If `omega` is a complex cube roots of unity, then find the value of the `(1+ omega)(1+ omega^(2))(1+ omega^(4)) (1+ omega^(8))`… to 2n factors.

Explanation

To solve the problem, we need to evaluate the expression \( (1 + \omega)(1 + \omega^2)(1 + \omega^4)(1 + \omega^8) \ldots \) up to \( 2n \) factors, where \( \omega \) is a complex cube root of unity.

### Step-by-Step Solution:

1. **Understanding Cube Roots of Unity**:
The cube roots of unity are the solutions to the equation \( x^3 = 1 \). They are given by:
\[ 1, \quad \omega = e^{2\pi i / 3}, \quad \omega^2 = e^{4\pi i / 3} \]
where \( \omega^3 = 1 \) and \( 1 + \omega + \omega^2 = 0 \).

2. **Identifying the Pattern**:
We can express the terms in the product:
\[ (1 + \omega^k) \text{ for } k = 1, 2, 4, 8, \ldots, 2n \]
Notice that \( \omega^3 = 1 \), so the powers of \( \omega \) will repeat every three terms.

3. **Grouping Terms**:
We can group the terms based on the periodicity of \( \omega \):
- For \( k = 1 \): \( 1 + \omega \)
- For \( k = 2 \): \( 1 + \omega^2 \)
- For \( k = 3 \): \( 1 + \omega^3 = 1 + 1 = 2 \)

Since \( \omega^3 = 1 \), the terms will repeat every three factors. Thus, we can group the terms into sets of three until we reach \( 2n \).

4. **Calculating the Product**:
Each group of three contributes:
\[ (1 + \omega)(1 + \omega^2)(1 + 1) = (1 + \omega)(1 + \omega^2)(2) \]
The product \( (1 + \omega)(1 + \omega^2) \) can be simplified using the identity \( 1 + \omega + \omega^2 = 0 \):
\[ (1 + \omega)(1 + \omega^2) = 1 + \omega + \omega^2 + \omega^3 = 1 + 0 + 1 = 1 \]

5. **Final Calculation**:
Each group of three contributes \( 2 \) (from \( 1 + 1 \)) and \( 1 \) from \( (1 + \omega)(1 + \omega^2) \):
\[ \text{Total product} = 1 \cdot 2^{\frac{2n}{3}} \text{ (number of complete groups of 3)} \]
Therefore, the total number of terms \( 2n \) can be divided by 3, leading to:
\[ \text{Final value} = 2^{\frac{2n}{3}} \]

### Conclusion:
The value of \( (1 + \omega)(1 + \omega^2)(1 + \omega^4)(1 + \omega^8) \ldots \) up to \( 2n \) factors is:
\[ \boxed{2^{\frac{2n}{3}}} \]