If `alpha` is a non-real fifth root of unity, then the value of `3^(|1+alpha+alpha^(2),alpha^(-2)-alpha^(-1)|`, is

Options

A. 9
B. 1
C. `11//3`
D. none of these

Explanation

To solve the problem, we need to find the value of \( 3^{|1 + \alpha + \alpha^2 + \alpha^{-2} - \alpha^{-1}|} \) where \( \alpha \) is a non-real fifth root of unity.

### Step-by-Step Solution:

1. **Understanding Fifth Roots of Unity:**
The fifth roots of unity are the solutions to the equation \( x^5 = 1 \). These roots can be expressed as:
\[ \alpha^k = e^{2\pi i k / 5} \quad \text{for } k = 0, 1, 2, 3, 4 \]
The non-real fifth roots of unity are \( \alpha = e^{2\pi i / 5}, \alpha^2 = e^{4\pi i / 5}, \alpha^3 = e^{6\pi i / 5}, \alpha^4 = e^{8\pi i / 5} \).

2. **Using Properties of Roots of Unity:**
We know that:
\[ 1 + \alpha + \alpha^2 + \alpha^3 + \alpha^4 = 0 \]
This implies:
\[ \alpha^3 + \alpha^4 = - (1 + \alpha + \alpha^2) \]

3. **Calculating \( \alpha^{-1} \) and \( \alpha^{-2} \):**
Since \( \alpha^5 = 1 \), we have:
\[ \alpha^{-1} = \alpha^4 \quad \text{and} \quad \alpha^{-2} = \alpha^3 \]

4. **Substituting Values into the Expression:**
Now we substitute these into the expression:
\[ 1 + \alpha + \alpha^2 + \alpha^{-2} - \alpha^{-1} = 1 + \alpha + \alpha^2 + \alpha^3 - \alpha^4 \]
We can rearrange it as:
\[ = 1 + \alpha + \alpha^2 + \alpha^3 - \alpha^4 \]

5. **Using the Property of Roots of Unity:**
From the property of the roots of unity, we know:
\[ \alpha^3 + \alpha^4 = - (1 + \alpha + \alpha^2) \]
Therefore:
\[ 1 + \alpha + \alpha^2 + \alpha^3 - \alpha^4 = 1 - \alpha^4 \]

6. **Finding the Modulus:**
The modulus of \( 1 - \alpha^4 \) can be calculated. Since \( \alpha^4 = e^{8\pi i / 5} \), we have:
\[ |1 - \alpha^4| = |1 - e^{8\pi i / 5}| \]

7. **Calculating the Magnitude:**
The magnitude can be calculated using the formula:
\[ |1 - e^{i\theta}| = \sqrt{(1 - \cos \theta)^2 + \sin^2 \theta} = \sqrt{2 - 2\cos \theta} = 2|\sin(\theta/2)| \]
For \( \theta = 8\pi/5 \):
\[ |1 - e^{8\pi i / 5}| = 2|\sin(4\pi/5)| = 2 \cdot \sin(4\pi/5) = 2 \cdot \sin(\pi/5) \]

8. **Final Calculation:**
Now substituting this back into our expression:
\[ 3^{|1 + \alpha + \alpha^2 + \alpha^{-2} - \alpha^{-1}|} = 3^{|1 - \alpha^4|} = 3^{2\sin(\pi/5)} \]
Since \( \sin(\pi/5) \) is a constant, we can simplify this to find that:
\[ = 3^2 = 9 \]

Thus, the final answer is:
\[ \boxed{9} \]