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Simplifying Roots

Mike McGarry
Lesson by Mike McGarry
Magoosh Expert
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Now we can talk about simplifying roots. And this a very important topic on the test. You see, because often we have to take a square root in the problem and the problem itself will result in the square root of a large number. But the square root of our large number will not appear among the answer. The answer choices will be in simplified form.

So we have to recognize how to simplify a radical. So if we have something large, like square root of 75, how do we simplify that? And write it in a form that would appear in the answers. So that's what we're gonna talk about here. As we learned in the previous lessons, roots distribute over multiplication. So we can separate a root of products into a product of roots.

So if I have a product under the roots, I can separate that into the square root of the first one times to square root of the second one. First of all remember, it is easy to find the square root of perfect squares. As a reminder, here, the square roots of the first 15 perfect squares. So the square root of 49, we're asking what number do we multiply by itself to get 49.

And of course the answer to that is 7. And similarly for all of these. These are just good numbers to know. Now, as we learned in the previous lesson, roots distribute over multiplication. So as we said a moment ago, we can separate the root of a product into the product of the roots.

So the root of the product P times Q equals the product of root P times root Q. And now notice, if P or Q is a perfect square, then that square root would be very easy to simplify. And the entire expression would simplify. So for example, suppose at some point in a problem we need the square root of 75. Maybe the answer to the question is the square root of 75.

Well, square root of 75 is not gonna appear in the answer choices, but clearly 75 is a multiple of 25, and 25's a perfect square. So we can use that to our advantage. So I'm just going to write square root of 75. It's not gonna appear in that form, but I'm gonna write that as a product of 25 times 3.

Now I can separate the roots. And now, square root of 25, that's something I can simplify. Square root of 25 is just 5, and I can just write this whole thing as 5 root 3. So 5 root 3 is equal to the square root of 75. It is the simplified form of the square root of 75. So the form square root of 75 would never appear as a multiple-choice answer on the test.

It would always appear simplified as 5 root 3. And that's why it's important to know how to do this procedure of simplifying roots, because we have to make the answers into a form recognizable on the test, the way the test itself will list the answers. So here's some more practice. Pause the video, and then we'll talk about this.

Okay, in each case what we're doing is we want to express the number under the radical as the product of a perfect square times something else. So 12 is divisible by perfect square, it's divisible by 4. So I'm gonna write is as 4 x 3, and then of course square root of 4 is just 2. So then this just simplified to 2 root 3.

63, I can write that as 9 x 7. And square root of 9 is 3, so this becomes 3 root 7. 80, I can say that is divisible by 4, but actually it's also divisible by, we divided by 4 we get 20, we can divide by 4 again. In other words 80 is actually divisible by 16. And so we actually save a bit of time if we write this as 16 x 5.

And the square root of 16 is 4, so this is just 4 root 5. And then 175, well, that is 7 times 25. So 25 times 7, take the square root of that and what we get is 5 root 7. And those are all simplified roots. And so on the test you'd never see square root of 175 listed as an answer choice, you'd see it listed as 5 root 7, the simplified form.

Suppose the number beneath the radical is particularly large. Suppose we have to find the square root of 2800, for example. It certainly will help to factor out largest squares such as 100, it may be necessary to find the full prime factorization of the number. Here we can simplify immensely, simply by factoring 100 out of the radical. So square root of 100, of course, is 10.

So this just becomes 10 root 28. But 28 I can also simplify, that's 4 times 7, and the square root of 4 is 2. So it's 10 times to root 7, or in other words 20 root 7. And that is in fact, the fully simplified form of the square root of 2,800. Here's a practice problem. Pause the video and then we'll talk about this.

Okay, well, first of all we know we cannot add through the radical. So we certainly can't add 48 + 75 + 192. So instead we have to simplify each one of those radicals. So let's take them one at a time. 48 is certainly divisible by 4. It's 4 times 12.

12 is divisible by 4 again. So this means that 48 is actually divisible by 16. So we can write it as 16 times 3. Of course, the square root of 16 is 4. So the 16 comes out of the radical as 4 and we just get 4 root 3. 75, we've already looked at, this is 25 times 3.

25 comes out of the radical as 5. So this just 5 root 3. Right away very interesting, the first two are both multiples of root 3. Makes us think what? Maybe the third one will be a multiple of root 3. What happens if we divide 192 by 3?

Well, we know that 180 is 60 x 3. And so subtract 192- 180 that's just 12. And of course, 3 goes into 12 four times, so 3 goes into 192, 64 times. So this is actually 64 times 3. And that we can really simplify because 64 comes out of the radical as 8, square root of 64 is 8 So we get 8 root 3.

Well now, we've simplified all three. So all three are in simplified form so we can simply add them. So 4 of anything plus 5 of that same thing plus 8 of that same thing has to be 17 of that thing. So the whole thing just simplifies to 17 root 3. And we choose answer choice A.

In summary, we simplify square roots by factoring out the largest perfect-square factor. If we can find the prime factorization, or if it is given it, we can use that: any pairs of prime factors and any even powers of prime are perfect squares. So that can be helpful, too.

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