Answer
$-20\sqrt{3}+18\sqrt{6}$
Work Step by Step
Using the properties of radicals, the given expression, $
5\sqrt{27}-3\sqrt{24}+8\sqrt{54}-7\sqrt{75}
,$ simplifies to
\begin{array}{l}\require{cancel}
5\sqrt{9\cdot3}-3\sqrt{4\cdot6}+8\sqrt{9\cdot6}-7\sqrt{25\cdot3}
\\\\=
5\sqrt{(3)^2\cdot3}-3\sqrt{(2)^2\cdot6}+8\sqrt{(3)^2\cdot6}-7\sqrt{(5)^2\cdot3}
\\\\=
5\cdot3\sqrt{3}-3\cdot2\sqrt{6}+8\cdot3\sqrt{6}-7\cdot5\sqrt{3}
\\\\=
15\sqrt{3}-6\sqrt{6}+24\sqrt{6}-35\sqrt{3}
\\\\=
(15\sqrt{3}-35\sqrt{3})+(-6\sqrt{6}+24\sqrt{6})
\\\\=
-20\sqrt{3}+18\sqrt{6}
.\end{array}
