Write the conjugate of `(6+5i)^(2)`

Answer

To find the conjugate of the complex number \((6 + 5i)^2\), we will follow these steps:

### Step 1: Expand the expression \((6 + 5i)^2\)
Using the formula for the square of a binomial, \((a + b)^2 = a^2 + 2ab + b^2\), we can expand \((6 + 5i)^2\):

\[ (6 + 5i)^2 = 6^2 + 2 \cdot 6 \cdot 5i + (5i)^2 \]

### Step 2: Calculate each term
Now, we calculate each term in the expansion:

1. \(6^2 = 36\)
2. \(2 \cdot 6 \cdot 5i = 60i\)
3. \((5i)^2 = 25i^2 = 25(-1) = -25\)

### Step 3: Combine the terms
Now, we combine the results from Step 2:

\[ (6 + 5i)^2 = 36 + 60i - 25 \]

This simplifies to:

\[ (6 + 5i)^2 = 11 + 60i \]

### Step 4: Find the conjugate
The conjugate of a complex number \(a + bi\) is given by \(a - bi\). Therefore, the conjugate of \(11 + 60i\) is:

\[ 11 - 60i \]

### Final Answer
Thus, the conjugate of \((6 + 5i)^2\) is:

\[ \boxed{11 - 60i} \]

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