 We often take a radical expression and simplify it. So square root of 3 times square root of 5 was simplified into a single square root. Square root of 18 was simplified into the square root of a smaller number. And so we can try the simplification process in general. So remember that in a simplified radical expression, the radicand is as small as possible. For example, suppose we want to simplify square root of 75. Now suppose you're walking along one day and all of a sudden out of the sky the fact that 75 is equal to 3 times 25 falls and hits you on the head. Now we could use this fact to simplify our radical expression. Equals means replaceable, so we'll replace 75 with 3 times 25. Our theorem about the product of roots says that anytime we see a product of square roots we can replace it with the square root of the product. But it also works backwards. Anytime we see the square root of a product, we can replace it with the product of the square roots. So that means we can replace this square root of 3 times 25 with square root of 3 times square root of 25. Again, since we are genetically programmed from birth with the knowledge that square root of 25 is the same as 5, we can replace it. And typically we write things so that the radical comes second, so we'll rewrite this as 5 square roots of 3. Now this is all well and good if you happen to be walking in areas where factors drop out of the sky and if you were born with the knowledge of all the perfect squares. But what if you're not? Our process suggests that we can try to remove perfect square factors from the radicand and it helps if you know the perfect squares. 1 squared equals 1, 2 squared equals 4, 3 squared equals 9, 4 squared equals 16, and so on. But you don't have to know them. You can work around it. So for example, suppose we want to simplify the square root of 800. Now it's useful to keep in mind that our simplification relies on our ability to write a square root of a product as the product of the square roots where at least one of the square roots can be reduced to a whole number. And what this means is that when simplifying a radical, a factor only matters if it's a perfect square factor. So maybe, just maybe, you know that 10 squared is equal to 100. If you do, then you can factor 800 is 100 times 8. Now if we take a look at this 8, we also note that 4 is 2 squared and we can factor 8 as 4 times 2. And so 800 will factor as 100 times 4 times 2. Equals means replaceable, so anywhere I see 800, I can replace it with 100 times 4 times 2. So in our expression, square root of 800, we can rewrite it. Because this is the square root of a product, we can rewrite it as the product of the square roots. Because we know 10 squared equals 100, then the square root of 100 is 10. Because we know that 4 is 2 squared, we know the square root of 4 is 2. And square root of 2 was there and it's still there. And again, while factored form is best, there's nothing else we can do with this expression. So we can multiply 10 by 2 to get 20 and our square root of 2 gets carried along for the ride for our final answer 20 square roots of 2. What if you didn't know that 10 squared equals 100? We can still factor. So let's try to factor 800. So the first thing we might observe is that because 800 is an even number, we know that it's 2 times something and in fact it will be 2 times 400. But wait, 400 is also an even number, so we know that 400 is going to be 2 times something. In fact, 2 times 200. And we can continue in this way until we find the complete factorization of 800 into prime factors. Now remember, we do have a useful theorem that says for n greater than or equal to 0, the square root of n squared is equal to n. And so what we might do is we might start to group these factors and rewrite them so they are squares of things. So these two 2s, well, that's really a 2 squared. These two 2s give us a 2 squared. Arithmetic is bookkeeping. We have a leftover 2 here. We still have that leftover 2. And these two 5s, well, those can be written as 5 squared. Equals means replaceable, and so 800 can be replaced with 2 squared times 2 squared times 2 times 5 squared. We have a square root of a product, so we can rewrite this as the product of the square roots. Square root of 2 squared is just 2. We have a square root of 2. We still have a square root of 2. And square root of 5 squared is 5. And again, we like rearranging our factors so the square root goes last, so we'll rearrange them. And again, factored form is best. This is a good answer, but since we're not going to do anything else with this number, we'll go ahead and multiply out the factors outside the square root.