The next theorem provides an important tool to calculate the -th power of a complex number:
Theorem (DeMoivre's Formula)
we have that:
The computation of the -th roots of a complex number is a bit more difficult. That is, given The number is called the n-th root of z.
By considering the polar form of a complex number we have:
So, we have that every complex number admits distinct roots.
We compute the -th roots of the unity. So, we consider , and we want to find
From this example we see that the distinct roots of a complex number represent, in the plane defined via the isomorphism between and , the vertices of a regular polygon.