Problem Statement
Let $w = -8 + 8\sqrt{3}i$.
**(a)** [3 marks] Express $w$ in the modulus-argument form $r e^{i\theta}$, where $r > 0$ and $-\pi < \theta \le \pi$.
**(b)** [4 marks] Solve the equation $z^4 = w$, expressing all roots in the form $R e^{i\phi}$ where $-\pi < \phi \le \pi$.
Verified Solution & Marking Scheme
Part (a) [M1]
Calculate modulus: $r = \sqrt{(-8)^2 + (8\sqrt{3})^2} = \sqrt{64 + 192} = \sqrt{256} = 16$
Part (a) [M1]
Identify quadrant: $x < 0, y > 0$ (Quadrant II). $\theta = \pi - \arctan\left(\frac{8\sqrt{3}}{8}\right) = \pi - \frac{\pi}{3} = \frac{2\pi}{3}$
Part (a) [A1]
$w = 16 e^{i \frac{2\pi}{3}}$
Part (b) [M1]
Apply De Moivre theorem for roots: $z_k = 16^{1/4} e^{i \left(\frac{2\pi/3 + 2k\pi}{4}\right)}$ for $k = 0, 1, 2, 3$
Part (b) [A1]
Modulus of each root is $R = 16^{1/4} = 2$
Part (b) [A1]
Calculate arguments: for $k = 0: \phi = \frac{2\pi}{12} = \frac{\pi}{6}$; for $k = 1: \phi = \frac{\pi}{6} + \frac{\pi}{2} = \frac{2\pi}{3}$
Part (b) [A1]
For $k = 2: \phi = \frac{2\pi}{3} + \frac{\pi}{2} = \frac{7\pi}{6} \equiv -\frac{5\pi}{6}$; for $k = 3: \phi = -\frac{5\pi}{6} + \frac{\pi}{2} = -\frac{\pi}{3}$.
Roots: $2e^{i\pi/6}, 2e^{i2\pi/3}, 2e^{-i5\pi/6}, 2e^{-i\pi/3}$