`[(3sqrt2)/(sqrt3 + sqrt6) - (4sqrt3)/(sqrt6 + sqrt2) + (sqrt6)/(sqrt2 + sqrt3)]` is simplified to

Options

A. `sqrt6`
B. `sqrt3`
C. `sqrt2`
D. 0

Answer

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

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

1. For the first term \(\frac{3\sqrt{2}}{\sqrt{3} + \sqrt{6}}\), multiply the numerator and denominator by \(\sqrt{3} - \sqrt{6}\):
\[ \frac{3\sqrt{2}(\sqrt{3} - \sqrt{6})}{(\sqrt{3} + \sqrt{6})(\sqrt{3} - \sqrt{6})} = \frac{3\sqrt{2}(\sqrt{3} - \sqrt{6})}{3 - 6} = \frac{3\sqrt{2}(\sqrt{3} - \sqrt{6})}{-3} = -\sqrt{2}(\sqrt{3} - \sqrt{6}) \]

2. For the second term \(\frac{4\sqrt{3}}{\sqrt{6} + \sqrt{2}}\), multiply the numerator and denominator by \(\sqrt{6} - \sqrt{2}\):
\[ \frac{4\sqrt{3}(\sqrt{6} - \sqrt{2})}{(\sqrt{6} + \sqrt{2})(\sqrt{6} - \sqrt{2})} = \frac{4\sqrt{3}(\sqrt{6} - \sqrt{2})}{6 - 2} = \frac{4\sqrt{3}(\sqrt{6} - \sqrt{2})}{4} = \sqrt{3}(\sqrt{6} - \sqrt{2}) \]

3. For the third term \(\frac{\sqrt{6}}{\sqrt{2} + \sqrt{3}}\), multiply the numerator and denominator by \(\sqrt{2} - \sqrt{3}\):
\[ \frac{\sqrt{6}(\sqrt{2} - \sqrt{3})}{(\sqrt{2} + \sqrt{3})(\sqrt{2} - \sqrt{3})} = \frac{\sqrt{6}(\sqrt{2} - \sqrt{3})}{2 - 3} = -\sqrt{6}(\sqrt{2} - \sqrt{3}) \]

### Step 2: Combine the terms
Now we can rewrite the expression with the simplified terms:
\[ -\sqrt{2}(\sqrt{3} - \sqrt{6}) - \sqrt{3}(\sqrt{6} - \sqrt{2}) - \sqrt{6}(\sqrt{2} - \sqrt{3}) \]

### Step 3: Distribute and combine like terms
Distributing each term:
1. \(-\sqrt{2}\sqrt{3} + \sqrt{2}\sqrt{6}\)
2. \(-\sqrt{3}\sqrt{6} + \sqrt{3}\sqrt{2}\)
3. \(-\sqrt{6}\sqrt{2} + \sqrt{6}\sqrt{3}\)

Combining these gives:
\[ (-\sqrt{2}\sqrt{3} + \sqrt{3}\sqrt{2}) + (\sqrt{2}\sqrt{6} - \sqrt{6}\sqrt{2}) + (-\sqrt{3}\sqrt{6} + \sqrt{6}\sqrt{3}) = 0 \]

### Final Result
Thus, the expression simplifies to:
\[ 0 \]