It is easy to simplify a radical that is a perfect square, such as sqrt(81). We can either use a calculator, or we can use our knowledge of perfect squares to get an answer of 9, since 9² equals 9. We must remember that -9 is also a solution to the problem, although it would be discarded in the context of a geometry problem involving length, or if we were only asked to provide the principal square root.
Simplifying a non-perfect square radical such as sqrt(20) involves a bit more work. We could use a calculator to get a lengthy decimal approximation of the answer, but that is not what is meant by simplifying the radical. What we are being asked to do, in essence, is to break the radical apart such that we are left with the product of an integer times the square root of a prime number.
To do this, it is essential to know the particular property of radicals shown above. In simplest terms, the equation tells us that we can split the radical of a product into the product of the radicals. To apply the formula to the sqrt(20) example above, we would break 20 into factors of 4 and 5. We then have sqrt(4 times 5), which can be split up into sqrt(4) times sqrt(5). Sqrt(4) we know is 2, so our final simplified answer is 2 times sqrt(5). That is the answer that would be expected on an exam. Notice how we cannot break down sqrt(5), since 5 is a prime number whose only factors are 1 and itself.
Sometimes students ask if they could have broken 20 into other factors, such as 2 and 10. The answer is that we could, but then we would have sqrt(2 times 10), which would break into sqrt(2) times sqrt(10). Since neither of those is a perfect square, we won't end up with an integer component in our answer, which is what we need to have.
Let's get back to example of sqrt(80) in the introduction. 80 can be broken up into many factor pairs such as 2 and 40, 4 and 20, 8 and 10, etc. What we need to look for is the largest perfect square factor of 80, and use that. 4 is a perfect square factor of 80, but there is a larger one: 16. That means we should use 16 and 5 as our factor pair. We now have sqrt(16 times 5) = sqrt(16) times sqrt(5) = 4 times sqrt(5), which is our answer.
In the above example, if we had used 4 and 20 as our factor pair, we would have lots of extra work to do. We'd have sqrt(4) times sqrt(20). That becomes 2 times sqrt(20), but then we'd have to break down sqrt(20) as we did before. By using the largest perfect square factor, 16, we got our answer in fewer steps.
One last example: sqrt(200). There are many factors, several of which are perfect squares. We want the largest perfect square factor, which is 100. That gives us sqrt(100) times sqrt(2) which equals 10 times sqrt(2).
Note that we have no way of reducing the square root of a number which is either prime, or the product of two primes. For example, we cannot simplify sqrt(13). It's a prime number with no perfects square factors. We just have to leave our answer as is.
Another example would be sqrt(6). 6 is not prime. We could break it into sqrt(2) times sqrt(3), but neither of those is a perfect square, so it won't simplify. We would just leave our answer as sqrt(6). It doesn't have any perfect square factors.
A final example is sqrt(77). 77 is not prime since it has factors other than 1 and itself, but those other factors are both primes. Since it doesn't have any perfect square factors, we just have to leave the answer alone, and it is correct to do so.
Algebra students should make sure that they are comfortable with this process. It comes up quite frequently in math, and there is no reason to do a problem perfectly but then lose partial or full credit just because you didn't simplify your square root answer.