If `1, omega` and `omega^(2)` are the cube roots of unity, then the value of `( 1+ omega) ( 1+ omega^(2)) ( 1+ omega^(4)) ( 1+ omega^(8))` is

Options

A. `-1`
B. 0
C. 1
D. 2

Answer

To solve the problem, we need to evaluate the expression \( (1 + \omega)(1 + \omega^2)(1 + \omega^4)(1 + \omega^8) \), 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 the solutions to the equation \( x^3 = 1 \). The roots are \( 1, \omega, \omega^2 \) where:
- \( \omega = e^{2\pi i / 3} \)
- \( \omega^2 = e^{4\pi i / 3} \)
Additionally, they satisfy the relations:
- \( \omega^3 = 1 \)
- \( 1 + \omega + \omega^2 = 0 \)

2. **Simplifying the Expression**:
We need to simplify the expression \( (1 + \omega)(1 + \omega^2)(1 + \omega^4)(1 + \omega^8) \).

3. **Finding \( \omega^4 \) and \( \omega^8 \)**:
Since \( \omega^3 = 1 \):
- \( \omega^4 = \omega \)
- \( \omega^8 = \omega^2 \)

Thus, we can rewrite the expression as:
\[ (1 + \omega)(1 + \omega^2)(1 + \omega)(1 + \omega^2) \]

4. **Combining Terms**:
This can be simplified to:
\[ (1 + \omega)^2 (1 + \omega^2)^2 \]

5. **Calculating \( (1 + \omega)(1 + \omega^2) \)**:
We can compute:
\[ (1 + \omega)(1 + \omega^2) = 1 + \omega + \omega^2 + \omega \cdot \omega^2 \]
Since \( \omega \cdot \omega^2 = \omega^3 = 1 \), we have:
\[ = 1 + \omega + \omega^2 + 1 = 2 + (\omega + \omega^2) \]
Using \( 1 + \omega + \omega^2 = 0 \), we find:
\[ \omega + \omega^2 = -1 \]
Thus:
\[ (1 + \omega)(1 + \omega^2) = 2 - 1 = 1 \]

6. **Final Calculation**:
Now substituting back:
\[ (1 + \omega)^2 (1 + \omega^2)^2 = 1^2 = 1 \]

### Conclusion:
The value of \( (1 + \omega)(1 + \omega^2)(1 + \omega^4)(1 + \omega^8) \) is \( 1 \).