Simplifying square root expressions

Overview

• Perfect Squares
• The Product Property of Square Roots
• The Quotient Property of Square Roots
• Square Roots Not Involving Fractions
• Square Roots Involving Fractions

To begin our study of the process of simplifying a square root expression, we must note three facts: one fact concerning perfect squares and two concerning properties of square roots.

Perfect squares

Perfect squares

Although we will not make a detailed study of irrational numbers, we will make the following observation:

Any indicated square root whose radicand is not a perfect square is an irrational number.

The numbers $\sqrt{6},\sqrt{15},$ and $\sqrt{\frac{3}{4}}$ are each irrational since each radicand $\left(6,15,\frac{3}{4}\right)$ is not a perfect square.

The product property of square roots

Notice that

$\sqrt{9·4}=\sqrt{36}=6$      and
$\sqrt{9}\sqrt{4}=3·2=6$

Since both $\sqrt{9·4}$ and $\sqrt{9}\sqrt{4}$ equal 6, it must be that

$\sqrt{9·4}=\sqrt{9}\sqrt{4}$

The product property $\sqrt{xy}=\sqrt{x}\sqrt{y}$

The quotient property of square roots

We can suggest a similar rule for quotients. Notice that

$\sqrt{\frac{36}{4}}=\sqrt{9}=3$      and
$\frac{\sqrt{36}}{\sqrt{4}}=\frac{6}{2}=3$

Since both $\frac{36}{4}$ and $\frac{\sqrt{36}}{\sqrt{4}}$ equal 3, it must be that

$\sqrt{\frac{36}{4}}=\frac{\sqrt{36}}{\sqrt{4}}$

The quotient property $\sqrt{\frac{x}{y}}=\frac{\sqrt{x}}{\sqrt{y}}$

CAUTION
It is extremely important to remember that

$\begin{array}{lllll}\sqrt{x+y}\ne \sqrt{x}+\sqrt{y}\hfill & \hfill & \text{or}\hfill & \hfill & \sqrt{x-y}\ne \sqrt{x}-\sqrt{y}\hfill \end{array}$

For example, notice that $\sqrt{16+9}=\sqrt{25}=5,$ but $\sqrt{16}+\sqrt{9}=4+3=7.$

We shall study the process of simplifying a square root expression by distinguishing between two types of square roots: square roots not involving a fraction and square roots involving a fraction.

Square roots not involving fractions

A square root that does not involve fractions is in simplified form if there are no perfect square in the radicand.

The square roots $\sqrt{x,}\sqrt{ab},\sqrt{5mn,}\sqrt{2\left(a+5\right)}$ are in simplified form since none of the radicands contains a perfect square.

The square roots $\sqrt{x^2,}\sqrt{a^3}=\sqrt{a^{2}a}$ are not in simplified form since each radicand contains a perfect square.