To solve the expression [tex]\(\frac{\sqrt{2}}{\sqrt{6}-\sqrt{2}}\)[/tex], we will rationalize the denominator. When rationalizing the denominator, we aim to eliminate the square root terms in the denominator.
Here's the step-by-step process:
1. Identify the conjugate of the denominator:
The denominator is [tex]\(\sqrt{6} - \sqrt{2}\)[/tex]. Its conjugate is [tex]\(\sqrt{6} + \sqrt{2}\)[/tex].
2. Multiply the numerator and the denominator by the conjugate of the denominator:
[tex]\[
\frac{\sqrt{2}}{\sqrt{6} - \sqrt{2}} \times \frac{\sqrt{6} + \sqrt{2}}{\sqrt{6} + \sqrt{2}} = \frac{\sqrt{2}(\sqrt{6} + \sqrt{2})}{(\sqrt{6} - \sqrt{2})(\sqrt{6} + \sqrt{2})}
\][/tex]
3. Simplify the numerator:
[tex]\[
\sqrt{2}(\sqrt{6} + \sqrt{2}) = \sqrt{2} \cdot \sqrt{6} + \sqrt{2} \cdot \sqrt{2}
\][/tex]
[tex]\[
= \sqrt{12} + \sqrt{4}
\][/tex]
[tex]\[
= 2\sqrt{3} + 2
\][/tex]
4. Simplify the denominator using the difference of squares:
[tex]\[
(\sqrt{6} - \sqrt{2})(\sqrt{6} + \sqrt{2}) = (\sqrt{6})^2 - (\sqrt{2})^2
\][/tex]
[tex]\[
= 6 - 2
\][/tex]
[tex]\[
= 4
\][/tex]
5. Combine the simplified numerator and the denominator:
[tex]\[
\frac{2\sqrt{3} + 2}{4}
\][/tex]
6. Simplify by dividing both terms in the numerator by the denominator:
[tex]\[
\frac{2\sqrt{3}}{4} + \frac{2}{4}
\][/tex]
[tex]\[
= \frac{\sqrt{3}}{2} + \frac{1}{2}
\][/tex]
Therefore, the simplified expression is:
[tex]\[
\frac{\sqrt{2}}{\sqrt{6} - \sqrt{2}} = \frac{1}{2} + \frac{\sqrt{3}}{2}
\][/tex]
