","Text":"The second bit multiplied by the 5,"},{"Start":"01:42.315 ","End":"01:44.010","Text":"but it\u0027s a minus 5,"},{"Start":"01:44.010 ","End":"01:48.300","Text":"and we\u0027ll need to put it in brackets because the whole thing is being subtracted."},{"Start":"01:48.300 ","End":"01:50.730","Text":"So 5 times 6 is 30,"},{"Start":"01:50.730 ","End":"01:54.600","Text":"5 times 2i is 10i."},{"Start":"01:54.600 ","End":"01:56.805","Text":"Now we have a subtraction problem,"},{"Start":"01:56.805 ","End":"02:02.205","Text":"so 6 minus 30 for the real part is minus 24,"},{"Start":"02:02.205 ","End":"02:07.335","Text":"and 3i, minus 10i is minus 7i,"},{"Start":"02:07.335 ","End":"02:09.790","Text":"and that\u0027s our answer."},{"Start":"02:10.310 ","End":"02:14.820","Text":"In this one, we multiply not by a real number,"},{"Start":"02:14.820 ","End":"02:17.070","Text":"but by a pure imaginary number,"},{"Start":"02:17.070 ","End":"02:19.690","Text":"and that changes things a bit."},{"Start":"02:19.690 ","End":"02:23.330","Text":"The way to do this is to multiply each bit,"},{"Start":"02:23.330 ","End":"02:24.800","Text":"but also reverse the order,"},{"Start":"02:24.800 ","End":"02:31.875","Text":"because notice that if I multiply i by i that gives me i^2 which is minus 1,"},{"Start":"02:31.875 ","End":"02:33.840","Text":"and that\u0027s the real part,"},{"Start":"02:33.840 ","End":"02:38.770","Text":"and i times 4 is 4i."},{"Start":"02:38.930 ","End":"02:42.930","Text":"When you have an imaginary in front just be prepared to change the order"},{"Start":"02:42.930 ","End":"02:46.890","Text":"and then that will be the standard representation."},{"Start":"02:46.890 ","End":"02:48.660","Text":"We still have the other part."},{"Start":"02:48.660 ","End":"02:52.920","Text":"So again, do the 3i by the 2i first,"},{"Start":"02:52.920 ","End":"02:55.170","Text":"and that gives us 6i ^2,"},{"Start":"02:55.170 ","End":"03:00.330","Text":"and 6i ^2 is minus 6."},{"Start":"03:00.330 ","End":"03:04.170","Text":"Let\u0027s just make a reminder,"},{"Start":"03:04.170 ","End":"03:07.935","Text":"i^2 is minus 1."},{"Start":"03:07.935 ","End":"03:14.500","Text":"Then the 3i times the minus 5 is minus 15i."},{"Start":"03:14.900 ","End":"03:18.135","Text":"Because it\u0027s a plus, I don\u0027t need brackets here,"},{"Start":"03:18.135 ","End":"03:21.870","Text":"and we now just collect together the real part is minus 1,"},{"Start":"03:21.870 ","End":"03:24.855","Text":"minus 6 is minus 7."},{"Start":"03:24.855 ","End":"03:31.540","Text":"4i minus 15i is minus 11i."},{"Start":"03:32.540 ","End":"03:35.150","Text":"Now the last one."},{"Start":"03:35.150 ","End":"03:38.010","Text":"This part has an imaginary multiplier,"},{"Start":"03:38.010 ","End":"03:40.365","Text":"this is one is a real multiplier."},{"Start":"03:40.365 ","End":"03:42.450","Text":"For the imaginary one, once again,"},{"Start":"03:42.450 ","End":"03:47.610","Text":"I do it the 2i times the i first, that\u0027s 2i^2,"},{"Start":"03:47.610 ","End":"03:52.110","Text":"and therefore, we have minus 2,"},{"Start":"03:52.110 ","End":"03:57.090","Text":"and then the 2i times the 3 is 6i."},{"Start":"03:57.090 ","End":"04:00.035","Text":"The other one, because the 4 is real,"},{"Start":"04:00.035 ","End":"04:02.045","Text":"you don\u0027t have to change the order or anything,"},{"Start":"04:02.045 ","End":"04:07.775","Text":"so we get plus 4 times 2 is 8 and minus 12i and then collect together."},{"Start":"04:07.775 ","End":"04:10.310","Text":"Minus 2 and 8 is 6,"},{"Start":"04:10.310 ","End":"04:14.810","Text":"6 minus 12i is minus 6i."},{"Start":"04:14.810 ","End":"04:20.410","Text":"This is just warm-up exercises and we\u0027re done."}],"ID":4923},{"Watched":false,"Name":"Exercise 2","Duration":"10m 27s","ChapterTopicVideoID":4920,"CourseChapterTopicPlaylistID":3860,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.074","Text":"In this exercise, we\u0027ll be practicing multiplication of complex numbers."},{"Start":"00:06.074 ","End":"00:12.940","Text":"There\u0027s a long way and a short way and I\u0027ll start with the long way of doing things."},{"Start":"00:13.820 ","End":"00:22.530","Text":"I guess we\u0027ll start with number 1 and we just expand using the general distributive law,"},{"Start":"00:22.530 ","End":"00:25.215","Text":"meaning each 1 of these multiplied with each 1 of these."},{"Start":"00:25.215 ","End":"00:34.305","Text":"Let\u0027s say 4 times 3 is 12 and then 4 times minus 2i is minus 8i,"},{"Start":"00:34.305 ","End":"00:38.115","Text":"i times 3 is 3i,"},{"Start":"00:38.115 ","End":"00:41.025","Text":"you don\u0027t write i3, you write 3i."},{"Start":"00:41.025 ","End":"00:46.480","Text":"Then i times minus 2i is minus 2i^2."},{"Start":"00:46.850 ","End":"00:49.385","Text":"Then in the next step,"},{"Start":"00:49.385 ","End":"00:53.580","Text":"you collect the real part."},{"Start":"00:53.620 ","End":"00:57.205","Text":"I just want to remind you"},{"Start":"00:57.205 ","End":"01:05.150","Text":"that i^2 is minus 1, that\u0027s very basic."},{"Start":"01:05.150 ","End":"01:08.990","Text":"It\u0027s the most essential thing there is in complex numbers."},{"Start":"01:08.990 ","End":"01:13.050","Text":"This i^2 is minus 1,"},{"Start":"01:13.050 ","End":"01:15.030","Text":"so what I have here is plus 2."},{"Start":"01:15.030 ","End":"01:23.680","Text":"So 12 plus 2 is 14 and minus 8i plus 3i is minus 5i."},{"Start":"01:23.990 ","End":"01:27.930","Text":"Next one. Just expand,"},{"Start":"01:27.930 ","End":"01:31.275","Text":"minus 2 times 4 is minus 8,"},{"Start":"01:31.275 ","End":"01:36.345","Text":"minus 2 times minus 5i is plus 10i,"},{"Start":"01:36.345 ","End":"01:43.330","Text":"and then plus 12i and minus 15i^2."},{"Start":"01:43.330 ","End":"01:45.740","Text":"Then I collect this with this."},{"Start":"01:45.740 ","End":"01:51.035","Text":"This is minus 8 plus 15 and"},{"Start":"01:51.035 ","End":"01:56.385","Text":"that gives us 7 and 10 and 12,"},{"Start":"01:56.385 ","End":"02:02.220","Text":"that makes it 22i."},{"Start":"02:02.220 ","End":"02:08.520","Text":"Next one. Let\u0027s see if we can get all 3 of them in, that\u0027s it."},{"Start":"02:08.520 ","End":"02:15.090","Text":"Minus 4 times minus 2 is 8 and then this with this minus"},{"Start":"02:15.090 ","End":"02:20.805","Text":"12i plus 4i"},{"Start":"02:20.805 ","End":"02:28.485","Text":"and minus 6i^2."},{"Start":"02:28.485 ","End":"02:31.680","Text":"So 8 plus 6, I\u0027ll write it again,"},{"Start":"02:31.680 ","End":"02:35.670","Text":"it\u0027s so important, i^2 is minus 1,"},{"Start":"02:35.670 ","End":"02:44.610","Text":"is 14 and minus 12 plus 4 is minus 8i."},{"Start":"02:44.610 ","End":"02:47.580","Text":"That\u0027s that one and the next,"},{"Start":"02:47.580 ","End":"02:54.750","Text":"2 minus 6i minus"},{"Start":"02:54.750 ","End":"03:00.960","Text":"4i plus 12i^2,"},{"Start":"03:00.960 ","End":"03:09.720","Text":"2 minus 12 is minus 10 and minus 10i."},{"Start":"03:09.720 ","End":"03:13.215","Text":"Then 4 times 4, 16,"},{"Start":"03:13.215 ","End":"03:16.270","Text":"4 times 2 plus 8i"},{"Start":"03:16.730 ","End":"03:25.510","Text":"minus 8i and then minus 4i^2."},{"Start":"03:26.600 ","End":"03:30.540","Text":"This cancels with this,"},{"Start":"03:30.540 ","End":"03:36.360","Text":"so we get just 16 plus 4,"},{"Start":"03:36.360 ","End":"03:38.520","Text":"that makes it 20."},{"Start":"03:38.520 ","End":"03:41.979","Text":"These 2 cancel, so 0i, so just 20."},{"Start":"03:41.979 ","End":"03:49.630","Text":"Of course, we could have done this also with the formula for a minus b,"},{"Start":"03:49.630 ","End":"03:58.610","Text":"a plus b and said straight away that this is a^2 minus b^2,"},{"Start":"03:58.610 ","End":"04:08.875","Text":"which in our case came out to be 4^2 minus 2i^2."},{"Start":"04:08.875 ","End":"04:14.470","Text":"But 2i^2 squared is 4i^2,"},{"Start":"04:14.470 ","End":"04:19.470","Text":"which is minus 4,"},{"Start":"04:19.470 ","End":"04:21.633","Text":"so minus minus 4 is plus 4."},{"Start":"04:21.633 ","End":"04:26.910","Text":"Anyway, you get 16 plus 4 and that\u0027s the same as the 20."},{"Start":"04:26.910 ","End":"04:30.190","Text":"I\u0027m just observing that we could have used difference of squares here."},{"Start":"04:30.190 ","End":"04:34.550","Text":"We\u0027re not saying it\u0027s easier, just noting it."},{"Start":"04:34.550 ","End":"04:38.290","Text":"What I\u0027d like to do now is go back to the beginning and do it with"},{"Start":"04:38.290 ","End":"04:41.650","Text":"the short way once you practice with this."},{"Start":"04:41.650 ","End":"04:45.740","Text":"I\u0027m going to go back up to number 1."},{"Start":"04:46.320 ","End":"04:52.270","Text":"Just clean up a bit."},{"Start":"04:52.270 ","End":"04:54.160","Text":"I\u0027ll use a different color."},{"Start":"04:54.160 ","End":"05:00.185","Text":"In this technique, we just can do everything in our heads without any intermediate steps."},{"Start":"05:00.185 ","End":"05:05.960","Text":"You say we get a real part from the first times the first and the last times the last."},{"Start":"05:05.960 ","End":"05:12.415","Text":"4 times 3 is 12 and then minus 2i^2,"},{"Start":"05:12.415 ","End":"05:15.660","Text":"so you just change the sign, it\u0027s plus 2."},{"Start":"05:15.660 ","End":"05:20.490","Text":"Basically you say this times this minus the 1 times minus 2,"},{"Start":"05:20.490 ","End":"05:23.860","Text":"so it\u0027s 12 plus 2 is 14."},{"Start":"05:23.860 ","End":"05:26.990","Text":"I\u0027m just reminding you this is 12."},{"Start":"05:26.990 ","End":"05:34.850","Text":"In our heads, we say 12 minus the product of these 2 is minus 2, for the first bit."},{"Start":"05:34.850 ","End":"05:41.670","Text":"For the i bit we just say 1 times 3 is 3,"},{"Start":"05:41.670 ","End":"05:45.225","Text":"4 times minus 2 is minus 8,"},{"Start":"05:45.225 ","End":"05:47.670","Text":"so we say minus 5i,"},{"Start":"05:47.670 ","End":"05:51.515","Text":"but mentally we did the exercise of"},{"Start":"05:51.515 ","End":"05:59.200","Text":"3 minus 8 to get to the minus 5."},{"Start":"05:59.200 ","End":"06:02.010","Text":"I\u0027m just going to erase that."},{"Start":"06:02.010 ","End":"06:06.705","Text":"That was just what takes place mentally."},{"Start":"06:06.705 ","End":"06:09.680","Text":"Let me be a bit more precise in this one,"},{"Start":"06:09.680 ","End":"06:11.495","Text":"I was a bit vague here."},{"Start":"06:11.495 ","End":"06:13.190","Text":"You took the real part,"},{"Start":"06:13.190 ","End":"06:15.170","Text":"you multiply this times this,"},{"Start":"06:15.170 ","End":"06:17.735","Text":"minus this times this, ignore the i,"},{"Start":"06:17.735 ","End":"06:23.180","Text":"say minus 2 times 4 is 8 and then subtract 3 times minus 5,"},{"Start":"06:23.180 ","End":"06:24.815","Text":"which is minus 15."},{"Start":"06:24.815 ","End":"06:35.010","Text":"We get 8 minus minus 15 is 7."},{"Start":"06:35.840 ","End":"06:41.520","Text":"Then for the imaginary bit you just take this times this plus this times this,"},{"Start":"06:41.520 ","End":"06:45.285","Text":"12 plus 10 is 22i."},{"Start":"06:45.285 ","End":"06:52.160","Text":"In fact, I think it will help if I just write a little formula in general, when we have,"},{"Start":"06:52.160 ","End":"06:59.950","Text":"let\u0027s say 1 of them is a plus bi and the other 1 is c plus di."},{"Start":"06:59.980 ","End":"07:10.040","Text":"What we get the real part is this times this minus this times this."},{"Start":"07:10.040 ","End":"07:15.385","Text":"In other words, it\u0027s ac minus bd."},{"Start":"07:15.385 ","End":"07:18.890","Text":"The reason for the minus is because that\u0027s what takes"},{"Start":"07:18.890 ","End":"07:22.370","Text":"the place of the i^2 if we were multiplying it out."},{"Start":"07:22.370 ","End":"07:26.050","Text":"That\u0027s the real part and the imaginary part,"},{"Start":"07:26.050 ","End":"07:28.245","Text":"we do this with this,"},{"Start":"07:28.245 ","End":"07:31.000","Text":"the inners and the outers."},{"Start":"07:31.520 ","End":"07:36.930","Text":"That would give us ad plus bc or bc plus ad."},{"Start":"07:36.930 ","End":"07:39.945","Text":"Let\u0027s say ad plus bc,"},{"Start":"07:39.945 ","End":"07:43.290","Text":"that\u0027s the i part."},{"Start":"07:43.290 ","End":"07:45.110","Text":"If I wanted to phrase that,"},{"Start":"07:45.110 ","End":"07:49.100","Text":"I say that for the real part"},{"Start":"07:49.100 ","End":"07:53.540","Text":"we get the first and the first minus the second and the second."},{"Start":"07:53.540 ","End":"07:55.775","Text":"Then for the i part,"},{"Start":"07:55.775 ","End":"08:00.765","Text":"the inner pair plus the outer pair."},{"Start":"08:00.765 ","End":"08:05.925","Text":"Let\u0027s practice that on this one, on the third one."},{"Start":"08:05.925 ","End":"08:09.255","Text":"Minus 4 and minus 2,"},{"Start":"08:09.255 ","End":"08:12.750","Text":"we multiply it by, is 8,"},{"Start":"08:12.750 ","End":"08:16.245","Text":"minus 2 times 3 is minus 6,"},{"Start":"08:16.245 ","End":"08:22.095","Text":"so I do 8 subtract minus 6 and that gives me 14."},{"Start":"08:22.095 ","End":"08:24.620","Text":"I can think of that 14 as being from here,"},{"Start":"08:24.620 ","End":"08:30.930","Text":"this times this minus this times this and for the i part,"},{"Start":"08:30.930 ","End":"08:34.540","Text":"this times this plus this times this."},{"Start":"08:34.690 ","End":"08:39.575","Text":"Sorry, plus 4 minus 12,"},{"Start":"08:39.575 ","End":"08:44.040","Text":"that\u0027s minus 8 and that gets the i stuck onto it."},{"Start":"08:45.100 ","End":"08:49.980","Text":"I\u0027ll continue. I\u0027ll just copy this down there."},{"Start":"08:52.090 ","End":"08:54.680","Text":"The formula, let\u0027s see,"},{"Start":"08:54.680 ","End":"08:58.530","Text":"I copied it, I\u0027ll paste it, there we are."},{"Start":"08:58.870 ","End":"09:01.310","Text":"Applying it to this one,"},{"Start":"09:01.310 ","End":"09:04.670","Text":"it doesn\u0027t hurt actually to draw the arcs in, I see."},{"Start":"09:04.670 ","End":"09:06.950","Text":"The first with the first, the second with the second,"},{"Start":"09:06.950 ","End":"09:08.330","Text":"then we do a subtraction."},{"Start":"09:08.330 ","End":"09:10.280","Text":"2 times 1 is 2,"},{"Start":"09:10.280 ","End":"09:13.400","Text":"minus 4 times minus 3 is 12,"},{"Start":"09:13.400 ","End":"09:17.685","Text":"2 minus 12 minus 10."},{"Start":"09:17.685 ","End":"09:22.450","Text":"Then this and this and this time we add."},{"Start":"09:22.450 ","End":"09:26.120","Text":"2 times minus 3 is minus 6,"},{"Start":"09:26.120 ","End":"09:27.590","Text":"this is minus 4,"},{"Start":"09:27.590 ","End":"09:34.790","Text":"minus 6 add minus 4 gives us minus 10."},{"Start":"09:34.790 ","End":"09:39.380","Text":"Same as this, but this time it\u0027s with an i on the second bit."},{"Start":"09:41.200 ","End":"09:44.970","Text":"Don\u0027t need this now."},{"Start":"09:45.100 ","End":"09:49.795","Text":"Once again, this and this and this and this and subtract."},{"Start":"09:49.795 ","End":"09:51.790","Text":"4 times 4 is 16,"},{"Start":"09:51.790 ","End":"09:54.690","Text":"minus 2 times 2 is minus 4,"},{"Start":"09:54.690 ","End":"09:58.695","Text":"16 less minus 4 is 20."},{"Start":"09:58.695 ","End":"10:02.190","Text":"Then this with this, this with this."},{"Start":"10:02.190 ","End":"10:05.370","Text":"Minus 2 times 4 is minus 8,"},{"Start":"10:05.370 ","End":"10:06.990","Text":"4 times 2 is plus 8,"},{"Start":"10:06.990 ","End":"10:09.450","Text":"minus 8 and 8 is 0."},{"Start":"10:09.450 ","End":"10:12.830","Text":"I could put plus 0 and then the i and of course,"},{"Start":"10:12.830 ","End":"10:15.740","Text":"we don\u0027t actually need this part,"},{"Start":"10:15.740 ","End":"10:19.320","Text":"just drop it, get the same as before."},{"Start":"10:19.460 ","End":"10:22.595","Text":"We did this exercise twice,"},{"Start":"10:22.595 ","End":"10:27.420","Text":"long and short, and you can choose whichever one you like. We\u0027re done."}],"ID":4924},{"Watched":false,"Name":"Exercise 3","Duration":"9m 41s","ChapterTopicVideoID":4921,"CourseChapterTopicPlaylistID":3860,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.390","Text":"In this exercise, there are six separate computations."},{"Start":"00:05.390 ","End":"00:09.050","Text":"They all have exponents or powers,"},{"Start":"00:09.050 ","End":"00:13.125","Text":"and mostly they\u0027re based on powers of i."},{"Start":"00:13.125 ","End":"00:16.575","Text":"Now the most fundamental thing that we could"},{"Start":"00:16.575 ","End":"00:20.145","Text":"use in all of these exercises is the following."},{"Start":"00:20.145 ","End":"00:23.850","Text":"That i to the power of 2,"},{"Start":"00:23.850 ","End":"00:32.250","Text":"i^2 rather is = minus 1."},{"Start":"00:32.250 ","End":"00:36.810","Text":"Also useful is i^3 is = minus i,"},{"Start":"00:36.810 ","End":"00:41.815","Text":"and very useful also is i^4 is =1."},{"Start":"00:41.815 ","End":"00:48.110","Text":"Just for completeness, I\u0027ll throw also i^1 =i in here."},{"Start":"00:48.110 ","End":"00:51.445","Text":"So we have i^1, 2, 3, and 4."},{"Start":"00:51.445 ","End":"00:55.730","Text":"It turns out that it goes periodically that i^5 is i,"},{"Start":"00:55.730 ","End":"00:59.747","Text":"i^6, i^7,"},{"Start":"00:59.747 ","End":"01:01.200","Text":"i^8, i^9, 10, 11, 12, and so on."},{"Start":"01:01.200 ","End":"01:05.413","Text":"There are only four combinations of i to the power of,"},{"Start":"01:05.413 ","End":"01:07.100","Text":"and this is the basis of it."},{"Start":"01:07.100 ","End":"01:10.670","Text":"Let\u0027s just assume that we remember these by heart and we\u0027ll use those in"},{"Start":"01:10.670 ","End":"01:15.270","Text":"all the six exercises that follow."},{"Start":"01:15.270 ","End":"01:19.570","Text":"We\u0027ll start with the first one."},{"Start":"01:20.660 ","End":"01:24.195","Text":"i^8, I can write,"},{"Start":"01:24.195 ","End":"01:26.200","Text":"this is one possibility,"},{"Start":"01:26.200 ","End":"01:30.765","Text":"as i^4 times i^4."},{"Start":"01:30.765 ","End":"01:33.570","Text":"Since i^4 is 1,"},{"Start":"01:33.570 ","End":"01:34.755","Text":"that is very important."},{"Start":"01:34.755 ","End":"01:36.525","Text":"This is 1 times 1,"},{"Start":"01:36.525 ","End":"01:38.295","Text":"which is just 1."},{"Start":"01:38.295 ","End":"01:42.480","Text":"Or I could have written (i^4)^2."},{"Start":"01:42.480 ","End":"01:46.840","Text":"Anyway, the answer is just 1."},{"Start":"01:47.300 ","End":"01:50.995","Text":"The other way what I meant was (i^4)^2."},{"Start":"01:50.995 ","End":"01:53.915","Text":"The reason I might do it this way is if it was a larger number,"},{"Start":"01:53.915 ","End":"01:57.770","Text":"say i^80, I wouldn\u0027t want to write i^4,"},{"Start":"01:57.770 ","End":"02:00.720","Text":"i^4, i^4 20 times."},{"Start":"02:00.720 ","End":"02:03.325","Text":"I would rather say (i^4)^20."},{"Start":"02:03.325 ","End":"02:06.110","Text":"In this case, I\u0027ll do it this way."},{"Start":"02:06.110 ","End":"02:07.610","Text":"But if it was a very large number,"},{"Start":"02:07.610 ","End":"02:09.350","Text":"I would maybe like,"},{"Start":"02:09.350 ","End":"02:11.625","Text":"I don\u0027t know i^400 here."},{"Start":"02:11.625 ","End":"02:13.775","Text":"Then I would say (i^4)^100,"},{"Start":"02:13.775 ","End":"02:16.880","Text":"and then it\u0027s 1^2 and it\u0027s 1."},{"Start":"02:16.880 ","End":"02:20.070","Text":"So this is just an alternative."},{"Start":"02:21.790 ","End":"02:25.415","Text":"I\u0027ll change the color. It\u0027s an alternative solution."},{"Start":"02:25.415 ","End":"02:28.700","Text":"Now here, i^15,"},{"Start":"02:28.700 ","End":"02:30.935","Text":"it doesn\u0027t divide by 4,"},{"Start":"02:30.935 ","End":"02:36.170","Text":"but I could write it as i to the power of the nearest thing"},{"Start":"02:36.170 ","End":"02:41.540","Text":"that divides by 4 that\u0027s below 15, is 12."},{"Start":"02:41.540 ","End":"02:48.120","Text":"I could write it then as i^12 times i^3."},{"Start":"02:48.120 ","End":"02:50.330","Text":"Then I could use this."},{"Start":"02:50.330 ","End":"02:53.150","Text":"I actually prefer to go with the exponent."},{"Start":"02:53.150 ","End":"02:54.875","Text":"This one would be the better way,"},{"Start":"02:54.875 ","End":"02:59.890","Text":"because in this case I can use it as (i^4)^3."},{"Start":"02:59.890 ","End":"03:08.255","Text":"I hope you remember your rules of exponents, so times i^3."},{"Start":"03:08.255 ","End":"03:10.340","Text":"Then this one,"},{"Start":"03:10.340 ","End":"03:15.750","Text":"i^4 is 1, 1^3 is 1."},{"Start":"03:16.070 ","End":"03:18.330","Text":"Write it as 1^3."},{"Start":"03:18.330 ","End":"03:22.810","Text":"An i^3 from here is minus i."},{"Start":"03:22.810 ","End":"03:28.380","Text":"Altogether the answer is minus i."},{"Start":"03:28.720 ","End":"03:34.175","Text":"We use this quite heavily that i^4 is 1."},{"Start":"03:34.175 ","End":"03:43.160","Text":"I will say you should remember that i^3 is minus i. P is fairly often i^2= minus 1,"},{"Start":"03:43.160 ","End":"03:44.900","Text":"of course is a very basic,"},{"Start":"03:44.900 ","End":"03:48.245","Text":"that\u0027s almost the definition of i."},{"Start":"03:48.245 ","End":"03:50.225","Text":"Of course, you know this."},{"Start":"03:50.225 ","End":"03:54.670","Text":"Continuing with the next one."},{"Start":"03:54.670 ","End":"04:01.120","Text":"The formulas, the stuff in red is gone but that\u0027s okay."},{"Start":"04:01.120 ","End":"04:05.710","Text":"Change my mind, I think I just copied them over."},{"Start":"04:05.710 ","End":"04:08.360","Text":"We\u0027ll have them right in front of us."},{"Start":"04:08.360 ","End":"04:11.930","Text":"Number 3 is not just i to the power of,"},{"Start":"04:11.930 ","End":"04:14.110","Text":"its minus i to the power of."},{"Start":"04:14.110 ","End":"04:20.720","Text":"Now minus i is minus 1 times i."},{"Start":"04:20.720 ","End":"04:23.630","Text":"When we take a product to the power of,"},{"Start":"04:23.630 ","End":"04:25.805","Text":"we take each bit to the power."},{"Start":"04:25.805 ","End":"04:29.700","Text":"So this thing will equal"},{"Start":"04:30.490 ","End":"04:37.870","Text":"minus 1^6 times i^6."},{"Start":"04:37.870 ","End":"04:43.880","Text":"Now minus 1^6 is because the 6 is an even number, this is =1."},{"Start":"04:43.880 ","End":"04:52.950","Text":"I^6, I can write as i^4 times i^2 because I want to use these."},{"Start":"04:52.960 ","End":"04:58.290","Text":"The i^4 is 1, i^2 is minus 1."},{"Start":"04:58.290 ","End":"05:04.177","Text":"So altogether I get 1 times 1 times minus 1,"},{"Start":"05:04.177 ","End":"05:07.920","Text":"and so the answer is minus 1."},{"Start":"05:08.600 ","End":"05:11.810","Text":"We have another one with minus i."},{"Start":"05:11.810 ","End":"05:13.555","Text":"Same technique."},{"Start":"05:13.555 ","End":"05:21.860","Text":"Write this as minus 1^9, i^9."},{"Start":"05:21.860 ","End":"05:26.015","Text":"This gives us, because 9 is an odd number,"},{"Start":"05:26.015 ","End":"05:29.345","Text":"minus 1^9 is minus 1."},{"Start":"05:29.345 ","End":"05:33.155","Text":"I^9, I can write as"},{"Start":"05:33.155 ","End":"05:41.105","Text":"i^8 times i. I take the multiple of 4 closest but below 9."},{"Start":"05:41.105 ","End":"05:44.795","Text":"Then this is equal to,"},{"Start":"05:44.795 ","End":"05:48.595","Text":"i^8 is also 1 because it\u0027s (i^4)^2,"},{"Start":"05:48.595 ","End":"05:51.735","Text":"or i^4 times i^4."},{"Start":"05:51.735 ","End":"05:58.469","Text":"This gives us minus 1 times 1 times i."},{"Start":"05:58.469 ","End":"06:02.635","Text":"This is equal to minus i."},{"Start":"06:02.635 ","End":"06:08.165","Text":"In general, i to the power of any number that\u0027s divisible by 4 is 1."},{"Start":"06:08.165 ","End":"06:10.805","Text":"So it\u0027s not just that i^4 is 1,"},{"Start":"06:10.805 ","End":"06:12.965","Text":"i^8 is 1,"},{"Start":"06:12.965 ","End":"06:18.010","Text":"i^12 is 1, i^16 is 1 and so on."},{"Start":"06:18.010 ","End":"06:25.975","Text":"In the next one, it\u0027s more involved."},{"Start":"06:25.975 ","End":"06:32.155","Text":"We have this whole thing to the power of 2 or squared."},{"Start":"06:32.155 ","End":"06:37.220","Text":"We just use the algebraic formula,"},{"Start":"06:38.060 ","End":"06:42.054","Text":"special binomial products or expansions."},{"Start":"06:42.054 ","End":"06:44.380","Text":"In general in algebra,"},{"Start":"06:44.380 ","End":"06:47.730","Text":"(a+b)^2"},{"Start":"06:47.730 ","End":"06:55.470","Text":"is a^2 plus 2ab plus b^2."},{"Start":"06:55.470 ","End":"06:58.205","Text":"Looking ahead at number 6,"},{"Start":"06:58.205 ","End":"07:00.400","Text":"we might need the same thing with a minus."},{"Start":"07:00.400 ","End":"07:03.345","Text":"Actually, I don\u0027t have to write another formula."},{"Start":"07:03.345 ","End":"07:05.690","Text":"Usually, I combine the 2 formulas in 1."},{"Start":"07:05.690 ","End":"07:07.175","Text":"If there\u0027s a minus here,"},{"Start":"07:07.175 ","End":"07:09.514","Text":"then the middle term gets the minus."},{"Start":"07:09.514 ","End":"07:11.600","Text":"So it\u0027s either plus here and plus here,"},{"Start":"07:11.600 ","End":"07:13.310","Text":"or minus here and minus here."},{"Start":"07:13.310 ","End":"07:15.930","Text":"That will be good for these two."},{"Start":"07:16.240 ","End":"07:22.385","Text":"In this case, using the formula with plus a is 2i, b is 3."},{"Start":"07:22.385 ","End":"07:28.170","Text":"I get (2i)^2 plus"},{"Start":"07:28.300 ","End":"07:35.000","Text":"twice 2i times 3 plus 3^2."},{"Start":"07:35.000 ","End":"07:41.510","Text":"This gives us 4i^2"},{"Start":"07:41.510 ","End":"07:46.226","Text":"plus 2 times 2 times 3 is 12i,"},{"Start":"07:46.226 ","End":"07:48.510","Text":"plus 3^2 is 9."},{"Start":"07:48.510 ","End":"07:52.170","Text":"Now i^2 is minus 1."},{"Start":"07:52.170 ","End":"07:58.490","Text":"So this is minus 4 plus 9 is 5."},{"Start":"07:58.490 ","End":"08:01.450","Text":"I\u0027m not going to write it again. i^2 is minus 1."},{"Start":"08:01.450 ","End":"08:05.215","Text":"That\u0027s the most basic thing that there is to know about complex numbers."},{"Start":"08:05.215 ","End":"08:08.185","Text":"Then plus 12i."},{"Start":"08:08.185 ","End":"08:11.900","Text":"This is our answer to this."},{"Start":"08:12.050 ","End":"08:17.770","Text":"Here, we\u0027re going to use the formula with a minus."},{"Start":"08:17.770 ","End":"08:20.559","Text":"But the thing is, I think we should first"},{"Start":"08:20.559 ","End":"08:23.920","Text":"simplify the powers of I and then use the formula,"},{"Start":"08:23.920 ","End":"08:25.810","Text":"but you could do it the other way round."},{"Start":"08:25.810 ","End":"08:28.335","Text":"So I suggest saying as follows."},{"Start":"08:28.335 ","End":"08:30.680","Text":"That for I expand it out the side,"},{"Start":"08:30.680 ","End":"08:36.275","Text":"I\u0027ll say i^5 is i^4 times i^1,"},{"Start":"08:36.275 ","End":"08:40.305","Text":"break up the 5 into power of 4,"},{"Start":"08:40.305 ","End":"08:43.605","Text":"multiple of 4 closest to 5."},{"Start":"08:43.605 ","End":"08:47.990","Text":"Then i^4 is 1, i^1 is i."},{"Start":"08:47.990 ","End":"08:50.045","Text":"So it is just i."},{"Start":"08:50.045 ","End":"08:52.985","Text":"In the case of i^13,"},{"Start":"08:52.985 ","End":"08:59.990","Text":"I\u0027ll write this as i to the multiple of 4 before 13 is 12."},{"Start":"08:59.990 ","End":"09:03.470","Text":"This is i^12 and also i^1 is 13,"},{"Start":"09:03.470 ","End":"09:05.030","Text":"is 12 plus 1."},{"Start":"09:05.030 ","End":"09:07.340","Text":"This is also equal to 1,"},{"Start":"09:07.340 ","End":"09:10.460","Text":"and this is also equal to i."},{"Start":"09:10.460 ","End":"09:13.040","Text":"This is interesting."},{"Start":"09:13.040 ","End":"09:15.080","Text":"These are both equal to i."},{"Start":"09:15.080 ","End":"09:16.280","Text":"Why is that interesting?"},{"Start":"09:16.280 ","End":"09:23.600","Text":"Because then I get i minus i^2 and I don\u0027t even care what power it is."},{"Start":"09:23.600 ","End":"09:30.174","Text":"0 to any power 0 happens to be 2, which is just 0."},{"Start":"09:30.174 ","End":"09:33.575","Text":"We don\u0027t even, I guess we don\u0027t even need this formula."},{"Start":"09:33.575 ","End":"09:39.410","Text":"I thought we would. Part 6 was the last."},{"Start":"09:39.410 ","End":"09:42.030","Text":"So we are done."}],"ID":4925},{"Watched":false,"Name":"Exercise 4","Duration":"14m 6s","ChapterTopicVideoID":4922,"CourseChapterTopicPlaylistID":3860,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.440","Text":"In this exercise,"},{"Start":"00:01.440 ","End":"00:04.050","Text":"there are 5 separate problems,"},{"Start":"00:04.050 ","End":"00:06.195","Text":"each one a division exercise,"},{"Start":"00:06.195 ","End":"00:07.380","Text":"except the last one,"},{"Start":"00:07.380 ","End":"00:08.970","Text":"whether there\u0027s something squared in,"},{"Start":"00:08.970 ","End":"00:11.775","Text":"but generally this is practice for division."},{"Start":"00:11.775 ","End":"00:16.275","Text":"I\u0027d like to remind you how we do division of complex numbers."},{"Start":"00:16.275 ","End":"00:20.625","Text":"We multiply top and bottom by what is called the conjugate."},{"Start":"00:20.625 ","End":"00:25.080","Text":"I\u0027ll just get ready for number 1 here."},{"Start":"00:25.080 ","End":"00:27.990","Text":"And I\u0027ll remind you what a conjugate is."},{"Start":"00:27.990 ","End":"00:33.945","Text":"If we have a complex number a plus bi,"},{"Start":"00:33.945 ","End":"00:38.490","Text":"then its conjugate is a minus bi,"},{"Start":"00:38.490 ","End":"00:40.260","Text":"or if this was a minus bi,"},{"Start":"00:40.260 ","End":"00:44.625","Text":"the conjugate would be with the plus it\u0027s the reverse sign."},{"Start":"00:44.625 ","End":"00:50.550","Text":"Let\u0027s write the word conjugate and as I say,"},{"Start":"00:50.550 ","End":"00:54.585","Text":"the trick is to multiply top and bottom"},{"Start":"00:54.585 ","End":"01:00.224","Text":"of the division by the conjugate of the denominator."},{"Start":"01:00.224 ","End":"01:02.280","Text":"How does that help us?"},{"Start":"01:02.280 ","End":"01:05.610","Text":"Because in general, if you multiply a number by"},{"Start":"01:05.610 ","End":"01:10.930","Text":"its conjugate a plus bi times a minus bi."},{"Start":"01:10.940 ","End":"01:16.605","Text":"What we get actually is a^2 plus b^2,"},{"Start":"01:16.605 ","End":"01:20.535","Text":"which is a real number without any imaginary part."},{"Start":"01:20.535 ","End":"01:24.750","Text":"If you use the difference of squares formula on this,"},{"Start":"01:24.750 ","End":"01:27.915","Text":"you actually get a^2 minus b^2 i^2,"},{"Start":"01:27.915 ","End":"01:31.035","Text":"but the i^2 is minus 1 and so this is what we get."},{"Start":"01:31.035 ","End":"01:41.310","Text":"I\u0027ll just make a note that we multiply top and bottom of"},{"Start":"01:41.310 ","End":"01:46.060","Text":"the fraction by conjugate"},{"Start":"01:46.850 ","End":"01:56.350","Text":"of the bottom denominator of course when I say top and bottom numerator and denominator."},{"Start":"01:57.320 ","End":"02:00.975","Text":"Let\u0027s apply that to the first one."},{"Start":"02:00.975 ","End":"02:03.645","Text":"We multiply top and bottom."},{"Start":"02:03.645 ","End":"02:06.405","Text":"What I\u0027m really saying is I\u0027m multiplying by 1,"},{"Start":"02:06.405 ","End":"02:12.525","Text":"but the conjugate of the bottom is 2 minus i."},{"Start":"02:12.525 ","End":"02:19.890","Text":"That\u0027s how we do it and then we have a multiplication on the top."},{"Start":"02:19.890 ","End":"02:21.870","Text":"So the top is easy enough,"},{"Start":"02:21.870 ","End":"02:27.300","Text":"5 times 2 minus i. I\u0027ll just write it in full the first time."},{"Start":"02:27.300 ","End":"02:34.080","Text":"5 times 2 minus i over 2 plus i,"},{"Start":"02:34.080 ","End":"02:44.560","Text":"2 minus i and this is equal to 5 times 2 minus i is 10 minus 5i."},{"Start":"02:45.050 ","End":"02:56.010","Text":"The bottom, using this formula is 2^2 plus 1^2 and 2^2 plus 1^2 is 5."},{"Start":"02:56.010 ","End":"02:57.450","Text":"Let\u0027s make a note of that."},{"Start":"02:57.450 ","End":"03:00.855","Text":"That is 2^2 plus 1^2,"},{"Start":"03:00.855 ","End":"03:03.580","Text":"1 being the 1i."},{"Start":"03:04.280 ","End":"03:12.180","Text":"I should have not multiplied it out if I would have left it like this,"},{"Start":"03:12.180 ","End":"03:13.440","Text":"it would have been easier to see."},{"Start":"03:13.440 ","End":"03:16.860","Text":"We get back to 2 minus i."},{"Start":"03:16.860 ","End":"03:19.995","Text":"Because this is 5 times 2 minus i,"},{"Start":"03:19.995 ","End":"03:25.360","Text":"I\u0027m canceling this with this and that\u0027s the answer."},{"Start":"03:26.270 ","End":"03:30.010","Text":"Onto the next one."},{"Start":"03:30.410 ","End":"03:37.260","Text":"As before, we multiply top and bottom by the conjugate of the denominator."},{"Start":"03:37.260 ","End":"03:39.075","Text":"Instead of 5 minus 2i,"},{"Start":"03:39.075 ","End":"03:41.100","Text":"it\u0027ll be 5 plus 2i."},{"Start":"03:41.100 ","End":"03:44.820","Text":"The same thing here, 5 plus 2i."},{"Start":"03:44.820 ","End":"03:50.955","Text":"This will equal we know how to multiply complex numbers."},{"Start":"03:50.955 ","End":"03:56.085","Text":"We do this times this minus this times this for the real part,"},{"Start":"03:56.085 ","End":"03:58.380","Text":"2 times 5 is 10,"},{"Start":"03:58.380 ","End":"04:00.390","Text":"3 times 2 is 6,"},{"Start":"04:00.390 ","End":"04:08.280","Text":"10 minus 6 is 4 and then the imaginary part is this times this plus this"},{"Start":"04:08.280 ","End":"04:16.875","Text":"times this so it\u0027s 15 plus 4 is 19i over,"},{"Start":"04:16.875 ","End":"04:23.670","Text":"and using the formula for multiplying by a conjugate it\u0027s 5^2 plus 2^2,"},{"Start":"04:23.670 ","End":"04:29.290","Text":"which we can mentally do is 25 plus 4 is 29."},{"Start":"04:29.480 ","End":"04:34.035","Text":"We would leave it like this except that it says in standard form."},{"Start":"04:34.035 ","End":"04:35.340","Text":"In standard form,"},{"Start":"04:35.340 ","End":"04:41.565","Text":"it\u0027s 4 over 29 plus 19 over 29i."},{"Start":"04:41.565 ","End":"04:48.310","Text":"It\u0027s always a plus bi in standard form and that\u0027s the answer to number 2."},{"Start":"04:48.410 ","End":"04:52.140","Text":"Next is number 3."},{"Start":"04:52.140 ","End":"04:55.095","Text":"And in number 3,"},{"Start":"04:55.095 ","End":"05:01.965","Text":"the usual multiply top and bottom by the conjugate of the bottom,"},{"Start":"05:01.965 ","End":"05:12.490","Text":"as we can think of it as 0 minus 4i and the conjugate of 0 minus 4i is 0 plus 4i."},{"Start":"05:13.810 ","End":"05:18.620","Text":"If it helps out a minus bi a plus bi."},{"Start":"05:18.620 ","End":"05:20.510","Text":"But we don\u0027t really need the 0s,"},{"Start":"05:20.510 ","End":"05:22.650","Text":"I guess I\u0027ll erase that."},{"Start":"05:27.440 ","End":"05:30.840","Text":"Of course, same thing on the numerator because we can\u0027t"},{"Start":"05:30.840 ","End":"05:34.740","Text":"change the exercise and multiply it by 1 essentially."},{"Start":"05:34.740 ","End":"05:42.315","Text":"On the numerator we get the real part comes from minus i times 4i,"},{"Start":"05:42.315 ","End":"05:44.760","Text":"which is minus 4i^2,"},{"Start":"05:44.760 ","End":"05:47.835","Text":"which is plus 4,"},{"Start":"05:47.835 ","End":"05:52.455","Text":"and 6 times 4 is 24i."},{"Start":"05:52.455 ","End":"05:54.330","Text":"That\u0027s the numerator,"},{"Start":"05:54.330 ","End":"05:59.745","Text":"on the denominator we could use the formula of a^2 plus b^2,"},{"Start":"05:59.745 ","End":"06:03.945","Text":"where a is 0 and it\u0027s just 4^2 is 16."},{"Start":"06:03.945 ","End":"06:09.075","Text":"Of course, you could also see it by just saying this times this is 4 times 4 is 16."},{"Start":"06:09.075 ","End":"06:16.590","Text":"It\u0027s minus 16i^2 minus 16i^2 squared is 16 because minus i^2 squared is 1."},{"Start":"06:16.590 ","End":"06:20.415","Text":"Let\u0027s write it in standard form."},{"Start":"06:20.415 ","End":"06:23.715","Text":"Standard form is 4 over 16,"},{"Start":"06:23.715 ","End":"06:32.460","Text":"4 over 16 mentally we can do that\u0027s a 1/4 and 24 over 16,"},{"Start":"06:32.460 ","End":"06:33.900","Text":"if you cancel by 8,"},{"Start":"06:33.900 ","End":"06:36.480","Text":"you get 3 over 2, which is 1/2."},{"Start":"06:36.480 ","End":"06:39.045","Text":"But I\u0027ll leave it as 3 over 2."},{"Start":"06:39.045 ","End":"06:44.010","Text":"Somehow looks better in this context as an improper fraction,"},{"Start":"06:44.010 ","End":"06:53.295","Text":"dividing by 8i and this is the answer to 3 onto number 4."},{"Start":"06:53.295 ","End":"06:57.165","Text":"As before, we multiply by one,"},{"Start":"06:57.165 ","End":"07:02.383","Text":"but just something over itself and that something is the conjugate of the denominator."},{"Start":"07:02.383 ","End":"07:07.390","Text":"To 4+2i here and 4+2i here."},{"Start":"07:07.640 ","End":"07:10.750","Text":"Let\u0027s see what we get."},{"Start":"07:11.780 ","End":"07:14.610","Text":"Trivial, I\u0027ll begin with the denominator."},{"Start":"07:14.610 ","End":"07:17.175","Text":"Using the a^2+b^2 formula,"},{"Start":"07:17.175 ","End":"07:21.330","Text":"4^2+2^2, 16+4, 20."},{"Start":"07:21.330 ","End":"07:24.550","Text":"Let\u0027s see, on the numerator."},{"Start":"07:24.830 ","End":"07:32.070","Text":"In fact, why don\u0027t we use the a+b^2 formula,"},{"Start":"07:32.070 ","End":"07:35.940","Text":"the special binomial, because this is 4+2i^2,"},{"Start":"07:35.940 ","End":"07:44.780","Text":"so it\u0027s 4^2+2*4*2i+2i^2,"},{"Start":"07:46.190 ","End":"07:52.470","Text":"then that will equal the real part is 16"},{"Start":"07:52.470 ","End":"08:01.545","Text":"and 4i^2 is -4, 16-4 is 12."},{"Start":"08:01.545 ","End":"08:11.925","Text":"The middle one, 2*4*2 is 16i/20."},{"Start":"08:11.925 ","End":"08:14.850","Text":"But we want to put it in standard form,"},{"Start":"08:14.850 ","End":"08:20.280","Text":"so it\u0027s 12/20 that can be canceled by 4 to get"},{"Start":"08:20.280 ","End":"08:27.270","Text":"3/5 and 16/20i is"},{"Start":"08:27.270 ","End":"08:31.510","Text":"also can be canceled by 4 and we get 4/5."},{"Start":"08:33.120 ","End":"08:36.305","Text":"This is the answer to Number 4."},{"Start":"08:36.305 ","End":"08:38.475","Text":"On to Number 5."},{"Start":"08:38.475 ","End":"08:40.895","Text":"I\u0027ll just scroll a bit."},{"Start":"08:40.895 ","End":"08:45.190","Text":"Here, it\u0027s not just a complex number over a complex number,"},{"Start":"08:45.190 ","End":"08:46.910","Text":"this one is squared."},{"Start":"08:46.910 ","End":"08:55.505","Text":"We can just modify the general method of multiplying by the conjugate of the denominator."},{"Start":"08:55.505 ","End":"09:03.490","Text":"What we can do is multiply by 2i-3,"},{"Start":"09:06.000 ","End":"09:09.730","Text":"take 2 on the last bit."},{"Start":"09:10.970 ","End":"09:13.725","Text":"Number 5 is a bit different."},{"Start":"09:13.725 ","End":"09:17.190","Text":"Well, for one thing, this is not written in the right order."},{"Start":"09:17.190 ","End":"09:20.659","Text":"Let me just do that first before I do anything."},{"Start":"09:20.659 ","End":"09:24.770","Text":"Let me write it as 3+2i^2."},{"Start":"09:24.770 ","End":"09:29.060","Text":"It\u0027s the same thing, but we usually write the real and then the imaginary."},{"Start":"09:29.060 ","End":"09:31.315","Text":"Now, there\u0027s two ways of doing this."},{"Start":"09:31.315 ","End":"09:33.130","Text":"One way is to square this,"},{"Start":"09:33.130 ","End":"09:35.570","Text":"and then we have a single complex number and we"},{"Start":"09:35.570 ","End":"09:38.645","Text":"multiply top and bottom by the conjugate of the denominator."},{"Start":"09:38.645 ","End":"09:41.570","Text":"But there\u0027s actually a slightly easier way to go,"},{"Start":"09:41.570 ","End":"09:44.765","Text":"and that is to multiply top and bottom."},{"Start":"09:44.765 ","End":"09:47.675","Text":"Actually, the conjugate turns out"},{"Start":"09:47.675 ","End":"09:50.420","Text":"that if you take the conjugate of a square of something,"},{"Start":"09:50.420 ","End":"09:54.910","Text":"it\u0027s the same as taking the conjugate and then squaring it."},{"Start":"09:54.910 ","End":"09:57.435","Text":"But even if you didn\u0027t know that,"},{"Start":"09:57.435 ","End":"09:59.365","Text":"if you see something to the power,"},{"Start":"09:59.365 ","End":"10:02.230","Text":"you multiply it by the conjugate to that same power,"},{"Start":"10:02.230 ","End":"10:04.170","Text":"and it will do the trick."},{"Start":"10:04.170 ","End":"10:09.410","Text":"Let\u0027s do it here and here; 3-2i^2."},{"Start":"10:10.940 ","End":"10:13.570","Text":"Now, what should I start with;"},{"Start":"10:13.570 ","End":"10:15.570","Text":"the numerator or the denominator?"},{"Start":"10:15.570 ","End":"10:18.420","Text":"I\u0027ll say let\u0027s start with the denominator."},{"Start":"10:18.420 ","End":"10:28.165","Text":"The denominator, I can rewrite it as (3+2i)(3-2i),"},{"Start":"10:28.165 ","End":"10:35.680","Text":"this whole thing squared because the square of a product is the product of the squares."},{"Start":"10:37.740 ","End":"10:41.350","Text":"We\u0027ll do that in a moment. I\u0027ll just write that."},{"Start":"10:41.350 ","End":"10:50.880","Text":"On the numerator we get (1-2i*3)-(2i^2)."},{"Start":"10:50.880 ","End":"10:56.435","Text":"Perhaps, the only real computational I\u0027ll do is this one."},{"Start":"10:56.435 ","End":"11:04.240","Text":"This comes out as using the binomial expansion,"},{"Start":"11:04.740 ","End":"11:13.210","Text":"(3^2-2*3*2i+2i^2)"},{"Start":"11:15.090 ","End":"11:19.490","Text":"Close the other bracket."},{"Start":"11:20.240 ","End":"11:23.630","Text":"I\u0027ll change that to a square bracket."},{"Start":"11:23.630 ","End":"11:25.780","Text":"Bit at a time."},{"Start":"11:25.780 ","End":"11:29.255","Text":"Let\u0027s see where we go."},{"Start":"11:29.255 ","End":"11:32.040","Text":"I would simplify this."},{"Start":"11:32.040 ","End":"11:36.130","Text":"Here we get (1-2i)*,"},{"Start":"11:36.130 ","End":"11:40.390","Text":"now let us see, this is 9 and this is 4i^2."},{"Start":"11:40.390 ","End":"11:46.595","Text":"4i^2 is-4, so 9-4 is 5 and then the middle one,"},{"Start":"11:46.595 ","End":"11:48.670","Text":"2*3*2 is 12,"},{"Start":"11:48.670 ","End":"11:53.010","Text":"so we have -12i for the numerator."},{"Start":"11:53.010 ","End":"11:55.040","Text":"For the denominator,"},{"Start":"11:55.040 ","End":"11:59.565","Text":"what I get is here I have a number times its conjugate."},{"Start":"11:59.565 ","End":"12:05.230","Text":"Using the formula that we is longer on screen, the a^2+b^2,"},{"Start":"12:05.230 ","End":"12:12.295","Text":"on the denominator I have 3^2+2^2,"},{"Start":"12:12.295 ","End":"12:16.090","Text":"but there\u0027s also this squared here."},{"Start":"12:20.300 ","End":"12:28.100","Text":"Let me continue, equals dividing line."},{"Start":"12:28.100 ","End":"12:31.745","Text":"Let\u0027s do the (1-2i)*(5-12i)."},{"Start":"12:31.745 ","End":"12:34.140","Text":"1*5 is 5,"},{"Start":"12:34.140 ","End":"12:42.555","Text":"and 5-24i^2 is 5+24"},{"Start":"12:42.555 ","End":"12:46.690","Text":"is 29 for the real part."},{"Start":"12:48.140 ","End":"12:57.880","Text":"Sorry. No. Take 2 on the last bit."},{"Start":"12:57.880 ","End":"13:06.115","Text":"1-5 is 5,-2i*-12i is+24i^2,"},{"Start":"13:06.115 ","End":"13:16.190","Text":"5+24i^2 is like 5-24, so-19."},{"Start":"13:16.190 ","End":"13:18.660","Text":"The i term comes out to"},{"Start":"13:18.660 ","End":"13:29.410","Text":"be -10i-12i, altogether,-22i."},{"Start":"13:30.120 ","End":"13:33.525","Text":"The denominator, 3^2 is 9,"},{"Start":"13:33.525 ","End":"13:40.410","Text":"2^2 is 4, 13^2, 169."},{"Start":"13:40.410 ","End":"13:43.940","Text":"Finally, we just have to put it in standard form."},{"Start":"13:43.940 ","End":"13:48.960","Text":"It\u0027s -19/169, that\u0027s"},{"Start":"13:48.960 ","End":"13:58.210","Text":"the real part, -22/169i."},{"Start":"14:00.840 ","End":"14:03.455","Text":"Just highlight that."},{"Start":"14:03.455 ","End":"14:04.830","Text":"That\u0027s the last part,"},{"Start":"14:04.830 ","End":"14:06.940","Text":"so we are done."}],"ID":4926},{"Watched":false,"Name":"Exercise 5","Duration":"2m 12s","ChapterTopicVideoID":4923,"CourseChapterTopicPlaylistID":3860,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.845","Text":"In this exercise,"},{"Start":"00:01.845 ","End":"00:07.770","Text":"we have to write each of these 3 square root of negative numbers in terms of i."},{"Start":"00:07.770 ","End":"00:12.165","Text":"It was once impossible before we had complex numbers and i."},{"Start":"00:12.165 ","End":"00:13.680","Text":"Now it becomes possible,"},{"Start":"00:13.680 ","End":"00:15.945","Text":"so we\u0027re going to simplify these."},{"Start":"00:15.945 ","End":"00:22.710","Text":"The main thing we\u0027re going to use is the fact that in general the square root of a,"},{"Start":"00:22.710 ","End":"00:26.580","Text":"b is the square root of a square root of b,"},{"Start":"00:26.580 ","End":"00:28.034","Text":"and in particular,"},{"Start":"00:28.034 ","End":"00:30.150","Text":"the square root of minus something,"},{"Start":"00:30.150 ","End":"00:33.180","Text":"say minus a, is going to be"},{"Start":"00:33.180 ","End":"00:38.745","Text":"the square root of a times the square root of minus 1, which is i."},{"Start":"00:38.745 ","End":"00:40.635","Text":"That\u0027s in general."},{"Start":"00:40.635 ","End":"00:44.220","Text":"That\u0027s the strategy."},{"Start":"00:44.220 ","End":"00:46.700","Text":"Let\u0027s get to the exercises."},{"Start":"00:46.700 ","End":"00:50.190","Text":"The first one,"},{"Start":"00:51.680 ","End":"00:54.395","Text":"I could even do this whole thing again."},{"Start":"00:54.395 ","End":"01:00.605","Text":"I\u0027ll write it as the square root of 4 times minus 1,"},{"Start":"01:00.605 ","End":"01:06.430","Text":"which is the square root of 4 times the square root of minus 1."},{"Start":"01:06.430 ","End":"01:09.855","Text":"The square root of 4 is 2,"},{"Start":"01:09.855 ","End":"01:12.570","Text":"square root of minus 1 is i,"},{"Start":"01:12.570 ","End":"01:14.660","Text":"so the answer is 2i,"},{"Start":"01:14.660 ","End":"01:19.950","Text":"and this one, very similar."},{"Start":"01:19.950 ","End":"01:24.170","Text":"I\u0027ll shorten it a bit."},{"Start":"01:24.170 ","End":"01:26.240","Text":"Whenever we have the square root of minus something,"},{"Start":"01:26.240 ","End":"01:31.290","Text":"we just take the square root of the positive part and write i."},{"Start":"01:31.490 ","End":"01:37.690","Text":"This is equal to 3i because square root of 9 is 3."},{"Start":"01:37.690 ","End":"01:42.965","Text":"Same here. This is square root of 5 times i."},{"Start":"01:42.965 ","End":"01:51.710","Text":"As opposed to previous cases where we could actually compute the square root,"},{"Start":"01:51.710 ","End":"01:54.470","Text":"here we can\u0027t, or at least we\u0027d need a calculator,"},{"Start":"01:54.470 ","End":"01:56.030","Text":"so to leave it precise,"},{"Start":"01:56.030 ","End":"01:58.675","Text":"we just leave the answer like this."},{"Start":"01:58.675 ","End":"02:02.870","Text":"Square root of 5i, it happens to be 2.26 something,"},{"Start":"02:02.870 ","End":"02:05.270","Text":"but don\u0027t bother doing it as a decimal."},{"Start":"02:05.270 ","End":"02:07.250","Text":"This is actually more precise."},{"Start":"02:07.250 ","End":"02:12.120","Text":"That\u0027s done with this exercise."}],"ID":4927},{"Watched":false,"Name":"Exercise 6","Duration":"12m 49s","ChapterTopicVideoID":4924,"CourseChapterTopicPlaylistID":3860,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.441","Text":"In this exercise, there are six quadratic equations to solve."},{"Start":"00:05.441 ","End":"00:10.080","Text":"I should mention that we assume the variable to be a complex variable,"},{"Start":"00:10.080 ","End":"00:14.190","Text":"meaning it can be a complex number with a real and imaginary part."},{"Start":"00:14.190 ","End":"00:21.375","Text":"It isn\u0027t clear until you see Part 6 that there are actually complex numbers here."},{"Start":"00:21.375 ","End":"00:26.910","Text":"Having said that, let\u0027s start with the first one."},{"Start":"00:26.910 ","End":"00:30.930","Text":"We could use the quadratic formula but in this case,"},{"Start":"00:30.930 ","End":"00:33.435","Text":"there\u0027s a missing x."},{"Start":"00:33.435 ","End":"00:38.090","Text":"We can solve it more simply by saying x^2 equals minus"},{"Start":"00:38.090 ","End":"00:46.415","Text":"36. x equals plus or minus the square root of minus 36."},{"Start":"00:46.415 ","End":"00:50.600","Text":"Like I said, we allow complex numbers."},{"Start":"00:50.600 ","End":"00:57.625","Text":"This is equal to plus or minus 6i."},{"Start":"00:57.625 ","End":"01:03.470","Text":"Because the square root of minus"},{"Start":"01:03.470 ","End":"01:13.140","Text":"36 is the square root of 36 times the square root of minus 1 which is 6i."},{"Start":"01:14.360 ","End":"01:16.685","Text":"Onto the next one."},{"Start":"01:16.685 ","End":"01:20.815","Text":"These are the two answers for the first highlighted."},{"Start":"01:20.815 ","End":"01:23.388","Text":"Anyway, next one."},{"Start":"01:23.388 ","End":"01:26.100","Text":"x^2 minus 2x plus 5 is 0."},{"Start":"01:26.100 ","End":"01:28.940","Text":"Let\u0027s use the formula here."},{"Start":"01:28.940 ","End":"01:33.350","Text":"x equals minus b which is 2 plus or"},{"Start":"01:33.350 ","End":"01:39.020","Text":"minus the square root of b^2 is 4 minus 4,"},{"Start":"01:39.020 ","End":"01:42.470","Text":"a is 1 times c,"},{"Start":"01:42.470 ","End":"01:46.900","Text":"and all this over 2a."},{"Start":"01:46.900 ","End":"01:51.165","Text":"What we get is 2 plus or minus."},{"Start":"01:51.165 ","End":"01:58.135","Text":"Let\u0027s see. 4 minus 20 is minus 16."},{"Start":"01:58.135 ","End":"02:01.370","Text":"Because we\u0027re working with complex numbers,"},{"Start":"02:01.370 ","End":"02:05.195","Text":"we can say that this is 2 plus or minus."},{"Start":"02:05.195 ","End":"02:08.720","Text":"The square root of minus 16 is 4i."},{"Start":"02:08.720 ","End":"02:17.405","Text":"Square root of minus 1 is i and square root of 16 is 4/2 which gives us the two answers,"},{"Start":"02:17.405 ","End":"02:23.625","Text":"either 2 plus 4i/2 which is 1 plus 2i."},{"Start":"02:23.625 ","End":"02:26.565","Text":"Dividing by 2 is easy."},{"Start":"02:26.565 ","End":"02:31.935","Text":"2 minus 4i divided by 2 so it\u0027s 1 minus 2i."},{"Start":"02:31.935 ","End":"02:34.800","Text":"These are the two solutions."},{"Start":"02:34.800 ","End":"02:38.610","Text":"For this one, we could combine them if you want to,"},{"Start":"02:38.610 ","End":"02:44.720","Text":"you could say 1 plus or minus 2i and use the plus-minus optional."},{"Start":"02:44.720 ","End":"02:48.230","Text":"Next one. Again, the formula."},{"Start":"02:48.230 ","End":"02:55.910","Text":"This time the variable is t. We got minus b plus or minus the square root of"},{"Start":"02:55.910 ","End":"03:05.615","Text":"b^2 minus 4 times 1 times 5/2."},{"Start":"03:05.615 ","End":"03:07.550","Text":"This time under the square root sign,"},{"Start":"03:07.550 ","End":"03:12.110","Text":"it\u0027s 16 minus 20 which is minus 4."},{"Start":"03:12.110 ","End":"03:19.260","Text":"What we get is 4 plus or minus the square root of minus 4/2."},{"Start":"03:19.660 ","End":"03:25.205","Text":"The square root of minus 4 is 2i."},{"Start":"03:25.205 ","End":"03:29.520","Text":"4 plus or minus 2i/2."},{"Start":"03:30.070 ","End":"03:35.060","Text":"This time I\u0027ll leave it as plus-minus and it\u0027s 2 plus or"},{"Start":"03:35.060 ","End":"03:42.095","Text":"minus i. I\u0027ll highlight this."},{"Start":"03:42.095 ","End":"03:44.620","Text":"For those that don\u0027t like the plus or minus,"},{"Start":"03:44.620 ","End":"03:50.810","Text":"we can say 2 plus i and 2 minus i are the solutions."},{"Start":"03:51.410 ","End":"03:54.977","Text":"Proceeding. Number 4,"},{"Start":"03:54.977 ","End":"03:57.250","Text":"let\u0027s see the rest of them."},{"Start":"03:57.250 ","End":"04:02.710","Text":"We have 4, 5, and 6. z or z"},{"Start":"04:02.710 ","End":"04:06.925","Text":"is actually a very common letter to use when"},{"Start":"04:06.925 ","End":"04:11.570","Text":"using complex numbers just like x for real numbers."},{"Start":"04:11.760 ","End":"04:14.910","Text":"Let\u0027s see. If I say z,"},{"Start":"04:14.910 ","End":"04:19.060","Text":"I mean z. The British way."},{"Start":"04:19.280 ","End":"04:30.226","Text":"z equals minus b plus or minus the square root of b^2,"},{"Start":"04:30.226 ","End":"04:37.310","Text":"that\u0027s 144 minus 4 times 4 times 25."},{"Start":"04:37.310 ","End":"04:41.730","Text":"All this over 2a is 8."},{"Start":"04:41.730 ","End":"04:44.355","Text":"Now, under the square root sign,"},{"Start":"04:44.355 ","End":"04:51.384","Text":"what we have is 4 times 4 times 25 is 400."},{"Start":"04:51.384 ","End":"04:56.570","Text":"144 minus 400 is 256."},{"Start":"04:56.570 ","End":"05:04.380","Text":"What I get is minus 12 plus or minus the square root of, did I say 256,"},{"Start":"05:04.380 ","End":"05:07.330","Text":"I meant minus 256/8"},{"Start":"05:09.380 ","End":"05:15.320","Text":"and this is equal to minus 12, plus or minus."},{"Start":"05:15.320 ","End":"05:21.110","Text":"Now, this comes out to be 16i because we\u0027ve got the square root of 256 is 16 and"},{"Start":"05:21.110 ","End":"05:26.950","Text":"the square root of minus 1 is i over 8."},{"Start":"05:26.950 ","End":"05:30.930","Text":"If I write it in standard form,"},{"Start":"05:30.930 ","End":"05:34.730","Text":"12/8 is 1/2 or 3/2."},{"Start":"05:34.730 ","End":"05:37.175","Text":"Let\u0027s write it as an improper fraction."},{"Start":"05:37.175 ","End":"05:41.060","Text":"We got minus 3/2."},{"Start":"05:41.060 ","End":"05:43.140","Text":"You could write this as 1 1/2 if you feel like,"},{"Start":"05:43.140 ","End":"05:51.330","Text":"plus or minus 16/8 is 2i."},{"Start":"05:51.330 ","End":"05:53.845","Text":"You can write the answer like this."},{"Start":"05:53.845 ","End":"05:56.440","Text":"Or if you prefer to write it separately,"},{"Start":"05:56.440 ","End":"06:05.390","Text":"then minus 3/2 plus 2i is one solution and the other solution is minus 3/2 minus 2i."},{"Start":"06:05.390 ","End":"06:11.140","Text":"Whatever you prefer. In the next one, again,"},{"Start":"06:11.140 ","End":"06:17.455","Text":"the formula Z equals minus b plus or minus"},{"Start":"06:17.455 ","End":"06:25.060","Text":"the square root of b^2 is 256 minus 4 times 4 times 25,"},{"Start":"06:25.060 ","End":"06:31.736","Text":"that looks familiar, over 8."},{"Start":"06:31.736 ","End":"06:35.990","Text":"Very similar to the previous one."},{"Start":"06:35.990 ","End":"06:42.480","Text":"What we get is minus 16 plus or minus the square root."},{"Start":"06:43.300 ","End":"06:45.950","Text":"This is 400."},{"Start":"06:45.950 ","End":"06:54.165","Text":"We computed that before and this minus this gives us the 144, 256 minus 400."},{"Start":"06:54.165 ","End":"06:58.570","Text":"All this over 8."},{"Start":"06:58.850 ","End":"07:05.420","Text":"We get minus 16 plus or minus."},{"Start":"07:05.420 ","End":"07:09.015","Text":"Did I forget to write the minus? Silly me."},{"Start":"07:09.015 ","End":"07:10.950","Text":"There it is, minus."},{"Start":"07:10.950 ","End":"07:13.980","Text":"Square root of 144 is 12,"},{"Start":"07:13.980 ","End":"07:16.425","Text":"square root of minus 1 is i,"},{"Start":"07:16.425 ","End":"07:18.790","Text":"all this over 8."},{"Start":"07:18.790 ","End":"07:21.485","Text":"This time again, similar to this,"},{"Start":"07:21.485 ","End":"07:24.215","Text":"it comes out to be minus 2,"},{"Start":"07:24.215 ","End":"07:28.080","Text":"plus or minus 3/2i."},{"Start":"07:28.300 ","End":"07:31.550","Text":"I\u0027ll leave the answer like this."},{"Start":"07:31.550 ","End":"07:33.890","Text":"Although like before, you can split it up with"},{"Start":"07:33.890 ","End":"07:37.170","Text":"a plus separately and the minus separately."},{"Start":"07:37.480 ","End":"07:41.480","Text":"Last one. This time"},{"Start":"07:41.480 ","End":"07:45.490","Text":"the coefficients are also complex numbers or at least some of them are."},{"Start":"07:45.490 ","End":"07:51.360","Text":"In other words, a is 1 plus i,"},{"Start":"07:51.360 ","End":"07:54.225","Text":"b is 2,"},{"Start":"07:54.225 ","End":"07:59.119","Text":"and c is 1 minus i. I\u0027m emphasizing"},{"Start":"07:59.119 ","End":"08:04.250","Text":"it because we haven\u0027t so far encountered one with complex number coefficients."},{"Start":"08:04.250 ","End":"08:14.740","Text":"We get that Z equals minus b plus or minus the square root of"},{"Start":"08:14.740 ","End":"08:21.075","Text":"b^2 is 4 minus 4 times a"},{"Start":"08:21.075 ","End":"08:28.620","Text":"which is 1 plus i times c which is 1 minus i,"},{"Start":"08:28.620 ","End":"08:37.240","Text":"all this over 2a twice 1 plus i."},{"Start":"08:38.870 ","End":"08:44.375","Text":"We\u0027ll need to do some maybe side computations."},{"Start":"08:44.375 ","End":"08:48.575","Text":"I\u0027d like to just do the bit under the square root sign at the side."},{"Start":"08:48.575 ","End":"08:52.355","Text":"What I have is 4 minus 4."},{"Start":"08:52.355 ","End":"08:55.129","Text":"Now, this is a number times its conjugate,"},{"Start":"08:55.129 ","End":"08:57.565","Text":"the 1 plus i times 1 minus i."},{"Start":"08:57.565 ","End":"09:04.085","Text":"It\u0027s 1^2 plus 1^2 using the rule for conjugates."},{"Start":"09:04.085 ","End":"09:06.230","Text":"But you can just do it directly."},{"Start":"09:06.230 ","End":"09:13.760","Text":"Difference of squares formula 1^2 minus I squared is 1 plus 1 is 2."},{"Start":"09:13.760 ","End":"09:21.990","Text":"Either way, what we get here is 4 minus 4 times 2 is 4 minus 8 is minus 4."},{"Start":"09:22.130 ","End":"09:26.170","Text":"Also note that the square root of minus 4"},{"Start":"09:26.170 ","End":"09:30.225","Text":"is square root of 4 times square root of minus 1."},{"Start":"09:30.225 ","End":"09:38.290","Text":"I\u0027ll just write that, square root of 4 and square root of minus one which is 2i."},{"Start":"09:38.290 ","End":"09:43.800","Text":"Back here, we get minus 2 plus or"},{"Start":"09:43.800 ","End":"09:50.770","Text":"minus 2i over twice 1 plus i."},{"Start":"09:51.610 ","End":"09:59.330","Text":"Now here we have a division of complex numbers but we can first of all,"},{"Start":"09:59.330 ","End":"10:01.895","Text":"divide top and bottom by 2."},{"Start":"10:01.895 ","End":"10:05.705","Text":"What we need actually are two computations."},{"Start":"10:05.705 ","End":"10:08.854","Text":"I mean, this thing itself is,"},{"Start":"10:08.854 ","End":"10:10.640","Text":"well, I could write it down here,"},{"Start":"10:10.640 ","End":"10:14.705","Text":"it\u0027s minus 1,"},{"Start":"10:14.705 ","End":"10:19.400","Text":"plus or minus i/1 plus I."},{"Start":"10:19.400 ","End":"10:23.020","Text":"That\u0027s after I got rid of the 2."},{"Start":"10:23.020 ","End":"10:24.410","Text":"I\u0027ll continue here anyway."},{"Start":"10:24.410 ","End":"10:26.090","Text":"If I take the plus,"},{"Start":"10:26.090 ","End":"10:33.610","Text":"I have minus 1 plus i over 1 plus i."},{"Start":"10:33.610 ","End":"10:35.760","Text":"That\u0027s taking the plus here."},{"Start":"10:35.760 ","End":"10:37.745","Text":"If I take the minus,"},{"Start":"10:37.745 ","End":"10:40.820","Text":"then it\u0027s minus 1,"},{"Start":"10:40.820 ","End":"10:46.435","Text":"minus i/1 plus i."},{"Start":"10:46.435 ","End":"10:52.325","Text":"I did the equal here but then I\u0027m moving over to the branch over here."},{"Start":"10:52.325 ","End":"10:55.715","Text":"This is the plus and this is the minus."},{"Start":"10:55.715 ","End":"10:58.675","Text":"One of them turns out to be easy."},{"Start":"10:58.675 ","End":"11:01.520","Text":"This one is the easy one because if you look at it,"},{"Start":"11:01.520 ","End":"11:04.970","Text":"the numerator is just minus 1 times the denominator."},{"Start":"11:04.970 ","End":"11:07.175","Text":"I mean just the same thing with opposite signs."},{"Start":"11:07.175 ","End":"11:09.110","Text":"This is minus 1."},{"Start":"11:09.110 ","End":"11:12.620","Text":"There\u0027s only one more thing left to compute and that\u0027s this."},{"Start":"11:12.620 ","End":"11:15.154","Text":"When we divide complex numbers,"},{"Start":"11:15.154 ","End":"11:19.580","Text":"then we multiply top and bottom by the conjugate of the denominator."},{"Start":"11:19.580 ","End":"11:23.870","Text":"Remember that 1 plus i so 1 minus i but top and"},{"Start":"11:23.870 ","End":"11:28.765","Text":"bottom to make it valid and legal so we\u0027ll multiply it by 1 effectively."},{"Start":"11:28.765 ","End":"11:31.885","Text":"Now, on the denominator,"},{"Start":"11:31.885 ","End":"11:35.540","Text":"the product of the conjugates 1 plus i,1 minus i. I"},{"Start":"11:35.540 ","End":"11:38.855","Text":"think we\u0027ve done this before already here and we got that as 2."},{"Start":"11:38.855 ","End":"11:43.055","Text":"It\u0027s 1^2 plus 1^2 is 2 here."},{"Start":"11:43.055 ","End":"11:50.260","Text":"On the numerator what we have is, let\u0027s see."},{"Start":"11:51.290 ","End":"11:53.745","Text":"Let\u0027s do the real part."},{"Start":"11:53.745 ","End":"12:02.385","Text":"Minus 1 times 1 is minus 1 and i times minus i"},{"Start":"12:02.385 ","End":"12:13.510","Text":"is minus i^2 so that plus 1."},{"Start":"12:14.890 ","End":"12:18.440","Text":"There\u0027s no real part, only an imaginary."},{"Start":"12:18.440 ","End":"12:24.535","Text":"Let\u0027s see, the imaginary part is i times 1,"},{"Start":"12:24.535 ","End":"12:30.080","Text":"and then minus 1 times minus i is also plus i."},{"Start":"12:30.080 ","End":"12:33.210","Text":"So it\u0027s i plus i so it\u0027s 2i/2."},{"Start":"12:34.390 ","End":"12:37.130","Text":"That is equal to just i,"},{"Start":"12:37.130 ","End":"12:39.085","Text":"the 2 is canceled."},{"Start":"12:39.085 ","End":"12:48.640","Text":"Our two answers are minus 1 and i for part 6 and we are done."}],"ID":4928}],"Thumbnail":null,"ID":3860}]

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Mattan Kedem10:37 03/08/2022Wow! that's really helpful Thanks!