WEBVTT Kind: captions Language: en 00:00:01.406 --> 00:00:05.936 >> Welcome to the Cypress College Math Review on Radicals. 00:00:07.786 --> 00:00:12.116 Objective one, taking nth roots with numbers. 00:00:13.946 --> 00:00:21.526 B is the nth root of a if b to the nth power is equal to a. Here's a notation that we use 00:00:21.526 --> 00:00:31.766 for radicals b is the nth root of a, n is called the index (the little number up to the left 00:00:31.906 --> 00:00:36.856 of the radical sign), a is the radicand (the thing inside the radical) 00:00:37.556 --> 00:00:41.616 and the whole thing nth root of a is called the radical expression. 00:00:42.426 --> 00:00:50.156 Here's an example, 2 is equal to the cube root or 3rd root of 8. 00:00:50.436 --> 00:00:54.326 Two is the 3rd root or cube root of 8. 00:00:55.776 --> 00:00:58.376 Three is the index of the radical. 00:00:59.016 --> 00:01:03.646 Eight is the radicand -- the thing inside the radical. 00:01:04.206 --> 00:01:08.796 And the cube root of 8 is the radical, the whole radical expression. 00:01:08.996 --> 00:01:17.596 In order to work with radicals, it's helpful to know as many perfect square numbers as possible. 00:01:18.596 --> 00:01:21.336 Try to memorize as many of these relationships as you can. 00:01:23.236 --> 00:01:29.836 Now, if b to the nth is equal to a then b is equal to the nth root of a. Well, 00:01:30.536 --> 00:01:32.986 let's put some numbers in here to make that a little more clear. 00:01:34.256 --> 00:01:43.126 If 3 squared is equal to 9, which it is, 3 times 3 is nine, then 3 is a 2nd root of 9. 00:01:44.466 --> 00:01:46.106 Well, how do we write 2nd root? 00:01:47.186 --> 00:01:49.536 Second root is the same thing as square root. 00:01:49.836 --> 00:01:53.476 In other words, when the index of the radical is a 2, we don't write it. 00:01:54.866 --> 00:01:57.186 So, 3 is the square root of 9. 00:01:57.996 --> 00:01:59.246 So, this is a 3. 00:02:00.686 --> 00:02:02.486 So, what would you square to get 4? 00:02:03.926 --> 00:02:06.506 Two. What would you square to get 1? 00:02:07.016 --> 00:02:09.666 One. What would you square to get 16? 00:02:10.416 --> 00:02:14.026 Four. What would you square to get 25? 00:02:15.096 --> 00:02:17.476 Five. These are the square roots of those numbers. 00:02:19.166 --> 00:02:22.006 Now, let's look at cube roots. 00:02:22.416 --> 00:02:26.246 Once again, it's helpful to know as many perfect cube numbers as possible. 00:02:27.076 --> 00:02:29.746 So, learn as many of these relationships as you can. 00:02:31.316 --> 00:02:34.786 What would you cube to get 1? 00:02:35.416 --> 00:02:36.446 Well, that's just 1. 00:02:37.046 --> 00:02:39.006 So, the cube root of 1 is just 1. 00:02:40.466 --> 00:02:42.796 What would you cube to get 8? 00:02:44.096 --> 00:02:47.646 Well, that's 2, so the cube root of 8 is 2. 00:02:49.036 --> 00:02:51.826 What would you cube to get 27? 00:02:52.876 --> 00:02:55.786 Well, that would be 3, 3 cubed is 27. 00:02:56.186 --> 00:02:58.406 So, the cube root of 27 is 3. 00:03:00.086 --> 00:03:02.866 What would you cube to get 64? 00:03:03.736 --> 00:03:07.666 Well, that's 4, so the cube root of 64 is 4. 00:03:09.096 --> 00:03:12.496 What would you cube to get 125? 00:03:13.366 --> 00:03:18.946 That's 5, so the cube root of 125 is 5. 00:03:20.256 --> 00:03:23.446 Example, simplify the following radical. 00:03:23.806 --> 00:03:25.766 The square root of 49. 00:03:26.336 --> 00:03:28.356 Well, the square root of 49 is 7. 00:03:29.306 --> 00:03:35.916 Why? Because 7 squared is equal to 49. 00:03:37.896 --> 00:03:38.826 Maybe you didn't know that. 00:03:38.826 --> 00:03:40.456 Maybe you didn't know that perfect square. 00:03:41.156 --> 00:03:42.086 What could you do then? 00:03:43.046 --> 00:03:44.246 You could do a factor tree. 00:03:44.496 --> 00:03:51.956 So, you take the number 49 and you do a factor tree and it turns out that it's 7 times 7. 00:03:53.246 --> 00:04:01.606 So, you write the square root of 49 as the square root of 7 times 7. 00:04:02.186 --> 00:04:07.496 Now, to be a perfect square, a number must have two of the same factor. 00:04:08.296 --> 00:04:10.886 We have two of the same factor. 00:04:11.646 --> 00:04:15.936 When this happens, you can take those two factors together 00:04:15.936 --> 00:04:18.456 and allow one to escape the square root. 00:04:19.536 --> 00:04:24.916 So, the square root of 7 times 7 is just 7. 00:04:24.916 --> 00:04:28.086 Now, note that you do not write the radical symbol anymore. 00:04:30.046 --> 00:04:36.966 Now we want to take the 3rd root or cube root of 216. 00:04:38.276 --> 00:04:41.186 Well, that's equal to 6. 00:04:41.716 --> 00:04:50.716 Why? Because if you take 6 and cube it, you get 216. 00:04:51.036 --> 00:04:59.116 We could rewrite 216 as 6 times 6 times 6. 00:05:00.096 --> 00:05:04.146 To be a perfect cube a number must have three of the same factor. 00:05:04.886 --> 00:05:08.996 When this happens, we can take those three factors together 00:05:09.356 --> 00:05:12.226 and allow one to escape the cube root. 00:05:12.886 --> 00:05:19.056 So, three factors together to escape a cube root so we get the number 6. 00:05:22.456 --> 00:05:24.796 Principal and negative square roots. 00:05:25.676 --> 00:05:28.946 If you take the number 5 and you square it. 00:05:29.846 --> 00:05:30.846 You get 25. 00:05:31.496 --> 00:05:39.416 If you take the number negative 5 and square that you also get 25. 00:05:40.946 --> 00:05:43.746 Well that means there's actually two different square roots 00:05:43.746 --> 00:05:46.246 of the number 25 or any positive number. 00:05:47.276 --> 00:05:52.386 So, 5 is a square root of 25 and negative 5 is a square root of 25. 00:05:52.686 --> 00:05:54.366 What does this symbol stand for then? 00:05:55.346 --> 00:05:59.836 This symbol stands for just the principal square root of 25. 00:06:00.666 --> 00:06:02.176 The non-negative square root. 00:06:02.716 --> 00:06:07.816 So, the symbol stands for just the non-negative or principal square root of 25. 00:06:08.476 --> 00:06:18.756 If you put a negative in front of that symbol, that gives you the other square root of 25. 00:06:20.236 --> 00:06:23.296 The square root of a negative number is not a real number. 00:06:23.996 --> 00:06:26.206 So, the square root of negative 9 is not real. 00:06:27.446 --> 00:06:32.456 The square root of 0 is 0 because if you square 0 you get 0. 00:06:33.256 --> 00:06:37.246 The square root of 1 is 1 because 1 squared is 1. 00:06:37.826 --> 00:06:45.506 If the index is even such as, square root which is the same thing as a 2nd root, 00:06:46.016 --> 00:06:52.956 4th root or 6th root, then the radicand must be non-negative for the root to be a real number. 00:06:53.286 --> 00:07:01.966 For example, the 4th root of 16 is equal to 2, because 2 to the 4th is equal to 16. 00:07:02.806 --> 00:07:06.776 But the 4th root of negative 16 is not a real number. 00:07:07.206 --> 00:07:10.706 A lot of people would say oh negative 2 works. 00:07:11.026 --> 00:07:17.726 But no, if you take negative 2 and raise it to the 4th power, you have an even number 00:07:17.726 --> 00:07:23.216 of negatives, which is going to give you positive 16, not negative 16. 00:07:23.786 --> 00:07:28.276 So, the 4th root of negative 16 is not a real number. 00:07:30.106 --> 00:07:36.566 The 6th root of 64 is equal to 2 because, 2 to the 6th is equal to 64. 00:07:36.856 --> 00:07:40.366 But the 6th root of negative 64 is not a real number. 00:07:41.336 --> 00:07:46.986 If the index is odd, such as cube root, which is 3rd root, 5th root or 7th root, 00:07:47.576 --> 00:07:49.956 then the radicand can be any real number. 00:07:51.076 --> 00:07:56.786 For example, the cube root of 64 is equal to 4 because 4 cubed is equal to 64. 00:07:57.756 --> 00:08:02.816 The cube root of negative 64 is negative because if you take negative 4 00:08:03.216 --> 00:08:11.376 and cube it that's an odd number of negatives, so that does give you a negative 64. 00:08:12.796 --> 00:08:18.276 The 5th root of 32 is equal to 2 because 2 to the 5th is equal to 32. 00:08:18.836 --> 00:08:26.006 And the 5th root of negative 32 is equal to negative 2 because if you take negative 2 00:08:26.006 --> 00:08:30.466 and raise it to an odd power you will get a negative number. 00:08:34.556 --> 00:08:39.286 If we do not get a perfect power that matches the index of the radical, 00:08:39.616 --> 00:08:42.526 we can still take out any double factors that exist 00:08:42.606 --> 00:08:45.296 and whatever's leftover will be stuck under the radical. 00:08:45.826 --> 00:08:48.236 So, we're working with 18. 00:08:49.116 --> 00:08:53.106 Eighteen is 2 times 9 and 9 is 3 times 3. 00:08:53.106 --> 00:09:00.216 So, we do have a double factor in the 3's so, we have 2 times 3 times 3. 00:09:01.486 --> 00:09:09.306 So, the pair of 3's will come out and give us one 3 on the outside and the 2 is left inside. 00:09:11.816 --> 00:09:15.816 Forty is 4 times 10. 00:09:18.116 --> 00:09:22.906 Four is 2 times 2 and 10 is 2 times 5. 00:09:23.556 --> 00:09:25.446 We're working with the cube root of 40. 00:09:26.716 --> 00:09:32.346 So, we have the cube root of 2 times 2 times 2 times 5. 00:09:33.056 --> 00:09:38.836 We need three like factors to be able to come out of a cube root. 00:09:39.436 --> 00:09:45.716 So, the cube root of 2 cubed is 2 it comes outside and we're left with a 5 on the inside. 00:09:45.816 --> 00:09:48.436 Be sure to write the index of the radical, the little 3. 00:09:48.956 --> 00:09:51.016 So, this is 2 times the cube root of 5. 00:09:52.266 --> 00:09:56.906 Well, 162 is 2 times 81. 00:09:57.996 --> 00:10:02.776 Eighty-one is 9 times 9 and 9 is 3 times 3. 00:10:03.426 --> 00:10:13.336 So, here we have the square root of 2 times 3 times 3 times 3 times 3. 00:10:14.226 --> 00:10:18.276 So, we have pair of 3's, so one 3 will escape. 00:10:19.106 --> 00:10:20.946 Then we have another pair of 3's 00:10:21.196 --> 00:10:22.836 so another 3 will escape. 00:10:23.346 --> 00:10:24.806 The 2 is left inside. 00:10:25.900 --> 00:10:28.440 So, the answer is 9 times the square root of 2. 00:10:31.200 --> 00:10:33.840 Pause the video and try these problems. 00:10:38.400 --> 00:10:41.700 Objective two, taking nth roots with variables. 00:10:42.606 --> 00:10:45.126 Variables follow the same rules that numbers do. 00:10:45.816 --> 00:10:49.996 It takes two like factors to escape the square root, three like factors 00:10:49.996 --> 00:10:52.146 to escape the cube root and so on and so forth. 00:10:53.916 --> 00:10:56.306 So, let's take the square root of x squared. 00:10:56.486 --> 00:10:59.326 Well, the square root of x squared is equal to x. Why? 00:10:59.966 --> 00:11:02.466 Because if you square x you get x squared. 00:11:03.076 --> 00:11:04.896 What's the square root of x to the 4th? 00:11:05.646 --> 00:11:07.606 Well that's x squared, why? 00:11:08.216 --> 00:11:11.476 Because when you square x squared you get x to the 4th. 00:11:12.036 --> 00:11:16.316 The square root of x to the 6th is x cubed 00:11:16.756 --> 00:11:21.246 because if you square x cubed you get x to the 6th. 00:11:21.926 --> 00:11:23.976 So, what's happening here? 00:11:24.576 --> 00:11:30.706 If you take the exponent and divide by the index of the radical 00:11:30.706 --> 00:11:33.916 and for a square root the index of the radical is 2. 00:11:34.706 --> 00:11:36.076 You get the new exponent. 00:11:37.026 --> 00:11:42.176 So, 8 divided by 2 is 4. 00:11:42.446 --> 00:11:45.496 So, the square root of x to the 8th is x to the 4th. 00:11:46.136 --> 00:11:50.046 Ten divided by 2 is 5. 00:11:50.626 --> 00:11:58.716 So, the square root of x to the 10th is x to the 5th. 00:12:00.006 --> 00:12:02.636 Let's take cube roots of variables. 00:12:03.186 --> 00:12:09.796 You can rewrite this as x times itself three times and since we're doing a cube root, 00:12:10.386 --> 00:12:13.476 it takes three like factors to pull it outside. 00:12:14.086 --> 00:12:18.526 So, that gives us just x to the 1st on the outside and nothing left inside. 00:12:19.236 --> 00:12:23.256 You can also think about this in terms of dividing the exponents. 00:12:23.256 --> 00:12:29.196 If you take the exponent that's inside divide by the index of the radical and we get 1. 00:12:29.946 --> 00:12:31.526 And that's the exponent you're left with. 00:12:31.976 --> 00:12:35.006 Let's take the cube root of x to the 6th. 00:12:35.376 --> 00:12:38.496 So, 6 divided by 3 is 2. 00:12:39.326 --> 00:12:42.996 Because not too many people want to keep writing out all these x's. 00:12:43.646 --> 00:12:48.206 So, there's one x that would come out. 00:12:48.606 --> 00:12:50.336 There's another x that would come out. 00:12:50.336 --> 00:12:55.716 Multiply times each other, that gives us x to the 2nd power. 00:12:58.800 --> 00:13:05.550 Nine divided by 3 is 3, so the cube root of x to the 9th is x to the 3rd. 00:13:06.526 --> 00:13:08.746 Twelve divided by 3 is 4. 00:13:10.756 --> 00:13:14.506 Fifteen divided by 3 is 5. 00:13:18.176 --> 00:13:23.236 Simplify the following radicals, assume variables represent positive real numbers. 00:13:24.106 --> 00:13:27.736 Since this is a square root the index of the radical is 2, 00:13:28.036 --> 00:13:30.726 so we divide by 4 by 2 and we get 2. 00:13:31.316 --> 00:13:35.286 On this problem, we divide 10 by 2 and we get 5. 00:13:36.506 --> 00:13:44.746 Now, we're going to divide 11 by 2, we get 5 with a remainder of 1. 00:13:45.646 --> 00:13:53.296 So, you will end up with 5 x's on the outside and one that's left on the inside. 00:13:53.736 --> 00:13:58.086 The remainder tells you the power of the variable left inside. 00:13:58.806 --> 00:14:02.086 The quotient tells you how many x's are on the outside. 00:14:03.236 --> 00:14:06.976 The other way to do this problem is to write out all of those 11 x's. 00:14:14.416 --> 00:14:20.376 Then, two like factors will bring us an x on the outside, 00:14:21.226 --> 00:14:30.626 another pair of like factors gives us another x, and another, and another, and another. 00:14:31.606 --> 00:14:33.776 Those are all on the outside of the radical. 00:14:34.376 --> 00:14:38.936 Left inside is that one x there. 00:14:40.076 --> 00:14:45.866 Once again, that gives us the answer we have above x to the 5th times the square root 00:14:46.596 --> 00:14:51.546 of x. Cube root of x to the 17th. 00:14:52.386 --> 00:14:58.396 So, we divide our exponent 17 by the index of the radical which is 3. 00:15:01.316 --> 00:15:04.816 We get 5 with the remainder of 2. 00:15:05.056 --> 00:15:08.696 So, we will end up with x to the 5th power on the outside 00:15:09.176 --> 00:15:11.616 and x to the 2nd power on the inside. 00:15:13.536 --> 00:15:18.596 In general, for any division problem the remainder is always smaller than the divisor. 00:15:18.756 --> 00:15:21.176 In this case, 2 of course is smaller than 3. 00:15:22.636 --> 00:15:25.196 For a radical to be in simplified form, 00:15:25.196 --> 00:15:30.966 the exponents on the inside must be smaller than index of the radical. 00:15:31.366 --> 00:15:33.646 Once again, 2 of course is smaller than 3. 00:15:34.426 --> 00:15:42.736 For this problem, we divide 23 by the index of the radical which is 5 00:15:43.366 --> 00:15:46.676 and we get 4 with the remainder of 3. 00:15:47.880 --> 00:15:56.500 So, the answer would be x to the 4th times the 5th root of x to the 3rd. 00:15:58.376 --> 00:16:01.366 This problem has numbers and variables. 00:16:01.866 --> 00:16:07.466 We've done the prime factorization of the coefficient that was there and we've written it 00:16:07.466 --> 00:16:09.996 as 3 to the 5th times 5 to the 6th. 00:16:10.436 --> 00:16:15.106 So, we can work with this just like the 3 and the 5 were variables. 00:16:16.166 --> 00:16:24.076 So, we divide 5 by 4 and get 1 with a remainder of 1. 00:16:25.326 --> 00:16:33.286 We divide 6 by 4 and we get 1 with a remainder of 2. 00:16:34.616 --> 00:16:44.296 We divide 11 by 4 and get 2 with a remainder of 3. 00:16:44.816 --> 00:16:55.556 We divide 12 by 4 and get 3 with no remainder, so no y's are left on the inside. 00:16:55.696 --> 00:17:04.396 So, our answer is 15 x squared, y cubed times the 4th root of - now what is this, 00:17:04.396 --> 00:17:11.986 5 squared is 25 and 25 times 3 is 75, so 75 x cubed. 00:17:12.976 --> 00:17:18.786 So, if you have your coefficient written in terms of a prime factorization, 00:17:18.786 --> 00:17:24.406 you have it all factored, you can work with it just like you can variables. 00:17:25.016 --> 00:17:32.156 We have 96 and we want to break it down into its prime factors. 00:17:32.626 --> 00:17:38.216 So, that 3 times 32 and 32 is made up all 2's. 00:17:38.546 --> 00:17:39.926 We've seen this number before. 00:17:40.546 --> 00:17:51.826 So, we have the 5th root of 2 to the 5th power times 3 times a 00:17:51.826 --> 00:17:56.116 to the 45th, times b to the 37th. 00:17:57.416 --> 00:17:59.826 Now we divide the exponents by the index. 00:17:59.826 --> 00:18:06.036 Five divided by 5 is 1 and there's no remainder, so there's no 2's left on the inside. 00:18:07.286 --> 00:18:10.976 Three is just to the exponent of 1, which is already smaller 00:18:10.976 --> 00:18:13.616 than the index, so the 3 stays on the inside. 00:18:14.916 --> 00:18:21.576 Forty-five divided by 5 is 9, with no remainder, so there's no a's left on the inside. 00:18:22.900 --> 00:18:30.720 Thirty-seven divided by 5 is 7, with a remainder of 2 and that's our answer. 00:18:33.440 --> 00:18:36.080 Pause the video and try these problems. 00:18:40.040 --> 00:18:42.760 Objective three, adding and subtracting radicals. 00:18:43.796 --> 00:18:47.926 Like radicals have the same index and the same radicand. 00:18:48.946 --> 00:18:53.666 We add or subtract radicals similar to the way that we combine like terms. 00:18:54.436 --> 00:18:57.786 Add or subtract the coefficients then keep the radical. 00:18:58.666 --> 00:19:02.226 Three x plus 7x is equal to 10x. 00:19:03.176 --> 00:19:06.936 Three times the square root of 5 plus 7 times the square root of 5 is equal 00:19:06.936 --> 00:19:09.156 to 10 times the square root of 5. 00:19:09.666 --> 00:19:15.466 Three of these plus 7 of these is 10 of these. 00:19:16.766 --> 00:19:18.476 If the radicals are not the same, 00:19:18.796 --> 00:19:22.116 then we simplify them first before we combine like terms. 00:19:22.656 --> 00:19:27.396 On this problem, the radicals are the same, they're both the square root of 7. 00:19:27.806 --> 00:19:32.496 Four of these minus 9 of these is negative 5 of these. 00:19:33.406 --> 00:19:37.976 On the next problem, the radicals are also the same, the cube root of 2y. 00:19:38.866 --> 00:19:45.276 Nine of these plus 15 of these is 24, of these. 00:19:49.226 --> 00:19:53.416 These radicals are not the same so first we have to simplify them. 00:19:54.376 --> 00:20:02.156 The prime factorization of 50 is 5 square times 2, because 25 times 2. 00:20:03.116 --> 00:20:08.946 The prime factorization of 18 is 3 square times 2, 9 times 2. 00:20:09.746 --> 00:20:14.146 Now we divide the exponents, these are square roots that we're going to divide by 2. 00:20:15.156 --> 00:20:20.476 Two divided by 2 is 1, with no remainder, so no 5's are left on the inside. 00:20:21.316 --> 00:20:22.976 The exponent for 2 is a 1, 00:20:23.206 --> 00:20:27.106 which is smaller than the index of the radical, which is 2. 00:20:27.886 --> 00:20:32.956 Therefore, the square root of 2 stays on the inside, because 1 is smaller than 2. 00:20:34.416 --> 00:20:41.386 We divide 2 by 2 and we get 3 to the 1st on the outside with no remainder. 00:20:42.076 --> 00:20:43.986 The 2 is left on the inside. 00:20:45.246 --> 00:20:50.606 So, we have 5 times the square root of 2 plus 15 times the square root 2. 00:20:51.046 --> 00:20:58.426 Well 5 of these plus 15 of these is 20 of these. 00:20:59.026 --> 00:21:02.006 For this problem, we're not going to do prime factorizations. 00:21:02.776 --> 00:21:07.886 Here we recognize that 4 is a perfect square and it is a factor of 20. 00:21:08.646 --> 00:21:11.556 That leaves us 5x on the inside. 00:21:13.066 --> 00:21:17.446 Sixteen is a perfect square, leaves x on the inside. 00:21:19.626 --> 00:21:24.076 Nine is a perfect square and that is a factor of 45 which leaves us 00:21:24.076 --> 00:21:29.366 with 5 times x. The square root of 4 is 2. 00:21:29.896 --> 00:21:34.106 The square root of 16 is 4. 00:21:35.736 --> 00:21:37.546 The square root of 9 is 3. 00:21:38.146 --> 00:21:45.936 So, we have 2 times the square root of 5x minus 24 times the square root 00:21:45.936 --> 00:21:49.976 of x plus 3 times the square root of 5x. 00:21:50.556 --> 00:21:52.176 Which of these are like terms? 00:21:52.766 --> 00:21:54.316 Well the first and the last. 00:21:55.086 --> 00:22:00.076 So, we have 5 times the square root of 5x the other term is not a 00:22:00.076 --> 00:22:02.486 like term and it cannot be combined. 00:22:03.156 --> 00:22:10.236 Here we recognize that the cube root of 8 is 2. 00:22:10.236 --> 00:22:11.306 So, we bring that out. 00:22:11.746 --> 00:22:14.866 You could also write that out as 2 cubed and then once again you end 00:22:14.866 --> 00:22:16.426 up with 2 to the 1st on the outside. 00:22:17.696 --> 00:22:25.366 Divide the exponent 5 by 3 and we get 1 with a remainder of 2. 00:22:25.996 --> 00:22:31.776 The cube root of 27 is 3 because 27 is 3 cubed. 00:22:33.516 --> 00:22:39.876 We divide 5 by 3 and we get 1 with a remainder of 2. 00:22:40.266 --> 00:22:41.716 These are like terms. 00:22:42.396 --> 00:22:46.396 Not only does the radical have to be the same but since this has a variable 00:22:46.396 --> 00:22:48.716 in front, that has to be the same also. 00:22:48.716 --> 00:22:50.706 This whole expression has to be the same. 00:22:51.300 --> 00:22:59.806 We have two of these plus three of these which gives us five of these. 00:23:01.980 --> 00:23:03.860 Pause the video and try these problems. 00:23:07.300 --> 00:23:10.840 Objective four, rationalizing monomial denominators. 00:23:11.556 --> 00:23:15.186 First let's look at the rules for a radical to be in simplified form. 00:23:15.616 --> 00:23:17.116 So, we're taking the nth root. 00:23:18.676 --> 00:23:22.426 First you can't have any perfect nth powers in the radicand. 00:23:23.386 --> 00:23:32.576 So, if we had 2 to the 4th times x squared, we would have to rewrite that as 2 00:23:32.576 --> 00:23:38.176 to the 1st times the cube root of 2 times x squared. 00:23:38.826 --> 00:23:41.986 Notice that the exponents that are left on the inside 00:23:41.986 --> 00:23:45.956 of the radical are smaller than the index of the radical. 00:23:47.366 --> 00:23:52.986 That's making sure that we don't have any perfect cubes inside a cube root. 00:23:54.486 --> 00:23:58.276 Next, you can't have any fractions in the radicand. 00:23:58.966 --> 00:24:01.506 So, this is not simplified form. 00:24:02.776 --> 00:24:05.376 You can't leave any radicals in the denominator. 00:24:06.136 --> 00:24:08.436 This is also not simplified form. 00:24:09.376 --> 00:24:12.786 We're going to work on these two processes next. 00:24:13.546 --> 00:24:20.916 We'll use the product rule and quotient rule for radicals to do these problems. 00:24:21.036 --> 00:24:25.476 Rationalizing the denominator of a fraction with one term in the denominator. 00:24:26.386 --> 00:24:28.216 The process of eliminating a radical 00:24:28.216 --> 00:24:31.216 in the denominator is called rationalizing the denominator. 00:24:31.906 --> 00:24:37.366 If the denominator is a single term, we multiply both the numerator and the denominator 00:24:37.366 --> 00:24:43.296 so that the denominator is a perfect square, or a perfect cube, or perfect 4th root, 00:24:43.336 --> 00:24:45.076 depending upon the index of the radical. 00:24:46.336 --> 00:24:50.986 Examples, simplify, since each of these expressions have a radical 00:24:50.986 --> 00:24:55.066 in the denominator simplify includes rationalizing the denominator. 00:24:55.866 --> 00:24:58.426 Assume variables represent positive real numbers. 00:24:59.046 --> 00:25:01.976 On the first problem, we have the square root of 5 in the denominator. 00:25:02.786 --> 00:25:08.646 If we multiply both the top and the bottom by the square root of 5, 00:25:08.866 --> 00:25:14.396 we end up with by the product rule, the square root of 5 squared in the denominator. 00:25:14.946 --> 00:25:19.806 Well, the square root of 5 squared is simply 5. 00:25:20.906 --> 00:25:24.876 So, the 5 is no longer in the radical and we don't have a radical in the denominator. 00:25:25.636 --> 00:25:30.646 We check to see if 2/5 reduces and it doesn't, so we're done. 00:25:31.296 --> 00:25:34.776 For the next problem, we have the cube root of 1/5. 00:25:34.846 --> 00:25:40.756 First of all, since we have a radical of a fraction, we see if the fraction can be reduced. 00:25:41.216 --> 00:25:43.436 It can't, 1/5 cannot be reduced. 00:25:43.676 --> 00:25:46.606 So, we split it using the quotient rule. 00:25:46.956 --> 00:25:50.256 The cube root of 1 over the cube root of 5. 00:25:51.356 --> 00:25:53.006 The cube root of 1 is just one. 00:25:54.916 --> 00:25:58.036 Now we have a radical in the denominator and we can't leave that. 00:25:58.626 --> 00:26:02.296 We have 5 to the 1st power under a cube root. 00:26:03.266 --> 00:26:09.846 We will need two more 5's, 5 to the 2nd power so that we have the 3 00:26:10.176 --> 00:26:12.536 that we need to get out of the cube root. 00:26:13.516 --> 00:26:18.286 So, the numerator is the cube root of 5 squared and 5 squared is simply 25. 00:26:18.836 --> 00:26:25.456 The denominator is the cube root of 5 cubed. 00:26:26.146 --> 00:26:29.706 What's the cube root of 5 cubed? 00:26:29.946 --> 00:26:30.876 That's just 5. 00:26:32.956 --> 00:26:38.646 And that gives us our answer, no more radical in the denominator. 00:26:39.226 --> 00:26:44.216 For the next problem, we're doing a 4th root. 00:26:44.826 --> 00:26:52.276 So, we have x to the 1st power, how many more x's do we need so we have x to the 4th power? 00:26:52.806 --> 00:26:54.436 Well, that would be x cubed. 00:26:54.496 --> 00:27:01.326 So, we have 6 times the 4th root of x cubed in the numerator 00:27:01.826 --> 00:27:05.296 and the denominator we have the 4th root of x to the 4th. 00:27:06.506 --> 00:27:16.086 But the 4th root of x to the 4th is simply x. For the next problem, we have the 7th root 00:27:16.086 --> 00:27:18.136 of x to the 5th in the denominator. 00:27:18.776 --> 00:27:29.626 We need two more x's to make it so that we will have a 7th root 00:27:30.046 --> 00:27:32.646 of x to the 7th in the denominator. 00:27:36.856 --> 00:27:51.006 Now our denominator is simply x. No more radical and the problem's done. 00:27:51.366 --> 00:27:55.906 You could combine this into one radical and you would do this 00:27:55.906 --> 00:27:58.716 if anything canceled out and simplified. 00:27:59.396 --> 00:28:01.596 But nothing does, so there's no reason to do it that way. 00:28:02.206 --> 00:28:06.046 Let's start working on that denominator. 00:28:06.566 --> 00:28:10.876 Well, 32 is 2 to the 5th power. 00:28:13.916 --> 00:28:19.846 Now, what do we need to multiply by to make this radical in the denominator disappear? 00:28:20.446 --> 00:28:26.636 Well, the 5th root of 2 to the 5th is just 2, so there aren't going to be any 2's left inside. 00:28:26.636 --> 00:28:27.986 So, we don't need any more 2's. 00:28:29.486 --> 00:28:31.586 The power for b is 17. 00:28:32.036 --> 00:28:34.276 The new power needs to be a multiple of 5. 00:28:34.896 --> 00:28:37.406 Five, 10, 15, 20. 00:28:37.406 --> 00:28:40.976 Ah... 20, so I need 20 b's altogether. 00:28:40.976 --> 00:28:42.086 So, I need three more. 00:28:42.716 --> 00:28:47.436 We multiply the numerators times each other. 00:28:49.486 --> 00:28:56.556 In the denominator we have the 5th root of 2 to the 5th times b to the 20th. 00:28:57.666 --> 00:29:00.016 In the numerator, can that be simplified? 00:29:01.106 --> 00:29:01.796 Always check. 00:29:02.886 --> 00:29:08.666 Two is smaller than 5, 3 is smaller than 5, so the numerator cannot be simplified. 00:29:08.666 --> 00:29:14.926 How about the denominator? 00:29:15.246 --> 00:29:19.516 The 5th root of 2 to the 5th is just 2, so 2 to the 1st comes outside. 00:29:20.186 --> 00:29:23.596 And 20 divided by 5 is 4. 00:29:24.166 --> 00:29:26.246 No more radical. 00:29:26.746 --> 00:29:34.696 On the next problem, you should combine them into one radical because that fraction reduces. 00:29:35.206 --> 00:29:40.846 Divide the numerator and denominator each by 3 and we get 6/25. 00:29:41.296 --> 00:29:44.526 Now we split it using the quotient rule. 00:29:45.196 --> 00:29:51.856 And the denominator is simply 5 because the square root of 25 is 5. 00:29:52.216 --> 00:29:53.986 No more radical in the denominator. 00:29:54.566 --> 00:29:58.226 And the square root of 6 is simplified. 00:29:58.540 --> 00:30:01.100 There's no perfect squares that can come out of that. 00:30:03.780 --> 00:30:06.120 Pause the video and try these problems. 00:30:10.040 --> 00:30:13.620 Objective five, rationalizing binomial denominators. 00:30:14.616 --> 00:30:20.186 Recall from factoring the formula the quantity a plus b times the quantity a minus b is equal 00:30:20.186 --> 00:30:22.076 to a squared minus b squared. 00:30:23.276 --> 00:30:28.116 This formula plays an integral part in allowing us to rationalize 00:30:28.426 --> 00:30:32.386 if our denominator is a binomial involving square roots. 00:30:33.236 --> 00:30:38.116 Where a plus b and a minus b are called conjugates of each other. 00:30:38.826 --> 00:30:41.876 First, let's multiply a couple conjugates and see what we get. 00:30:43.046 --> 00:30:49.026 When you multiply 4 minus the square root of 3 times 4 plus the square root of 3, 00:30:49.706 --> 00:30:56.626 you get 4 squared minus the square root of 3 squared, a squared minus b squared. 00:30:56.626 --> 00:30:58.916 The outer and inner terms in F.O.I.L. cancel out. 00:30:59.336 --> 00:31:04.496 So, we end up with 16 minus 3 which is 13. 00:31:04.946 --> 00:31:06.066 No more radical. 00:31:09.856 --> 00:31:14.106 Simplify. And since these expressions have a radical in the denominator, 00:31:14.406 --> 00:31:17.646 simplify includes rationalizing the denominator. 00:31:18.696 --> 00:31:22.186 We have two terms in the denominator and at least one 00:31:22.186 --> 00:31:25.226 of them has a square root, so we multiply by the conjugate. 00:31:25.586 --> 00:31:28.886 The conjugate of the square root of 5 plus square root 00:31:28.886 --> 00:31:32.136 of 3 is the square root of 5 minus the square root of 3. 00:31:32.996 --> 00:31:35.136 What you do to the top, you must do to the bottom. 00:31:35.536 --> 00:31:38.386 Because this expression has to equal the number 1. 00:31:38.866 --> 00:31:41.926 You can only multiply an expression by the number 1. 00:31:42.806 --> 00:31:47.576 Since this 6 does not have a radical in it, I leave it factored out. 00:31:48.016 --> 00:31:50.146 So, I have 6 times that quantity. 00:31:51.256 --> 00:31:54.616 In the denominator, I have a squared minus b squared. 00:31:55.216 --> 00:32:01.456 So, I'm going to have the square root of 5 squared minus the square root of 3 squared, 00:32:01.616 --> 00:32:05.006 because the outer and the inner terms once again in F.O.I.L., cancel out. 00:32:05.116 --> 00:32:11.016 So, we will end up with a squared minus b squared. 00:32:11.046 --> 00:32:15.006 We leave the 6 out there just in case there will be any canceling that we can do. 00:32:15.576 --> 00:32:18.916 So, the square root of 5 squared is just 5. 00:32:19.366 --> 00:32:22.206 The square root of 3 squared is 3. 00:32:22.206 --> 00:32:33.746 So, we end up with 2 in the denominator and that does cancel with our 6, 00:32:34.636 --> 00:32:40.406 and since our denominator is now just 1 we go ahead and distribute, 00:32:41.566 --> 00:32:50.596 and our final answer is 3 times the square root of 5 minus 3 times the square root of 3. 00:32:50.846 --> 00:32:53.646 Here we also have two terms in the denominator 00:32:54.136 --> 00:32:57.736 with at least one radical, so we multiply by the conjugate. 00:32:59.106 --> 00:33:05.916 So, 5 times the square root of 3 plus 4 times the square root of 2 over itself. 00:33:06.086 --> 00:33:08.526 Once again, you can only multiply by the number 1. 00:33:10.556 --> 00:33:16.156 This numerator has a radical in it so, in this case we do go ahead and distribute. 00:33:16.886 --> 00:33:19.826 So, we have 3 times the square root of 6 00:33:20.306 --> 00:33:23.846 and that's multiplied times 5 times the square root of 3. 00:33:25.256 --> 00:33:28.606 Plus 3 times the square root of 6 00:33:28.866 --> 00:33:34.236 and that's multiplied times 4 times the square root of 2, I distributed. 00:33:35.046 --> 00:33:39.676 In the denominator, we have a squared minus b squared. 00:33:39.706 --> 00:33:48.246 So, 5 rad 3 quantity squared minus 4 rad 2 quantity squared. 00:33:49.946 --> 00:33:54.726 In the numerator, we have on the outside 3 times 5 which gives us 15, 00:33:55.116 --> 00:33:58.636 on the inside we have 6 times three which is 18. 00:33:59.206 --> 00:34:04.446 For the next term in the numerator, we have 3 times 4 which is 12. 00:34:05.136 --> 00:34:08.486 Inside the radical we have 6 times 2 which is also 12. 00:34:10.306 --> 00:34:13.816 In the denominator 5 squared is 25. 00:34:14.666 --> 00:34:17.766 The square root of 3 times the square root of 3 is 3. 00:34:19.326 --> 00:34:22.846 Minus 4 squared is 16. 00:34:23.806 --> 00:34:26.696 The square root of 2 times the square root of 2 is 2. 00:34:26.916 --> 00:34:28.696 No more radical in the denominator. 00:34:30.856 --> 00:34:34.306 Eighteen is 9 times 2. 00:34:35.616 --> 00:34:42.886 Twelve is 4 times 3, because we have to get perfect squares out of those square roots. 00:34:51.096 --> 00:35:02.306 So, we have 15 times 3, square root of 2 plus 12 times 2 square root of 3. 00:35:03.106 --> 00:35:05.506 All over 43. 00:35:06.186 --> 00:35:15.706 So, we have 45 square root of 2 plus 24 times the square root of 3. 00:35:16.356 --> 00:35:19.486 All over 43. 00:35:20.096 --> 00:35:21.806 Can we factor anything out? 00:35:21.926 --> 00:35:23.476 Yes, we can factor out a 3. 00:35:24.296 --> 00:35:25.256 What are we left with? 00:35:25.946 --> 00:35:33.766 Well, 15 times the square root of 2 plus 8 times the square root of 3 and that's over 43. 00:35:34.486 --> 00:35:37.616 Nothing cancels, so we're done. 00:35:40.096 --> 00:35:45.386 We once again have two terms in the denominator with at least one square root, 00:35:45.386 --> 00:35:49.976 so we multiply by the conjugate of the denominator, over itself. 00:35:55.046 --> 00:35:58.316 In our numerator, we are doing F.O.I.L. So, 00:35:58.316 --> 00:36:01.036 the first terms times each other would be the square root 00:36:01.036 --> 00:36:04.096 of x times the square root of x or squared. 00:36:04.936 --> 00:36:07.606 The outer terms would be 10 times the square root 00:36:07.606 --> 00:36:12.556 of x. The inner terms would be 1 times the square root of x 00:36:13.226 --> 00:36:15.976 and the last terms would be 1 times 10. 00:36:17.406 --> 00:36:19.956 In the denominator, we have a squared minus b squared. 00:36:20.676 --> 00:36:25.876 So, we have the square root of x squared minus 10 squared. 00:36:27.800 --> 00:36:31.100 The square root of x squared those cancel 00:36:31.120 --> 00:36:38.740 and you just x. Ten rad x plus 1 rad x is 11 rad x. We added like terms. 00:36:40.660 --> 00:36:43.580 There aren't any other like terms in the numerator. 00:36:43.740 --> 00:36:48.000 In the denominator, we have x minus 100. 00:36:48.400 --> 00:36:54.400 Nothing factors, so we can't cancel anything and we're done. 00:36:56.680 --> 00:36:58.920 Pause the video and try these problems.