# How do you simplify (2sqrt5 - 4sqrt6) (3sqrt3 + 8sqrt2)?

Mar 5, 2018

(2sqrt5-4sqrt6)(3sqrt3+8sqrt2)=6sqrt15+16sqrt10-36sqrt2-64sqrt3)

#### Explanation:

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

Expand using the FOIL method.

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

color(blue)((2sqrt5*3sqrt3))+color(teal)((2sqrt5*8sqrt2))+color(red)((-4sqrt6*3sqrt3))+color(green)((-4sqrt6*8sqrt2)

Multiply integers and square roots.

$\textcolor{b l u e}{\left(2 \cdot 3 \cdot \sqrt{5} \sqrt{3}\right)} + \textcolor{t e a l}{\left(2 \cdot 8 \cdot \sqrt{5} \sqrt{2}\right)} + \textcolor{red}{\left(- 4 \cdot 3 \cdot \sqrt{6} \sqrt{3}\right)} + \textcolor{g r e e n}{\left(- 4 \cdot 8 \cdot \sqrt{6} \sqrt{2}\right)}$

Simplify.

$\textcolor{b l u e}{\left(6 \sqrt{15}\right)} + \textcolor{t e a l}{\left(16 \sqrt{10}\right)} + \textcolor{red}{\left(- 12 \sqrt{18}\right)} + \textcolor{g r e e n}{\left(- 32 \sqrt{12}\right)}$

The first two sets of parentheses are cannot be simplified further. The second two sets of parentheses need further simplification.

Prime factorize $\textcolor{red}{\left(- 12 \sqrt{18}\right)}$.

color(blue)((6sqrt15))+color(teal)((16sqrt10))+color(red)((-12sqrt(2*3*3))+color(green)((-32sqrt12)

Simplify.

$\textcolor{b l u e}{\left(6 \sqrt{15}\right)} + \textcolor{t e a l}{\left(16 \sqrt{10}\right)} + \textcolor{red}{\left(- 12 \cdot 3 \sqrt{2}\right)} + \textcolor{g r e e n}{\left(- 32 \sqrt{12}\right)}$

Simplify.

$\textcolor{b l u e}{\left(6 \sqrt{15}\right)} + \textcolor{t e a l}{\left(16 \sqrt{10}\right)} + \textcolor{red}{\left(- 36 \sqrt{2}\right)} + \textcolor{g r e e n}{\left(- 32 \sqrt{12}\right)}$

Prime factorize $\textcolor{g r e e n}{\left(- 32 \sqrt{12}\right)}$.

$\textcolor{b l u e}{\left(6 \sqrt{15}\right)} + \textcolor{t e a l}{\left(16 \sqrt{10}\right)} + \textcolor{red}{\left(- 36 \sqrt{2}\right)} + \textcolor{g r e e n}{\left(- 32 \sqrt{2 \cdot 2 \cdot 3}\right)}$

Simplify.

$\textcolor{b l u e}{\left(6 \sqrt{15}\right)} + \textcolor{t e a l}{\left(16 \sqrt{10}\right)} + \textcolor{red}{\left(- 36 \sqrt{2}\right)} + \textcolor{g r e e n}{\left(- 32 \cdot 2 \sqrt{3}\right)}$

Simplify.

$\textcolor{b l u e}{\left(6 \sqrt{15}\right)} + \textcolor{t e a l}{\left(16 \sqrt{10}\right)} + \textcolor{red}{\left(- 36 \sqrt{2}\right)} + \textcolor{g r e e n}{\left(- 64 \sqrt{3}\right)}$

Simplify.

$6 \sqrt{15} + 16 \sqrt{10} - 36 \sqrt{2} - 64 \sqrt{3}$