# How do I find the fourth root of a complex number?

Sep 2, 2015

If you express your complex number in polar form as $r \left(\cos \theta + i \sin \theta\right)$, then it has fourth roots:

$\alpha = \sqrt[4]{r} \left(\cos \left(\frac{\theta}{4}\right) + i \sin \left(\frac{\theta}{4}\right)\right)$, $i \alpha$, $- \alpha$ and $- i \alpha$

#### Explanation:

Given $a + i b$, let $r = \sqrt{{a}^{2} + {b}^{2}}$, $\theta = \text{atan2} \left(b , a\right)$

Then $a + i b = r \left(\cos \theta + i \sin \theta\right)$

This has one $4 t h$ root $\alpha = \sqrt[4]{r} \left(\cos \left(\frac{\theta}{4}\right) + i \sin \left(\frac{\theta}{4}\right)\right)$

There are three other $4 t h$ roots: $i \alpha$, $- \alpha$ and $- i \alpha$