# How do you simplify (5sqrt[2] + 3)(sqrt[2] – 2sqrt[3])?

Jul 5, 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.

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

(color(red)(5sqrt(2)) xx color(blue)(sqrt(2))) - (color(red)(5sqrt(2)) xx color(blue)(2sqrt(3)))) + (color(red)(3) xx color(blue)(sqrt(2))) - (color(red)(3) xx color(blue)(2sqrt(3))))

$\left(5 \cdot 4\right) - \left(10 \sqrt{2} \sqrt{3}\right) + 3 \sqrt{2} - 6 \sqrt{3}$

$20 - \left(10 \sqrt{2} \sqrt{3}\right) + 3 \sqrt{2} - 6 \sqrt{3}$

We can use this rule of radicals to complete the simplification:

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

$20 - \left(10 \sqrt{\textcolor{red}{2}} \sqrt{\textcolor{b l u e}{3}}\right) + 3 \sqrt{2} - 6 \sqrt{3}$

$20 - \left(10 \sqrt{\textcolor{red}{2} \cdot \textcolor{b l u e}{3}}\right) + 3 \sqrt{2} - 6 \sqrt{3}$

$20 - 10 \sqrt{6} + 3 \sqrt{2} - 6 \sqrt{3}$