`((1+sqrt2)/(sqrt5+sqrt3)+(1-sqrt2)/(sqrt5-sqrt3))` Simplify

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 can follow these steps:

### Step 1: Write down the expression
The given expression is:
\[ \frac{1+\sqrt{2}}{\sqrt{5}+\sqrt{3}} + \frac{1-\sqrt{2}}{\sqrt{5}-\sqrt{3}} \]

### Step 2: Find a common denominator
The common denominator for the two fractions is \((\sqrt{5}+\sqrt{3})(\sqrt{5}-\sqrt{3})\). Therefore, 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 need to expand the numerators:
1. For \((1+\sqrt{2})(\sqrt{5}-\sqrt{3})\):
\[ = \sqrt{5} + \sqrt{10} - \sqrt{3} - \sqrt{6} \]

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

Adding these two results together:
\[ (\sqrt{5} + \sqrt{10} - \sqrt{3} - \sqrt{6}) + (\sqrt{5} + \sqrt{3} - \sqrt{10} - \sqrt{6}) \]
This simplifies to:
\[ 2\sqrt{5} - 2\sqrt{6} \]

### Step 4: Simplify the denominator
The denominator simplifies as follows:
\[ (\sqrt{5}+\sqrt{3})(\sqrt{5}-\sqrt{3}) = 5 - 3 = 2 \]

### Step 5: Combine the results
Now we can combine the results:
\[ \frac{2\sqrt{5} - 2\sqrt{6}}{2} \]
This simplifies to:
\[ \sqrt{5} - \sqrt{6} \]

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

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