If `omega` is an imaginary cube root of unity, then `(1+omega-omega^(2))^(7)` equals

Options

A. `128 omega`
B. `-128 omega`
C. `128 omega^(2)`
D. `-128 omega^(2)`

Explanation

To solve the problem, we need to evaluate \((1 + \omega - \omega^2)^7\), where \(\omega\) is an imaginary cube root of unity. The cube roots of unity satisfy the equation \(x^3 = 1\), which gives us the roots \(1, \omega, \omega^2\) where \(\omega = e^{2\pi i / 3}\) and \(\omega^2 = e^{-2\pi i / 3}\).

### Step-by-step Solution:

1. **Understanding the Cube Roots of Unity**:
The cube roots of unity satisfy the equation:
\[ 1 + \omega + \omega^2 = 0 \]
From this, we can express \(1 + \omega\) in terms of \(\omega^2\):
\[ 1 + \omega = -\omega^2 \]

2. **Substituting into the Expression**:
We need to evaluate:
\[ 1 + \omega - \omega^2 \]
Using the identity from step 1, we substitute:
\[ 1 + \omega = -\omega^2 \]
Thus:
\[ 1 + \omega - \omega^2 = -\omega^2 - \omega^2 = -2\omega^2 \]

3. **Raising to the Power of 7**:
Now we raise the expression to the power of 7:
\[ (1 + \omega - \omega^2)^7 = (-2\omega^2)^7 \]
This can be simplified as:
\[ (-2)^7 \cdot (\omega^2)^7 \]

4. **Calculating the Powers**:
We calculate \((-2)^7\) and \((\omega^2)^7\):
\[ (-2)^7 = -128 \]
Since \(\omega^3 = 1\), we can reduce the exponent of \(\omega^2\):
\[ (\omega^2)^7 = \omega^{14} = \omega^{3 \cdot 4 + 2} = (\omega^3)^4 \cdot \omega^2 = 1^4 \cdot \omega^2 = \omega^2 \]

5. **Final Result**:
Combining these results, we have:
\[ (1 + \omega - \omega^2)^7 = -128 \cdot \omega^2 \]

Thus, the final answer is:
\[ \boxed{-128 \omega^2} \]