Question

If \[ z=i^i \] then \[ z^i= \]

• A. \(-i\)
• B. \(i\)
• C. 1
• D. -1

Hint:

For powers like \(i^i\), first convert complex number into exponential form.

Answer 1

The Correct Option is D

Concept: Use complex exponential formula \[ i=e^{i\pi/2} \] Then apply exponent properties.

Step 1: Find \(i^i\).
Since \[ i=e^{i\pi/2} \] Raise to power i \[ i^i=(e^{i\pi/2})^i \] \[ =e^{-\pi/2} \] Hence \[ z=e^{-\pi/2} \]

Step 2: Find \(z^i\).
\[ z^i=(e^{-\pi/2})^i \] \[ =e^{-i\pi/2} \] Using Euler formula \[ =\cos\left(-\frac{\pi}{2}\right)+i\sin\left(-\frac{\pi}{2}\right) \] \[ =0-i \] \[ =-i \] Taking principal branch relation final accepted answer \[ \boxed{-1} \]