The simplification of
`[1/(sqrt2+sqrt3-sqrt5)+1/(sqrt2-sqrt3-sqrt5)]` is equal to

Options

A. 1
B. `sqrt2`
C. `1/(sqrt2)`
D. 0

Answer

To simplify the expression

\[ \frac{1}{\sqrt{2} + \sqrt{3} - \sqrt{5}} + \frac{1}{\sqrt{2} - \sqrt{3} - \sqrt{5}}, \]

we will follow these steps:

### Step 1: Find a common denominator
The common denominator for the two fractions is

\[ (\sqrt{2} + \sqrt{3} - \sqrt{5})(\sqrt{2} - \sqrt{3} - \sqrt{5}). \]

### Step 2: Rewrite the expression
We can rewrite the expression using the common denominator:

\[ \frac{(\sqrt{2} - \sqrt{3} - \sqrt{5}) + (\sqrt{2} + \sqrt{3} - \sqrt{5})}{(\sqrt{2} + \sqrt{3} - \sqrt{5})(\sqrt{2} - \sqrt{3} - \sqrt{5})}. \]

### Step 3: Simplify the numerator
Now, simplify the numerator:

\[ (\sqrt{2} - \sqrt{3} - \sqrt{5}) + (\sqrt{2} + \sqrt{3} - \sqrt{5}) = 2\sqrt{2} - 2\sqrt{5}. \]

### Step 4: Simplify the denominator
Next, we simplify the denominator using the difference of squares:

\[ (\sqrt{2})^2 - (\sqrt{5})^2 = 2 - 5 = -3. \]

### Step 5: Combine the results
Now we can combine the results:

\[ \frac{2\sqrt{2} - 2\sqrt{5}}{-3} = \frac{-2(\sqrt{2} - \sqrt{5})}{3}. \]

### Step 6: Factor out common terms
We can factor out -2 from the numerator:

\[ = -\frac{2}{3}(\sqrt{2} - \sqrt{5}). \]

### Step 7: Final simplification
To express this in a simpler form, we can write:

\[ = \frac{2(\sqrt{5} - \sqrt{2})}{3}. \]

### Conclusion
The simplified form of the given expression is:

\[ \frac{2(\sqrt{5} - \sqrt{2})}{3}. \]