If ` 1 , omega and omega^(2)` are the cube roots of unity, then the value of `[ 1 -omega + omega^(2)][1 - omega^(2) + omega^(4)]` . . . . upto 8 terms is

Options

A. `2^(6)`
B. ` 2^(10)`
C. `2^(12)`
D. `2^(8)`

Answer

To solve the problem, we need to evaluate the expression \([1 - \omega + \omega^2][1 - \omega^2 + \omega^4]\) up to 8 terms, where \(1\), \(\omega\), and \(\omega^2\) 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:
\[ \omega^2 = -1 - \omega \quad \text{and} \quad \omega = -1 - \omega^2 \]
Also, we know that \(\omega^3 = 1\).

2. **Evaluating the First Term**:
The first term is:
\[ 1 - \omega + \omega^2 \]
Substituting \(\omega^2 = -1 - \omega\):
\[ 1 - \omega + (-1 - \omega) = 1 - \omega - 1 - \omega = -2\omega \]

3. **Evaluating the Second Term**:
The second term is:
\[ 1 - \omega^2 + \omega^4 \]
Since \(\omega^4 = \omega\) (because \(\omega^3 = 1\)):
\[ 1 - \omega^2 + \omega = 1 - (-1 - \omega) + \omega = 1 + 1 + \omega + \omega = 2 + 2\omega \]

4. **Continuing the Pattern**:
The pattern continues with the terms alternating between \(-2\omega\) and \(2 + 2\omega\). We can observe that:
- The 1st term is \(-2\omega\)
- The 2nd term is \(2 + 2\omega\)
- The 3rd term is \(-2\omega\)
- The 4th term is \(2 + 2\omega\)
- The 5th term is \(-2\omega\)
- The 6th term is \(2 + 2\omega\)
- The 7th term is \(-2\omega\)
- The 8th term is \(2 + 2\omega\)

5. **Calculating the Product**:
The product of these 8 terms can be expressed as:
\[ (-2\omega)^4 \cdot (2 + 2\omega)^4 \]
Simplifying each part:
\[ (-2\omega)^4 = 16\omega^4 = 16\omega \]
\[ (2 + 2\omega)^4 = 2^4(1 + \omega)^4 = 16(1 + \omega)^4 \]
Since \(1 + \omega = -\omega^2\):
\[ (1 + \omega)^4 = (-\omega^2)^4 = \omega^8 = \omega^2 \]
Thus:
\[ (2 + 2\omega)^4 = 16\omega^2 \]

6. **Final Product**:
Now, combining the results:
\[ 16\omega \cdot 16\omega^2 = 256\omega^3 = 256 \cdot 1 = 256 \]
Since \(256 = 2^8\), the final answer is:
\[ \boxed{2^8} \]