If `alpha and beta` are roots of equation `x^2 - 7x -1 =0` , then the value of `((alpha^(10)-beta^(10))-(alpha^(8)-beta^(8)))/(alpha^((alpha+beta+2))-beta^((alpha+beta+2)))` is equal to

Options

A. 1
B. 0
C. 7
D. `1/7`

Answer

To solve the problem, we need to find the value of

\[ \frac{(\alpha^{10} - \beta^{10}) - (\alpha^{8} - \beta^{8})}{\alpha^{(\alpha + \beta + 2)} - \beta^{(\alpha + \beta + 2)}} \]

where \(\alpha\) and \(\beta\) are the roots of the equation \(x^2 - 7x - 1 = 0\).

### Step 1: Find the roots \(\alpha\) and \(\beta\)
Using the quadratic formula, the roots of the equation \(ax^2 + bx + c = 0\) are given by:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

For our equation \(x^2 - 7x - 1 = 0\):

- \(a = 1\)
- \(b = -7\)
- \(c = -1\)

Calculating the discriminant:

\[ D = b^2 - 4ac = (-7)^2 - 4 \cdot 1 \cdot (-1) = 49 + 4 = 53 \]

Now, substituting back into the quadratic formula:

\[ \alpha, \beta = \frac{7 \pm \sqrt{53}}{2} \]

### Step 2: Use the properties of the roots
From Vieta's formulas, we know:

- \(\alpha + \beta = 7\)
- \(\alpha \beta = -1\)

### Step 3: Simplify the numerator
We can rewrite the numerator:

\[ \alpha^{10} - \beta^{10} - (\alpha^{8} - \beta^{8}) = (\alpha^{10} - \alpha^{8}) - (\beta^{10} - \beta^{8}) = \alpha^{8}(\alpha^{2} - 1) - \beta^{8}(\beta^{2} - 1) \]

### Step 4: Express \(\alpha^2 - 1\) and \(\beta^2 - 1\)
Using the original equation, we can express \(\alpha^2\) and \(\beta^2\):

\[ \alpha^2 = 7\alpha + 1 \quad \text{and} \quad \beta^2 = 7\beta + 1 \]

Thus,

\[ \alpha^2 - 1 = 7\alpha \quad \text{and} \quad \beta^2 - 1 = 7\beta \]

### Step 5: Substitute back into the numerator
Substituting these back, we get:

\[ \alpha^{8}(7\alpha) - \beta^{8}(7\beta) = 7(\alpha^{9} - \beta^{9}) \]

### Step 6: Simplify the denominator
Now, we simplify the denominator:

\[ \alpha^{(\alpha + \beta + 2)} - \beta^{(\alpha + \beta + 2)} = \alpha^{9} - \beta^{9} \]

### Step 7: Combine results
Now we can substitute back into our original expression:

\[ \frac{7(\alpha^{9} - \beta^{9})}{\alpha^{9} - \beta^{9}} = 7 \]

### Final Answer
Thus, the value of the expression is:

\[ \boxed{7} \]