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

Step 1: Understanding the Question:
The goal is to find the conjugate of a complex number raised to the third power. We can either compute the power first then conjugate, or conjugate first then compute the power.
Step 2: Key Formula or Approach:
1. Binomial expansion: \( (a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 \).
2. Conjugate property: \( \overline{z^n} = (\overline{z})^n \).
Step 3: Detailed Explanation:

Let \( z = (1 + i)^3 \).

First, expand \( (1 + i)^3 \):
\( (1 + i)^3 = 1^3 + 3(1^2)(i) + 3(1)(i^2) + i^3 \)
\( = 1 + 3i + 3(-1) + (-i) \)
\( = 1 + 3i - 3 - i \)
\( = -2 + 2i \).

Now, find the conjugate \( \overline{z} \).
The conjugate of \( a + bi \) is \( a - bi \).
Therefore, the conjugate of \( -2 + 2i \) is \( -2 - 2i \).

Step 4: Final Answer:
The conjugate is \( -2 - 2i \).