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 Depdendent Variable

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 Dependent Variable

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# 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: because 3 and represent the same number by the rule for multiplying square roots together 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. If possible, separate or factor 45 into a product of two numbers, one of which is the square of a whole number. (Recall, we called such numbers “perfect squares” earlier.)  Use the rule for multiplying two square roots. since the square root of a square is the original number. The multiplication symbol can be omitted.

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.