If `1 , omega, omega^(2)` are cube roots of unity then the value of `(3 + 5omega+3omega^(2))^(3)` is

Options

A. 6
B. 8
C. 12
D. 16

Answer

To solve the problem, we need to find the value of \( (3 + 5\omega + 3\omega^2)^3 \), where \( 1, \omega, \omega^2 \) are the cube roots of unity.

### Step-by-Step Solution:

1. **Understanding Cube Roots of Unity**:
The cube roots of unity are defined as:
\[ 1 + \omega + \omega^2 = 0 \]
From this, we can derive that:
\[ \omega + \omega^2 = -1 \]

2. **Rearranging the Expression**:
We need to simplify the expression \( 3 + 5\omega + 3\omega^2 \). We can rewrite it using the identity we derived:
\[ 3 + 5\omega + 3\omega^2 = 3 + 5\omega + 3(-1 - \omega) = 3 + 5\omega - 3 - 3\omega \]
This simplifies to:
\[ 5\omega - 3\omega = 2\omega \]

3. **Calculating the Cube**:
Now we need to find the cube of \( 2\omega \):
\[ (2\omega)^3 = 2^3 \cdot \omega^3 = 8 \cdot \omega^3 \]
Since \( \omega^3 = 1 \), we have:
\[ 8 \cdot 1 = 8 \]

4. **Final Answer**:
Therefore, the value of \( (3 + 5\omega + 3\omega^2)^3 \) is:
\[ \boxed{8} \]