Evaluate : `(sqrt(7)+sqrt(5))/(sqrt(7)+sqrt(20)+sqrt(28)-sqrt(5)-sqrt(80))`

Answer

To evaluate the expression \((\sqrt{7} + \sqrt{5}) / (\sqrt{7} + \sqrt{20} + \sqrt{28} - \sqrt{5} - \sqrt{80})\), we will simplify both the numerator and the denominator step by step.

### Step-by-Step Solution:

1. **Simplify the Denominator:**
- Start with the denominator: \(\sqrt{7} + \sqrt{20} + \sqrt{28} - \sqrt{5} - \sqrt{80}\).
- Rewrite \(\sqrt{20}\), \(\sqrt{28}\), and \(\sqrt{80}\):
- \(\sqrt{20} = \sqrt{4 \cdot 5} = 2\sqrt{5}\)
- \(\sqrt{28} = \sqrt{4 \cdot 7} = 2\sqrt{7}\)
- \(\sqrt{80} = \sqrt{16 \cdot 5} = 4\sqrt{5}\)

Therefore, the denominator becomes:
\[ \sqrt{7} + 2\sqrt{5} + 2\sqrt{7} - \sqrt{5} - 4\sqrt{5} \]

2. **Combine Like Terms in the Denominator:**
- Combine \(\sqrt{7}\) terms:
- \(\sqrt{7} + 2\sqrt{7} = 3\sqrt{7}\)
- Combine \(\sqrt{5}\) terms:
- \(2\sqrt{5} - \sqrt{5} - 4\sqrt{5} = (2 - 1 - 4)\sqrt{5} = -3\sqrt{5}\)

So, the denominator simplifies to:
\[ 3\sqrt{7} - 3\sqrt{5} \]

3. **Factor the Denominator:**
- Factor out the common term \(3\):
\[ 3(\sqrt{7} - \sqrt{5}) \]

4. **Rewrite the Expression:**
- Now the expression can be rewritten as:
\[ \frac{\sqrt{7} + \sqrt{5}}{3(\sqrt{7} - \sqrt{5})} \]

5. **Multiply by the Conjugate:**
- To simplify further, multiply the numerator and denominator by the conjugate of the denominator:
\[ \frac{(\sqrt{7} + \sqrt{5})(\sqrt{7} + \sqrt{5})}{3(\sqrt{7} - \sqrt{5})(\sqrt{7} + \sqrt{5})} \]

6. **Calculate the Numerator:**
- The numerator becomes:
\[ (\sqrt{7} + \sqrt{5})^2 = 7 + 5 + 2\sqrt{35} = 12 + 2\sqrt{35} \]

7. **Calculate the Denominator:**
- The denominator becomes:
\[ 3(\sqrt{7}^2 - \sqrt{5}^2) = 3(7 - 5) = 3 \cdot 2 = 6 \]

8. **Final Expression:**
- Now, the expression simplifies to:
\[ \frac{12 + 2\sqrt{35}}{6} \]

9. **Simplify Further:**
- Divide each term in the numerator by 6:
\[ 2 + \frac{1}{3}\sqrt{35} \]

### Final Answer:
\[ 2 + \frac{1}{3}\sqrt{35} \]