解$：(1)$原式$=27 \sqrt {5} \div \frac {3 \sqrt {5}}{5} ×\frac {3}{2} \sqrt {\frac {8}{3}} $
$=27 \sqrt {5} \div \frac {3 \sqrt {5}}{5} ×\sqrt {6} $
$=45 \sqrt {6}$
$(2)$原式$=(\sqrt {5})^2-2 ×\sqrt {5} ×\sqrt {3}+(\sqrt {3})^2 $
$=5-2 \sqrt {15}+3 $
$=8-2 \sqrt {15}$
