`(3 + omega + 3 omega ^(2) ) ^(4)` equals

Options

A. 16
B. `16 omega`
C. `10 omega^(2)`
D. none of these

Answer

To solve the expression \( (3 + \omega + 3 \omega^2)^4 \), where \( \omega \) is a cube root of unity, we can follow these steps:

### Step 1: Understand the properties of cube roots of unity
The cube roots of unity are given by:
- \( 1 \)
- \( \omega \)
- \( \omega^2 \)

They satisfy the following properties:
- \( 1 + \omega + \omega^2 = 0 \)
- \( \omega^3 = 1 \)

### Step 2: Simplify the expression inside the parentheses
We need to simplify \( 3 + \omega + 3\omega^2 \):
\[ 3 + \omega + 3\omega^2 = 3 + \omega + 3(-1 - \omega) \quad \text{(using } \omega^2 = -1 - \omega\text{)} \]
\[ = 3 + \omega - 3 - 3\omega = -2\omega \]

### Step 3: Raise the simplified expression to the power of 4
Now we raise \( -2\omega \) to the power of 4:
\[ (-2\omega)^4 = (-2)^4 \cdot \omega^4 = 16 \cdot \omega^4 \]

### Step 4: Simplify \( \omega^4 \)
Since \( \omega^3 = 1 \), we can express \( \omega^4 \) as:
\[ \omega^4 = \omega^3 \cdot \omega = 1 \cdot \omega = \omega \]

### Step 5: Combine the results
Now substituting back, we have:
\[ (-2\omega)^4 = 16 \cdot \omega \]

### Final Result
Thus, the final result is:
\[ (3 + \omega + 3\omega^2)^4 = 16\omega \]

---