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

Explanation

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

### Step 1: Find the roots of the quadratic equation
The roots of the quadratic equation \( x^2 - x + 1 = 0 \) can be found using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = -1 \), and \( c = 1 \).

Substituting these values into the formula:
\[ x = \frac{1 \pm \sqrt{(-1)^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} = \frac{1 \pm \sqrt{1 - 4}}{2} = \frac{1 \pm \sqrt{-3}}{2} \]
This simplifies to:
\[ x = \frac{1 \pm i\sqrt{3}}{2} \]
Thus, the roots are:
\[ \alpha = \frac{1 + i\sqrt{3}}{2}, \quad \beta = \frac{1 - i\sqrt{3}}{2} \]

### Step 2: Express powers of the roots in terms of \( \omega \)
Notice that \( \alpha \) and \( \beta \) can be expressed in terms of the cube roots of unity. The cube roots of unity are \( 1, \omega, \omega^2 \), where \( \omega = e^{2\pi i / 3} = \frac{-1 + i\sqrt{3}}{2} \).

We can observe that:
\[ \alpha = \omega^2, \quad \beta = \omega \]

### Step 3: Calculate \( \alpha^{101} + \beta^{107} \)
Now we can compute \( \alpha^{101} + \beta^{107} \):
\[ \alpha^{101} = (\omega^2)^{101} = \omega^{202} \]
\[ \beta^{107} = \omega^{107} \]

### Step 4: Simplify using properties of \( \omega \)
Since \( \omega^3 = 1 \), we can reduce the exponents modulo 3:
\[ 202 \mod 3 = 1 \quad \text{(since } 202 = 3 \cdot 67 + 1\text{)} \]
\[ 107 \mod 3 = 2 \quad \text{(since } 107 = 3 \cdot 35 + 2\text{)} \]
Thus:
\[ \alpha^{101} = \omega^{202} = \omega^1 = \omega \]
\[ \beta^{107} = \omega^{107} = \omega^2 \]

### Step 5: Combine the results
Now we can combine the results:
\[ \alpha^{101} + \beta^{107} = \omega + \omega^2 \]

### Step 6: Use the property of cube roots of unity
From the property of cube roots of unity, we know:
\[ 1 + \omega + \omega^2 = 0 \implies \omega + \omega^2 = -1 \]

### Final Result
Thus, we find:
\[ \alpha^{101} + \beta^{107} = -1 \]

The final answer is:
\[ \boxed{-1} \]