a: \(\frac{\sqrt{15}-\sqrt{27}}{\sqrt5-3}+\frac{4}{2+\sqrt3}-\sqrt{\left(2\sqrt3-4\right)^2}\)
\(=\frac{\sqrt3\left(\sqrt5-3\right)}{\sqrt5-3}+\frac{4\left(2-\sqrt3\right)}{\left(2+\sqrt3\right)\left(2-\sqrt3\right)}-\left|4-2\sqrt3\right|\)
\(=\sqrt3+4\left(2-\sqrt3\right)-4+2\sqrt3=3\sqrt3-4+8-4\sqrt3=-\sqrt3+4\)
b: \(6\sqrt{\frac12}-3\sqrt8+\sqrt{19+6\sqrt2}\)
\(=3\sqrt2-3\cdot2\sqrt2+\sqrt{\left(3\sqrt2+1\right)^2}\)
\(=3\sqrt2-6\sqrt2+3\sqrt2+1=1\)
c: \(\frac{8}{\sqrt3-1}+\frac{26}{4+\sqrt3}-\frac{\sqrt6-\sqrt3}{\sqrt2-1}\)
\(=\frac{8\left(\sqrt3+1\right)}{\left(\sqrt3-1\right)\left(\sqrt3+1\right)}+\frac{26\left(4-\sqrt3\right)}{\left(4+\sqrt3\right)\left(4-\sqrt3\right)}-\frac{\sqrt3\left(\sqrt2-1\right)}{\sqrt2-1}\)
\(=4\left(\sqrt3+1\right)+2\left(4-\sqrt3\right)-\sqrt3=4\sqrt3+4+8-2\sqrt3-\sqrt3=\sqrt3+12\)
d: \(\frac{3\sqrt{15}-\sqrt5}{3\sqrt3-1}-\frac{2}{\sqrt5+\sqrt3}+\sqrt{12-6\sqrt3}\)
\(=\frac{\sqrt5\left(3\sqrt3-1\right)}{3\sqrt3-1}-\frac{2\left(\sqrt5-\sqrt3\right)}{\left(\sqrt5+\sqrt3\right)\left(\sqrt5-\sqrt3\right)}+\sqrt{\left(3-\sqrt3\right)^2}\)
\(=\sqrt5-\sqrt5+\sqrt3+3-\sqrt3=3\)
