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:
The method to be used here has been well illustrated already,
so you should use this example as a practice problem – try
to solve it yourself before looking at the solution that follows.
Since
x^{ 7} + x^{ 5} = x^{ 5} (x^{ 2}
+ 1) = (x^{ 4} )(x)(x^{ 2} + 1) = (x^{ 2}
)^{ 2} (x)(x^{ 2} + 1)
we can write
as the most simplified form. Probably most practitioners would
leave the expression in the radical factored in this way even
though this factored form doesn’t help simplify the overall
expression any further.
Example 2:
Simplify
solution:
Although this looks like a somewhat more complicated problem
than in the previous examples, the same strategy is still
applied. First, we factor the expression in the radical as much
as possible.
Here, we recognized that after removal of the monomial
factors, y^{ 2} and z, the remaining expression, x^{ 2}
– 6x + 9, was a trinomial in x, and so might be factorizable
into the form (x + a)(x + b). Using the systematic approach for
deducing such factors as we described and illustrated it in an
earlier section of these notes, we were able to determine that
x^{ 2} – 6x + 9 = (x – 3)(x – 3) = (x
– 3)^{ 2}
Thus, we can now write
as the final, most simplified, result.
