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

Step 1: Understanding the Concept:
The problem involves simplifying a complex number and then calculating its power. Key properties of the imaginary unit `i` are used.
Step 2: Key Formula or Approach:
1. Simplify the complex number `z`. Remember that \(\frac{1}{i} = -i\).
2. Calculate z\(^2\).
3. Calculate z\(^4\) by squaring the result of z\(^2\).
Key properties: `i\(^2\) = -1`.
Step 3: Detailed Explanation:
First, simplify `z`. To rationalize the fraction \(\frac{1}{i}\), multiply the numerator and denominator by `i`:
\[ \frac{1}{i} = \frac{1 \cdot i}{i \cdot i} = \frac{i}{i^2} = \frac{i}{-1} = -i \] So, the complex number `z` is:
\[ z = 1 + (-i) = 1 - i \] Next, we calculate z\(^2\):
\[ z^2 = (1 - i)^2 = 1^2 - 2(1)(i) + i^2 \] \[ z^2 = 1 - 2i - 1 = -2i \] Finally, we calculate z\(^4\), which is (z\(^2\))\(^2\):
\[ z^4 = (z^2)^2 = (-2i)^2 = (-2)^2 \cdot i^2 \] \[ z^4 = 4 \cdot (-1) = -4 \] Step 4: Final Answer:
The value of z\(^4\) is -4.