If `x=(sqrt(5)+sqrt(3))/(sqrt(5)-sqrt(3))` and `y=(sqrt(5)-sqrt(3))/(sqrt(5)+sqrt(3))` then `(x+y)` equals

Options

A. `8`
B. `16`
C. `2sqrt(15)`
D. `2(sqrt(5)+sqrt(3))`

Answer

To solve the problem, we need to find the value of \( x + y \) where:

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

### Step 1: Write the expression for \( x + y \)

We can write:

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

### Step 2: Find a common denominator

The common denominator for the two fractions is \( (\sqrt{5} - \sqrt{3})(\sqrt{5} + \sqrt{3}) \). Thus, we rewrite \( x + y \):

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

### Step 3: Expand the numerators

Now, we need to expand \( (\sqrt{5} + \sqrt{3})^2 \) and \( (\sqrt{5} - \sqrt{3})^2 \):

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

### Step 4: Combine the numerators

Now, we can combine these results:

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

This simplifies to:

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

### Final Answer

Thus, the value of \( x + y \) is:

\[ \boxed{8} \]