解$：(1)$原式$=(4\sqrt {3}+2\sqrt {5})+(\sqrt {3}-\sqrt {5})$
$=4\sqrt {3}+\sqrt {3}+2\sqrt {5}-\sqrt {5}$
$=5\sqrt {3}+\sqrt {5}$
$(2)$原式$=2\sqrt {6}-\frac {1}{2}\sqrt {3}+\frac {2\sqrt {6}}{3}-\frac {1}{3}\sqrt {2}$
$=\frac {8\sqrt {6}}{3}-\frac {5\sqrt {3}}{6}$
$(3)$原式$=2\sqrt {3}-\frac {4\sqrt {3}}{3}-\frac {3\sqrt {3}}{10}+\frac {\sqrt {3}}{3}$
$=\frac {7\sqrt {3}}{10}$
$(4)$原式$=4\sqrt {2}+2\sqrt {3}-\frac {\sqrt {2}}{2}-3\sqrt {3}$
$=\frac {7\sqrt {2}}{2}-\sqrt {3}$
