Step 5: Double-check
One easy way to go wrong here is to choose the wrong perfect square. What if the perfect square you use isn't the largest? You will still get a tidy and finished-looking solution, but it won't be reduced completely. Here's how you check to make sure you've used the right perfect square:
Let's say you need to reduce √72. Go back through the steps...First, find all the factors: 2, 4, 6, 8, 9, 12, 18, 24 and 36. There are three perfect squares on that list: 4, 9 and 36. The one you need is 36, because it's the largest. Let's say that you miss that one, though, and choose 9 instead.
When you rewrite your square root, you will write it as √(9×8). It will pull apart to √9 × √8, and finally it will reduce to 3√8. It looks like a finished answer, but it isn't fully reduced.
If you choose 36, the largest perfect square on your list of factors, your steps will go like this:
√(36×2) = √36 × √2 = 6√2. Now it is fully reduced, and you have found your answer.
So how can you tell whether or not you have used the correct factor? When you think you've finished the problem, look at the number you have inside the root sign. In our not-fully-reduced example, that number was 8. Find the factors of that number – 2 and 4. Are any of those numbers also a perfect square? Yes. 4 is a perfect square.
If the number you end up with inside the root sign has a factor that is a perfect square, your radical is not fully reduced.
Let's look at our correct-answer example. In 6√2, let's consider the number 2. Does it have any factors that are perfect squares? No. This means that the square root is fully reduced.