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One of the key steps in simplifying mathematical expressions involving square roots—or 'radicals' as they are often called—is to break the numbers down into their prime factors and separate the perfect squares from the non-perfect squares. A perfect square is simply a number that has a whole number as its square root, such as 4, 9, or 16.

For example, to simplify the square root of 20, you look for the largest perfect square that is a factor of 20. In this case, that's four because 20 can be expressed as 4 times 5, or \( 4 \times 5 \) and four is a perfect square. Hence, \( \sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5} \). Simplifying radicals involves breaking down the number under the square root in this way, finding the perfect square factors, and taking their square root outside the radical.

After simplifying the radicals, the next step is to combine like terms to simplify the expression further. 'Like terms' are terms that contain the same variables raised to the same power, or in the case of square roots, the same radical part. The coefficients (the number in front of the radical) can be different, but as long as the radical part is the same, they can be added (or subtracted) just like regular numbers.

Using the previous simplification, once we have \( 2\sqrt{5} \) from \( \sqrt{20} \), \( 3\sqrt{5} \) from \( \sqrt{45} \) and \( 4\sqrt{5} \) from \( \sqrt{80} \), they are clearly like terms because of the common \(5\) under the square root. Now we can simply add the coefficients together, as you would do with simple numbers.

Perfect squares play a significant role in the simplification of square roots because they are essentially square roots that resolve neatly into integers. Factorization of perfect squares involves expressing a number as a product of its prime factors and then pairing the primes that form the perfect squares. Doing this allows us to take the square roots of those perfect squares outside the radical for further simplification.

For instance, 80 is not a perfect square, but it has 16 (which is 4 squared) as a factor. Factorize 80 into \( 16 \times 5 \), take the square root of the perfect square, which is 16, to get 4, and place it outside the radical. This process leaves you with \( 4\sqrt{5} \) and makes the square roots more manageable, usually leading to a more simplified expression overall.