If `omega` is a complex cube root of unity, then `(1-omega+omega^(2))^(6)+(1-omega^(2)+omega)^(6)=`

Options

A. 0
B. 6
C. 64
D. 128

Answer

To solve the problem, we need to evaluate the expression \((1 - \omega + \omega^2)^6 + (1 - \omega^2 + \omega)^6\), where \(\omega\) is a complex cube root of unity.

### Step-by-Step Solution:

1. **Understanding the properties of cube roots of unity**:
- The cube roots of unity are \(1\), \(\omega\), and \(\omega^2\) where \(\omega = e^{2\pi i / 3}\).
- The properties we need are:
\[ 1 + \omega + \omega^2 = 0 \quad \text{and} \quad \omega^3 = 1. \]

2. **Simplifying \(1 - \omega + \omega^2\)**:
- We can rewrite \(1 - \omega + \omega^2\) using the property \(1 + \omega + \omega^2 = 0\):
\[ 1 - \omega + \omega^2 = 1 + \omega^2 - \omega = (1 + \omega + \omega^2) - 2\omega = -2\omega. \]

3. **Calculating \((1 - \omega + \omega^2)^6\)**:
- Now we can compute:
\[ (1 - \omega + \omega^2)^6 = (-2\omega)^6 = (-2)^6 \cdot \omega^6 = 64 \cdot 1 = 64. \]

4. **Simplifying \(1 - \omega^2 + \omega\)**:
- Similarly, we simplify \(1 - \omega^2 + \omega\):
\[ 1 - \omega^2 + \omega = 1 + \omega - \omega^2 = (1 + \omega + \omega^2) - 2\omega^2 = -2\omega^2. \]

5. **Calculating \((1 - \omega^2 + \omega)^6\)**:
- Now we compute:
\[ (1 - \omega^2 + \omega)^6 = (-2\omega^2)^6 = (-2)^6 \cdot (\omega^2)^6 = 64 \cdot 1 = 64. \]

6. **Final Calculation**:
- Now we can sum the two results:
\[ (1 - \omega + \omega^2)^6 + (1 - \omega^2 + \omega)^6 = 64 + 64 = 128. \]

### Final Answer:
Thus, the value of \((1 - \omega + \omega^2)^6 + (1 - \omega^2 + \omega)^6\) is \(\boxed{128}\).