Express as an equivalent fraction with rational denominator.
`(11)/(2+sqrt((3))+sqrt((5)))`

Answer

To express the fraction \(\frac{11}{2 + \sqrt{3} + \sqrt{5}}\) as an equivalent fraction with a rational denominator, we will use the method of rationalization.

### Step-by-Step Solution:

1. **Identify the Denominator**: The denominator is \(2 + \sqrt{3} + \sqrt{5}\).

2. **Multiply by the Conjugate**: To rationalize the denominator, we will multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate in this case is \(2 - (\sqrt{3} + \sqrt{5})\).

\[ \frac{11}{2 + \sqrt{3} + \sqrt{5}} \cdot \frac{2 - (\sqrt{3} + \sqrt{5})}{2 - (\sqrt{3} + \sqrt{5})} \]

3. **Simplify the Numerator**: The numerator becomes:

\[ 11 \cdot (2 - (\sqrt{3} + \sqrt{5})) = 22 - 11\sqrt{3} - 11\sqrt{5} \]

4. **Simplify the Denominator**: The denominator is:

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

Now, calculating the squares:

\[ 2^2 = 4 \]
\[ (\sqrt{3} + \sqrt{5})^2 = 3 + 5 + 2\sqrt{15} = 8 + 2\sqrt{15} \]

Therefore, the denominator becomes:

\[ 4 - (8 + 2\sqrt{15}) = 4 - 8 - 2\sqrt{15} = -4 - 2\sqrt{15} \]

5. **Combine the Results**: Now we have:

\[ \frac{22 - 11\sqrt{3} - 11\sqrt{5}}{-4 - 2\sqrt{15}} \]

6. **Simplify the Fraction**: We can factor out \(-2\) from the denominator:

\[ = \frac{22 - 11\sqrt{3} - 11\sqrt{5}}{-2(2 + \sqrt{15})} \]

This simplifies to:

\[ = -\frac{11 - \frac{11}{2}\sqrt{3} - \frac{11}{2}\sqrt{5}}{2 + \sqrt{15}} \]

7. **Final Form**: Thus, the equivalent fraction with a rational denominator is:

\[ = -\frac{11 - \frac{11}{2}\sqrt{3} - \frac{11}{2}\sqrt{5}}{2 + \sqrt{15}} \]