Question

If \( z = \dfrac{\sqrt{3}}{2} + \dfrac{i}{2} \), then the value of \[ \left(z^{201} - i\right)^8 \] is:

• A. 0
• B. 256
• C. 1
• D. \( -1 \)

Hint:

When powers of complex numbers are involved, always convert to polar form and reduce angles modulo \(2\pi\).

Answer 1

The Correct Option is B

Step 1: Write \( z \) in polar form.
\[ z = \cos\frac{\pi}{6} + i\sin\frac{\pi}{6} \] Hence, \[ z = \operatorname{cis}\!\left(\frac{\pi}{6}\right) \]
Step 2: Evaluate \( z^{201} \).
Using De Moivre’s theorem, \[ z^{201} = \operatorname{cis}\!\left(\frac{201\pi}{6}\right) = \operatorname{cis}\!\left(\frac{67\pi}{2}\right) \] \[ = \operatorname{cis}\!\left(\frac{3\pi}{2}\right) = -i \]
Step 3: Substitute into the expression.
\[ z^{201} - i = -i - i = -2i \]
Step 4: Compute the final value.
\[ (-2i)^8 = 2^8 = 256 \]