If `1, omega, 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

Explanation

To solve the problem, we need to evaluate the expression \( (1 + \omega)(1 + \omega^2)(1 + \omega^4)(1 + \omega^8) \), where \( \omega \) is a primitive cube root of unity. The cube roots of unity are \( 1, \omega, \omega^2 \), and they satisfy the equation \( \omega^3 = 1 \) and \( 1 + \omega + \omega^2 = 0 \).

### Step-by-Step Solution:

1. **Identify the cube roots of unity**:
The cube roots of unity are \( 1, \omega, \omega^2 \) where \( \omega = e^{2\pi i / 3} \) and \( \omega^2 = e^{4\pi i / 3} \).

2. **Simplify \( \omega^4 \) and \( \omega^8 \)**:
Since \( \omega^3 = 1 \), we can reduce higher powers of \( \omega \):
- \( \omega^4 = \omega^{3+1} = \omega \)
- \( \omega^8 = \omega^{6+2} = \omega^2 \)

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

3. **Combine like terms**:
This can be simplified to:
\[ (1 + \omega)^2(1 + \omega^2)^2 \]

4. **Calculate \( (1 + \omega)(1 + \omega^2) \)**:
We can use the identity \( 1 + \omega + \omega^2 = 0 \):
\[ 1 + \omega^2 = -\omega \]
Therefore,
\[ (1 + \omega)(1 + \omega^2) = 1 + \omega + \omega^2 + \omega \cdot \omega^2 = 1 + 0 + \omega^3 = 1 + 1 = 2 \]

5. **Square the result**:
Now we have:
\[ (1 + \omega)(1 + \omega^2) = 2 \]
Therefore,
\[ (1 + \omega)^2(1 + \omega^2)^2 = (2)^2 = 4 \]

### Final Answer:
Thus, the value of \( (1 + \omega)(1 + \omega^2)(1 + \omega^4)(1 + \omega^8) \) is \( 4 \).