Question

Let \( z = 1 - i \). Then the value of \( z^4 \) is equal to

• A. \(4 \)
• B. \(-4 \)
• C. \(1 - i \)
• D. \(1 + i \)
• E. \( i \)

Hint:

For powers of complex numbers, convert to polar form and apply De Moivre’s theorem for faster calculation.

Answer 1

The Correct Option is B

Concept: Use \( i = \sqrt{-1} \) and simplify complex numbers before exponentiation.

Step 1: Simplify given expression.
\[ z = 1 + \frac{1}{i} \] \[ \frac{1}{i} = -i \Rightarrow z = 1 - i \]

Step 2: Convert to polar form (optional shortcut).
\[ z = \sqrt{2}\left(\cos\left(-\frac{\pi}{4}\right) + i\sin\left(-\frac{\pi}{4}\right)\right) \]

Step 3: Apply De Moivre’s theorem.
\[ z^4 = (\sqrt{2})^4 \left[\cos(-\pi) + i\sin(-\pi)\right] \] \[ = 4(-1 + 0i) = -4 \]