Finding nth Roots of a Complex Number

de Moivre's formula can be used to find the roots of a complex number, not just its powers. Suppose $n$ is a positive integer and a complex number $\omega$ is an $n$th root of $z$ (written $\omega=z^{1/n}$), so that

Let $\omega=\rho(\cos\phi+i\sin\phi)$, and write $z$ in polar form, but crucially using its general argument (not just the principal value), since a single value of $\theta$ would only capture one root:

Since $\omega$ is an $n$th root of $z$, $\omega^n=z$, and by de Moivre's theorem $\omega^n=\rho^n(\cos n\phi+i\sin n\phi)$. So

Comparing moduli and arguments: $\rho^n=r$ and $n\phi=\theta+2k\pi$, so

Therefore the values of $\omega$ are

Although $k$ ranges over all integers, the distinct values occur only for $k=0,1,2,\dots,n-1$ — for $k=n,n+1,n+2,\dots$ the same $n$ roots repeat cyclically (since incrementing $k$ by $n$ adds a full $2\pi$ to the angle). So the $n$th roots of $z=r(\cos\theta+i\sin\theta)$ are

Geometric picture. Writing $z=re^{i(\theta+2k\pi)}$, since $|\rho|=r^{1/n}$ is the same for every $k$, all $n$ roots lie on a circle of radius $r^{1/n}$ centred at the origin. Furthermore, since successive roots' arguments differ by exactly $\dfrac{2\pi}{n}$, the $n$ roots are equally spaced around that circle — the vertices of a regular $n$-gon inscribed in the circle of radius $r^{1/n}$. …