# How do you simplify Square root 18(5 square root 2 - square root 18)?

Jul 24, 2015

Probable intended answer: $36 \sqrt{2}$
It is possible to argue that the answer might be any of $\pm 36 \sqrt{2}$ or $\pm 144 \sqrt{2}$

#### Explanation:

If "square root" is intended to mean only non-negative square root:

$\sqrt{18} = \sqrt{2 \cdot {3}^{2}} = \sqrt{2} \cdot 3$

$18 \left(5 \sqrt{2} - \sqrt{18}\right)$
$\textcolor{w h i t e}{\text{XXXX}}$$= 18 \left(5 \sqrt{2} - 3 \sqrt{2}\right)$

$\textcolor{w h i t e}{\text{XXXX}}$$= 18 \left(2 \sqrt{2}\right)$

$\textcolor{w h i t e}{\text{XXXX}}$$= 36 \sqrt{2}$

Alternate answers are possible if we interpret
$\textcolor{w h i t e}{\text{XXXX}}$"square root of 2" to mean $\pm \sqrt{2}$
and
$\textcolor{w h i t e}{\text{XXXX}}$"square root of 18" to mean $\pm \sqrt{18}$