Question

If \( \left( \frac{1 - i}{1 + i} \right)^{10} = a + ib \), then the values of \( a \) and \( b \) are, respectively:

• A. 1 and 0
• B. 0 and 1
• C. -1 and 0
• D. 0 and -1
• E. 1 and -1

Hint:

To simplify complex expressions, use the conjugate of the denominator and apply exponent rules for powers of \( i \).

Answer 1

The Correct Option is C

We simplify the complex number \( \frac{1 - i}{1 + i} \) by multiplying both the numerator and denominator by the conjugate of the denominator: \[ \frac{1 - i}{1 + i} \times \frac{1 - i}{1 - i} = \frac{(1 - i)^2}{(1 + i)(1 - i)} = \frac{1 - 2i - 1}{1 + 1} = \frac{-2i}{2} = -i \] Now, we compute \( (-i)^{10} \): \[ (-i)^{10} = (i^2)^5 = (-1)^5 = -1 \] Thus, \( a = -1 \) and \( b = 0 \).