what is the following quotient?\n$\frac{sqrt{6}+sqrt{11}}{sqrt{5}+sqrt{3}}$\n$\frac{sqrt{30}+3sqrt{2}+sqrt{55…

Explanation:

Step1: Rationalize the denominator

Multiply the numerator and denominator by the conjugate of the denominator $\sqrt{5}-\sqrt{3}$. [ \begin{align*} &\frac{\sqrt{6}+\sqrt{11}}{\sqrt{5}+\sqrt{3}}\times\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})} \end{align*} ]

Step2: Expand the numerator and denominator

For the numerator: $(\sqrt{6}+\sqrt{11})(\sqrt{5}-\sqrt{3})=\sqrt{6}\times\sqrt{5}-\sqrt{6}\times\sqrt{3}+\sqrt{11}\times\sqrt{5}-\sqrt{11}\times\sqrt{3}=\sqrt{30}- \sqrt{18}+\sqrt{55}-\sqrt{33}=\sqrt{30}-3\sqrt{2}+\sqrt{55}-\sqrt{33}$. For the denominator: $(\sqrt{5}+\sqrt{3})(\sqrt{5}-\sqrt{3})=(\sqrt{5})^2-(\sqrt{3})^2 = 5 - 3=2$.

Step3: Write the final quotient

The quotient is $\frac{\sqrt{30}-3\sqrt{2}+\sqrt{55}-\sqrt{33}}{2}$.

Answer:

$\frac{\sqrt{30}-3\sqrt{2}+\sqrt{55}-\sqrt{33}}{2}$