 What if we multiply two roots, like principal square root of 6 times principal square root of 27, or square root negative 4 times square root 2, or square root negative 10 times square root negative 10? So let's try to find square root of 6 times square root of 27. Now if we want to approach this problem as a mathematician would, we might begin with the following. First, it's useful to remember, if you don't know what something is, give it a name. I'll call square root of 6 times square root of 27, n, and the question you want to know is, what can we say about n? Now seeing as how we have a bunch of principal square roots, we might ask ourselves, what is the square root? Remember, definitions are the whole of mathematics, all else is commentary. So let's pull in our definition of principal square root, and so we know that principal square root of 6 is going to be the non-negative number whose square is equal to 6, and principal square root of 27 is the non-negative number whose square is equal to 27. Now it's important that we know this, but another useful idea to remember, paper is cheap, write stuff down. We may know it, but it's a lot easier to work with if we actually see it in front of us written down. So we know that square root of 6 is the positive number whose square is equal to 6, and so that tells us square root of 6 squared is equal to 6. Also, square root of 27 is the positive number whose square is equal to 27, so square root of 27 squared is equal to 27. So here's how we can make use of this. We have square root of 6, square root of 27 over here on the left, we know something about the squares of the square roots. So let's take our equation and square both sides. Remember, definitions are the whole of mathematics, all else is commentary, and paper is cheap, we can write stuff down. When we write square root of 6 times square root of 27 squared, what we really mean is we're going to take square root of 6, square root of 27, and multiply it by itself. Now because this is just a multiplication, we can rearrange the factors in any order that we want to, so let's reorganize them so that our square root of 6's are together and our square root of 27's are together. But wait, definitions are the whole of mathematics, all else is commentary. When I multiply something by itself, that's the same as that thing squared. So square root of 6 times square root of 6, well that's square root of 6 squared. And similarly, square root of 27 times square root of 27, well that's square root of 27 squared. But I know something about square root of 6 squared and square root of 27 squared. Since equals means replaceable, I can replace square root of 6 squared with 6 and square root of 27 squared with 27. Now let's think about this. Square root of 6 and square root of 27 are non-negative numbers. So this means that n is a product of non-negative numbers, so n must be non-negative. But we see here from the last line that n is the non-negative number whose square is 6 times 27. And so by our definition, this means that n is the principal square root of 6 times 27. And since we wanted to find square root of 6 times square root of 27, and that's equal to n, and n is equal to square root of 6 times 27, so square root of 6 times square root of 27 is square root of 6 times 27. So the proceedings suggest the following theorem. For a and b non-negative numbers, square root of a times square root of b is equal to the square root of a times b. So let's find square root of 5 times square root of 3. Our theorem says that the product of the square roots is the square root of the product. So we have square root 5 times square root of 3. Well that's really the square root of 5 times 3. Now a useful idea to keep in mind as we move along is that factored form is best. It's actually most useful to leave this in the form square root of 5 times 3 if we're going to do anything further with this. But in this particular case, we're not, and we might, but don't have to, multiply out 5 times 3 to get our final answer, square root of 15. How about square root 2 times square root of 9? By our theorem, we have square root of 2 times square root of 9 is equal to square root of 2 times 9, which will multiply out square root of 18. But in life, and in mathematics, it's always useful to remember, always ask, what else could we do? In this case, we might try to find the square roots of 2 and square roots of 9 first. And so if we find the squares of 1, 2, 3, and so on, we find. And so we know that the square root of 9 is equal to 3. And so what if we tried square root of 2 times square root of 9? Well, we know what square root of 9 is. It's equal to 3. And so this is square root of 2 times 3. And conventionally, we like to write the radical parts as the second factor. So we'll rewrite this as 3 times square root of 2. And so we found that square root 2 times square root of 9 is equal to square root of 18. But it could also be equal to 3 square root of 2. Equals means replaceable. And consequently, this tells us that square root of 18 is the same as 3 square root of 2. Now, because they're equal, we can use either expression, but which is preferable? It really depends on the situation. But as a general rule, in a simplified radical expression, the radicand is as small as possible. So in this expression, we have radicand 18. In this expression, we have radicand 2. And so this expression on the right, 3 square root of 2, is a preferred simplification about square root of 1,957 times square root of 1,957. So our theorem tells us that if we have a product of square roots, we can rewrite that as the square root of the product. The only problem here is our radicand has gotten a lot larger because it used to be 1,957 here, 1,957 here. And here, it's a much larger number, the product of 1,957 with itself. Which means we haven't actually simplified this. Now, we could multiply this out, but let's procrastinate and not multiply 1,957 by 1,957. Instead, we'll observe that this product is equal to 1,957 squared. And remember, definitions are the whole of mathematics, all else is commentary. Our definition of the principal square root is that it's a non-negative number whose square is equal to the radicand. So square root of 1,957 squared is the non-negative number whose square is 1,957 squared. And we can write this as follows, something squared equals 1,957 squared. And so the only question is what goes inside our set of parentheses. And after thinking about this for a little while, you realize that it has to be 1,957. And so that means the square root of 1,957 squared is equal to 1,957. And if we do this generally, this suggests the following, for n greater than or equal to zero, the square root of n squared is just n. Next, we'll take a look at how we can use this to simplify more radical expressions.