`(3 + 2i)^(3) = ?`

Options

A. `9 + 46i`
B. `9 + 46i`
C. `-9 + 46i`
D. none of these

Answer

To solve the expression \((3 + 2i)^3\), we will use the formula for the cube of a binomial, which is given by:

\[ (a + b)^3 = a^3 + b^3 + 3a^2b + 3ab^2 \]

### Step 1: Identify \(a\) and \(b\)
In our case, we have:
- \(a = 3\)
- \(b = 2i\)

### 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 \(3a^2b\)
Next, we calculate \(3a^2b\):
- \(a^2 = 3^2 = 9\)
- \(3a^2b = 3 \cdot 9 \cdot (2i) = 54i\)

### Step 4: Calculate \(3ab^2\)
Now, we calculate \(3ab^2\):
- \(b^2 = (2i)^2 = 4i^2 = 4(-1) = -4\)
- \(3ab^2 = 3 \cdot 3 \cdot (-4) = -36\)

### Step 5: Combine all parts
Now we combine all parts together:
\[ (3 + 2i)^3 = a^3 + b^3 + 3a^2b + 3ab^2 \]
Substituting the values we calculated:
\[ = 27 - 8i + 54i - 36 \]

### Step 6: Simplify the expression
Now, we simplify:
\[ = (27 - 36) + (-8i + 54i) \]
\[ = -9 + 46i \]

### Final Answer
Thus, the final result is:
\[ (3 + 2i)^3 = -9 + 46i \]

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