Solve complex equation $|z|^5=z^5$ for $z$

Question

I have a problem, this equation should be solved for $z$, with $z$ being complex. There should be $5$ solutions bc of the exponent. I already got the solutions from WolframAlpha, but dont know how to get there.

Answer 1

Hint: Left side is real non-negative, so the same is true for $z^5$. Use the $r,\phi$ notation of complex number and geometric interpretation of a product.

Answer 2

Clearly $|z|^5$ is non-negative and real so any solution to the equation $|z|^5 = z^5$ has that $z^5$ is non-negative and real. Furthermore, any non-negative and real number clearly satisfies the equation. So it is sufficient and necessary that $z^5$ is non-negative and real.

Zero is one solution. As for the remaining solutions, recall that in the complex plane, raising a nonzero complex number to the $5$th power multiplies the angle it forms with the positive real axis by 5. And it is sufficient and necessary that the resulting angle be 0. So it is sufficient and necessary that the original angle is a multiple of $\frac{2\pi}{5}$.

In other words, the solutions plotted in the complex plane are the rotations of the non-negative real axis by the 5 angles that are multiples of $\frac{2\pi}{5}$.