If ` x= ( sqrt5- sqrt3)/ ( sqrt5+ sqrt3) and y= ( sqrt5+ sqrt3)/( sqrt5- sqrt3) ` find the value of ` x^(2) + y ^(2)`

Answer

To find the value of \( x^2 + y^2 \) where \( x = \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}} \) and \( y = \frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}} \), we can follow these steps:

### Step 1: Calculate \( x + y \)
We start by adding \( x \) and \( y \):

\[ x + y = \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}} + \frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}} \]

To add these fractions, we need a common denominator:

\[ x + y = \frac{(\sqrt{5} - \sqrt{3})^2 + (\sqrt{5} + \sqrt{3})^2}{(\sqrt{5} + \sqrt{3})(\sqrt{5} - \sqrt{3})} \]

### Step 2: Simplify the Numerator
Now, we will simplify the numerator:

\[ (\sqrt{5} - \sqrt{3})^2 = 5 + 3 - 2\sqrt{15} = 8 - 2\sqrt{15} \]
\[ (\sqrt{5} + \sqrt{3})^2 = 5 + 3 + 2\sqrt{15} = 8 + 2\sqrt{15} \]

Adding these results together:

\[ (\sqrt{5} - \sqrt{3})^2 + (\sqrt{5} + \sqrt{3})^2 = (8 - 2\sqrt{15}) + (8 + 2\sqrt{15}) = 16 \]

### Step 3: Simplify the Denominator
Now, we simplify the denominator:

\[ (\sqrt{5} + \sqrt{3})(\sqrt{5} - \sqrt{3}) = 5 - 3 = 2 \]

### Step 4: Combine Results
Now we can combine the results from the numerator and denominator:

\[ x + y = \frac{16}{2} = 8 \]

### Step 5: Calculate \( xy \)
Next, we calculate \( xy \):

\[ xy = \left(\frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}}\right) \left(\frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}}\right) = 1 \]

### Step 6: Use the Identity
We can now use the identity:

\[ x^2 + y^2 = (x + y)^2 - 2xy \]

Substituting the values we found:

\[ x^2 + y^2 = 8^2 - 2 \cdot 1 = 64 - 2 = 62 \]

### Final Answer
Thus, the value of \( x^2 + y^2 \) is:

\[ \boxed{62} \]