# How do you simplify 0.5sqrt200 - sqrt(-32)?

Jul 3, 2017

$\frac{5 \sqrt{8}}{2} - 4 \sqrt{- 2}$ or $\frac{5 \sqrt{8}}{2} - 4 i \sqrt{2}$

#### Explanation:

$200$ can be rewritten as $25 x 8$, so $\sqrt{200} = \sqrt{25 x 8} = \sqrt{25} \sqrt{8} = 5 \sqrt{8}$

$5 \sqrt{8} \cdot \frac{1}{2} = \frac{5 \sqrt{8}}{2}$

$\sqrt{- 32}$ is impossible to write as a real number so it is taken as $\frac{5 \sqrt{8}}{2} - \sqrt{- 32}$, however $\sqrt{x} , x < 0 , = \sqrt{\left\mid x \right\mid} \cdot i$.

$- 32 = - 2 \cdot 16 \mathmr{and} 2 \cdot - 16$. Therefore $\sqrt{- 32}$ is equal to:
$\sqrt{- 2 \cdot 16} = \sqrt{- 2} \sqrt{16} = 4 \sqrt{- 2} = 4 \cdot \sqrt{2} i = 4 i \sqrt{2}$, or
$\sqrt{2 \cdot - 16} = \sqrt{2} \sqrt{- 16} = \sqrt{2} 4 i = 4 i \sqrt{2}$

So it can be written as $\frac{5 \sqrt{8}}{2} - 4 i \sqrt{2}$