WEBVTT 00:21.280 --> 00:28.662 In this video, we'll be discussing simplifying square roots and rationalizing denominators. 00:28.720 --> 00:36.337 A square root is simplified when the radicand has no perfect square factors other than one. 00:36.400 --> 00:44.979 For example, the square root of twenty is a radical expression that is not simplified 00:45.040 --> 00:52.586 because I can write the radicand twenty as the product of two factors one of which is a perfect 00:52.620 --> 01:01.629 square other than one. Twenty is the same as four times five. So the square root of twenty 01:01.680 --> 01:09.937 can be written as the square root of four times five, and four is a perfect square. 01:10.000 --> 01:16.911 The Product Rule for Square Roots says that if the square root of a and the square root of b 01:16.960 --> 01:24.418 are real numbers, then the square root of a times b is equal to the square root of a times 01:24.480 --> 01:32.827 the square root of b. We can use this product rule to continue simplifying the square root of twenty. 01:32.880 --> 01:39.400 Now that we have rewritten the radicand as four times five, we can use the product rule 01:39.440 --> 01:48.576 to separate this radical expression into two radical factors, the square root of four times 01:48.640 --> 01:57.885 the square root of five. And we know that the square root of four is two. So this gives us two 01:57.920 --> 02:06.527 times the square root of five. Now the radicand five only has factors of one and five, so it 02:06.560 --> 02:14.668 is simplified. Again, as long as the radicand has no perfect square factors other than one, 02:14.720 --> 02:24.578 it is simplified. Two root five is equivalent to root twenty, and it is in the simplified form. 02:24.640 --> 02:33.020 Now let's look at some examples. Here we want to simplify these radical expressions. We'll begin with the 02:33.053 --> 02:41.695 square root of fifty-four. We need to think of two numbers that can multiply together to give us fifty-four, one of which is a 02:41.760 --> 02:49.303 perfect square, and ideally, we'd like to choose the largest perfect square that divides into 02:49.360 --> 02:57.311 this radicand. So let's think of the ways that we can multiply two numbers to get fifty-four. 02:57.360 --> 03:07.955 Well, we can multiply nine and six to get fifty-four, and nine is a perfect square. 03:08.000 --> 03:15.863 If we tried other factors, for example two and twenty-seven, those are factors of fifty-four, but 03:15.920 --> 03:24.271 two is not a perfect square nor is twenty-seven. Now let's use the product rule to simplify this. 03:24.320 --> 03:30.844 The square root of nine times six can be written as the square root of nine 03:30.880 --> 03:39.887 times the square root of six, and the square root of nine is three. This gives us three 03:39.920 --> 03:47.461 times the square root of six. Now ask yourself are there any perfect square factors other 03:47.520 --> 04:00.641 than one that divide evenly into the radicand six. There are not. So we are simplified here. The next expression is the 04:00.674 --> 04:08.182 square root of five hundred. Again, we're looking for two numbers that multiply together to give me five hundred. 04:08.240 --> 04:14.822 One of them needs to be a perfect square, and we'd like to choose the largest perfect square 04:14.880 --> 04:23.797 that divides evenly into five hundred. That's one hundred. Five hundred can be written as 04:23.840 --> 04:31.305 one hundred times five. So the square root of five hundred can be written as the square root 04:31.360 --> 04:40.748 of one hundred times five which is the square root of one hundred times the square root of five. 04:40.800 --> 04:50.024 The square root of one hundred is ten, so this gives us ten times 04:50.057 --> 04:55.162 the square root of five. We know that we are completely simplified 04:55.200 --> 05:02.603 when the radicand has no perfect square factors other than one. 05:02.640 --> 05:10.677 And let's look at our third example. We have three times the square root of twelve. The radicand is 05:10.720 --> 05:19.553 twelve. Twelve can be broken down into two factors, one of which is a perfect square. 05:19.600 --> 05:29.163 Three times four is twelve. So we can write this as three times the square root of four times 05:29.200 --> 05:37.571 three, and this can be rewritten using the product rule as three times the square root of four 05:37.604 --> 05:49.583 times the square root of three. The square root of four is two. So this gives us three times two 05:49.616 --> 05:57.958 times the square root of three. And our final step here will be multiplying the two factors that are 05:58.000 --> 06:06.266 outside of the radical. Three times two is six, and so this gives us six times the square root of 06:06.320 --> 06:19.313 three as our simplified radical expression that is equivalent to three square root of twelve. 06:19.360 --> 06:24.918 Next, we're going to examine the square root of a quotient. The Quotient Rule 06:24.960 --> 06:32.025 for Square Roots says that if a and b are real numbers and b is not equal to zero, 06:32.080 --> 06:39.633 then the square root of the quotient a divided by b is equal to the square root of a 06:39.680 --> 06:46.607 divided by the square root of b. Notice that b can't be zero because then you would have zero 06:46.640 --> 06:53.313 in the denominator of a fraction which is undefined. Let's work a couple of examples. 06:53.360 --> 07:02.189 We'll begin by simplifying the square root of five over forty-nine, five forty-ninths. Let's rewrite 07:02.240 --> 07:13.567 this using the quotient rule as the square root of five divided by the square root of forty-nine. 07:13.600 --> 07:19.640 And now let's see if either of these can be simplified or evaluated. 07:19.680 --> 07:24.444 The square root of five cannot be simplified. There are no factors 07:24.480 --> 07:32.686 other than one that are perfect squares of five. The square root of forty-nine is seven. 07:32.720 --> 07:42.563 So this can be written as the square root of five over seven. In our next example, we have the square root 07:42.596 --> 07:48.268 of one over one twenty-one. Using the quotient rule, we can rewrite this 07:48.320 --> 07:55.876 as the square root of one divided by the square root of one twenty-one. Now you may 07:55.920 --> 08:03.317 notice that both the numerator and denominator are perfect squares, and each can be evaluated. 08:03.360 --> 08:08.922 The square root of one is one, and the square root of one twenty-one is 08:08.960 --> 08:20.367 eleven. So the square root of one over one twenty-one is one-eleventh. 08:20.400 --> 08:26.907 Sometimes it is easier to work with a radical expression if the denominator does not contain 08:26.960 --> 08:33.480 a radical. Rewriting a radical expression to eliminate a radical in the denominator 08:33.520 --> 08:40.921 is called rationalizing the denominator. For example, the square root of five divided by 08:40.960 --> 08:46.927 the square root of two. We can see that in this fraction we have a radical expression 08:46.960 --> 08:53.467 both in the numerator and the denominator. If we would like to rationalize the denominator, 08:53.520 --> 08:59.806 meaning that we want a rational value as opposed to an irrational number in the denominator, 08:59.840 --> 09:06.680 then we need a way to eliminate this radical expression in the denominator. 09:06.720 --> 09:15.322 What could I multiply this radicand by such that the value in the denominator would be a rational 09:15.360 --> 09:25.232 value? We know that four is a perfect square. So the square root of four would give us a rational 09:25.280 --> 09:32.606 value. In order to accomplish that, we would multiply the denominator 09:32.640 --> 09:37.944 by the square root of two, and if I multiply the denominator of a fraction by the square 09:38.000 --> 09:42.582 root of two, I must also multiply the numerator by the square root of two 09:42.640 --> 09:47.854 so that I'm not changing the value of this expression. You can see here that all I'm 09:47.920 --> 09:56.830 doing is multiplying by one. Now using a version of the product rule for radicals 09:56.880 --> 10:02.335 in the opposite direction of what we used it in the previous examples, I can combine these 10:02.400 --> 10:09.142 two radicands under a single radical sign. So the square root of two times the square root of 10:09.200 --> 10:19.953 two gives me the square root of two times two which we know to be the square root of four. 10:20.000 --> 10:25.325 And the numerator, the square root of five times the square root of two, 10:25.360 --> 10:33.400 gives us the square root of five times two which gives us the square root of ten. 10:33.440 --> 10:39.005 And now we can evaluate this denominator, the square root of four, 10:39.040 --> 10:49.316 which is two. This gives us the square root of ten over two. The square root of ten in the numerator 10:49.360 --> 10:57.057 is a simplified radical expression. It has no factors that are perfect squares other than one. 10:57.120 --> 11:04.664 And the denominator is a rational number. So we have rationalized the denominator 11:04.720 --> 11:11.705 of our original problem. These expressions, the square root of five over the square root of two 11:11.760 --> 11:18.678 and the square root of ten over two, are equivalent expressions. Let's work a 11:18.712 --> 11:27.854 few examples. We want to rationalize the denominator of these expressions and simplify. 11:27.920 --> 11:32.025 We'll begin with the square root of thirteen divided by the square root of 11:32.080 --> 11:40.834 fifteen. The denominator of this fraction is an irrational number, the square root of fifteen. 11:40.880 --> 11:49.476 We want to be able to have a rational number, so we will multiply by the square root of fifteen in 11:49.520 --> 12:00.020 the denominator which means we must also multiply by the square root of fifteen in the numerator. 12:00.080 --> 12:05.859 Using the product rule now allows us to combine these radicands. 12:05.920 --> 12:13.233 The denominator is the square root of fifteen times fifteen which can be written as the square 12:13.280 --> 12:28.982 root of fifteen squared, and the numerator is the square root of thirteen times fifteen. 12:29.040 --> 12:33.787 The square root of thirteen times the square root of fifteen is the square root 12:33.840 --> 12:43.697 of one hundred ninety-five, and the square root of fifteen squared, well, that's just fifteen. And now 12:43.760 --> 12:50.437 we have a rational denominator. The square root of one ninety-five over fifteen 12:50.480 --> 13:01.081 is equivalent to the original problem square root of thirteen divided by square root of fifteen. 13:01.120 --> 13:06.686 And in our last example, we have one divided by the square root of seventy-two. 13:06.720 --> 13:13.326 The square root of seventy-two is an irrational number in the denominator of our fraction. 13:13.360 --> 13:18.765 We have two options in simplifying this expression. We could multiply 13:18.800 --> 13:24.204 both the numerator and denominator by root seventy-two to rationalize this denominator, 13:24.240 --> 13:29.075 or we can simplify the square root of seventy-two first which will make our 13:29.120 --> 13:34.814 multiplication a little bit more efficient because the numbers will be smaller. 13:34.880 --> 13:40.820 Let's do that. Let's begin by simplifying the square root of seventy-two. 13:40.880 --> 13:48.094 What two numbers multiply together to give me seventy-two, one of which is a perfect square? And 13:48.160 --> 13:55.635 ideally we'd like to choose the largest perfect square that divides evenly into seventy-two. 13:55.680 --> 14:02.509 Well, those numbers would be thirty-six and two. So we'll rewrite the square root of seventy-two 14:02.560 --> 14:12.185 as the square root of thirty-six times two. Now using our product rule, we know that this 14:12.240 --> 14:17.057 can be written as the square root of thirty-six times the square root of two 14:17.090 --> 14:28.435 And the square root of thirty-six is six. That gives us six times the square root of two. 14:28.480 --> 14:34.674 So our denominator is simplified, but it is still an irrational number. 14:34.720 --> 14:42.749 So to rationalize this denominator, we want this radical expression or the radicand rather to be a 14:42.800 --> 14:53.960 perfect square. So we will multiply the numerator and denominator by the square root of two. 14:54.000 --> 14:58.465 Now using the product rule, we know that the square root of two 14:58.498 --> 15:07.640 times the square root of two is the square root of four, and the square root of four is two. That 15:07.674 --> 15:19.040 gives us six times two in our denominator. And one times the square root of two in the numerator 15:19.052 --> 15:28.595 is just the square root of two. So our final simplified answer will be the square root of two 15:28.640 --> 15:37.537 divided by twelve. The square root of two divided by twelve has a rational value in the denominator, 15:37.600 --> 15:53.760 and it is equivalent to one over the square root of seventy-two.