# How do you simplify sqrt(-2x^5)?

Jul 3, 2015

If $x \le 0$ then $\sqrt{- 2 {x}^{5}} = {x}^{2} \sqrt{- 2 x}$

If $x > 0$ then $- 2 {x}^{5} < 0$ so $\sqrt{- 2 {x}^{5}} \notin \mathbb{R}$

#### Explanation:

If $a , b \ge 0$ then $\sqrt{a b} = \sqrt{a} \sqrt{b}$

If $x \le 0$ then $- 2 x \ge 0$ and ${x}^{4} = {\left({x}^{2}\right)}^{2} \ge 0$

So:

$\sqrt{- 2 {x}^{5}}$

$= \sqrt{- 2 x \cdot {x}^{4}}$

$= \sqrt{- 2 x} \cdot \sqrt{{x}^{4}}$

$= \sqrt{- 2 x} \cdot {x}^{2}$

$= {x}^{2} \sqrt{- 2 x}$