General Solution for a Complex Root Formula

For any complex number $w=a+ib$, the $n$-th roots of $w$ are:

$$z_k=r^{1/n}\left[\cos \left(\frac{\theta +2\pi k}{n}\right)+i\sin \left(\frac{\theta +2\pi k}{n}\right)\right]$$

where:

• $r=\sqrt{a^2+b^2}$ is the magnitude of $w$,
• $\theta =\arg (w)$ is the argument of $w$,
• $k=0,1,2,\dots ,n-1$.