Question

Which expression is a cube root of -1+i√3?

Answer 1

Hint:
I can rewrite your complex number, as follows:
\[z = - 1 + i\sqrt 3 = 2\left\{ {\cos \left( {\frac{{2\pi }}{3} + 2k\pi } \right) + i\sin \left( {\frac{{2\pi }}{3} + 2k\pi } \right)} \right\},\quad k \in \mathbb{Z}\]

Answer 2

so we have:
\[\sqrt[3]{z} = \sqrt[3]{2}\left\{ {\cos \left( {\frac{{2\pi }}{9} + \frac{{2k\pi }}{3}} \right) + i\sin \left( {\frac{{2\pi }}{9} + \frac{{2k\pi }}{3}} \right)} \right\},\quad k = 0,\;1,\;2\]