Simplify : `(3sqrt(2))/(sqrt(6)-sqrt(3))-(4sqrt(3))/(sqrt(6)-sqrt(2))+(2sqrt(3))/(sqrt(6)+2)`

Options

A. `sqrt(2)`
B. `sqrt(3)`
C. `1`
D. `0`

Answer

To simplify the expression \(\frac{3\sqrt{2}}{\sqrt{6}-\sqrt{3}} - \frac{4\sqrt{3}}{\sqrt{6}-\sqrt{2}} + \frac{2\sqrt{3}}{\sqrt{6}+2}\), we will follow these steps:

### Step 1: Rationalize the denominators
We will rationalize each term in the expression.

1. For the first term \(\frac{3\sqrt{2}}{\sqrt{6}-\sqrt{3}}\):
- Multiply the numerator and denominator by the conjugate of the denominator, \(\sqrt{6} + \sqrt{3}\):
\[ \frac{3\sqrt{2}(\sqrt{6}+\sqrt{3})}{(\sqrt{6}-\sqrt{3})(\sqrt{6}+\sqrt{3})} \]
- The denominator becomes:
\[ (\sqrt{6})^2 - (\sqrt{3})^2 = 6 - 3 = 3 \]
- So, the first term simplifies to:
\[ \frac{3\sqrt{2}(\sqrt{6}+\sqrt{3})}{3} = \sqrt{2}(\sqrt{6}+\sqrt{3}) = \sqrt{12} + \sqrt{6} \]

2. For the second term \(\frac{4\sqrt{3}}{\sqrt{6}-\sqrt{2}}\):
- Multiply the numerator and denominator by the conjugate of the denominator, \(\sqrt{6} + \sqrt{2}\):
\[ \frac{4\sqrt{3}(\sqrt{6}+\sqrt{2})}{(\sqrt{6}-\sqrt{2})(\sqrt{6}+\sqrt{2})} \]
- The denominator becomes:
\[ (\sqrt{6})^2 - (\sqrt{2})^2 = 6 - 2 = 4 \]
- So, the second term simplifies to:
\[ \frac{4\sqrt{3}(\sqrt{6}+\sqrt{2})}{4} = \sqrt{3}(\sqrt{6}+\sqrt{2}) = \sqrt{18} + \sqrt{6} \]

3. For the third term \(\frac{2\sqrt{3}}{\sqrt{6}+2}\):
- Multiply the numerator and denominator by the conjugate of the denominator, \(\sqrt{6}-2\):
\[ \frac{2\sqrt{3}(\sqrt{6}-2)}{(\sqrt{6}+2)(\sqrt{6}-2)} \]
- The denominator becomes:
\[ (\sqrt{6})^2 - (2)^2 = 6 - 4 = 2 \]
- So, the third term simplifies to:
\[ \frac{2\sqrt{3}(\sqrt{6}-2)}{2} = \sqrt{3}(\sqrt{6}-2) = \sqrt{18} - 2\sqrt{3} \]

### Step 2: Combine all terms
Now we can combine all the simplified terms:
\[ (\sqrt{12} + \sqrt{6}) - (\sqrt{18} + \sqrt{6}) + (\sqrt{18} - 2\sqrt{3}) \]

### Step 3: Simplify the expression
Combine like terms:
1. The \(\sqrt{6}\) terms:
\[ \sqrt{6} - \sqrt{6} = 0 \]
2. The \(\sqrt{18}\) terms:
\[ -\sqrt{18} + \sqrt{18} = 0 \]
3. The remaining terms:
\[ \sqrt{12} - 2\sqrt{3} \]

### Step 4: Final simplification
Now, we know that \(\sqrt{12} = 2\sqrt{3}\):
\[ 2\sqrt{3} - 2\sqrt{3} = 0 \]

### Final Answer
Thus, the simplified expression is:
\[ \boxed{0} \]