a: \(\sqrt3+\sqrt{32}-\frac{\sqrt3}{2}-\sqrt{\frac13}-\sqrt{\frac{1}{12}}\)
\(=\sqrt3-\frac{\sqrt3}{2}-\frac{\sqrt3}{3}-\sqrt{\frac{3}{36}}+4\sqrt2\)
\(=\frac{6\sqrt3}{6}-\frac{3\sqrt3}{6}-\frac{2\sqrt3}{6}-\frac{\sqrt3}{6}+4\sqrt2=4\sqrt2\)
b: \(\frac12\cdot\sqrt{\left(\sqrt3-2\right)^2}-\sqrt{\frac23}\cdot\sqrt{2-\sqrt3}\)
\(=\frac12\left|\sqrt3-2\right|-\sqrt{\frac13}\cdot\sqrt{4-2\sqrt3}\)
\(=\frac12\left(2-\sqrt3\right)-\frac{\sqrt3}{3}\cdot\sqrt{\left(\sqrt3-1\right)^2}\)
\(=1-\frac12\sqrt3-\frac{\sqrt3}{3}\left(\sqrt3-1\right)=1-\frac12\sqrt3-3+\frac{\sqrt3}{3}=-\frac12\sqrt3+\frac13\sqrt3=-\frac16\sqrt3\)
c: \(\frac{2\sqrt3+3\sqrt2}{\sqrt2+\sqrt3}-\frac{\sqrt3-1}{\sqrt3-\sqrt2}-2\sqrt{\frac12}-\frac{3}{\sqrt3}\)
\(=\frac{\sqrt6\left(\sqrt2+\sqrt3\right)}{\sqrt2+\sqrt3}-\frac{\left(\sqrt3-1\right)\left(\sqrt3+\sqrt2\right)}{\left(\sqrt3-\sqrt2\right)\left(\sqrt3+\sqrt2\right)}-\sqrt2-\sqrt3\)
\(=\sqrt6-\left(\sqrt3-1\right)\left(\sqrt3+\sqrt2\right)-\sqrt2-\sqrt3\)
\(=\sqrt6-3-\sqrt6+\sqrt3+\sqrt2-\sqrt2-\sqrt3=-3\)
d: \(\left(\sqrt3-1\right)\left\lbrack3+\sqrt3-\frac{\left(1-\sqrt3\right)^2}{\sqrt{7-\sqrt{48}}}\right\rbrack+\frac{4}{1-\sqrt3}\)
\(=\sqrt3\left(\sqrt3-1\right)\left(\sqrt3+1\right)-\frac{\left(\sqrt3-1\right)^3}{\sqrt{7-4\sqrt3}}-\frac{4\left(\sqrt3+1\right)}{\left(\sqrt3-1\right)\left(\sqrt3+1\right)}\)
\(=\sqrt3\left(3-1\right)-\frac{\left(\sqrt3-1\right)^3}{2-\sqrt3}-2\left(\sqrt3+1\right)\)
\(=2\sqrt3-2\sqrt3-2-\frac{2\left(\sqrt3-1\right)^3}{4-2\sqrt3}=-2-\frac{2\left(\sqrt3-1\right)^3}{\left(\sqrt3-1\right)^2}=-2-2\left(\sqrt3-1\right)\)
\(=-2-2\sqrt3+2=-2\sqrt3\)
