Q12 (d)

Criteria	Marks
Provides correct solution	3
Finds one correct cube root, or equivalent merit	2
Writes $2-2i$ in exponential form, or equivalent merit	1

Sample answer:

Let $z^3=2-2i$ where $z=re^{i\theta}$, $r\in\mathbb R$.

$r^3e^{3i\theta}=2-2i$
$2-2i=\sqrt8e^{-i\pi/4}$
$\therefore r^3=\sqrt8\text{ and }3\theta=-\frac\pi4\text{ or }\frac{7\pi}4\text{ or }\frac{15\pi}4$
$r=\sqrt2\text{ and }\theta=-\frac\pi{12}\text{ or }\frac{7\pi}{12}\text{ or }\frac{15\pi}{12}$
$\therefore z=\sqrt2e^{-i\pi/12}\text{ or }\sqrt2e^{7i\pi/12}\text{ or }\sqrt2e^{5i\pi/4}$