Question

The conjugate of \( (1+i)^3 \) is

• A. \(1+2i\)
• B. \(-2+2i\)
• C. \(-2-2i\)
• D. \(1-2i\)

Hint:

First simplify the complex expression completely, then change the sign of the imaginary part to find its conjugate.

Answer 1

The Correct Option is C

Concept: The conjugate of a complex number \(a+bi\) is \(a-bi\).

Step 1: First calculate: \[ (1+i)^3 \]

Step 2: Find \((1+i)^2\). \[ (1+i)^2=1+2i+i^2 \] Since \(i^2=-1\), \[ (1+i)^2=1+2i-1=2i \]

Step 3: Now multiply by \((1+i)\). \[ (1+i)^3=(1+i)^2(1+i) \] \[ =2i(1+i) \] \[ =2i+2i^2 \] \[ =2i-2 \] \[ =-2+2i \]

Step 4: Find the conjugate. \[ \overline{-2+2i}=-2-2i \] Therefore, \[ \boxed{-2-2i} \]