Question

find the all cubic roots of -1+i

Answer 1

Hello, are you still there?

Answer 2

Here is an example if you ever come back...
Say we had the question find all cubic roots of \[\frac{1}{2}+\frac{\sqrt{3}}{2} i\]
\[\text{ Let } z^3=\frac{1}{2}+\frac{\sqrt{3}}{2}i \\ z^3=e^{i (\frac{\pi}{3}+2 \pi n)} \\ \text{ we are looking for 3 roots so n=0,1,2 } \\ z=e^{\frac{1}{3} i (\frac{\pi}{3}+2n \pi)} \\ z=e^{i(\frac{\pi}{9}+\frac{2 n \pi}{3})}\]
\[z=e^{i \frac{\pi}{9}} , z=e^{i(\frac{\pi}{9}+\frac{2\pi}{3})}=e^{i \frac{7\pi}{9}} , z=e^{i(\frac{\pi}{9}+\frac{4\pi}{3})}=e^{i \frac{13 \pi}{9}}\]