Simiplify : `(3-7i)^(2)`

Answer

To simplify \((3 - 7i)^2\), we can use the formula for the square of a binomial, which states that:

\[ (a - b)^2 = a^2 - 2ab + b^2 \]

In this case, let \(a = 3\) and \(b = 7i\).

### Step-by-Step Solution:

1. **Identify \(a\) and \(b\)**:
- Here, \(a = 3\) and \(b = 7i\).

2. **Apply the formula**:
\[ (3 - 7i)^2 = 3^2 - 2(3)(7i) + (7i)^2 \]

3. **Calculate \(3^2\)**:
\[ 3^2 = 9 \]

4. **Calculate \((7i)^2\)**:
\[ (7i)^2 = 49i^2 \]
Since \(i^2 = -1\), we have:
\[ 49i^2 = 49(-1) = -49 \]

5. **Calculate \(-2(3)(7i)\)**:
\[ -2(3)(7i) = -42i \]

6. **Combine all the terms**:
\[ (3 - 7i)^2 = 9 - 42i - 49 \]

7. **Combine the real parts**:
\[ 9 - 49 = -40 \]
So we have:
\[ -40 - 42i \]

### Final Result:
Thus, the simplified form of \((3 - 7i)^2\) is:
\[ -40 - 42i \]