If `(x^(2) + y^(2))/(x^(2) - y^(2)) = 2(1)/(8)`, find :
(i) `(x)/(y)`
(ii) `(x^(3) + y^(3))/(x^(3) - y^(3))`

Answer

To solve the problem step by step, we will follow the approach outlined in the video transcript.

### Given:
\[ \frac{x^2 + y^2}{x^2 - y^2} = 2 \frac{1}{8} \]

### Step 1: Convert the mixed fraction to an improper fraction.
\[ 2 \frac{1}{8} = \frac{16 + 1}{8} = \frac{17}{8} \]
Thus, we rewrite the equation as:
\[ \frac{x^2 + y^2}{x^2 - y^2} = \frac{17}{8} \]

### Step 2: Apply the component and dividendo rule.
According to the component and dividendo rule:
\[ \frac{A + B}{A - B} = \frac{C + D}{C - D} \]
We can apply this rule here:
\[ \frac{x^2 + y^2 + x^2 - y^2}{x^2 + y^2 - (x^2 - y^2)} = \frac{17 + 8}{17 - 8} \]

### Step 3: Simplify the left-hand side.
The left-hand side simplifies to:
\[ \frac{2x^2}{2y^2} = \frac{x^2}{y^2} \]

### Step 4: Simplify the right-hand side.
The right-hand side simplifies to:
\[ \frac{25}{9} \]

### Step 5: Set the two sides equal.
Now we have:
\[ \frac{x^2}{y^2} = \frac{25}{9} \]

### Step 6: Take the square root of both sides.
Taking the square root gives:
\[ \frac{x}{y} = \frac{5}{3} \]

### Answer for (i):
\[ \frac{x}{y} = \frac{5}{3} \]

### Step 7: Now find \(\frac{x^3 + y^3}{x^3 - y^3}\).
Using the identity:
\[ \frac{x^3 + y^3}{x^3 - y^3} = \frac{\frac{x}{y}^3 + 1}{\frac{x}{y}^3 - 1} \]

### Step 8: Substitute \(\frac{x}{y} = \frac{5}{3}\).
Calculating \(\left(\frac{x}{y}\right)^3\):
\[ \left(\frac{5}{3}\right)^3 = \frac{125}{27} \]

### Step 9: Substitute into the identity.
Now substituting into the identity:
\[ \frac{\frac{125}{27} + 1}{\frac{125}{27} - 1} \]

### Step 10: Simplify the numerator and denominator.
Convert 1 to a fraction:
\[ 1 = \frac{27}{27} \]
Thus, the numerator becomes:
\[ \frac{125 + 27}{27} = \frac{152}{27} \]
And the denominator becomes:
\[ \frac{125 - 27}{27} = \frac{98}{27} \]

### Step 11: Simplify the entire fraction.
Now we have:
\[ \frac{\frac{152}{27}}{\frac{98}{27}} = \frac{152}{98} \]

### Step 12: Reduce the fraction.
Dividing both the numerator and denominator by 2 gives:
\[ \frac{76}{49} \]

### Answer for (ii):
\[ \frac{x^3 + y^3}{x^3 - y^3} = \frac{76}{49} \]

### Summary of Answers:
(i) \(\frac{x}{y} = \frac{5}{3}\)
(ii) \(\frac{x^3 + y^3}{x^3 - y^3} = \frac{76}{49}\)