`((1+sqrt2)/(sqrt5+sqrt3) + (1-sqrt2)/(sqrt5-sqrt3))` simplifies to :

Options

A. `sqrt5 + sqrt6`
B. `2sqrt5 + sqrt6`
C. `sqrt5-sqrt6`
D. `2sqrt5 - 3sqrt6`

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 expression
We can 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 expanded numerators
Now, we combine both results:
\[ (\sqrt{5} + \sqrt{10} - \sqrt{3} - \sqrt{6}) + (\sqrt{5} + \sqrt{3} - \sqrt{10} - \sqrt{6}) \]
This simplifies to:
\[ 2\sqrt{5} - 2\sqrt{6} \]

### Step 5: Write the full expression
Now we can write the full expression:
\[ \frac{2\sqrt{5} - 2\sqrt{6}}{(\sqrt{5}+\sqrt{3})(\sqrt{5}-\sqrt{3})} \]

### Step 6: Simplify the denominator
The denominator simplifies to:
\[ (\sqrt{5})^2 - (\sqrt{3})^2 = 5 - 3 = 2 \]

### Step 7: Final simplification
Thus, we can simplify the entire expression:
\[ \frac{2(\sqrt{5} - \sqrt{6})}{2} = \sqrt{5} - \sqrt{6} \]

### Final Answer
The expression simplifies to:
\[ \sqrt{5} - \sqrt{6} \]