Rewriting square root of fraction

So we have here the square root, the principal root of one two hundredths. What I want to do is simplify this. When I say simplify, I really mean I want to, if there's any perfect squares here that I can factor out to take it out from under the radical. I encourage you to pause the video and see if you can do that.

All right, so there's a couple of ways that you could approach this. One way is to say, well, this is going to be the same thing as the square root of 1 over the square root of 200. The square root of 1 is just 1 over the square root of 200. There are a couple of ways to try to simplify the square root of 200. I'll do it a couple of ways here.

The square root of 200, you could realize that, okay, look, 100 is a perfect square and it goes into 200. So this is the same thing as 2 times 100. The square root of 200 is the square root of 2 times 100, which is the same thing as the square root of 2 times the square root of 100. We know that the square root of 100 is 10, so it's the square root of 2 times 10, or we could write this as 10 square roots of 2.

That's one way to approach it. But if it didn't jump out at you immediately that you have this large perfect square that is a factor of 200, you could just start with small numbers. You could say, all right, let me do this alternate method in a different color. You could say that the square root of 200, say well, it's divisible by 2, so it's 2 times 100. And if 100 didn't jump out at you as a perfect square, you could say, well, that's just going to be 2 times 50.

Well, I could still divide 2 into that. That's 2 times 25. Let's see. And 25, if that doesn't jump out at you as a perfect square, you could say that, well, see, that's not divisible by 2, not divisible by 3, 4, but it is divisible by 5. That is 5 times 5.

To identify the perfect squares, you would say, all right, are there any factors where I have at least two of them? Well, I have two times two here, and I also have five times five here. So I can rewrite the square root of 200 as being equal to the square root of 2 times 2 times 5 times 5.

Let me just write it all out so to check the common amount of space. So, give myself more space under the radical, the square root of 2 times 2 times 5 times 5 times 5 times 5 times 2 times 2. When I wrote it in this order, you can see the perfect squares here.

Well, this is going to be the same thing as the square root of 2 times 2. This second method is a little bit more monotonous, but hopefully you see that it works. This is one way to think about it, and they really boil down to the same method. We're still going to get to the same answer.

So, the square root of 2 times 2 times the square root of 5 times 5 times the square root of 5 times 5 times the square root of 2 times the square root of 2. Well, the square root of 2 times 2 is just going to be, this is just 2. The square root of 5 times 5, well, that's just going to be 5. So, you have 2 times 5 times the square root of 2, which is 10 times the square root of 2.

So, this right over here is square root of 200. We can rewrite as 10 square roots of 2. So, this is going to be equal to 1 over 10 square roots of 2. Now, some people don't like having a radical in the denominator, and if you wanted to get rid of that, you could multiply both the numerator and the denominator by square root of 2.

Because notice, we're just multiplying by 1, we're expressing 1 as square root of 2 over square root of 2. And then what that does is we rewrite this as the square root of 2 over 10 times the square root of 2 times the square root of 2. Well, the square root of 2 times square root of 2 is just going to be 2, so it's going to be 10 times 2, which is 20.

So, it could also be written like that. Hopefully, you found that helpful. In fact, even this one you could write, if you want to visualize it slightly differently, you could view it as one twentieth times the square root of two. So, these are all the same thing.