# Simplifying Radical Expressions with Variables

When you need to simplify a radical expression that has variables under the radical sign, first see if you can factor out a square.

Since a negative number times a negative number is always a positive number, you need to remember when taking a square root that the answer will be both a positive and a negative number or expression. For example $a\cdot a={a}^{2}$ , and also $\left(-a\right)\cdot \left(-a\right)={a}^{2}$ . We usually will denote such dual answers as $±a$ .

Example 1:

Simplify.

$c\sqrt{{a}^{3}{c}^{4}}$

Factor the radicand as the product of $a$ and a squared expression.

$c\sqrt{{a}^{3}{c}^{4}}=c\sqrt{{\left(a{c}^{2}\right)}^{2}\cdot a}$

Use the product property of square roots :

$=c\sqrt{{\left(a{c}^{2}\right)}^{2}}\cdot \sqrt{a}$

Simplify.

$\begin{array}{l}=c\left(±a{c}^{2}\right)\sqrt{a}\\ =±a{c}^{3}\sqrt{a}\end{array}$

Example 2:

Simplify. $\sqrt{{a}^{3}{c}^{2}}$

Rewrite the radicand using squared expressions where possible.

$=\sqrt{a\cdot {a}^{2}\cdot {c}^{2}}$

Simplify.  The square roots of ${a}^{2}$ and ${c}^{2}$ can be negative or positive, so use the sign $±$ .

$=±ac\sqrt{a}$