Express as an equivalent fraction with rational denominator.
`(15)/(sqrt((10))+sqrt((20))+sqrt((40))-sqrt((5))-sqrt((80)))`.
Simplify

Answer

To express the given fraction \(\frac{15}{\sqrt{10} + \sqrt{20} + \sqrt{40} - \sqrt{5} - \sqrt{80}}\) as an equivalent fraction with a rational denominator, we will follow these steps:

### Step 1: Simplify the square roots in the denominator
We start by simplifying each square root term in the denominator:

- \(\sqrt{20} = \sqrt{4 \cdot 5} = 2\sqrt{5}\)
- \(\sqrt{40} = \sqrt{4 \cdot 10} = 2\sqrt{10}\)
- \(\sqrt{80} = \sqrt{16 \cdot 5} = 4\sqrt{5}\)

Now, substituting these simplifications back into the denominator, we have:
\[ \sqrt{10} + \sqrt{20} + \sqrt{40} - \sqrt{5} - \sqrt{80} = \sqrt{10} + 2\sqrt{5} + 2\sqrt{10} - \sqrt{5} - 4\sqrt{5} \]

### Step 2: Combine like terms in the denominator
Now, we combine the like terms in the denominator:
\[ \sqrt{10} + 2\sqrt{10} + 2\sqrt{5} - \sqrt{5} - 4\sqrt{5} = 3\sqrt{10} - 3\sqrt{5} \]

Thus, the expression simplifies to:
\[ \frac{15}{3\sqrt{10} - 3\sqrt{5}} \]

### Step 3: Factor out the common term in the denominator
We can factor out the common factor of 3 from the denominator:
\[ \frac{15}{3(\sqrt{10} - \sqrt{5})} \]
This simplifies to:
\[ \frac{15}{3} \cdot \frac{1}{\sqrt{10} - \sqrt{5}} = 5 \cdot \frac{1}{\sqrt{10} - \sqrt{5}} = \frac{5}{\sqrt{10} - \sqrt{5}} \]

### Step 4: Rationalize the denominator
To rationalize the denominator, we multiply the numerator and denominator by the conjugate of the denominator, which is \(\sqrt{10} + \sqrt{5}\):
\[ \frac{5(\sqrt{10} + \sqrt{5})}{(\sqrt{10} - \sqrt{5})(\sqrt{10} + \sqrt{5})} \]

### Step 5: Simplify the denominator using the difference of squares
The denominator simplifies as follows:
\[ (\sqrt{10})^2 - (\sqrt{5})^2 = 10 - 5 = 5 \]

### Step 6: Final simplification
Now, substituting back, we have:
\[ \frac{5(\sqrt{10} + \sqrt{5})}{5} = \sqrt{10} + \sqrt{5} \]

Thus, the final answer is:
\[ \sqrt{10} + \sqrt{5} \]