` 1 , omega , omega ^(2)` are the cube roots of unity, then the value of `(1 + omega )^(3) - (1+ omega ^(2))^(3)` is

Options

A. `2 omega`
B. 2
C. `-2`
D. 0

Answer

To solve the problem, we need to find the value of \( (1 + \omega)^3 - (1 + \omega^2)^3 \), 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 the solutions to the equation \( z^3 = 1 \). These roots are:
\[ 1, \quad \omega = e^{2\pi i / 3}, \quad \omega^2 = e^{4\pi i / 3} \]
where \( \omega^3 = 1 \) and \( 1 + \omega + \omega^2 = 0 \).

2. **Expressing \( 1 + \omega \) and \( 1 + \omega^2 \)**:
From the identity \( 1 + \omega + \omega^2 = 0 \), we can express:
\[ 1 + \omega = -\omega^2 \]
and
\[ 1 + \omega^2 = -\omega \]

3. **Calculating \( (1 + \omega)^3 \)**:
Substitute \( 1 + \omega \):
\[ (1 + \omega)^3 = (-\omega^2)^3 = -(\omega^2)^3 = -\omega^6 \]
Since \( \omega^3 = 1 \), we have \( \omega^6 = (\omega^3)^2 = 1^2 = 1 \). Therefore:
\[ (1 + \omega)^3 = -1 \]

4. **Calculating \( (1 + \omega^2)^3 \)**:
Substitute \( 1 + \omega^2 \):
\[ (1 + \omega^2)^3 = (-\omega)^3 = -\omega^3 \]
Again, since \( \omega^3 = 1 \):
\[ (1 + \omega^2)^3 = -1 \]

5. **Finding the Final Value**:
Now we can substitute back into the original expression:
\[ (1 + \omega)^3 - (1 + \omega^2)^3 = -1 - (-1) = -1 + 1 = 0 \]

### Conclusion:
The value of \( (1 + \omega)^3 - (1 + \omega^2)^3 \) is \( 0 \).