# How do you simplify (4 sqrt7 - 8 sqrt 5) (5sqrt7 + 10sqrt 5)?

Jun 9, 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.

$\left(\textcolor{red}{4 \sqrt{7}} - \textcolor{red}{8 \sqrt{5}}\right) \left(\textcolor{b l u e}{5 \sqrt{7}} + \textcolor{b l u e}{10 \sqrt{5}}\right)$ becomes:

$\left(\textcolor{red}{4 \sqrt{7}} \times \textcolor{b l u e}{5 \sqrt{7}}\right) + \left(\textcolor{red}{4 \sqrt{7}} \times \textcolor{b l u e}{10 \sqrt{5}}\right) - \left(\textcolor{red}{8 \sqrt{5}} \times \textcolor{b l u e}{5 \sqrt{7}}\right) - \left(\textcolor{red}{8 \sqrt{5}} \times \textcolor{b l u e}{10 \sqrt{5}}\right)$

$\left(20 \times 7\right) + \left(40 \left(\textcolor{red}{\sqrt{7}} \times \textcolor{b l u e}{\sqrt{5}}\right)\right) - \left(40 \left(\textcolor{red}{\sqrt{5}} \times \textcolor{b l u e}{\sqrt{7}}\right)\right) - \left(80 \times 5\right)$

$140 + \left(40 \left(\textcolor{red}{\sqrt{7}} \times \textcolor{b l u e}{\sqrt{5}}\right)\right) - \left(40 \left(\textcolor{red}{\sqrt{5}} \times \textcolor{b l u e}{\sqrt{7}}\right)\right) - 400$

We can now group and combine like terms:

$140 - 400 + \left(40 \left(\textcolor{red}{\sqrt{7}} \times \textcolor{b l u e}{\sqrt{5}}\right)\right) - \left(40 \left(\textcolor{red}{\sqrt{5}} \times \textcolor{b l u e}{\sqrt{7}}\right)\right)$

$\left(140 - 400\right) + \left(40 - 40\right) \left(\textcolor{red}{\sqrt{7}} \times \textcolor{b l u e}{\sqrt{5}}\right)$

$- 260 + 0 \left(\textcolor{red}{\sqrt{7}} \times \textcolor{b l u e}{\sqrt{5}}\right)$

$- 260 + 0$

$- 260$