If `omega` is the imaginary cube root of unity, then what is `( 2 -omega + 2 omega^(2))^(27)` equal to ?

Options

A. `3^(27) omega`
B. `- 3 ^(27) omega^(2)`
C. `3^(27)`
D. `- 3^(27)`

Answer

To solve the problem, we need to evaluate the expression \( (2 - \omega + 2\omega^2)^{27} \), where \( \omega \) is the imaginary cube root of unity. The cube roots of unity satisfy the equation \( 1 + \omega + \omega^2 = 0 \).

### Step-by-Step Solution:

1. **Identify the properties of \( \omega \)**:
- Since \( \omega \) is a cube root of unity, we have:
\[ \omega^3 = 1 \quad \text{and} \quad 1 + \omega + \omega^2 = 0 \]
- From \( 1 + \omega + \omega^2 = 0 \), we can express \( \omega^2 \) in terms of \( \omega \):
\[ \omega^2 = -1 - \omega \]

2. **Substitute \( \omega^2 \) into the expression**:
- We substitute \( \omega^2 \) into the expression \( 2 - \omega + 2\omega^2 \):
\[ 2 - \omega + 2(-1 - \omega) = 2 - \omega - 2 - 2\omega = -3\omega \]

3. **Raise the expression to the power of 27**:
- Now we need to evaluate \( (-3\omega)^{27} \):
\[ (-3\omega)^{27} = (-3)^{27} \cdot \omega^{27} \]

4. **Simplify \( \omega^{27} \)**:
- Since \( \omega^3 = 1 \), we can simplify \( \omega^{27} \):
\[ \omega^{27} = (\omega^3)^9 = 1^9 = 1 \]

5. **Combine the results**:
- Therefore, we have:
\[ (-3\omega)^{27} = (-3)^{27} \cdot 1 = -3^{27} \]

### Final Answer:
Thus, the value of \( (2 - \omega + 2\omega^2)^{27} \) is:
\[ \boxed{-3^{27}} \]