`(sqrt5)/(sqrt3+sqrt2)-(3sqrt3)/(sqrt5+sqrt2)+(2sqrt2)/(sqrt5+sqrt3)` 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 simplify each term step by step.

### Step 1: Simplify the first term
We start with the first term:

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

To simplify this, we multiply the numerator and denominator by the conjugate of the denominator, \(\sqrt{3}-\sqrt{2}\):

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

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: Simplify the second term
Now, we simplify the second term:

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

Again, we multiply the numerator and denominator by the conjugate of the denominator, \(\sqrt{5}-\sqrt{2}\):

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

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: Simplify the third term
Now we simplify the third term:

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

We multiply the numerator and denominator by the conjugate of the denominator, \(\sqrt{5}-\sqrt{3}\):

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

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 \(\frac{\sqrt{5}}{\sqrt{3}+\sqrt{2}} - \frac{3\sqrt{3}}{\sqrt{5}+\sqrt{2}} + \frac{2\sqrt{2}}{\sqrt{5}+\sqrt{3}}\) is equal to \(0\).