解$：(1)$原式$=3\sqrt {3}-4\sqrt {3}+\sqrt {2}×\sqrt {6}$
$=-\sqrt {3}+2\sqrt {3}$
$=\sqrt {3}$
$(2)$原式$=4+\sqrt {\frac {1}{2}×12}-2\sqrt {6}$
$=4+\sqrt {6}-2\sqrt {6}$
$=4-\sqrt {6}$
$(3)$原式$=2\sqrt {3}×(2\sqrt {3}-15\sqrt {3}+\frac {1}{3}×6\sqrt {3})$
$=2\sqrt {3}×(-11\sqrt {3})$
$=-66$
$(4)$原式$=4\sqrt {3}×\frac {\sqrt {3}}{4}÷\sqrt {2}-2\sqrt {2}$
$=3÷\sqrt {2}-2\sqrt {2}$
$=-\frac {\sqrt {2}}{2}$
