$z$ is a complex number, what is the solution of $z^n=-1$ for $n$ an interger and $\geq 2$

Question

$z$ is a complex number, what is the solution of $z^n=-1$? For $n$ an interger and $\geq 2$. How can we expand $z^n$?

Thanks in advance.

Answer 1

You can simply solve it by $$z^n=-1=\text{cis}(\pi)$$ then one of the roots is $\text{cis}\left({\pi\over n}\right)$ and the other roots are in the form $$\text{cis}\left({(2k+1)\pi\over n}\right)$$ for $k=0,1, \ldots ,(n-1)$.