To simplify the expression [tex]\(\frac{1}{2-\sqrt{2}}\)[/tex] by rationalizing the denominator, we follow these steps:
1. Write down the expression:
[tex]\[\frac{1}{2-\sqrt{2}}\][/tex]
2. Identify the conjugate of the denominator. The conjugate of [tex]\(2 - \sqrt{2}\)[/tex] is [tex]\(2 + \sqrt{2}\)[/tex].
3. Multiply the numerator and the denominator by the conjugate of the denominator to eliminate the square root:
[tex]\[
\frac{1 \cdot (2 + \sqrt{2})}{(2 - \sqrt{2}) \cdot (2 + \sqrt{2})}
\][/tex]
4. Simplify the numerator:
[tex]\[
1 \cdot (2 + \sqrt{2}) = 2 + \sqrt{2}
\][/tex]
5. Simplify the denominator using the difference of squares formula [tex]\( (a - b)(a + b) = a^2 - b^2 \)[/tex]:
[tex]\[
(2 - \sqrt{2})(2 + \sqrt{2}) = 2^2 - (\sqrt{2})^2 = 4 - 2 = 2
\][/tex]
6. Combine the simplified numerator and denominator:
[tex]\[
\frac{2 + \sqrt{2}}{2}
\][/tex]
Thus, the expression [tex]\(\frac{1}{2-\sqrt{2}}\)[/tex] simplifies to:
[tex]\[
\frac{2 + \sqrt{2}}{2}
\][/tex]
So, the correct answer is:
[tex]\[ \boxed{\frac{2+\sqrt{2}}{2}} \][/tex]
