Simplifying square-root expressions: no variables

- [Voiceover] Let's get some practice rewriting and simplifying radical expressions. So in this first exercise, and these are all from Khan 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 negative 40 (laughs), the negative square root of 40 I should say. Let me write a little bit bigger so you can see that. So the negative square root of 40 plus the square root of 90. So let's see, what perfect squares are in 40? So, what immediately jumps out at me is that this, it's divisible by four and four is a perfect square. So this is the negative square root of four times 10, plus the square root of, well what jumps out at me is that this is divisible by nine. Nine is a perfect square, so nine times 10. And if we look at the 10s 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 two times five. 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 four times the square root of 10, plus the square root of nine, times the square root of 10. And 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, so that is, so let me do this is 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 negative two square roots of 10 plus three square roots of 10. So if I have negative two of something and I add three of that same something to it, that's going to be what? Well that's going to be one square root of 10. Now 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 three square roots of 10 minus two square roots of 10. That might jump out at you a little bit clear. If I have three of something and I were to take away two of that something, and that case it's squares of 10s, well, I'm going to be left with just one of that something. I'm just gonna 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 three minus two, which is of course, this is just one. So you're just left with the square root of 10. So all of this simplifies to 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 combine 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 1/2. So 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 1/2, this is the same thing as the square root of 180 times 1/2. And 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 1/2 to the 1/2 power, is going to be equal to 180 times 1/2 to the 1/2 power, taking the square root, the principal root is the same thing as raising something to the 1/2 power. And so this is the square root of 80 times 1/2 which is going to be the square root of 90, which is equal to the square root of nine times 10, and we just simplified square root of 90 in the last problem, that's equal to the square root of nine times the square root of, principle root of 10, which is equal to three times the square root of 10. Three times 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 square, okay, these are just same directions that we've seen the last few times. And so let's see. If I wanted to do, if I wanted to simplify this, this is equal to the square root of, well, 64 times two is 128, and 64 is a perfect square, so I'm gonna write it as 64 times two, over 27 is nine times three. Nine is a perfect square. So this is going to be the same thing. And there's a couple of ways that we can think about it. We could say this is the same thing as the square root of 64 times two, over the square root of nine times three, which is the same thing as the square root of 64 times the square root of two, over square root of nine times the square root of three, which is equal to, this is eight, this is three, so it would be eight times the square root of two, over three times the square root of three. That's one way to say it. Or we could even view the square root of two over the square root of three as a square root of 2/3. So we could say this is eight over three times the square root of 2/3. So 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's other ways that you could express this that would be equivalent but hopefully this makes some sense.