Question

If $ (1 - 4i)^3 = a + ib $, then the value of $ a $ and $ b $ is

• A. \( -47, 52 \)
• B. \( 49, -74 \)
• C. \( -74, 49 \)
• D. \( -48, -52 \)

Hint:

For complex number expansions, use the binomial expansion and simplify using known powers of \( i \). Pay attention to the real and imaginary parts.

Answer 1

The Correct Option is A

We are given that: \[ (1 - 4i)^3 = a + ib \] To expand \( (1 - 4i)^3 \), we use the binomial expansion. The general form of the expansion is: \[ (a - bi)^3 = a^3 - 3a^2bi + 3ab^2i^2 - b^3i^3 \] Since \( i^2 = -1 \) and \( i^3 = -i \), we get: \[ (1 - 4i)^3 = 1^3 - 3(1)^2(4i) + 3(1)(4i)^2 - (4i)^3 \] First, compute each term: \[ 1^3 = 1 \] \[ -3(1)^2(4i) = -12i \] \[ 3(1)(4i)^2 = 3 \times 1 \times 16(-1) = -48 \] \[ -(4i)^3 = -64i^3 = 64i \] Now, add the real and imaginary parts: \[ (1 - 4i)^3 = (1 - 48) + (-12i + 64i) = -47 + 52i \] Thus, the values of \( a \) and \( b \) are: \[ a = -47, \quad b = 52 \]