# What is the square root of 108 in simplest radical form?

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12
Dec 20, 2016

$\sqrt{108} = \textcolor{b l u e}{6 \sqrt{3}}$

#### Explanation:

Decomposing $108$ into factors one step at a time:
$108$
$\textcolor{w h i t e}{\text{XXX}} = 2 \times 54$
$\textcolor{w h i t e}{\text{XXX}} = 2 \times 2 \times 27$
$\textcolor{w h i t e}{\text{XXX}} = 2 \times 2 \times 3 \times 9$
$\textcolor{w h i t e}{\text{XXX}} = 2 \times 2 \times 3 \times 3 \times 3$
$\textcolor{w h i t e}{\text{XXX}} = {2}^{2} \times {3}^{2} \times 3$

$\sqrt{108} = \sqrt{{2}^{2} \times {3}^{2} \times 3}$
$\textcolor{w h i t e}{\text{XXX}} = \sqrt{{2}^{2}} \times \sqrt{{3}^{2}} \times \sqrt{3}$
$\textcolor{w h i t e}{\text{XXX}} = 2 \times 3 \times \sqrt{3}$
$\textcolor{w h i t e}{\text{XXX}} = 6 \sqrt{3}$

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