The simplified form of `( 2)/(sqrt(7) + sqrt(5)) + (7)/(sqrt(12)- sqrt(5)) - (5)/(sqrt(12) - sqrt(7))` is

Options

A. 5
B. 2
C. 1
D. 0

Answer

To simplify the expression

\[ \frac{2}{\sqrt{7} + \sqrt{5}} + \frac{7}{\sqrt{12} - \sqrt{5}} - \frac{5}{\sqrt{12} - \sqrt{7}}, \]

we will follow these steps:

### Step 1: Rationalize the denominators
We will rationalize each term by multiplying the numerator and denominator by the conjugate of the denominator.

1. For the first term \(\frac{2}{\sqrt{7} + \sqrt{5}}\), multiply by \(\sqrt{7} - \sqrt{5}\):

\[ \frac{2(\sqrt{7} - \sqrt{5})}{(\sqrt{7} + \sqrt{5})(\sqrt{7} - \sqrt{5})} = \frac{2(\sqrt{7} - \sqrt{5})}{7 - 5} = \frac{2(\sqrt{7} - \sqrt{5})}{2} = \sqrt{7} - \sqrt{5}. \]

2. For the second term \(\frac{7}{\sqrt{12} - \sqrt{5}}\), multiply by \(\sqrt{12} + \sqrt{5}\):

\[ \frac{7(\sqrt{12} + \sqrt{5})}{(\sqrt{12} - \sqrt{5})(\sqrt{12} + \sqrt{5})} = \frac{7(\sqrt{12} + \sqrt{5})}{12 - 5} = \frac{7(\sqrt{12} + \sqrt{5})}{7} = \sqrt{12} + \sqrt{5}. \]

3. For the third term \(-\frac{5}{\sqrt{12} - \sqrt{7}}\), multiply by \(\sqrt{12} + \sqrt{7}\):

\[ -\frac{5(\sqrt{12} + \sqrt{7})}{(\sqrt{12} - \sqrt{7})(\sqrt{12} + \sqrt{7})} = -\frac{5(\sqrt{12} + \sqrt{7})}{12 - 7} = -\frac{5(\sqrt{12} + \sqrt{7})}{5} = -(\sqrt{12} + \sqrt{7}). \]

### Step 2: Combine the terms
Now we can combine the simplified terms:

\[ (\sqrt{7} - \sqrt{5}) + (\sqrt{12} + \sqrt{5}) - (\sqrt{12} + \sqrt{7}). \]

### Step 3: Simplify further
Combining like terms:

1. The \(\sqrt{7}\) terms: \(\sqrt{7} - \sqrt{7} = 0\).
2. The \(\sqrt{12}\) terms: \(\sqrt{12} - \sqrt{12} = 0\).
3. The \(\sqrt{5}\) terms: \(-\sqrt{5} + \sqrt{5} = 0\).

Thus, the entire expression simplifies to:

\[ 0. \]

### Final Answer:
The simplified form of the expression is

\[ \boxed{0}. \]