`((1+sqrt(2))/(sqrt(5)+sqrt(3))+(1-sqrt(2))/(sqrt(5)-sqrt(3)))` simplifies to :

Options

A. `sqrt(5)+sqrt(6)`
B. `2sqrt(5)+sqrt(6)`
C. `sqrt(5)-sqrt(6)`
D. `2sqrt(5)-3sqrt(6)`

Answer

To simplify the expression \(\frac{1+\sqrt{2}}{\sqrt{5}+\sqrt{3}} + \frac{1-\sqrt{2}}{\sqrt{5}-\sqrt{3}}\), we will follow these steps:

### Step 1: Find a common denominator
The common denominator for the two fractions is \((\sqrt{5}+\sqrt{3})(\sqrt{5}-\sqrt{3})\).

### Step 2: Rewrite the fractions
We rewrite the expression as:
\[ \frac{(1+\sqrt{2})(\sqrt{5}-\sqrt{3}) + (1-\sqrt{2})(\sqrt{5}+\sqrt{3})}{(\sqrt{5}+\sqrt{3})(\sqrt{5}-\sqrt{3})} \]

### Step 3: Expand the numerators
Now we will expand the numerators:
1. For \((1+\sqrt{2})(\sqrt{5}-\sqrt{3})\):
\[ = \sqrt{5} + \sqrt{10} - \sqrt{3} - \sqrt{6} \]

2. For \((1-\sqrt{2})(\sqrt{5}+\sqrt{3})\):
\[ = \sqrt{5} + \sqrt{3} - \sqrt{10} - \sqrt{6} \]

### Step 4: Combine the numerators
Now we combine the results from the expansions:
\[ (\sqrt{5} + \sqrt{10} - \sqrt{3} - \sqrt{6}) + (\sqrt{5} + \sqrt{3} - \sqrt{10} - \sqrt{6}) \]
This simplifies to:
\[ 2\sqrt{5} - 2\sqrt{6} \]

### Step 5: Simplify the denominator
The denominator simplifies using the difference of squares:
\[ (\sqrt{5})^2 - (\sqrt{3})^2 = 5 - 3 = 2 \]

### Step 6: Combine the results
Putting it all together, we have:
\[ \frac{2\sqrt{5} - 2\sqrt{6}}{2} \]
This simplifies to:
\[ \sqrt{5} - \sqrt{6} \]

### Final Answer
Thus, the simplified form of the expression is:
\[ \sqrt{5} - \sqrt{6} \]