`(3sqrt(2))/(sqrt(6)+sqrt(3))-(2sqrt(6))/(sqrt(3)+1)+(2sqrt(3))/(sqrt(6)+2)` is equal to

Options

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

Answer

To solve the expression \((3\sqrt{2})/(\sqrt{6}+\sqrt{3}) - (2\sqrt{6})/(\sqrt{3}+1) + (2\sqrt{3})/(\sqrt{6}+2)\), we will follow these steps:

### Step 1: Rationalize the first term
The first term is \(\frac{3\sqrt{2}}{\sqrt{6}+\sqrt{3}}\). To rationalize it, we multiply the numerator and denominator by the conjugate of the denominator, which is \(\sqrt{6}-\sqrt{3}\):

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

Calculating the denominator:

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

So, 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
The second term is \(-\frac{2\sqrt{6}}{\sqrt{3}+1}\). We rationalize it by multiplying the numerator and denominator by the conjugate of the denominator, \(\sqrt{3}-1\):

\[ -\frac{2\sqrt{6}(\sqrt{3}-1)}{(\sqrt{3}+1)(\sqrt{3}-1)} \]

Calculating the denominator:

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

So, the second term becomes:

\[ -\frac{2\sqrt{6}(\sqrt{3}-1)}{2} = -\sqrt{6}(\sqrt{3}-1) = -\sqrt{18} + \sqrt{6} \]

### Step 3: Rationalize the third term
The third term is \(\frac{2\sqrt{3}}{\sqrt{6}+2}\). We rationalize it by multiplying the numerator and denominator by the conjugate of the denominator, \(\sqrt{6}-2\):

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

Calculating the denominator:

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

So, the third term becomes:

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

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

1. From Step 1: \(\sqrt{12} - \sqrt{6}\)
2. From Step 2: \(-\sqrt{18} + \sqrt{6}\)
3. From Step 3: \(\sqrt{18} - 2\sqrt{3}\)

Combining these:

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

Notice that \(-\sqrt{6} + \sqrt{6}\) cancels out:

\[ \sqrt{12} - 2\sqrt{3} \]

### Step 5: Simplify the expression
Now we simplify \(\sqrt{12}\):

\[ \sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3} \]

So, we have:

\[ 2\sqrt{3} - 2\sqrt{3} = 0 \]

### Final Answer
The expression simplifies to \(0\).