How do you simplify: the square root of -32?

Jul 25, 2015

The square roots of $- 32$ are $\pm 4 \sqrt{2} i$

Explanation:

The identity $\sqrt{a b} = \sqrt{a} \sqrt{b}$ is only valid for $a , b \ge 0$.

About the best we can do more generally is $\sqrt{a b} = \pm \sqrt{a} \sqrt{b}$

To see why, consider the invalid reasoning:

$- 1 = \sqrt{- 1} \sqrt{- 1} = \sqrt{- 1 \cdot - 1} = \sqrt{1} = 1$

With this in mind:

$\sqrt{- 32} = \sqrt{{4}^{2} \cdot 2 \cdot - 1} = \pm \sqrt{{4}^{2}} \sqrt{2} \sqrt{- 1} = \pm 4 \sqrt{2} i$