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

Answer

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

### Step 1: Identify the common denominator
The denominators of the two fractions are \(\sqrt{6} - \sqrt{2}\) and \(\sqrt{6} + \sqrt{2}\). The common denominator will be the product of these two denominators:
\[ (\sqrt{6} - \sqrt{2})(\sqrt{6} + \sqrt{2}) \]

### Step 2: Rewrite the expression with the common denominator
We can rewrite the expression as:
\[ \frac{\sqrt{2}(\sqrt{6} + \sqrt{2})}{(\sqrt{6} - \sqrt{2})(\sqrt{6} + \sqrt{2})} - \frac{\sqrt{3}(\sqrt{6} - \sqrt{2})}{(\sqrt{6} - \sqrt{2})(\sqrt{6} + \sqrt{2})} \]
This simplifies to:
\[ \frac{\sqrt{2}(\sqrt{6} + \sqrt{2}) - \sqrt{3}(\sqrt{6} - \sqrt{2})}{(\sqrt{6} - \sqrt{2})(\sqrt{6} + \sqrt{2})} \]

### Step 3: Simplify the numerator
Now we will simplify the numerator:
\[ \sqrt{2}(\sqrt{6} + \sqrt{2}) - \sqrt{3}(\sqrt{6} - \sqrt{2}) \]
Expanding both terms:
\[ = \sqrt{2}\sqrt{6} + \sqrt{2}\sqrt{2} - \sqrt{3}\sqrt{6} + \sqrt{3}\sqrt{2} \]
This gives us:
\[ = \sqrt{12} + 2 - \sqrt{18} + \sqrt{6} \]

### Step 4: Simplify the square roots
Now we simplify the square roots:
\[ = 2\sqrt{3} + 2 - 3\sqrt{2} + \sqrt{6} \]

### Step 5: Combine the terms
Now we combine the terms:
\[ = (2 - 3\sqrt{2} + 2\sqrt{3} + \sqrt{6}) \]

### Step 6: Simplify the denominator
The denominator simplifies using the difference of squares:
\[ (\sqrt{6})^2 - (\sqrt{2})^2 = 6 - 2 = 4 \]

### Step 7: Final expression
Putting it all together, we have:
\[ \frac{2 - 3\sqrt{2} + 2\sqrt{3} + \sqrt{6}}{4} \]

### Step 8: Final simplification
We can express this as:
\[ \frac{1}{4}(2 - 3\sqrt{2} + 2\sqrt{3} + \sqrt{6}) \]

### Final Answer:
\[ \frac{2 - 3\sqrt{2} + 2\sqrt{3} + \sqrt{6}}{4} \]