# How do you simplify sqrt216?

Apr 30, 2016

$\sqrt{216} = 6 \sqrt{6}$

#### Explanation:

The quick version is:

$\sqrt{216} = \sqrt{{6}^{2} \cdot 6} = 6 \sqrt{6}$

How do we find out that $216 = {6}^{2} \cdot 6$ ?

One way is to split off prime factors one at a time, then recombine them.

Here's a factor tree for $216$:

$\textcolor{w h i t e}{0000} 216$
$\textcolor{w h i t e}{0000} \text{/"color(white)(0)"\}$
$\textcolor{w h i t e}{000} 2 \textcolor{w h i t e}{00} 108$
$\textcolor{w h i t e}{000000} \text{/"color(white)(0)"\}$
$\textcolor{w h i t e}{00000} 2 \textcolor{w h i t e}{00} 54$
$\textcolor{w h i t e}{0000000} \text{/"color(white)(00)"\}$
$\textcolor{w h i t e}{000000} 2 \textcolor{w h i t e}{000} 27$
$\textcolor{w h i t e}{000000000} \text{/"color(white)(00)"\}$
$\textcolor{w h i t e}{00000000} 3 \textcolor{w h i t e}{0000} 9$
$\textcolor{w h i t e}{000000000000} \text{/"color(white)(0)"\}$
$\textcolor{w h i t e}{00000000000} 3 \textcolor{w h i t e}{000} 3$

So we find:

$216 = 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 3 = \left(2 \cdot 3 \cdot 2 \cdot 3\right) \cdot \left(2 \cdot 3\right) = {6}^{2} \cdot 6$

By definition:

$\sqrt{{6}^{2}} = 6$

For any positive numbers $a$ and $b$ we have:

$\sqrt{a b} = \sqrt{a} \sqrt{b}$

Hence:

$\sqrt{216} = \sqrt{{6}^{2} \cdot 6} = \sqrt{{6}^{2}} \sqrt{6} = 6 \sqrt{6}$