What is the following quotient? \( \frac{2}{\sqrt{13}+\sqrt{11}} \)

To simplify the quotient \( \frac{2}{\sqrt{13}+\sqrt{11}} \), we can rationalize the denominator by multiplying the numerator and the denominator by the conjugate of the denominator, which is \( \sqrt{13}-\sqrt{11} \): \[ \frac{2}{\sqrt{13}+\sqrt{11}} \cdot \frac{\sqrt{13}-\sqrt{11}}{\sqrt{13}-\sqrt{11}} = \frac{2(\sqrt{13}-\sqrt{11})}{(\sqrt{13}+\sqrt{11})(\sqrt{13}-\sqrt{11})} \] Now, simplify the denominator: \[ (\sqrt{13}+\sqrt{11})(\sqrt{13}-\sqrt{11}) = \sqrt{13}^2 - \sqrt{11}^2 = 13 - 11 = 2 \] So the expression now looks like this: \[ \frac{2(\sqrt{13}-\sqrt{11})}{2} \] The 2's cancel out: \[ \sqrt{13} - \sqrt{11} \] Thus, the simplified form of the quotient \( \frac{2}{\sqrt{13}+\sqrt{11}} \) is: \[ \sqrt{13} - \sqrt{11} \]