Express `(2 + 3i)^(3)` in the form `(a + ib).`

Options

A. `(6 + 9i)`
B. `(-4 + 9i)`
C. `(-46 + 9i)`
D. `(-6 + 5i)`

Explanation

To express \( (2 + 3i)^3 \) in the form \( a + ib \), we will use the binomial expansion formula. The formula for the cube of a binomial \( (a + b)^3 \) is given by:

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

In our case, let \( a = 2 \) and \( b = 3i \).

### Step 1: Calculate \( a^3 \) and \( b^3 \)

1. Calculate \( a^3 \):
\[ a^3 = 2^3 = 8 \]

2. Calculate \( b^3 \):
\[ b^3 = (3i)^3 = 27i^3 \]
Since \( i^2 = -1 \), we have \( i^3 = -i \). Thus,
\[ b^3 = 27(-i) = -27i \]

### Step 2: Calculate \( 3a^2b \) and \( 3ab^2 \)

1. Calculate \( 3a^2b \):
\[ 3a^2b = 3(2^2)(3i) = 3(4)(3i) = 36i \]

2. Calculate \( 3ab^2 \):
\[ 3ab^2 = 3(2)(3i)^2 = 3(2)(9i^2) = 3(2)(9)(-1) = -54 \]

### Step 3: Combine all parts

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

### Step 4: Simplify the expression

Combine the real and imaginary parts:
- Real part: \( 8 - 54 = -46 \)
- Imaginary part: \( -27i + 36i = 9i \)

Thus, we have:
\[ (2 + 3i)^3 = -46 + 9i \]

### Final Answer:
The expression \( (2 + 3i)^3 \) in the form \( a + ib \) is:
\[ \boxed{-46 + 9i} \]