Question

The complex conjugate of \( (4 - 3i)(2 + 3i)(1 + 4i) \) is:

• A. \( 7 + 74i \)
• B. \( -7 + 74i \)
• C. \( -7 - 74i \)
• D. \( 7 - 74i \)

Hint:

The complex conjugate of \( a + bi \) is \( a - bi \). Use this property to find the conjugate of a product of complex numbers.

Answer 1

The Correct Option is C

We are given the complex expression \( (4 - 3i)(2 + 3i)(1 + 4i) \), and we need to find its complex conjugate. 
Step 1: First, we simplify the product \( (4 - 3i)(2 + 3i) \). Using the distributive property (FOIL method): \[ (4 - 3i)(2 + 3i) = 4(2) + 4(3i) - 3i(2) - 3i(3i) \] \[ = 8 + 12i - 6i - 9i^2. \] Since \( i^2 = -1 \), this becomes: \[ = 8 + 12i - 6i + 9 = 17 + 6i. \]
Step 2: Now, multiply \( (17 + 6i) \) by \( (1 + 4i) \): \[ (17 + 6i)(1 + 4i) = 17(1) + 17(4i) + 6i(1) + 6i(4i) \] \[ = 17 + 68i + 6i + 24i^2. \] Again, using \( i^2 = -1 \): \[ = 17 + 68i + 6i - 24 = -7 + 74i. \] 
Step 3: The result of the multiplication is \( -7 + 74i \). The complex conjugate of a complex number \( a + bi \) is \( a - bi \). Therefore, the complex conjugate of \( -7 + 74i \) is: \[ -7 - 74i. \] Thus, the complex conjugate of \( (4 - 3i)(2 + 3i)(1 + 4i) \) is \( -7 - 74i \).