Solving a complex equation $z^4=-\sqrt 3 + i$

Question

Find all the solutions of the equation : $$z^4=-\sqrt 3 + i$$

I tried solving this by using the 4th root and then, using sin and cos to find a final answer, but I don't think that this is how you solve it. Can you help?

Answer 1

Hint: $\displaystyle-\sqrt3+i=2\left(-\frac{\sqrt3}2+\frac12i\right)=2\left(\cos\left(\frac{5\pi}6\right)+\sin\left(\frac{5\pi}6\right)i\right)$

Answer 2

Hint:

The general method to determine the $n$-th roots of a complex number $Z$ consists in first writing it in complex exponential notation: $$z=r\,\mathrm e^{i\theta}\qquad(0\le\theta<2\pi, r\in\mathbf R^+)$$ and solving for $u^n=Z$ with $u=\rho\,\mathrm e^{i\varphi}$, leads to $$\begin{cases} \rho^n=r \\ n\varphi\equiv \theta\mod 2\pi \end{cases}\iff \begin{cases} \rho=r^{1/n} \\ \varphi\equiv \frac\theta n\mod \frac{2\pi}n \end{cases}$$