# How do you evaluate 3\sqrt { 3} ( 6\sqrt { 6} - 9\sqrt { 9} )?

Jun 2, 2018

See a solution process below:

#### Explanation:

First, expand the term in parenthesis by multiplying each term within the parenthesis by the term outside the parenthesis and then use this rule for radicals to combine terms:

$\sqrt{\textcolor{red}{a}} \cdot \sqrt{\textcolor{b l u e}{b}} = \sqrt{\textcolor{red}{a} \cdot \textcolor{b l u e}{b}}$

$\textcolor{red}{3 \sqrt{3}} \left(6 \sqrt{6} - 9 \sqrt{9}\right) \implies$

$\left(\textcolor{red}{3 \sqrt{3}} \cdot 6 \sqrt{6}\right) - \left(\textcolor{red}{3 \sqrt{3}} \cdot 9 \sqrt{9}\right) \implies$

$\left(\textcolor{red}{3} \cdot 6 \cdot \textcolor{red}{\sqrt{3}} \cdot \sqrt{6}\right) - \left(\textcolor{red}{3} \cdot 9 \cdot \textcolor{red}{\sqrt{3}} \cdot \sqrt{9}\right) \implies$

$\left(18 \cdot \sqrt{\textcolor{red}{3} \cdot 6}\right) - \left(27 \cdot \sqrt{\textcolor{red}{3} \cdot 9}\right) \implies$

$18 \sqrt{18} - 27 \sqrt{27}$

$18 \sqrt{9 \cdot 2} - 27 \sqrt{9 \cdot 3}$

$\sqrt{\textcolor{red}{a} \cdot \textcolor{b l u e}{b}} = \sqrt{\textcolor{red}{a}} \cdot \sqrt{\textcolor{b l u e}{b}}$
$\left(18 \sqrt{9} \cdot \sqrt{2}\right) - \left(27 \sqrt{9} \sqrt{3}\right) \implies$
$\left(18 \cdot 3 \cdot \sqrt{2}\right) - \left(27 \cdot 3 \cdot \sqrt{3}\right) \implies$
$54 \sqrt{2} - 81 \sqrt{3}$