 So we can add, subtract, and multiply radical expressions. What happens if we try to divide them? And so the idea is that we can have a fraction whose numerator and denominator are radical expressions. For example, we might have a radical in the numerator, a radical in the denominator, or we could have other things. So let's see if we can simplify square root of 75 over square root of 15. So let's simplify the numerator and denominator if we can. Square root of 75, we can try to simplify as square root of 25 times 3, the square root of a product is the product of the square roots, and the square root of 25 is 5, so this simplifies to 5 radical 3. We'll try the same thing with square root of 15. 15 is 5 times 3, and the square root of a product is the product of the square roots. Unfortunately, neither 5 nor 3 is a perfect square, and so we have to leave them in this product of square roots form. E equals means replaceable, so instead of square root of 75, I could write 5 square root of 3. And instead of square root of 15, I could write square root of 5 times square root of 3. But now, numerator and denominator have a common factor, square root of 3, so we can remove that common factor and get a simplified form. And this leads to the idea of rationalizing. When we rationalize, we eliminate the radical expression and replace it with something else. For example, we eliminate a square root of 2 and replace it with, I don't know what, we'll figure that out, or we'll eliminate 5 plus square root of 3 and we'll replace it with, again, I don't quite know what we're going to replace it with, but we'll figure it out. Here's the important thing, you can always eliminate a radical, but it will show up again elsewhere. Given a fraction, a over b, we can try to make the numerator or the denominator rational. And rationalizing a fraction is based on two ideas. First, if n is non-negative, the square root of n times the square root of n is just n. The other important idea is that for any fraction, a over b and any number n, a over b is na over nb. So for example, let's try to rewrite 5 divided by square root of 5, so it does not have a radical in the denominator. So let's think about this. One of the things we do know about getting rid of the square root is that if n is non-negative, the square root of n times the square root of n is equal to n itself without the square root. And so that means we can eliminate the square root of 5 in the denominator by multiplying the denominator by square root of 5. But we've got to pay a price. We can't just multiply the denominator by square root of 5. We also have to multiply the numerator by the same thing. And so now I'm willing to write equals in the denominator, square root of 5 times square root of 5 is just 5. And in the numerator, 5 times the square root of 5 is 5 times the square root of 5. We can't really do anything with it. But wait, there's more. Here we have a fraction where our numerator and denominator have a common factor of 5. And so we can remove this common factor, arithmetic is bookkeeping, and the thing that's left after we remove the common factor is just the square root of 5.