Question

If $(-\sqrt{3} - i)^{30} = -4^k$, then the value of $k$ is

• A. $15$
• B. $20$
• C. $25$
• D. $30$
• E. $60$

Hint:

Always reduce complex powers using polar form and De Moivre’s theorem.

Answer 1

The Correct Option is A

Step 1: Convert to polar form.
\[ z = -\sqrt{3} - i \] Magnitude: \[ r = \sqrt{3 + 1} = 2 \] Argument: \[ \tan \theta = \frac{-1}{-\sqrt{3}} = \frac{1}{\sqrt{3}} \Rightarrow \theta = \frac{\pi}{6} \] Since point lies in 3rd quadrant: \[ \theta = \pi + \frac{\pi}{6} = \frac{7\pi}{6} \]

Step 2: Apply De Moivre's theorem. \[ z^{30} = 2^{30} \left[\cos\left(30 \cdot \frac{7\pi}{6}\right) + i\sin\left(30 \cdot \frac{7\pi}{6}\right)\right] \] \[ = 2^{30} \left[\cos(35\pi) + i\sin(35\pi)\right] \] \[ \cos(35\pi) = -1, \sin(35\pi)=0 \] \[ z^{30} = -2^{30} \]

Step 3: Compare with given expression. \[ -4^k = - (2^2)^k = -2^{2k} \] \[ 2^{30} = 2^{2k} \Rightarrow 2k = 30 \Rightarrow k = 15 \] \[ \boxed{15} \]