Question

The conjugate of $(1+i)^3$ is:

• A. $1+2i$
• B. $-2+2i$
• C. $-2-2i$
• D. $1-2i$

Hint:

$(1+i)^2 = 2i$. So, $(1+i)^3 = 2i(1+i) = 2i - 2$.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The conjugate of a complex number \( z = a + bi \) is \( \bar{z} = a - bi \). We first need to expand the cubic expression into the standard form \( a + bi \).
Step 2: Key Formula or Approach:
1. \( (a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 \).
2. \( i^2 = -1, i^3 = -i \).
Step 3: Detailed Explanation:
Expand \( z = (1+i)^3 \): \[ z = 1^3 + 3(1)^2(i) + 3(1)(i)^2 + i^3 \] \[ z = 1 + 3i + 3(-1) + (-i) \] \[ z = 1 + 3i - 3 - i = -2 + 2i \] The conjugate is obtained by changing the sign of the imaginary part: \[ \bar{z} = -2 - 2i \]
Step 4: Final Answer:
The conjugate is \( -2 - 2i \).