# How do you multiply (2sqrt7+sqrt5)(sqrt3+sqrt2)(2sqrt7-sqrt5)?

Jun 5, 2015

It's easiest to multiply $\left(2 \sqrt{7} + \sqrt{5}\right) \left(2 \sqrt{7} - \sqrt{5}\right)$ first...

$\left(2 \sqrt{7} + \sqrt{5}\right) \left(2 \sqrt{7} - \sqrt{5}\right)$ is of the form

$\left(a + b\right) \left(a - b\right) = {a}^{2} - {b}^{2}$ with $a = 2 \sqrt{7}$ and $b = \sqrt{5}$

So:

$\left(2 \sqrt{7} + \sqrt{5}\right) \left(2 \sqrt{7} - \sqrt{5}\right)$

$= {\left(2 \sqrt{7}\right)}^{2} - {\sqrt{5}}^{2} = \left(4 \cdot 7\right) - 5 = 28 - 5 = 23$

Then

$\left(2 \sqrt{7} + \sqrt{5}\right) \left(\sqrt{3} + \sqrt{2}\right) \left(2 \sqrt{7} - \sqrt{5}\right)$

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

$= 23 \left(\sqrt{3} + \sqrt{2}\right) = 23 \sqrt{3} + 23 \sqrt{2}$