Answer
$\dfrac{30+25\sqrt{6}-2\sqrt{33}-5\sqrt{22}}{-38}$
Work Step by Step
The given expression, $
\dfrac{5\sqrt{3}-\sqrt{11}}{2\sqrt{3}-5\sqrt{2}}
,$ simplifies to
\begin{array}{l}\require{cancel}
\dfrac{5\sqrt{3}-\sqrt{11}}{2\sqrt{3}-5\sqrt{2}}\cdot\dfrac{2\sqrt{3}+5\sqrt{2}}{2\sqrt{3}+5\sqrt{2}}
\\\\=
\dfrac{5\sqrt{3}(2\sqrt{3})+5\sqrt{3}(5\sqrt{2})-\sqrt{11}(2\sqrt{3})-\sqrt{11}(5\sqrt{2})}{(2\sqrt{3})^2-(5\sqrt{2})^2}
\\\\=
\dfrac{5(2)\sqrt{3(3)}+5(5)\sqrt{3(2)}-2\sqrt{11(3)}-5\sqrt{11(2)}}{4\cdot3-25\cdot2}
\\\\=
\dfrac{10\sqrt{(3)^2}+25\sqrt{6}-2\sqrt{33}-5\sqrt{22}}{12-50}
\\\\=
\dfrac{10\cdot3+25\sqrt{6}-2\sqrt{33}-5\sqrt{22}}{-38}
\\\\=
\dfrac{30+25\sqrt{6}-2\sqrt{33}-5\sqrt{22}}{-38}
.\end{array}
