IAL 2022 Oct Q4

(a) Express the complex number $18\sqrt{3}-18\mathrm{i}$ in the form

\begin{align*} r(\cos \theta + \mathrm{i}\sin \theta) \qquad -\pi < \theta \leqslant \pi\\[2mm] \end{align*}

(3)

(b) Solve the equation

\begin{align*} z^4 =\,& 18\sqrt{3}-18\mathrm{i}\\[2mm] \end{align*}

giving your answers in the form $r\mathrm{e}^{\mathrm{i}\theta}$ where $-\pi < \theta \leqslant \pi$

(5)