DeMoivre's Formula and roots of a complex number

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.

Example

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.

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