`[(sqrt3 + sqrt2 )/(sqrt3 - sqrt2) - (sqrt3 - sqrt2)/(sqrt3 + sqrt2)]` simplifies to

Options

A. `2 sqrt6`
B. `4 sqrt6`
C. ` 2 sqrt3`
D. `3 sqrt2`

Answer

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

### Step 1: Find a common denominator
The common denominator for the two fractions is \((\sqrt{3} - \sqrt{2})(\sqrt{3} + \sqrt{2})\).

### Step 2: Rewrite the expression with the common denominator
The expression can be rewritten as:
\[ \frac{(\sqrt{3} + \sqrt{2})^2 - (\sqrt{3} - \sqrt{2})^2}{(\sqrt{3} - \sqrt{2})(\sqrt{3} + \sqrt{2})} \]

### Step 3: Expand the numerators
Using the formula \((a + b)^2 = a^2 + 2ab + b^2\) and \((a - b)^2 = a^2 - 2ab + b^2\):
- For \((\sqrt{3} + \sqrt{2})^2\):
\[ (\sqrt{3})^2 + 2(\sqrt{3})(\sqrt{2}) + (\sqrt{2})^2 = 3 + 2\sqrt{6} + 2 = 5 + 2\sqrt{6} \]
- For \((\sqrt{3} - \sqrt{2})^2\):
\[ (\sqrt{3})^2 - 2(\sqrt{3})(\sqrt{2}) + (\sqrt{2})^2 = 3 - 2\sqrt{6} + 2 = 5 - 2\sqrt{6} \]

### Step 4: Subtract the expanded numerators
Now we subtract the two results:
\[ (5 + 2\sqrt{6}) - (5 - 2\sqrt{6}) = 5 + 2\sqrt{6} - 5 + 2\sqrt{6} = 4\sqrt{6} \]

### Step 5: Simplify the denominator
The denominator simplifies as follows:
\[ (\sqrt{3} - \sqrt{2})(\sqrt{3} + \sqrt{2}) = (\sqrt{3})^2 - (\sqrt{2})^2 = 3 - 2 = 1 \]

### Step 6: Combine the results
Thus, the entire expression simplifies to:
\[ \frac{4\sqrt{6}}{1} = 4\sqrt{6} \]

### Final Answer
The simplified form of the expression is:
\[ \boxed{4\sqrt{6}} \]