If `x=(sqrt(7)-sqrt(5))/(sqrt((7))+sqrt((5)))`, `y=(sqrt(7)+sqrt(5))/(sqrt((7))-sqrt((5)))` find the value of `x^(3)+y^(3)`.

Answer

To find the value of \( x^3 + y^3 \) given \( x = \frac{\sqrt{7} - \sqrt{5}}{\sqrt{7} + \sqrt{5}} \) and \( y = \frac{\sqrt{7} + \sqrt{5}}{\sqrt{7} - \sqrt{5}} \), we can follow these steps:

### Step 1: Recognize the relationship between \( x \) and \( y \)
Notice that \( y \) is the reciprocal of \( x \):
\[ y = \frac{1}{x} \]

### Step 2: Use the identity for \( x^3 + y^3 \)
We can use the identity:
\[ x^3 + y^3 = (x + y)(x^2 - xy + y^2) \]
Since \( y = \frac{1}{x} \), we have \( xy = 1 \).

### Step 3: Find \( x + y \)
To find \( x + y \):
\[ x + y = \frac{\sqrt{7} - \sqrt{5}}{\sqrt{7} + \sqrt{5}} + \frac{\sqrt{7} + \sqrt{5}}{\sqrt{7} - \sqrt{5}} \]
Taking the LCM:
\[ x + y = \frac{(\sqrt{7} - \sqrt{5})^2 + (\sqrt{7} + \sqrt{5})^2}{(\sqrt{7} + \sqrt{5})(\sqrt{7} - \sqrt{5})} \]

### Step 4: Simplify the numerator
Calculating the squares:
\[ (\sqrt{7} - \sqrt{5})^2 = 7 - 2\sqrt{35} + 5 = 12 - 2\sqrt{35} \]
\[ (\sqrt{7} + \sqrt{5})^2 = 7 + 2\sqrt{35} + 5 = 12 + 2\sqrt{35} \]
Adding these:
\[ (\sqrt{7} - \sqrt{5})^2 + (\sqrt{7} + \sqrt{5})^2 = (12 - 2\sqrt{35}) + (12 + 2\sqrt{35}) = 24 \]

### Step 5: Simplify the denominator
The denominator becomes:
\[ (\sqrt{7} + \sqrt{5})(\sqrt{7} - \sqrt{5}) = 7 - 5 = 2 \]

### Step 6: Combine results for \( x + y \)
Thus:
\[ x + y = \frac{24}{2} = 12 \]

### Step 7: Calculate \( x^2 + y^2 \)
Using the identity:
\[ x^2 + y^2 = (x + y)^2 - 2xy = 12^2 - 2 \cdot 1 = 144 - 2 = 142 \]

### Step 8: Substitute into \( x^3 + y^3 \)
Now substituting back into the identity:
\[ x^3 + y^3 = (x + y)((x^2 + y^2) - xy) = 12(142 - 1) = 12 \cdot 141 = 1692 \]

### Final Answer
Thus, the value of \( x^3 + y^3 \) is:
\[ \boxed{1692} \]