How does one plot $z=(-2)^{3/5}$?

Question

How does one plot $z=(-2)^{3/5}$?

I see in Wolfram Alpha:

https://www.wolframalpha.com/input/?i=z%5E5%3D-8

But I don't understand how that plot is created.

Answer 1

In the complex numbers a number has five fifth roots. If we use the polar form, we have $z=re^{i \theta}$ and we want $z^{5}=-8$ so $$\left(re^{i \theta}\right)^{5}=-8\\ r^{5}=8\\r=8^{1/5}=2^{3/5}\\ \left(e^{i\theta}\right)^{5}=-1\\ 5\theta=\pi+2k\pi\\ \theta=\frac \pi 5,\frac {3\pi}5, \frac {5\pi}5, \frac {7\pi}5,\frac {9\pi}5$$ The complex solutions are shown on the page. You can get approximate values by clicking the box.