If `alpha and beta` are the complex cube roots of unity, then what is the vlaue of `(1+alpha)(1+beta)(1+alpha^(2))(1+beta^(2))`?

Options

A. `-1`
B. 0
C. 1
D. 4

Explanation

To solve the problem, we need to find the value of the expression \((1+\alpha)(1+\beta)(1+\alpha^2)(1+\beta^2)\), where \(\alpha\) and \(\beta\) are the complex cube roots of unity.

### Step-by-Step Solution:

1. **Identify the Cube Roots of Unity**:
The complex cube roots of unity are given by:
\[ 1, \quad \omega = e^{2\pi i / 3}, \quad \omega^2 = e^{-2\pi i / 3} \]
where \(\omega\) and \(\omega^2\) are the non-real cube roots of unity.

2. **Properties of Cube Roots of Unity**:
The cube roots of unity satisfy the equation:
\[ 1 + \omega + \omega^2 = 0 \]
and also:
\[ \omega^3 = 1 \]
Therefore, we can denote \(\alpha = \omega\) and \(\beta = \omega^2\).

3. **Substituting Values**:
We need to evaluate:
\[ (1+\omega)(1+\omega^2)(1+\omega^2)(1+\omega) \]
This can be rewritten as:
\[ (1+\omega)(1+\omega^2)^2 \]

4. **Calculating \(1+\omega\) and \(1+\omega^2\)**:
We know:
\[ 1+\omega = 1 + e^{2\pi i / 3} \quad \text{and} \quad 1+\omega^2 = 1 + e^{-2\pi i / 3} \]

5. **Using the Properties**:
From the property \(1 + \omega + \omega^2 = 0\), we can express:
\[ 1 + \omega^2 = -\omega \]
and
\[ 1 + \omega = -\omega^2 \]

6. **Substituting Back**:
Now substituting these into our expression:
\[ (1+\omega)(1+\omega^2) = (-\omega^2)(-\omega) = \omega^3 = 1 \]

7. **Final Calculation**:
Therefore, we have:
\[ (1+\alpha)(1+\beta)(1+\alpha^2)(1+\beta^2) = (1+\omega)(1+\omega^2)(1+\omega)(1+\omega^2) = 1 \cdot 1 = 1 \]

### Final Answer:
The value of \((1+\alpha)(1+\beta)(1+\alpha^2)(1+\beta^2)\) is \(\boxed{1}\).