# How do you simplify (3sqrt7 - 6sqrt5)(4sqrt7 + 8sqrt5)?

May 18, 2018

$\left(3 \sqrt{7} - 6 \sqrt{5}\right) \left(4 \sqrt{7} + 8 \sqrt{5}\right) = - 156$

#### Explanation:

Simplifying would mean getting rid of as many square roots as possible. We should, therefore, multiply it out.

First we note that 3 is common in the first parenthesis, and 4 is common in the second. Therefore:

$\left(3 \sqrt{7} - 6 \sqrt{5}\right) \left(4 \sqrt{7} + 8 \sqrt{5}\right)$
$= 3 \left(\sqrt{7} - 2 \sqrt{5}\right) \cdot 4 \left(\sqrt{7} + 2 \sqrt{5}\right)$

You should know that $\left(a - b\right) \left(a + b\right) = {a}^{2} - {b}^{2}$,
which means that $\left(\sqrt{7} - 2 \sqrt{5}\right) \left(\sqrt{7} + 2 \sqrt{5}\right) = 7 - {2}^{2} \cdot 5 = - 13$

Therefore $= 3 \left(\sqrt{7} - 2 \sqrt{5}\right) \cdot 4 \left(\sqrt{7} + 2 \sqrt{5}\right) = - 3 \cdot 4 \cdot 13 = - 156$

Ergo $\left(3 \sqrt{7} - 6 \sqrt{5}\right) \left(4 \sqrt{7} + 8 \sqrt{5}\right) = - 156$