Complex Numbers
Powers of Complex Number — Evaluating Expression
nta_pyq_2026_jan
Grade 11
Question:
If $z=\dfrac{\sqrt{3}}{2}+\dfrac{i}{2}$, $i=\sqrt{-1}$, then $(z^{201}-i)^8$ is equal to
Step-by-Step Solution
Key Concept: $z=e^{i\pi/6}$. $z^{201}=e^{i\cdot201\pi/6}=e^{i\cdot33.5\pi}=e^{i(33\pi+\pi/2)}$. Since $e^{i33\pi}=e^{i\pi}=-1$: $z^{201}=-e^{i\pi/2}=-i$.
$z^{201}=-i$. $(z^{201}-i)^8=(-2i)^8=256$.
Correct Answer: 2