# How do you simplify sqrt(-4/5)?

Dec 27, 2015

$\sqrt{- \frac{4}{5}} = i \sqrt{\frac{4}{5}} = \frac{2 \sqrt{5}}{5} i$

#### Explanation:

By definition and convention, if $x < 0$ then $\sqrt{x} = i \sqrt{- x}$ where $i$ is the imaginary unit.

Also, if $a , b > 0$ then $\sqrt{a b} = \sqrt{a} \sqrt{b}$

So:

$\sqrt{- \frac{4}{5}} = i \sqrt{\frac{4}{5}} = i \sqrt{\frac{4}{25} \cdot 5} = i \sqrt{\frac{4}{25}} \sqrt{5} = i \sqrt{{\left(\frac{2}{5}\right)}^{2}} \sqrt{5}$

$= \frac{2 \sqrt{5}}{5} i$