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 first term
We start with the first term \(\frac{2}{\sqrt{7}+\sqrt{5}}\). To rationalize the denominator, we multiply the numerator and denominator by the conjugate of the denominator, which is \(\sqrt{7}-\sqrt{5}\):

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

**Hint for Step 1:** Use the conjugate of the denominator to eliminate the square roots in the denominator.

### Step 2: Rationalize the second term
Next, we simplify the second term \(\frac{7}{\sqrt{12}-\sqrt{5}}\) by multiplying by the conjugate \(\sqrt{12}+\sqrt{5}\):

\[ \frac{7}{\sqrt{12}-\sqrt{5}} \cdot \frac{\sqrt{12}+\sqrt{5}}{\sqrt{12}+\sqrt{5}} = \frac{7(\sqrt{12}+\sqrt{5})}{(\sqrt{12})^2-(\sqrt{5})^2} = \frac{7(\sqrt{12}+\sqrt{5})}{12-5} = \frac{7(\sqrt{12}+\sqrt{5})}{7} = \sqrt{12}+\sqrt{5} \]

**Hint for Step 2:** Again, use the conjugate to rationalize the denominator.

### Step 3: Rationalize the third term
Now, we simplify the third term \(-\frac{5}{\sqrt{12}-\sqrt{7}}\) by multiplying by the conjugate \(\sqrt{12}+\sqrt{7}\):

\[ -\frac{5}{\sqrt{12}-\sqrt{7}} \cdot \frac{\sqrt{12}+\sqrt{7}}{\sqrt{12}+\sqrt{7}} = -\frac{5(\sqrt{12}+\sqrt{7})}{(\sqrt{12})^2-(\sqrt{7})^2} = -\frac{5(\sqrt{12}+\sqrt{7})}{12-7} = -\frac{5(\sqrt{12}+\sqrt{7})}{5} = -(\sqrt{12}+\sqrt{7}) \]

**Hint for Step 3:** Use the conjugate to rationalize the denominator, and remember to distribute the negative sign.

### Step 4: Combine all the terms
Now we can combine all the simplified terms:

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

This simplifies to:

\[ \sqrt{7}-\sqrt{5} + \sqrt{12}+\sqrt{5} - \sqrt{12} - \sqrt{7} \]

### Step 5: Cancel out like terms
Notice that \(\sqrt{7}\) and \(-\sqrt{7}\) cancel out, \(\sqrt{12}\) and \(-\sqrt{12}\) cancel out, and \(-\sqrt{5}\) and \(+\sqrt{5}\) cancel out:

\[ 0 \]

Thus, the simplified form of the expression is:

\[ \boxed{0} \]