Given that `i = sqrt(-1),` what is the sum of the complex numbers `(21 i^(2) -12i) + (3-5i)` ?

Options

A. 7
B. 7i
C. `-18-17i`
D. `24-17i`

Answer

To solve the problem, we need to find the sum of the complex numbers \( (21 i^2 - 12i) + (3 - 5i) \).

### Step-by-Step Solution:

1. **Identify the value of \( i^2 \)**:
We know that \( i = \sqrt{-1} \), which implies that:
\[ i^2 = -1 \]

2. **Substitute \( i^2 \) in the expression**:
Substitute \( i^2 \) in the first part of the expression:
\[ 21 i^2 - 12i = 21(-1) - 12i = -21 - 12i \]

3. **Combine the two complex numbers**:
Now, we can rewrite the original expression:
\[ (-21 - 12i) + (3 - 5i) \]

4. **Combine the real parts**:
Combine the real parts:
\[ -21 + 3 = -18 \]

5. **Combine the imaginary parts**:
Combine the imaginary parts:
\[ -12i - 5i = -17i \]

6. **Write the final result**:
Therefore, the sum of the complex numbers is:
\[ -18 - 17i \]

### Final Answer:
The sum of the complex numbers is \( -18 - 17i \).