Simplifying Square Roots

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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

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solution:

The expression inside the square root is nearly completely factored already, but we do need to write it so that perfect square factors show up more explicitly to complete step (i) in applying the general strategy given above. We have

50 = 2 · 5 · 5 = 2 · 5 ² and x ¹³ = x ¹² · x = (x ⁶) ² · x

Thus our original square root can be rewritten as

	after rearranging the order of the factors. Now we can do step (ii) of the strategy: separate the perfect square factors in this square root into individual square root factors.
	now we can apply step (iii) of the strategy to the first two factors …
	to get our final result.

Notice that the remaining square root factor in this final expression, , contains no further perfect square factors, and so there is no way to reduce the expression inside this remaining square root any further. Thus, our final answer is:

Even though you can probably do the first step of this strategy – the factorization to show perfect square factors – without writing down a lot of intermediate steps, we suggest that you do write down enough of the details to reliably keep track of what you are doing. It is particularly important to ensure that the perfect squares that you identify are accurate, and the best way to do that is to show at least as much work as we did in the solution to Example 1 above. For powers of symbols, even powers are always perfect squares. Odd powers are a product of a perfect square and the first power of that symbol.

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Example 2:

Simplify

solution:

First, we factor the expression inside the square root to display all possible perfect square factors explicitly. Methods for ensuring a complete factorization of numbers into a product of prime factors have been covered elsewhere in these notes.

400 = 2 · 200 = 2 · 2 · 100 = 2 · 2 · 2 · 50 = 2 · 2 · 2 · 2 · 25 = 2 · 2 · 2 · 2 · 5 · 5 = 2 ⁴ · 5 ² = (2 ² ) ² · 5 ²

and

y ⁷ = y ⁶ · y = (y ³ ) ² y

So,

as the final answer.

Example 3:

Simplify

solution:

Here

32 = 2 ⁵ = (2 ² ) ² · 2

and

ab ³c ⁴ = a · b ² · b · (c ²) ² = b ²(c ²) ² · ab

So