# 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 $(-a)\cdot (-a)={a}^{2}$ . We usually will denote such dual answers as $\pm a$ .

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Example 1:
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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.

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

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Example 2:
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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 $\pm $ .

$=\pm ac\sqrt{a}$