`(sqrt(5))/(sqrt(3)+sqrt(2))-(3sqrt(3))/(sqrt(5)+sqrt(2))+(2sqrt(2))/(sqrt(5)+sqrt(3))` is equal to :

Options

A. `0`
B. `2sqrt(15)`
C. `2sqrt(10)`
D. `2sqrt(6)`

Answer

To solve the expression

\[ \frac{\sqrt{5}}{\sqrt{3} + \sqrt{2}} - \frac{3\sqrt{3}}{\sqrt{5} + \sqrt{2}} + \frac{2\sqrt{2}}{\sqrt{5} + \sqrt{3}}, \]

we will rationalize each term individually.

### Step 1: Rationalize the first term

For the first term \(\frac{\sqrt{5}}{\sqrt{3} + \sqrt{2}}\), we multiply the numerator and the denominator by the conjugate of the denominator, which is \(\sqrt{3} - \sqrt{2}\):

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

The denominator simplifies as follows:

\[ (\sqrt{3})^2 - (\sqrt{2})^2 = 3 - 2 = 1. \]

Thus, the first term simplifies to:

\[ \sqrt{5}(\sqrt{3} - \sqrt{2}) = \sqrt{15} - \sqrt{10}. \]

### Step 2: Rationalize the second term

Now, for the second term \(-\frac{3\sqrt{3}}{\sqrt{5} + \sqrt{2}}\), we multiply the numerator and denominator by the conjugate of the denominator, which is \(\sqrt{5} - \sqrt{2}\):

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

The denominator simplifies as follows:

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

Thus, the second term simplifies to:

\[ -\frac{3\sqrt{3}(\sqrt{5} - \sqrt{2})}{3} = -\sqrt{3}(\sqrt{5} - \sqrt{2}) = -\sqrt{15} + \sqrt{6}. \]

### Step 3: Rationalize the third term

For the third term \(\frac{2\sqrt{2}}{\sqrt{5} + \sqrt{3}}\), we multiply the numerator and denominator by the conjugate of the denominator, which is \(\sqrt{5} - \sqrt{3}\):

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

The denominator simplifies as follows:

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

Thus, the third term simplifies to:

\[ \frac{2\sqrt{2}(\sqrt{5} - \sqrt{3})}{2} = \sqrt{2}(\sqrt{5} - \sqrt{3}) = \sqrt{10} - \sqrt{6}. \]

### Step 4: Combine all terms

Now we combine all the simplified terms:

\[ (\sqrt{15} - \sqrt{10}) + (-\sqrt{15} + \sqrt{6}) + (\sqrt{10} - \sqrt{6}). \]

Combining like terms:

- The \(\sqrt{15}\) terms: \(\sqrt{15} - \sqrt{15} = 0\).
- The \(\sqrt{10}\) terms: \(-\sqrt{10} + \sqrt{10} = 0\).
- The \(\sqrt{6}\) terms: \(\sqrt{6} - \sqrt{6} = 0\).

Thus, the entire expression simplifies to:

\[ 0. \]

### Final Answer

The expression is equal to \(0\).