Problem Statement
Consider $z^3 = 8i$.
(a) Write $8i$ in form $r e^{i\theta}$. [2 marks]
(b) Find the three roots in exponential form. [3 marks]
(c) Find the area of the triangle formed by these roots. [3 marks]
Verified Solution & Marking Scheme
Part (a)
$$8i = 8 e^{i\pi/2}$$
Part (b)
$$z_k = 2 e^{i(\frac{\pi}{6} + \frac{2k\pi}{3})} \implies z_1 = 2 e^{i\pi/6}, \, z_2 = 2 e^{i5\pi/6}, \, z_3 = 2 e^{-i\pi/2}$$
Part (c)
Equilateral triangle with circumradius $R = 2$:
$$\text{Area} = 3 \left(\frac{1}{2} \times 2^2 \times \sin\frac{2\pi}{3}\right) = 6 \left(\frac{\sqrt{3}}{2}\right) = 3\sqrt{3}$$