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# Simplifying Square Roots

An expression containing a square root is considered to be as simple as possible when the expression inside the square root is as simple or small as possible. The reduction of the contents inside the square root is accomplished (when possible) by a very straightforward strategy:

(i) Factor the expression inside the square root completely. Write factors which are perfect squares as explicit squares.

(ii) Use the property that the square root of a product is equal to the product of the square roots of the factors to rewrite the square root from step (i) as a product of square roots of factors which are perfect squares and a single square root of an expression which contains no perfect square factors. The pattern is:

where u, v, etc. are perfect squares, and w is an expression containing no perfect square factors. (This may seem a bit abstract, but the meaning of this pattern should become more obvious after you have studied a few of the examples below. It is important in mathematics not only to study specific examples of a type of operation, but to eventually understand an overall general strategy or pattern for similar types of problems.

(iii) Replace the square roots of perfect squares by factors which are not square roots using the property

We now illustrate this general strategy with a series of specific examples.

Example 1:

Simplify

solution:

There is a strong temptation here to simply take the square root term by term to get

However, you should see immediately that the first step violates the previously stated properties of radicals and so is invalid. The only way we can simplify this expression is if we are able to first factor it into a product in which one or more factors are perfect squares. Here, the expression x 2 + y 2 cannot be factored using any of the techniques available to us, and so no progress can be made as far as simplifying this square root. We are forced to conclude that the given expression is already in simplest radical form, and nothing can be done to reduce the expression inside the radical to a simpler algebraic form.

Example 2:

Simplify

solution:

There seems to be a lot of perfect squares here – in fact, x 4, 9, and x 2 are all perfect squares. However, only perfect square factors of the entire expression in the radical are of any use to us here. Proceeding using the methods for factoring algebraic expressions that were covered in detail earlier in these notes, we get x 4 + 9x 2 = x 2(x 2 + 9) as the most complete factorization possible. Thus

Again, the expression left in the square root in this last line cannot be factored further at all, and so we cannot extract any further perfect square factors inside the square root to allow further simplification of the square root. Thus

must be the final answer here.