Complex Number Written Answer Question

[Relz]

Explain what is meant by the statement, "the n^th roots of a complex number form a regular n-polygon on the complex plane"

I have an idea of what is meant by this but I can't really explain it.

 

[pka]

Explain what is meant by the statement, "the n^th roots of a complex number form a regular n-polygon on the complex plane"

If \(\displaystyle z\) is a non-zero complex number, write \(\displaystyle \xi=\sqrt[n]{|z|}\exp\left(\dfrac{i\theta}{n}\right)\) where \(\displaystyle \theta=\text{Arg}(z).\)
Let \(\displaystyle \omega=\exp\left(\dfrac{i\pi}{n}\right).\)
The n nth roots of \(\displaystyle z\) are \(\displaystyle \xi\cdot \omega^k,~k=0,1,\cdots n-1\).

Those n roots are equally spaced on a circle centered at the origin with radius \(\displaystyle \sqrt[n]{|z|}\).
As such, they are the vertices of a regular n-polygon on the complex plane.