If `alpha, beta in C` are distinct roots of the equation `x^2+1=0` then `alpha^(101)+beta^(107)` is equal to

Options

A. 2
B. `-1`
C. 0
D. 1

Explanation

To solve the problem, we need to find the values of \( \alpha^{101} + \beta^{107} \) where \( \alpha \) and \( \beta \) are distinct roots of the equation \( x^2 + 1 = 0 \).

### Step-by-step Solution:

1. **Identify the Roots**:
The equation \( x^2 + 1 = 0 \) can be rewritten as:
\[ x^2 = -1 \]
The roots of this equation are:
\[ \alpha = i \quad \text{and} \quad \beta = -i \]

2. **Calculate \( \alpha^{101} \)**:
We can express \( \alpha \) in exponential form:
\[ \alpha = i = e^{i\frac{\pi}{2}} \]
Therefore, we can compute \( \alpha^{101} \):
\[ \alpha^{101} = (e^{i\frac{\pi}{2}})^{101} = e^{i\frac{101\pi}{2}} \]
To simplify \( e^{i\frac{101\pi}{2}} \), we can reduce \( \frac{101\pi}{2} \) modulo \( 2\pi \):
\[ \frac{101\pi}{2} = 50\pi + \frac{\pi}{2} \quad \text{(since } 50\pi \text{ is a multiple of } 2\pi\text{)} \]
Therefore:
\[ e^{i\frac{101\pi}{2}} = e^{i\frac{\pi}{2}} = i \]

3. **Calculate \( \beta^{107} \)**:
Similarly, for \( \beta \):
\[ \beta = -i = e^{-i\frac{\pi}{2}} \]
Thus, we compute \( \beta^{107} \):
\[ \beta^{107} = (e^{-i\frac{\pi}{2}})^{107} = e^{-i\frac{107\pi}{2}} \]
Reducing \( -\frac{107\pi}{2} \) modulo \( 2\pi \):
\[ -\frac{107\pi}{2} = -53\pi - \frac{\pi}{2} \quad \text{(since } -53\pi \text{ is a multiple of } 2\pi\text{)} \]
Therefore:
\[ e^{-i\frac{107\pi}{2}} = e^{-i\frac{\pi}{2}} = -i \]

4. **Combine the Results**:
Now we can combine the results:
\[ \alpha^{101} + \beta^{107} = i + (-i) = 0 \]

### Final Answer:
\[ \alpha^{101} + \beta^{107} = 0 \]