If `1, omega, omega^2` be the cube roots of unity, then the value of `(1 - omega + omega^2)^(5) + (1 + omega - omega^2)^5` is :

Options

A. a) 0
B. b )16
C. c) 32
D. d) 64

Answer

To solve the problem, we need to evaluate the expression \( (1 - \omega + \omega^2)^5 + (1 + \omega - \omega^2)^5 \), where \( 1, \omega, \omega^2 \) are the cube roots of unity.

### Step-by-Step Solution:

1. **Understanding Cube Roots of Unity**:
The cube roots of unity are defined as:
\[ 1 + \omega + \omega^2 = 0 \]
This implies that:
\[ \omega + \omega^2 = -1 \]

2. **Simplifying the First Term**:
We start with the first term:
\[ 1 - \omega + \omega^2 \]
Substituting \( \omega + \omega^2 = -1 \):
\[ 1 - \omega + \omega^2 = 1 - \omega + (-1 - \omega) = 1 - \omega - 1 - \omega = -2\omega \]

3. **Calculating the First Power**:
Now we raise this to the power of 5:
\[ (1 - \omega + \omega^2)^5 = (-2\omega)^5 = -32\omega^5 \]
Since \( \omega^3 = 1 \), we have:
\[ \omega^5 = \omega^{3+2} = \omega^2 \]
Thus:
\[ (1 - \omega + \omega^2)^5 = -32\omega^2 \]

4. **Simplifying the Second Term**:
Now, we simplify the second term:
\[ 1 + \omega - \omega^2 \]
Again substituting \( \omega + \omega^2 = -1 \):
\[ 1 + \omega - \omega^2 = 1 + \omega + (-1 - \omega) = 1 + \omega - 1 - \omega = 0 \]
Thus:
\[ (1 + \omega - \omega^2)^5 = 0^5 = 0 \]

5. **Combining Both Terms**:
Now we combine both results:
\[ (1 - \omega + \omega^2)^5 + (1 + \omega - \omega^2)^5 = -32\omega^2 + 0 = -32\omega^2 \]

6. **Final Evaluation**:
To find the final value, we note that \( \omega^2 \) is one of the cube roots of unity. We can express the final result in terms of \( \omega \):
\[ -32\omega^2 \]

### Conclusion:
The value of \( (1 - \omega + \omega^2)^5 + (1 + \omega - \omega^2)^5 \) is \( -32\omega^2 \).