`[(1)/(sqrt(2)+sqrt(3)-sqrt(5))+(1)/(sqrt(2)-sqrt(3)-sqrt(5))]` in simplified form equals to :

Options

A. `1`
B. `sqrt(2)`
C. `(1)/(sqrt(2))`
D. `0`

Answer

To simplify the expression \(\left[\frac{1}{\sqrt{2}+\sqrt{3}-\sqrt{5}}+\frac{1}{\sqrt{2}-\sqrt{3}-\sqrt{5}}\right]\), we can 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 as:
\[ \frac{(\sqrt{2}-\sqrt{3}-\sqrt{5}) + (\sqrt{2}+\sqrt{3}-\sqrt{5})}{(\sqrt{2}+\sqrt{3}-\sqrt{5})(\sqrt{2}-\sqrt{3}-\sqrt{5})} \]

### 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:
\[ (\sqrt{2})^2 - (\sqrt{3}+\sqrt{5})^2 = 2 - (3 + 5 + 2\sqrt{15}) = 2 - 8 - 2\sqrt{15} = -6 - 2\sqrt{15} \]

### Step 5: Combine the results
Now, we can combine the results:
\[ \frac{2(\sqrt{2}-\sqrt{5})}{-6 - 2\sqrt{15}} \]

### Step 6: Factor out common terms
Factor out -2 from the denominator:
\[ \frac{2(\sqrt{2}-\sqrt{5})}{-2(3+\sqrt{15})} = \frac{\sqrt{2}-\sqrt{5}}{-(3+\sqrt{15})} \]

### Step 7: Final simplification
This can be rewritten as:
\[ -\frac{\sqrt{2}-\sqrt{5}}{3+\sqrt{15}} \]

### Step 8: Rationalize the denominator
To rationalize the denominator, multiply the numerator and denominator by the conjugate of the denominator:
\[ -\frac{(\sqrt{2}-\sqrt{5})(3-\sqrt{15})}{(3+\sqrt{15})(3-\sqrt{15})} \]

### Step 9: Simplify the denominator
The denominator simplifies to:
\[ 9 - 15 = -6 \]

### Step 10: Final expression
Thus, the final expression becomes:
\[ \frac{(\sqrt{2}-\sqrt{5})(3-\sqrt{15})}{6} \]

### Conclusion
The simplified form of the original expression is:
\[ \frac{\sqrt{2}-\sqrt{5}}{3+\sqrt{15}} \]