What does taking the $n^{\text{th}}$ root of a complex number geometrically mean?

Question

What are the geometrical implications of taking the $n^{\text{th}}$ root of a complex number, say $3+4i$.

What is the geometrical implication of $\sqrt[n] {3+4i}$ in the complex plane?

Answer 1

When a complex number $z$ is written in polar form $z = e^r (\cos(\theta) + i \sin(\theta))$, then the polar forms of the $n$ different $n^{th}$ roots of $z$ are all obtained by multiplying $e^{r/n}$ by $\cos((\theta + 2 \pi k)/n) + i \sin((\theta + 2 \pi k)/n)$, $k=0,…,n-1$.

In other words: take the $n$th root of the radius $e^r$; and divide the angle $\theta$ (and all equivalent angles by adding multiples of $2\pi$) by $n$.