a: \(\frac12\cdot\sqrt{48}-2\sqrt{75}+\frac{\sqrt{33}}{\sqrt{11}}+5\cdot\sqrt{1\frac13}\)
\(=\frac12\cdot4\sqrt3-2\cdot5\sqrt3+\sqrt3+5\cdot\sqrt{\frac43}\)
\(=2\sqrt3-10\sqrt3+\sqrt3+5\cdot\frac{2\sqrt3}{3}=-7\sqrt3+\frac{10}{3}\sqrt3=-\frac{11}{3}\sqrt3\)
b: \(\sqrt{\left(2\sqrt3-5\right)^2}-2\cdot\sqrt{7+4\sqrt3}\)
\(=5-2\sqrt3-2\cdot\sqrt{\left(2+\sqrt3\right)^2}\)
\(=5-2\sqrt3-2\left(2+\sqrt3\right)=5-2\sqrt3-4-2\sqrt3=1-4\sqrt3\)
c: \(\left(3+\sqrt8\right)\cdot\sqrt{17-12\sqrt2}\)
\(=\left(3+2\sqrt2\right)\cdot\sqrt{\left(3-2\sqrt2\right)^2}\)
\(=\left(3+2\sqrt2\right)\left(3-2\sqrt2\right)=9-8=1\)
d: \(\sqrt{32-3\cdot\sqrt{60}}-\sqrt{\frac{\left(2-\sqrt5\right)^2}{9}}\)
\(=\sqrt{32-6\sqrt{15}}-\frac{\left|2-\sqrt5\right|}{3}\)
\(=\sqrt{32-2\cdot\sqrt{135}}-\frac{\sqrt5-2}{3}=\sqrt{\left(3\sqrt3-\sqrt5\right)^2}-\frac{\sqrt5-2}{3}\)
\(=3\sqrt3-\sqrt5-\frac{\sqrt5-2}{3}=\frac{3\left(3\sqrt3-\sqrt5\right)-\sqrt5+2}{3}=\frac{9\sqrt3-4\sqrt5+2}{3}\)
e: \(10\sqrt{500}-\sqrt{\frac15}-\left(\sqrt{20}-\sqrt{25}\right)\)
\(=10\cdot10\sqrt5-\frac{\sqrt5}{5}-\left(2\sqrt5-5\right)\)
\(=100\sqrt5-\frac15\cdot\sqrt5-2\sqrt5+5=5-97,8\sqrt5\)
