What is the following quotient? \( \frac{\sqrt{6}+\sqrt{11}}{\sqrt{5}+\sqrt{3}} \)

To simplify the expression \( \frac{\sqrt{6}+\sqrt{11}}{\sqrt{5}+\sqrt{3}} \), we can rationalize the denominator. This involves multiplying the numerator and denominator by the conjugate of the denominator. The conjugate of \( \sqrt{5} + \sqrt{3} \) is \( \sqrt{5} - \sqrt{3} \). Let's perform this step: \[ \frac{\sqrt{6}+\sqrt{11}}{\sqrt{5}+\sqrt{3}} \cdot \frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}-\sqrt{3}} = \frac{(\sqrt{6}+\sqrt{11})(\sqrt{5}-\sqrt{3})}{(\sqrt{5}+\sqrt{3})(\sqrt{5}-\sqrt{3})} \] The denominator becomes: \[ (\sqrt{5})^2 - (\sqrt{3})^2 = 5 - 3 = 2 \] Now, for the numerator: \[ (\sqrt{6}+\sqrt{11})(\sqrt{5}-\sqrt{3}) = \sqrt{30} - \sqrt{18} + \sqrt{55} - \sqrt{33} \] which simplifies to: \[ \sqrt{30} - 3\sqrt{2} + \sqrt{55} - \sqrt{33} \] Incorporating this into our expression, we have: \[ \frac{\sqrt{30} + \sqrt{55} - \sqrt{18} - \sqrt{33}}{2} \] Thus, the final quotient simplifies to: \[ \frac{\sqrt{30} + \sqrt{55} - 3\sqrt{2} - \sqrt{33}}{2} \] So, \( \frac{\sqrt{6}+\sqrt{11}}{\sqrt{5}+\sqrt{3}} = \frac{\sqrt{30} + \sqrt{55} - 3\sqrt{2} - \sqrt{33}}{2} \).