Solving equations of the form $x^n+a=0$.

Question

I was wondering if there is a quick method to find all solutions to equations of the form $x^n+a=0$. In many cases I have seen solutions to such equations to be nth roots multiplied by nth roots of unity, what is the general method to solve such problems?

Answer 1

$x^n+a=0 \implies x^n=-a \implies x=\zeta_n\sqrt[n]{-a}$ where $n=0,1, \dots, n-1$. Here, $\zeta_n$ is the primitive $n$-th root of unity you're talking about in your question.