# How do you simplify (3sqrt6 + 2sqrt10) (2sqrt2 + 3sqrt5)?

Jul 17, 2017

See a solution process below:

#### Explanation:

To multiply these two terms you multiply each individual term in the left parenthesis by each individual term in the right parenthesis.

$\left(\textcolor{red}{3 \sqrt{6}} + \textcolor{red}{2 \sqrt{10}}\right) \left(\textcolor{b l u e}{2 \sqrt{2}} + \textcolor{b l u e}{3 \sqrt{5}}\right)$ becomes:

$\left(\textcolor{red}{3 \sqrt{6}} \times \textcolor{b l u e}{2 \sqrt{2}}\right) + \left(\textcolor{red}{3 \sqrt{6}} \times \textcolor{b l u e}{3 \sqrt{5}}\right) + \left(\textcolor{red}{2 \sqrt{10}} \times \textcolor{b l u e}{2 \sqrt{2}}\right) + \left(\textcolor{red}{2 \sqrt{10}} \times \textcolor{b l u e}{3 \sqrt{5}}\right)$

$\left(6 \textcolor{red}{\sqrt{6}} \textcolor{b l u e}{\sqrt{2}}\right) + \left(9 \textcolor{red}{\sqrt{6}} \textcolor{b l u e}{\sqrt{5}}\right) + \left(4 \textcolor{red}{\sqrt{10}} \textcolor{b l u e}{\sqrt{2}}\right) + \left(6 \textcolor{red}{\sqrt{10}} \textcolor{b l u e}{\sqrt{5}}\right)$

We can next use this rule for radicals to simplify the radical terms:

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

$\left(6 \sqrt{\textcolor{red}{6} \cdot \textcolor{b l u e}{2}}\right) + \left(9 \sqrt{\textcolor{red}{6} \cdot \textcolor{b l u e}{5}}\right) + \left(4 \sqrt{\textcolor{red}{10} \cdot \textcolor{b l u e}{2}}\right) + \left(6 \sqrt{\textcolor{red}{10} \cdot \textcolor{b l u e}{5}}\right)$

$6 \sqrt{12} + 9 \sqrt{30} + 4 \sqrt{20} + 6 \sqrt{50}$

We can now rewrite the terms in the radicals and use the above rule in reverse to further simplify the terms:

$6 \sqrt{4 \cdot 3} + 9 \sqrt{30} + 4 \sqrt{4 \cdot 5} + 6 \sqrt{25 \cdot 2}$

$\left(6 \sqrt{4} \sqrt{3}\right) + 9 \sqrt{30} + \left(4 \sqrt{4} \sqrt{5}\right) + \left(6 \sqrt{25} \sqrt{2}\right)$

$\left(6 \cdot 2 \sqrt{3}\right) + 9 \sqrt{30} + \left(4 \cdot 2 \sqrt{5}\right) + \left(6 \cdot 5 \sqrt{2}\right)$

$12 \sqrt{3} + 9 \sqrt{30} + 8 \sqrt{5} + 30 \sqrt{2}$