# How do you multiply (4sqrt5 + 3sqrt5)(2sqrt5 - 4sqrt2)?

Oct 8, 2015

The same way you would multiply non-radicals. Break it down into simpler steps.

The front numbers multiply, then the outer, then the inner, then the last numbers. You might have learned this as "FOIL".

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

$= \left(8 \sqrt{5} \cdot \sqrt{5}\right) + \left(- 16 \sqrt{5} \cdot \sqrt{2}\right) + \left(6 \sqrt{5} \cdot \sqrt{5}\right) + \left(- 12 \sqrt{5} \cdot \sqrt{2}\right)$

$= 8 \sqrt{5} \cdot \sqrt{5} - 16 \sqrt{5} \cdot \sqrt{2} + 6 \sqrt{5} \cdot \sqrt{5} - 12 \sqrt{5} \cdot \sqrt{2}$

$= 40 - 16 \sqrt{10} + 30 - 12 \sqrt{10}$

$= \textcolor{b l u e}{70 - 28 \sqrt{10}}$

Alternatively you could write this as:

$= 7 \left(\sqrt{10} \sqrt{10} - 4 \sqrt{10}\right)$

$= 7 \left(2 \sqrt{5} \textcolor{g r e e n}{\sqrt{5}} - 4 \sqrt{2} \textcolor{g r e e n}{\sqrt{5}}\right)$

$= \textcolor{b l u e}{7 \sqrt{5} \left(2 \sqrt{5} - 4 \sqrt{2}\right)}$