To simplify the expression ( \frac{1}{\sqrt{6} - 2} - \frac{1}{\sqrt{6} + 2} ), we can find a common denominator and combine the two fractions.
The denominators are ( \sqrt{6} - 2 ) and ( \sqrt{6} + 2 ). The common denominator is ( (\sqrt{6} - 2)(\sqrt{6} + 2) ).
Now we can rewrite the expression:
[
\frac{1}{\sqrt{6} - 2} - \frac{1}{\sqrt{6} + 2} = \frac{(\sqrt{6} + 2) - (\sqrt{6} - 2)}{(\sqrt{6} - 2)(\sqrt{6} + 2)}
]
Now let's simplify the numerator:
[
(\sqrt{6} + 2) - (\sqrt{6} - 2) = \sqrt{6} + 2 - \sqrt{6} + 2 = 4
]
So, the expression now looks like:
[
\frac{4}{(\sqrt{6} - 2)(\sqrt{6} + 2)}
]
Next, we can simplify the denominator. Notice that:
[
(\sqrt{6} - 2)(\sqrt{6} + 2) = \sqrt{6}^2 - 2^2 = 6 - 4 = 2
]
Putting this all together, we have:
[
\frac{4}{2} = 2
]
Thus, the simplified result of the original expression is:
[
\boxed{2}
]
