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 $\pm a$ .

Example 1:

Simplify.

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

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

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

Use the product property of square roots :

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

Simplify.

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

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 $\pm$ .

$= \pm a c \sqrt{a}$

Simplifying Radical Expressions with Variables

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