ANSWER
[tex]\frac{\sqrt{30} + \sqrt{55}- \sqrt{33} -3\sqrt{2} }{ 2 } [/tex]
EXPLANATION
The given radical expression is
[tex] \frac{ \sqrt{6} + \sqrt{11} }{ \sqrt{5} + \sqrt{3} } [/tex]
We rationalize the denominator to get:
[tex]\frac{ \sqrt{6} + \sqrt{11} }{ \sqrt{5} + \sqrt{3} } \times \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} - \sqrt{3}} [/tex]
This implies that:
[tex] \frac{( \sqrt{6} + \sqrt{11} )(\sqrt{5} - \sqrt{3})}{ (\sqrt{5} + \sqrt{3})(\sqrt{5} - \sqrt{3}) } [/tex]
We expand using the distributive property and also applying difference of two squares on the denominator.
[tex] \frac{( \sqrt{6} \times \sqrt{5} - \sqrt{6} \times \sqrt{3} + \sqrt{5} \times \sqrt{11} - \sqrt{3} \times \sqrt{11} )}{ {( \sqrt{5} )}^{2} - {( \sqrt{3} )}^{2} } [/tex]
This will give us,
[tex]\frac{\sqrt{30} - 3\sqrt{2} + \sqrt{55}-\sqrt{33} }{ 5 - 3 } [/tex]
[tex]\frac{\sqrt{30} + \sqrt{55}- \sqrt{33} -3\sqrt{2} }{ 2 } [/tex]
