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

Nov 1, 2015

Convert to polar form first, then...

#### Explanation:

A Complex number in the form $r \left(\cos \theta + i \sin \theta\right)$ has $5$th roots:

$\sqrt[5]{r} \left(\cos \left(\frac{\theta}{5}\right) + i \sin \left(\frac{\theta}{5}\right)\right)$

$\sqrt[5]{r} \left(\cos \left(\frac{\theta + 2 \pi}{5}\right) + i \sin \left(\frac{\theta + 2 \pi}{5}\right)\right)$

$\sqrt[5]{r} \left(\cos \left(\frac{\theta + 4 \pi}{5}\right) + i \sin \left(\frac{\theta + 4 \pi}{5}\right)\right)$

$\sqrt[5]{r} \left(\cos \left(\frac{\theta + 6 \pi}{5}\right) + i \sin \left(\frac{\theta + 6 \pi}{5}\right)\right)$

$\sqrt[5]{r} \left(\cos \left(\frac{\theta + 8 \pi}{5}\right) + i \sin \left(\frac{\theta + 8 \pi}{5}\right)\right)$

Conventionally your original $\theta$ is in the range $\left(- \pi , \pi\right]$ or the range $\left[0 , 2 \pi\right)$ according to your definition of $A r g \left(z\right)$ and the first of these five roots is the Principal Complex fifth root.