Simplifying radicals examples

Let's get some practice rewriting and simplifying radical expressions. So in this first exercise, and these are all from KH Academy, it says simplify the expression by removing all factors that are perfect squares from inside the radicals and combine the terms. If the expression cannot be simplified, enter it as given.

All right, let's see what we can do here. So we have -40, the negative square root of 40. I should say, let me write a little bit bigger so you can see that. So the square root of 40 plus the square root of 90.

So let's see, what perfect squares are in 40? Immediately, what jumps out at me is that it's divisible by four, and four is a perfect square. So this is the negative square root of 4 * 10 plus the square root of, well, what jumps out at me is this is divisible by 9. Nine is a perfect square, so 9 * 10.

If we look at the 10 here, 10 does not have any perfect squares in it anymore. If you wanted to do a full factorization of 10, a full prime factorization, it would be 2 * 5. So there's no perfect squares in 10, and so we can work it out from here.

This is the same thing as the negative of the square root of 4 times the square root of 10 plus the square root of 9 times the square root of 10. When I say square root, I'm really saying principal root, the positive square root. So it's the negative of the positive square root of four.

Let me do this in another color so it can be clear. So this right here is two; this right here is three. So it's going to be equal to 2 square roots of 10 plus 3 square roots of 10.

If I have -2 of something and I add 3 of that same something to it, that's going to be what? Well, that's going to be 1 square root of 10. Now, if this last step doesn't make full sense, actually, let me slow it down a little bit. I could rewrite it this way. I could write it as 3 square roots of 10 minus 2 square roots of 10.

That might jump out at you a little bit clearer. If I have three of something and I were to take away two of that something, in that case the square roots of 10, well, I'm going to be left with just one of that something. I'm just going to be left with one square root of 10, which we could just write as the square root of 10.

Another way to think about it is we could factor out a square root of 10 here. So you undistribute it; do the distributive property in reverse. That would be the square root of 10 times 3 - 2, which, of course, this is just one. So you're just left with the square root of 10.

So all of this simplifies to the square root of 10. Let's do a few more of these. So this says, simplify the expression by removing all factors that are perfect squares from inside the radicals and combining the terms.

So essentially the same idea. All right, let's see what we can do. So this is interesting; we have a square root of 2. Can I...? Well, actually, what could be interesting is since if I have a square root of something times the square root of something else, so the square root of 180 times the square root of 12, this is the same thing as the square root of 180 times 12.

This just comes straight out of our exponent properties. It might look a little bit more familiar if I wrote it as 180 to the 1/2 power times 12 to the 1/2 power. This is going to be equal to (180 * 12) to the 1/2 power. Taking the square root, the principal root is the same thing as raising something to the 1/2 power.

So this is the square root of 180 * 1/2, which is going to be the square root of 90, which is equal to the square root of 9 * 10. We just simplified the square root of 90 in the last problem. That's equal to the square root of 9 times the square root of 10, which is equal to 3 * the square root of 10.

All right, let's keep going. So I have one more of these examples, and like always, pause the video and see if you can work through these on your own before I work it out with you. Simplify the expression by removing all factors that are perfect squares.

Okay, these are the same directions that we've seen the last few times. So let's see if I wanted to simplify this. This is equal to the square root of 64 * 2, which is 128, and 64 is a perfect square. So I'm going to write it as 64 * 2 over 27, which is 9 * 3. Nine is a perfect square, so this is going to be the same thing.

There are a couple of ways that we could think about it. We could say this is the same thing as the square root of (64 * 2) over the square root of (9 * 3), which is the same thing as the square root of 64 times the square root of 2 over the square root of 9 times the square root of 3.

This is equal to... This is 8; this is 3, so it would be 8 * the square root of 2 over 3 * the square root of 3. That's one way to say it. Or we could even view the square root of 2 over the square root of 3 as a square root of (2/3). So we could say this is 8 over 3 times the square root of (2/3).

These are all possible ways of trying to tackle this. So we could just write it, let's see, have we removed all factors that are perfect squares? Yes, from inside the radicals, and we've combined terms. We weren't doing any adding or subtracting here, so it's really just removing the perfect squares from inside the radicals, and I think we've done that.

So we could say this is going to be (8/3) times the square root of (2/3), and there are other ways that you could express this that would be equivalent. But, hopefully, this makes some sense.