# How do I solve sqrt(-8)(sqrt(-3)-sqrt(5))?

Nov 21, 2016

$- 2 \sqrt{6} - \left(2 \sqrt{10}\right) i$
or
$- 2 \sqrt{2} \left(\sqrt{3} + i \sqrt{5}\right)$

#### Explanation:

"Simplify" is perhaps a better word than "solve", since there's no variable to solve for. First, we rewrite using $i$ notation:

$\sqrt{- 8} \left(\sqrt{- 3} - \sqrt{5}\right)$

$= i \sqrt{8} \left(i \sqrt{3} - \sqrt{5}\right)$

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

Next, distribute:

$= 2 {i}^{2} \sqrt{6} - 2 i \sqrt{10}$

Then, we remember that ${i}^{2} = - 1$ by definition, so we can write

$= - 2 \sqrt{6} - \left(2 \sqrt{10}\right) i$

This is now in $x + y i$ form. It can be "simplified" further, however, by factoring out -$2 \sqrt{2}$ from both terms:

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