Simplifying Square Roots
by Removing Perfect Square Factors
We can exploit the rules for multiplication of square roots to attempt to rewrite square roots in what is considered to be a simpler form. Here,
simpler square roots means the number inside the square root is smaller
Simplifying a square root means rewriting it as an expression of the same value, but with the number or expression inside the square root as small or simple as possible.
We will illustrate the technique here for square roots involving just numbers, but this method is most important in simplifying square roots containing algebraic expressions.
As an example, notice that we can do the following:
Thus has the same value as . But, we would consider to be a simpler form because the quantity in the square root is a smaller number. If we rewrite the above example with the steps in reverse order, we can see the strategy for simplifying a square root when that is possible.
Since the remaining number in the square root, the 5, obviously cannot be written as a product of a perfect square and another number, we have achieved as much simplification here as is possible.
This strategy for simplifying square root expressions requires us to develop a strategy for deducing how numbers can be rewritten as a product involving one or more perfect squares indeed, we need to be able to rewrite the original number in the square root as a product of perfect squares, and the one smallest value which is not a perfect square.