Write the conjugate of
`(3+4i)^(2)`

Answer

To find the conjugate of the complex number \((3 + 4i)^2\), we will follow these steps:

### Step 1: Expand the expression \((3 + 4i)^2\)
Using the formula \((a + b)^2 = a^2 + 2ab + b^2\), where \(a = 3\) and \(b = 4i\):

\[ (3 + 4i)^2 = 3^2 + 2 \cdot 3 \cdot 4i + (4i)^2 \]

### Step 2: Calculate each term
- \(3^2 = 9\)
- \(2 \cdot 3 \cdot 4i = 24i\)
- \((4i)^2 = 16i^2 = 16(-1) = -16\) (since \(i^2 = -1\))

### Step 3: Combine the results
Now, combine all the terms:

\[ (3 + 4i)^2 = 9 + 24i - 16 \]

### Step 4: Simplify the expression
Combine the real parts:

\[ 9 - 16 + 24i = -7 + 24i \]

### Step 5: Find the conjugate
The conjugate of a complex number \(a + bi\) is \(a - bi\). Therefore, the conjugate of \(-7 + 24i\) is:

\[ -7 - 24i \]

### Final Answer
The conjugate of \((3 + 4i)^2\) is \(-7 - 24i\).

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