`((sqrt(5)+sqrt(3))/(sqrt(5)-sqrt(3)))^(2)+((sqrt(5)-sqrt(3))/(sqrt(5)+sqrt(3)))^(2)` is equal to :

Options

A. A) `64`
B. B) `62`
C. C) `66`
D. D) `68`

Answer

To solve the expression

\[ \left(\frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}}\right)^{2} + \left(\frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}}\right)^{2} \]

let's break it down step by step.

### Step 1: Define the Variables
Let:
\[ x = \frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}} \]
Then, we can rewrite the expression as:
\[ x^{2} + \frac{1}{x^{2}} \]

### Step 2: Find \(x^{2} + \frac{1}{x^{2}}\)
We know that:
\[ x^{2} + \frac{1}{x^{2}} = \left(x + \frac{1}{x}\right)^{2} - 2 \]
So, we need to find \(x + \frac{1}{x}\).

### Step 3: Calculate \(x + \frac{1}{x}\)
First, calculate \(x\):
\[ x = \frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}} \]
Now find \(\frac{1}{x}\):
\[ \frac{1}{x} = \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}} \]
Now add \(x\) and \(\frac{1}{x}\):
\[ x + \frac{1}{x} = \frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}} + \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}} \]

### Step 4: Combine the Fractions
To combine these fractions, we find a common denominator:
\[ x + \frac{1}{x} = \frac{(\sqrt{5} + \sqrt{3})^2 + (\sqrt{5} - \sqrt{3})^2}{(\sqrt{5} - \sqrt{3})(\sqrt{5} + \sqrt{3})} \]

### Step 5: Simplify the Numerator
Calculating the numerator:
\[ (\sqrt{5} + \sqrt{3})^2 = 5 + 3 + 2\sqrt{15} = 8 + 2\sqrt{15} \]
\[ (\sqrt{5} - \sqrt{3})^2 = 5 + 3 - 2\sqrt{15} = 8 - 2\sqrt{15} \]
Now add these:
\[ (8 + 2\sqrt{15}) + (8 - 2\sqrt{15}) = 16 \]

### Step 6: Simplify the Denominator
The denominator is:
\[ (\sqrt{5} - \sqrt{3})(\sqrt{5} + \sqrt{3}) = 5 - 3 = 2 \]

### Step 7: Final Calculation for \(x + \frac{1}{x}\)
Thus,
\[ x + \frac{1}{x} = \frac{16}{2} = 8 \]

### Step 8: Calculate \(x^{2} + \frac{1}{x^{2}}\)
Now substitute back into the equation:
\[ x^{2} + \frac{1}{x^{2}} = (8)^{2} - 2 = 64 - 2 = 62 \]

### Final Answer
The value of the expression is:
\[ \boxed{62} \]