# How do you multiply (sqrt5-sqrt6)(sqrt5+sqrt2)?

May 19, 2018

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.

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

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

We can now use this rule for radicals to multiply the radicals:

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

$\sqrt{\textcolor{red}{5} \times \textcolor{b l u e}{5}} + \sqrt{\textcolor{red}{5} \times \textcolor{b l u e}{2}} - \sqrt{\textcolor{red}{6} \times \textcolor{b l u e}{5}} - \sqrt{\textcolor{red}{6} \times \textcolor{b l u e}{2}}$

$\sqrt{25} + \sqrt{10} - \sqrt{30} - \sqrt{12}$

$5 + \sqrt{10} - \sqrt{30} - \sqrt{12}$

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

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

$5 + \sqrt{10} - \sqrt{30} - \sqrt{12}$

$5 + \sqrt{10} - \sqrt{30} - \sqrt{\textcolor{red}{4} \cdot \textcolor{b l u e}{3}}$

$5 + \sqrt{10} - \sqrt{30} - \left(\sqrt{\textcolor{red}{4}} \cdot \sqrt{\textcolor{b l u e}{3}}\right)$

$5 + \sqrt{10} - \sqrt{30} - 2 \sqrt{3}$