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

Answer

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

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

Here, we can identify \(a = 3\) and \(b = 7i\). Let's proceed step by step:

### Step 1: Identify \(a\) and \(b\)
Let \(a = 3\) and \(b = 7i\).

### Step 2: Apply the formula
Using the binomial square formula, we have:

\[ (3 - 7i)^2 = 3^2 - 2(3)(7i) + (7i)^2 \]

### Step 3: Calculate \(3^2\)
Calculating \(3^2\):

\[ 3^2 = 9 \]

### Step 4: Calculate \(-2(3)(7i)\)
Calculating \(-2(3)(7i)\):

\[ -2(3)(7i) = -42i \]

### Step 5: Calculate \((7i)^2\)
Calculating \((7i)^2\):

\[ (7i)^2 = 49i^2 \]

Since \(i^2 = -1\), we have:

\[ 49i^2 = 49(-1) = -49 \]

### Step 6: Combine all parts
Now, we combine all the parts together:

\[ (3 - 7i)^2 = 9 - 42i - 49 \]

### Step 7: Simplify the expression
Combining the real parts:

\[ 9 - 49 = -40 \]

Thus, we have:

\[ (3 - 7i)^2 = -40 - 42i \]

### Final Answer
The simplified form of \((3 - 7i)^2\) is:

\[ \boxed{-40 - 42i} \]

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