How do you simplify 7sqrt3-4sqrt6+sqrt48-sqrt54?

Jan 5, 2017

$23 \sqrt{3} - 7 \sqrt{6}$

Explanation:

Note:
$\textcolor{w h i t e}{\text{XXX}} \textcolor{g r e e n}{\sqrt{6}} = \textcolor{g r e e n}{\sqrt{2}} \cdot \textcolor{m a \ge n t a}{\sqrt{3}}$
$\textcolor{w h i t e}{\text{XXX}} \textcolor{red}{\sqrt{48}} = \textcolor{red}{\sqrt{16}} \cdot \textcolor{m a \ge n t a}{\sqrt{3}} = \textcolor{red}{4} \cdot \textcolor{m a \ge n t a}{\sqrt{3}}$
$\textcolor{w h i t e}{\text{XXX}} \textcolor{b l u e}{\sqrt{54}} = \textcolor{b l u e}{\sqrt{9} \cdot \sqrt{2}} \cdot \textcolor{m a \ge n t a}{\sqrt{3}} = \textcolor{b l u e}{3 \sqrt{2}} \cdot \textcolor{m a \ge n t a}{\sqrt{3}}$

Therefore:
$7 \textcolor{m a \ge n t a}{\sqrt{3}} - 4 \textcolor{g r e e n}{\sqrt{6}} + 4 \textcolor{red}{\sqrt{48}} - \textcolor{b l u e}{\sqrt{54}}$

$= \textcolor{m a \ge n t a}{\sqrt{3}} \cdot \left(7 - 4 \textcolor{g r e e n}{\sqrt{2}} + 4 \cdot \textcolor{red}{4} - \textcolor{b l u e}{3 \sqrt{2}}\right)$

$= \textcolor{m a \ge n t a}{\sqrt{3}} \left(23 - 7 \sqrt{2}\right)$

or
$= 23 \sqrt{3} - 7 \sqrt{6}$