Question

Find all the roots of \[z^{3}=\sqrt{3}+i\]

Answer 1

\[z^3 = \sqrt3 + i \]
\[\theta=\tan^{-1}\frac{y}{x} \implies \frac{1}{\sqrt3}\]

Answer 2

\[\left\{z\to \left(i+\sqrt{3}\right)^{1/3},z\to -(-1)^{1/3} \left(i+\sqrt{3}\right)^{1/3},z\to (-1)^{2/3} \left(i+\sqrt{3}\right)^{1/3}\right\} \]