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$\begin{array}{l}
A = \dfrac{3}{{2 + \sqrt 3 }} + \dfrac{{13}}{{4 - \sqrt 3 }} + \dfrac{6}{{\sqrt 3 }}\\
A = 3(2 - \sqrt 3 ) + 4 + \sqrt 3  + 3\sqrt 3 \\
A = 6 - 3\sqrt 3  + 4 + 3\sqrt 3 \\
A = 10\\
b)\\
B = \dfrac{{x\sqrt y  - y\sqrt x }}{{\sqrt {xy} }} + \dfrac{{x - y}}{{\sqrt x  - \sqrt y }}\\
B = \sqrt x  - \sqrt y  + \sqrt x  + \sqrt y \\
B = 2\sqrt x \\
c)\\
C = \dfrac{{\sqrt {4 - 2\sqrt 3 } }}{{\sqrt 6  - \sqrt 2 }}\\
C = \dfrac{{\sqrt {{{(\sqrt 3 )}^2} - 2\sqrt 3  + 1} }}{{\sqrt 2 (\sqrt 3  - 1)}}\\
C = \dfrac{{\sqrt {{{(\sqrt 3  - 1)}^2}} }}{{\sqrt 2 (\sqrt 3  - 1)}}\\
C = \dfrac{1}{{\sqrt 2 }}
\end{array}$

$\begin{array}{l}
d)D = (3\sqrt 2  + \sqrt 6 )(\sqrt {6 - 3\sqrt 3 } )\\
D = \sqrt 6 (\sqrt 3  + 1)\sqrt {3(2 - \sqrt 3 )} \\
D = 3\sqrt {2(2 - \sqrt 3 )} (\sqrt 3  + 1)\\
D = 3\sqrt {4 - 2\sqrt 3 } (\sqrt 3  + 1)\\
D = 3\sqrt {{{(\sqrt 3 )}^2} - 2\sqrt 3  + 1} .(\sqrt 3  + 1)\\
D = 3\sqrt {{{(\sqrt 3  - 1)}^2}} .(\sqrt 3  + 1)\\
D = 3.(\sqrt 3  - 1)(\sqrt 3  + 1)\\
D = 3.(3 - 1)\\
D = 3.2 = 6
\end{array}$