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

Options

A. 0
B. 32
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 1: Understand the properties of \(\omega\)
The complex cube roots of unity satisfy the equation:
\[ 1 + \omega + \omega^2 = 0 \]
From this, we can derive:
\[ \omega^2 = -1 - \omega \quad \text{and} \quad \omega = -1 - \omega^2 \]

### Step 2: Simplify the first term
Let's simplify \(1 - \omega + \omega^2\):
\[ 1 - \omega + \omega^2 = 1 - \omega + (-1 - \omega) = 1 - \omega - 1 - \omega = -2\omega \]

### Step 3: Simplify the second term
Now simplify \(1 - \omega^2 + \omega\):
\[ 1 - \omega^2 + \omega = 1 - (-1 - \omega) + \omega = 1 + 1 + \omega + \omega = 2 + 2\omega \]

### Step 4: Raise both terms to the power of 6
Now we have:
\[ (1 - \omega + \omega^2)^6 = (-2\omega)^6 = (-2)^6 \cdot \omega^6 \]
Since \(\omega^3 = 1\), we have \(\omega^6 = (\omega^3)^2 = 1^2 = 1\). Therefore:
\[ (-2\omega)^6 = 64 \]

For the second term:
\[ (1 - \omega^2 + \omega)^6 = (2 + 2\omega)^6 = 2^6(1 + \omega)^6 \]
Using \(1 + \omega = -\omega^2\):
\[ (1 + \omega)^6 = (-\omega^2)^6 = (\omega^2)^6 = 1 \]
Thus:
\[ (2 + 2\omega)^6 = 64 \]

### Step 5: Combine the results
Now we can combine the two results:
\[ (1 - \omega + \omega^2)^6 + (1 - \omega^2 + \omega)^6 = 64 + 64 = 128 \]

### Final Answer
Therefore, the final answer is:
\[ \boxed{128} \]