Simplify `(3 + 2i)^(3)`. Also find its conjugate.

Answer

To simplify \((3 + 2i)^{3}\) and find its conjugate, we will follow these steps:

### Step 1: Use the Binomial Theorem
We can use the formula for the cube of a binomial, which states:
\[ (a + b)^{3} = a^{3} + b^{3} + 3ab(a + b) \]
Here, let \(a = 3\) and \(b = 2i\).

### Step 2: Calculate \(a^{3}\) and \(b^{3}\)
Now, we calculate \(a^{3}\) and \(b^{3}\):
\[ a^{3} = 3^{3} = 27 \]
\[ b^{3} = (2i)^{3} = 2^{3} \cdot i^{3} = 8 \cdot (-i) = -8i \]

### Step 3: Calculate \(3ab(a + b)\)
Next, we calculate \(3ab(a + b)\):
\[ ab = 3 \cdot 2i = 6i \]
\[ a + b = 3 + 2i \]
Thus,
\[ 3ab(a + b) = 3 \cdot 6i \cdot (3 + 2i) = 18i(3 + 2i) \]
Now, distribute:
\[ 18i \cdot 3 + 18i \cdot 2i = 54i + 36i^{2} \]
Since \(i^{2} = -1\), we have:
\[ 36i^{2} = 36(-1) = -36 \]
So,
\[ 3ab(a + b) = 54i - 36 \]

### Step 4: Combine All Parts
Now, we combine all parts:
\[ (3 + 2i)^{3} = a^{3} + b^{3} + 3ab(a + b) = 27 - 8i + (54i - 36) \]
Combine the real and imaginary parts:
\[ = (27 - 36) + (-8i + 54i) = -9 + 46i \]

### Step 5: Find the Conjugate
The conjugate of a complex number \(z = a + bi\) is given by \(z^{*} = a - bi\). Here, \(z = -9 + 46i\), so its conjugate is:
\[ z^{*} = -9 - 46i \]

### Final Answer
Thus, the simplified form of \((3 + 2i)^{3}\) is \(-9 + 46i\), and its conjugate is \(-9 - 46i\).

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