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

Options

A. `2`
B. `-1`
C. `0`
D. `1`

Answer

To simplify the expression

\[ \frac{6}{2\sqrt{3} - \sqrt{6}} + \frac{\sqrt{6}}{\sqrt{3} + \sqrt{2}} - \frac{4\sqrt{3}}{\sqrt{6} - \sqrt{2}}, \]

we will handle each term separately by rationalizing the denominators.

### Step 1: Simplify the first term

The first term is

\[ \frac{6}{2\sqrt{3} - \sqrt{6}}. \]

To rationalize the denominator, we multiply the numerator and the denominator by the conjugate of the denominator, which is \(2\sqrt{3} + \sqrt{6}\):

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

The denominator simplifies as follows:

\[ (2\sqrt{3})^2 - (\sqrt{6})^2 = 12 - 6 = 6. \]

Thus, the first term becomes:

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

### Step 2: Simplify the second term

The second term is

\[ \frac{\sqrt{6}}{\sqrt{3} + \sqrt{2}}. \]

We rationalize the denominator by multiplying by the conjugate \(\sqrt{3} - \sqrt{2}\):

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

The denominator simplifies as follows:

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

Thus, the second term simplifies to:

\[ \sqrt{6}(\sqrt{3} - \sqrt{2}) = \sqrt{18} - \sqrt{12} = 3\sqrt{2} - 2\sqrt{3}. \]

### Step 3: Simplify the third term

The third term is

\[ -\frac{4\sqrt{3}}{\sqrt{6} - \sqrt{2}}. \]

We rationalize the denominator by multiplying by the conjugate \(\sqrt{6} + \sqrt{2}\):

\[ -\frac{4\sqrt{3}(\sqrt{6} + \sqrt{2})}{(\sqrt{6} - \sqrt{2})(\sqrt{6} + \sqrt{2})}. \]

The denominator simplifies as follows:

\[ (\sqrt{6})^2 - (\sqrt{2})^2 = 6 - 2 = 4. \]

Thus, the third term simplifies to:

\[ -\frac{4\sqrt{3}(\sqrt{6} + \sqrt{2})}{4} = -\sqrt{3}(\sqrt{6} + \sqrt{2}) = -\sqrt{18} - \sqrt{6} = -3\sqrt{2} - \sqrt{6}. \]

### Step 4: Combine all terms

Now we combine all simplified terms:

\[ (2\sqrt{3} + \sqrt{6}) + (3\sqrt{2} - 2\sqrt{3}) - (3\sqrt{2} + \sqrt{6}). \]

Combining like terms:

- The \(\sqrt{3}\) terms: \(2\sqrt{3} - 2\sqrt{3} = 0\).
- The \(\sqrt{6}\) terms: \(\sqrt{6} - \sqrt{6} = 0\).
- The \(\sqrt{2}\) terms: \(3\sqrt{2} - 3\sqrt{2} = 0\).

Thus, the entire expression simplifies to:

\[ 0. \]

### Final Answer

The simplified expression is

\[ \boxed{0}. \]