To simplify the expression [tex]\(\frac{2}{\sqrt{13} + \sqrt{11}}\)[/tex], we need to rationalize the denominator.
1. Rationalizing the Denominator:
- The denominator is [tex]\(\sqrt{13} + \sqrt{11}\)[/tex]. To rationalize it, we multiply both the numerator and the denominator by the conjugate of the denominator, [tex]\(\sqrt{13} - \sqrt{11}\)[/tex].
2. Multiplying by the Conjugate:
- The conjugate of [tex]\(\sqrt{13} + \sqrt{11}\)[/tex] is [tex]\(\sqrt{13} - \sqrt{11}\)[/tex].
- We multiply the numerator and the denominator by [tex]\(\sqrt{13} - \sqrt{11}\)[/tex]:
[tex]\[
\frac{2}{\sqrt{13} + \sqrt{11}} \cdot \frac{\sqrt{13} - \sqrt{11}}{\sqrt{13} - \sqrt{11}} = \frac{2(\sqrt{13} - \sqrt{11})}{(\sqrt{13} + \sqrt{11})(\sqrt{13} - \sqrt{11})}
\][/tex]
3. Simplifying the Denominator:
- The product of [tex]\((\sqrt{13} + \sqrt{11})(\sqrt{13} - \sqrt{11})\)[/tex] is a difference of squares:
[tex]\[
(\sqrt{13})^2 - (\sqrt{11})^2 = 13 - 11 = 2
\][/tex]
So the denominator simplifies to [tex]\(2\)[/tex].
4. Simplifying the Numerator:
- The numerator is [tex]\(2(\sqrt{13} - \sqrt{11})\)[/tex].
5. Combining the Results:
- Now our expression is:
[tex]\[
\frac{2(\sqrt{13} - \sqrt{11})}{2}
\][/tex]
- The [tex]\(2\)[/tex]s in the numerator and denominator cancel each other out:
[tex]\[
\sqrt{13} - \sqrt{11}
\][/tex]
Therefore, the simplified form of [tex]\(\frac{2}{\sqrt{13} + \sqrt{11}}\)[/tex] is [tex]\(\sqrt{13} - \sqrt{11}\)[/tex].
Thus, the correct answer from the given options is:
[tex]\[
\boxed{\sqrt{13} - \sqrt{11}}
\][/tex]
