The simplified value of `(3 sqrt(2))/(sqrt(3) + sqrt(6)) - (4 sqrt(3))/(sqrt(6) + sqrt(2)) + (sqrt(6))/(sqrt(3) + sqrt(2))` is

Options

A. a) `sqrt(2)`
B. b) `(1)/(sqrt(2))`
C. c) `sqrt(3) - sqrt(2)`
D. d) 0

Answer

To simplify the expression

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

we will follow these steps:

### 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 the denominator by the conjugate of the denominator, which is \(\sqrt{6} - \sqrt{3}\):

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

The denominator simplifies as follows:

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

Thus, the first term becomes:

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

### Step 2: Rationalize the second term
Next, we consider the second term:

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

Again, we rationalize the denominator by multiplying by the conjugate:

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

The denominator simplifies to:

\[ 6 - 2 = 4. \]

Thus, the second term becomes:

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

### Step 3: Rationalize the third term
Now we look at the third term:

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

We rationalize the denominator again:

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

The denominator simplifies to:

\[ 3 - 2 = 1. \]

Thus, the third term becomes:

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

### Step 4: Combine all terms
Now we combine all three simplified terms:

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

This simplifies to:

\[ \sqrt{12} - \sqrt{6} - \sqrt{18} + \sqrt{6} + \sqrt{18} - \sqrt{12}. \]

### Step 5: Simplify the expression
Now we can see that:

- \(\sqrt{12}\) and \(-\sqrt{12}\) cancel each other.
- \(-\sqrt{6}\) and \(+\sqrt{6}\) cancel each other.
- \(-\sqrt{18}\) and \(+\sqrt{18}\) cancel each other.

Thus, the entire expression simplifies to:

\[ 0. \]

### Final Answer
The simplified value of the expression is:

\[ \boxed{0}. \]