The simplified value of `[(1)/(sqrt(2)+sqrt(3)-sqrt(5))+(1)/(sqrt(2)-sqrt(3)-sqrt(5))]` is

Options

A. 1
B. `(2)/(sqrt(2))`
C. `(sqrt(2))/(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 will follow these steps:

### Step 1: Combine the Fractions
We start by finding a common denominator for the two fractions. The common denominator will be the product of the two denominators:

\[ \text{Common Denominator} = (\sqrt{2} + \sqrt{3} - \sqrt{5})(\sqrt{2} - \sqrt{3} - \sqrt{5}) \]

Thus, we can write:

\[ \frac{1}{\sqrt{2} + \sqrt{3} - \sqrt{5}} + \frac{1}{\sqrt{2} - \sqrt{3} - \sqrt{5}} = \frac{(\sqrt{2} - \sqrt{3} - \sqrt{5}) + (\sqrt{2} + \sqrt{3} - \sqrt{5})}{(\sqrt{2} + \sqrt{3} - \sqrt{5})(\sqrt{2} - \sqrt{3} - \sqrt{5})} \]

### Step 2: 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 3: Simplify the Denominator
Next, we need to simplify the denominator:

\[ (\sqrt{2} + \sqrt{3} - \sqrt{5})(\sqrt{2} - \sqrt{3} - \sqrt{5}) = (\sqrt{2})^2 - (\sqrt{3})^2 - (\sqrt{5})^2 + \text{cross terms} \]

Calculating this gives:

\[ = 2 - 3 - 5 + \text{cross terms} \]

The cross terms will cancel out, leading to:

\[ = 2 - 3 - 5 = -6 \]

### Step 4: Combine the Results
Now we can combine the results:

\[ \frac{2\sqrt{2} - 2\sqrt{5}}{-6} = \frac{-1}{3}(\sqrt{2} - \sqrt{5}) \]

### Step 5: Final Simplification
We can further simplify this:

\[ = \frac{1}{3}(\sqrt{5} - \sqrt{2}) \]

Thus, the simplified value of the original expression is:

\[ \frac{\sqrt{5} - \sqrt{2}}{3} \]

### Final Answer
The simplified value is:

\[ \frac{\sqrt{5} - \sqrt{2}}{3} \]

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