I was given this question:
Use de Moivre's theorem to derive a formula for the $4^{th}$ roots of 8.
As far as my understanding of this theorem goes, it is only applicaple to complex numbers. How am I supposed to use it for 8?
My initial thought was use 8 to make $z = 8 + i0$. Determining the polar form:
$$z = 8(\cos \pi + i \sin \pi )$$
from the polar form a I then get that
$$z^4 = 8^4 (\cos (4\pi) + i \sin(4\pi))$$
This is as far as I can go. I only recently started working with de Moivre's theorem so I'm not sure about my calculations and would appreciate any clarification on how to go about answering this question.
Best Answer
Even though you were told to use the estimable Mr. de Moivre, it’s easiest without:
You know that if $\rho$ is one fourth root of a number, the others are $-\rho$ and $\pm i\rho$. Since one fourth root of $8$ is the real number $2^{3/4}$, you have them all.