# How do you simplify the expression sqrt3(2+3sqrt6)?

May 13, 2017

See a solution process below:

#### Explanation:

First, expand the parentheses by multiplying each term within the parentheses by the factor outside the parentheses:

$\textcolor{red}{\sqrt{3}} \left(2 + 3 \sqrt{6}\right) \implies$

$\left(\textcolor{red}{\sqrt{3}} \cdot 2\right) + \left(\textcolor{red}{\sqrt{3}} \cdot 3 \sqrt{6}\right) \implies$

$2 \sqrt{3} + 3 \sqrt{3} \sqrt{6}$

We can now use this rule for radicals to simplify the term on the right:

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

$2 \sqrt{3} + 3 \sqrt{3} \sqrt{6} \implies$

$2 \sqrt{3} + 3 \sqrt{3 \cdot 6} \implies$

$2 \sqrt{3} + 3 \sqrt{18}$

We can use this same rule in reverse to further simplify the term on the right:

$2 \sqrt{3} + 3 \sqrt{18} \implies$

$2 \sqrt{3} + 3 \sqrt{9 \cdot 2} \implies$

$2 \sqrt{3} + 3 \sqrt{9} \sqrt{2} \implies$

$2 \sqrt{3} + \left(3 \cdot 3\right) \sqrt{2} \implies$

$2 \sqrt{3} + 9 \sqrt{2}$