a) Ta có: \(A=\sqrt{20}-2\sqrt{45}+3\sqrt{18}+\sqrt{72}\)

\(=2\sqrt{5}-6\sqrt{5}+9\sqrt{2}+6\sqrt{2}\)

\(=-4\sqrt{5}+15\sqrt{2}\)

b) Ta có: \(B=4\sqrt{\left(\sqrt{3}-1\right)^2}+2\sqrt{12}+4\sqrt{\dfrac{1}{2}}\)

\(=4\left(\sqrt{3}-1\right)+2\cdot2\sqrt{3}+\dfrac{4}{\sqrt{2}}\)

\(=4\sqrt{3}-4+4\sqrt{3}+2\sqrt{2}\)

\(=8\sqrt{3}+2\sqrt{2}-4\)

c) Ta có: \(C=\left(3+\dfrac{3-\sqrt{3}}{\sqrt{3}-1}\right)\left(3-\dfrac{3+\sqrt{3}}{1+\sqrt{3}}\right)\)

\(=\left(3+\sqrt{3}\right)\left(3-\sqrt{3}\right)\)

=9-3

=6

d) Ta có: \(D=\dfrac{1}{2+\sqrt{3}}+\dfrac{1}{2-\sqrt{3}}\)

\(=2-\sqrt{3}+2+\sqrt{3}\)

=4

4: Ta có: \(\dfrac{1}{3+\sqrt{5}}-\dfrac{1}{3-\sqrt{5}}\)

\(=\dfrac{3-\sqrt{5}-3-\sqrt{5}}{4}\)

\(=\dfrac{-\sqrt{5}}{2}\)

5: Ta có: \(\dfrac{2-\sqrt{2}}{1-\sqrt{2}}+\dfrac{\sqrt{2}-\sqrt{6}}{\sqrt{3}-1}\)

\(=-\sqrt{2}-\sqrt{2}\)

\(=-2\sqrt{2}\)

Ta có: \(\dfrac{4}{3\sqrt{3}}-\dfrac{\sqrt{3}-\sqrt{15}}{1-\sqrt{5}}-\dfrac{\sqrt{3}}{1-\sqrt{3}}\)

\(=\dfrac{4\sqrt{3}}{9}-\sqrt{3}+\dfrac{\sqrt{3}\left(1+\sqrt{3}\right)}{2}\)

\(=\dfrac{4\sqrt{3}-9\sqrt{3}}{9}+\dfrac{\sqrt{3}\left(1+\sqrt{3}\right)}{2}\)

\(=\dfrac{-10\sqrt{3}}{18}+\dfrac{9\sqrt{3}\left(1+\sqrt{3}\right)}{18}\)

\(=\dfrac{-10\sqrt{3}+9\sqrt{3}+27}{18}\)

\(=\dfrac{27-\sqrt{3}}{18}\)

\(=\sqrt{3}-1-\sqrt{3}+1-\sqrt{3}=-\sqrt{3}\)

\(=\sqrt{3}-1-\sqrt{3}+1+2-2\sqrt{3}=2-2\sqrt{3}\)

\(\left(\sqrt[3]{\dfrac{1}{9}}+4\cdot\sqrt[3]{\dfrac{1}{72}}-\sqrt[3]{4}\right)\left(\sqrt[3]{72}+\sqrt[3]{96}+\sqrt[3]{128}\right)\)

\(=\left(\dfrac{1}{3}\cdot\sqrt[3]{3}+4\cdot\dfrac{1}{6}\cdot\sqrt[3]{3}-2\sqrt[3]{\dfrac{1}{2}}\right)\left(2\sqrt[3]{9}+2\sqrt[3]{12}+4\sqrt[3]{2}\right)\)

\(=\left(\sqrt[3]{3}-2\sqrt[3]{\dfrac{1}{2}}\right)\left(6\sqrt[3]{3}+2\sqrt[3]{12}+4\sqrt[3]{2}\right)\)

\(=6\cdot3+2\sqrt[3]{36}+4\sqrt[3]{6}-12\sqrt[3]{\dfrac{3}{2}}-4\sqrt[3]{6}-8\)

\(=10+12\sqrt[3]{\dfrac{1}{6}}-6\sqrt[3]{12}\)

a) \(\dfrac{1}{3\sqrt{2}-2\sqrt{3}}-\dfrac{1}{2\sqrt{3}+3\sqrt{2}}\)

\(=\dfrac{1}{\sqrt{6}\left(\sqrt{3}-\sqrt{2}\right)}-\dfrac{1}{\sqrt{6}\left(\sqrt{3}+\sqrt{2}\right)}\)

\(=\dfrac{\sqrt{3}+\sqrt{2}}{\sqrt{6}\left(\sqrt{3}+\sqrt{2}\right)\left(\sqrt{3}-\sqrt{2}\right)}-\dfrac{\sqrt{3}-\sqrt{2}}{\sqrt{6}\left(\sqrt{3}+\sqrt{2}\right)\left(\sqrt{3}-\sqrt{2}\right)}\)

\(=\dfrac{\sqrt{3}+\sqrt{2}}{\sqrt{6}}-\dfrac{\sqrt{3}-\sqrt{2}}{\sqrt{6}}\)

\(=\dfrac{\sqrt{3}+\sqrt{2}-\sqrt{3}+\sqrt{2}}{\sqrt{6}}\)

\(=\dfrac{2\sqrt{2}}{\sqrt{6}}\)

\(=\dfrac{2\sqrt{3}}{3}\)

b) \(\dfrac{4\sqrt{3}-8}{2\sqrt{3}-4}-\dfrac{1}{\sqrt{5}-2}\)

\(=\dfrac{4\left(\sqrt{3}-2\right)}{2\left(\sqrt{3}-2\right)}-\dfrac{\sqrt{5}+2}{\left(\sqrt{5}+2\right)\left(\sqrt{5}-2\right)}\)

\(=\dfrac{4}{2}-\dfrac{\sqrt{5}+2}{5-4}\)

\(=2-\sqrt{5}-2\)

\(=-\sqrt{5}\)

\(C=\sqrt{4+\sqrt{5\sqrt{3}+5\sqrt{48-10\sqrt{7+4\sqrt{3}}}}}\)

\(C=\sqrt{4+\sqrt{5\sqrt{3}+5\sqrt{48-10\sqrt{7+2\sqrt{12}}}}}\)

\(C=\sqrt{4+\sqrt{5\sqrt{3}+5\sqrt{48-10\left(2+\sqrt{3}\right)}}}\)

\(C=\sqrt{4+\sqrt{5\sqrt{3}+5\sqrt{28-10\sqrt{3}}}}\)

\(C=\sqrt{4+\sqrt{5\sqrt{3}+5\sqrt{28-2\sqrt{75}}}}\)

\(C=\sqrt{4+\sqrt{5\sqrt{3}+5\left(5-\sqrt{3}\right)}}\)

\(C=\sqrt{4+5}\)

\(C=3\)