Given that `alpha and beta` are the roots of the equation `x^(2)=x+7`.
(i) Prove that (a) `(1)/(alpha)=(alpha-1)/(7) and ` (b) `alpha^(3)=8alpha+7`.
(ii) Find the numerical value of `(alpha)/(beta)+(beta)/(alpha)`.

Answer

To solve the given problem step by step, we will follow the instructions provided in the question.

### Given:
The equation is \( x^2 = x + 7 \).

### Step 1: Rewrite the equation in standard form
We can rearrange the equation to standard quadratic form:
\[ x^2 - x - 7 = 0 \]
This is our equation (1).

### Step 2: Use the roots of the equation
Let \( \alpha \) and \( \beta \) be the roots of the equation. According to Vieta's formulas:
- The sum of the roots \( \alpha + \beta = -\frac{b}{a} = 1 \)
- The product of the roots \( \alpha \beta = \frac{c}{a} = -7 \)

### Part (i): Prove (a) \( \frac{1}{\alpha} = \frac{\alpha - 1}{7} \)

#### Step 3: Start from the quadratic equation
From the equation \( \alpha^2 - \alpha - 7 = 0 \), we can express \( \alpha^2 \) in terms of \( \alpha \):
\[ \alpha^2 = \alpha + 7 \]

#### Step 4: Divide both sides by \( \alpha \)
Now, divide both sides of the equation by \( \alpha \) (assuming \( \alpha \neq 0 \)):
\[ \frac{\alpha^2}{\alpha} = \frac{\alpha + 7}{\alpha} \]
This simplifies to:
\[ \alpha = 1 + \frac{7}{\alpha} \]

#### Step 5: Rearranging gives us the desired result
Rearranging gives:
\[ \frac{7}{\alpha} = \alpha - 1 \]
Taking the reciprocal of both sides:
\[ \frac{1}{\alpha} = \frac{\alpha - 1}{7} \]
Thus, part (a) is proved.

### Part (i): Prove (b) \( \alpha^3 = 8\alpha + 7 \)

#### Step 6: Use the expression for \( \alpha^2 \)
From the previous step, we have \( \alpha^2 = \alpha + 7 \).

#### Step 7: Multiply both sides by \( \alpha \)
Now multiply both sides by \( \alpha \):
\[ \alpha^3 = \alpha \cdot \alpha^2 = \alpha(\alpha + 7) \]
Expanding this gives:
\[ \alpha^3 = \alpha^2 + 7\alpha \]

#### Step 8: Substitute \( \alpha^2 \) from earlier
Now substitute \( \alpha^2 \) from our earlier result:
\[ \alpha^3 = (\alpha + 7) + 7\alpha \]
This simplifies to:
\[ \alpha^3 = 8\alpha + 7 \]
Thus, part (b) is proved.

### Part (ii): Find the numerical value of \( \frac{\alpha}{\beta} + \frac{\beta}{\alpha} \)

#### Step 9: Use the identity
We can use the identity:
\[ \frac{\alpha}{\beta} + \frac{\beta}{\alpha} = \frac{\alpha^2 + \beta^2}{\alpha \beta} \]

#### Step 10: Calculate \( \alpha^2 + \beta^2 \)
Using the formula \( \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \):
\[ \alpha^2 + \beta^2 = (1)^2 - 2(-7) = 1 + 14 = 15 \]

#### Step 11: Calculate \( \alpha \beta \)
From Vieta's, we know \( \alpha \beta = -7 \).

#### Step 12: Substitute values into the identity
Now substituting these values into our identity:
\[ \frac{\alpha^2 + \beta^2}{\alpha \beta} = \frac{15}{-7} = -\frac{15}{7} \]

### Final Answer
Thus, the numerical value of \( \frac{\alpha}{\beta} + \frac{\beta}{\alpha} \) is:
\[ -\frac{15}{7} \]