a: \(\left(3\sqrt8-\sqrt{18}+2\sqrt{72}\right)\cdot\sqrt2\)
\(=\left(3\cdot2\sqrt2-3\sqrt2+2\cdot6\sqrt2\right)\cdot\sqrt2\)
\(=\left(6\sqrt2-3\sqrt2+12\sqrt2\right)\cdot\sqrt2=15\sqrt2\cdot\sqrt2=30\)
b: \(2\sqrt{48}-2\sqrt{98}+\sqrt{50}-\sqrt{147}\)
\(=2\cdot4\sqrt3-2\cdot7\sqrt2+5\sqrt2-7\sqrt3\)
\(=\sqrt3-9\sqrt2\)
c: \(\sqrt{4-2\sqrt3}+\sqrt{7-4\sqrt3}\)
\(=\sqrt{\left(\sqrt3-1\right)^2}+\sqrt{\left(2-\sqrt3\right)^2}=\sqrt3-1+2-\sqrt3=1\)
d: \(\sqrt{12+6\sqrt3}+\sqrt{12-6\sqrt3}\)
\(=\sqrt{\left(3+\sqrt3\right)^2}+\sqrt{\left(3-\sqrt3\right)^2}\)
\(=3+\sqrt3+3-\sqrt3=6\)
e: \(\sqrt{8-2\sqrt{15}}-\sqrt{8+2\sqrt{15}}\)
\(=\sqrt{\left(\sqrt5-\sqrt3\right)^2}-\sqrt{\left(\sqrt5+\sqrt3\right)^2}\)
\(=\sqrt5-\sqrt3-\sqrt5-\sqrt3=-2\sqrt3\)
f: \(\sqrt{\left(\sqrt5-3\right)^2}+\sqrt{\left(2-\sqrt5\right)^2}\)
\(=\left|3-\sqrt5\right|+\left|\sqrt5-2\right|\)
\(=3-\sqrt5+\sqrt5-2=3-2=1\)
g: \(\frac{3}{\sqrt7-2}+\frac{-3}{\sqrt7+2}\)
\(=\frac{3\left(\sqrt7+2\right)-3\left(\sqrt7-2\right)}{\left(\sqrt7+2\right)\left(\sqrt7-2\right)}\)
\(=\frac{3\sqrt7+6-3\sqrt7+6}{7-4}=\frac{12}{3}=4\)
h: \(\frac{1}{5+2\sqrt3}-\frac{1}{5-2\sqrt3}\)
\(=\frac{5-2\sqrt3-5-2\sqrt3}{\left(5+2\sqrt3\right)\left(5-2\sqrt3\right)}\)
\(=\frac{-4\sqrt3}{25-12}=\frac{-4\sqrt3}{13}\)
i: \(\frac{5\sqrt2-2\sqrt5}{\sqrt5-\sqrt2}-\frac{9}{\sqrt{10}+1}\)
\(=\frac{\sqrt{10}\left(\sqrt5-\sqrt2\right)}{\sqrt5-\sqrt2}-\frac{9\left(\sqrt{10}-1\right)}{\left(\sqrt{10}+1\right)\left(\sqrt{10}-1\right)}\)
\(=\sqrt{10}-\left(\sqrt{10}-1\right)=1\)
k: \(3\sqrt2\left(\sqrt{50}-2\sqrt{18}+\sqrt{98}\right)\)
\(=3\sqrt2\left(5\sqrt2-2\cdot3\sqrt2+7\sqrt2\right)=3\sqrt2\cdot\left(12\sqrt2-6\sqrt2\right)\)
\(=3\sqrt2\cdot6\sqrt2=18\cdot2=36\)
i: \(\sqrt{27}-3\sqrt{48}+2\cdot\sqrt{108}-\sqrt{\left(2-\sqrt3\right)^2}\)
\(=3\sqrt3-3\cdot4\sqrt3+2\cdot6\sqrt3-\left|2-\sqrt3\right|\)
\(=3\sqrt3-\left(2-\sqrt3\right)=3\sqrt3-2+\sqrt3=4\sqrt3-2\)
m: \(\left(\sqrt{3+\sqrt5}-\sqrt{3-\sqrt5}\right)^2\)
\(=3+\sqrt5+3-\sqrt5-2\cdot\sqrt{\left(3+\sqrt5\right)\cdot\left(3-\sqrt5\right)}\)
\(=6-2\cdot\sqrt{9-5}=6-2\cdot2=6-4=2\)
