If `(x^(2)+y^(2))/(x^(2)-y^(2))=17/(8)`, using the properties of proportion find the value of :
(ii) `(x^(3)+y^(3))/(x^(3)-y^(3))`

Answer

To solve the problem, we start with the given equation:

\[ \frac{x^2 + y^2}{x^2 - y^2} = \frac{17}{8} \]

### Step 1: Apply Componendo and Dividendo
Using the properties of proportion, specifically the Componendo and Dividendo theorem, we can rewrite the equation. According to this theorem, if \(\frac{a}{b} = \frac{c}{d}\), then:

\[ \frac{a + b}{a - b} = \frac{c + d}{c - d} \]

Here, let \(a = x^2 + y^2\) and \(b = x^2 - y^2\). Thus, we have:

\[ \frac{(x^2 + y^2) + (x^2 - y^2)}{(x^2 + y^2) - (x^2 - y^2)} = \frac{17 + 8}{17 - 8} \]

### Step 2: Simplify the Left Side
The left side simplifies as follows:

\[ \frac{(x^2 + y^2 + x^2 - y^2)}{(x^2 + y^2 - x^2 + y^2)} = \frac{2x^2}{2y^2} = \frac{x^2}{y^2} \]

### Step 3: Simplify the Right Side
Now, simplify the right side:

\[ \frac{17 + 8}{17 - 8} = \frac{25}{9} \]

### Step 4: Set Up the Equation
Now we equate the two sides:

\[ \frac{x^2}{y^2} = \frac{25}{9} \]

### Step 5: Find the Ratio of \(x\) to \(y\)
Taking the square root of both sides gives:

\[ \frac{x}{y} = \frac{5}{3} \]

### Step 6: Cube Both Sides
Cubing both sides, we find:

\[ \frac{x^3}{y^3} = \left(\frac{5}{3}\right)^3 = \frac{125}{27} \]

### Step 7: Apply Componendo and Dividendo Again
Now we apply Componendo and Dividendo again to find \(\frac{x^3 + y^3}{x^3 - y^3}\):

Using the same theorem:

\[ \frac{x^3 + y^3}{x^3 - y^3} = \frac{125 + 27}{125 - 27} \]

### Step 8: Simplify the Right Side
Calculating the right side:

\[ \frac{125 + 27}{125 - 27} = \frac{152}{98} \]

### Step 9: Final Simplification
Now, we can simplify \(\frac{152}{98}\):

\[ \frac{152 \div 2}{98 \div 2} = \frac{76}{49} \]

Thus, the final answer is:

\[ \frac{x^3 + y^3}{x^3 - y^3} = \frac{76}{49} \]