The simplification of
`(3sqrt2)/(sqrt3+sqrt6)-(4sqrt3)/(sqrt6+sqrt2)+(sqrt6)/(sqrt2+sqrt3)` is

Options

A. a)`sqrt6`
B. b)`sqrt3`
C. c)`sqrt2`
D. d)6`sqrt2` - 2`sqrt6`

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 rationalize the denominators and combine the fractions step by step.

### Step 1: Rationalize the first term

We start with the first term:

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

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

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

Calculating the denominator:

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

Thus, the first term becomes:

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

### Step 2: Rationalize the second term

Next, we simplify the second term:

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

Again, we multiply by the conjugate \(\sqrt{6} - \sqrt{2}\):

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

Calculating the denominator:

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

Thus, the second term becomes:

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

### Step 3: Simplify the third term

Now, we simplify the third term:

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

We multiply by the conjugate \(\sqrt{2} - \sqrt{3}\):

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

Calculating the denominator:

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

Thus, the third term becomes:

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

### Step 4: Combine all terms

Now we combine all three terms:

1. From the first term: \(-\sqrt{6} + \sqrt{6}\)
2. From the second term: \(-3\sqrt{2} + \sqrt{6}\)
3. From the third term: \(-2\sqrt{3} + 3\sqrt{2}\)

Combining these, we have:

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

### Final Answer

Thus, the simplified expression is:

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