1.Trục căn thức ở mẫu

$\frac{\sqrt{a}-\sqrt{b}}{\sqrt{a}+\sqrt{b}}$

= \(\dfrac{a-2\sqrt{ab}+b}{a-b}\)

a, = \(\frac{\sqrt{7}-5}{2}-\frac{2\left(3-\sqrt{7}\right)}{4}+\frac{6\left(\sqrt{7}+2\right)}{\left(\sqrt{7}-2\right)\left(\sqrt{7}+2\right)}-\frac{5\left(4-\sqrt{7}\right)}{\left(4-\sqrt{7}\right)\left(4+\sqrt{7}\right)}\)

a, = \(=\frac{\sqrt{7}-5}{2}-\frac{3-\sqrt{7}}{2}+\frac{6\sqrt{7}+12}{7-4}-\frac{20-5\sqrt{7}}{16-7}=\frac{\sqrt{7}-5-3+\sqrt{7}}{2}+\frac{6\sqrt{7}+12}{3}-\frac{20-5\sqrt{7}}{9}\)

b. = \(\frac{\sqrt{3}+\sqrt{2}+\sqrt{5}}{\left(\sqrt{3}+\sqrt{2}+\sqrt{5}\right)\left(\sqrt{3}+\sqrt{2}-\sqrt{5}\right)}-\frac{\sqrt{3}+\sqrt{2}-\sqrt{5}}{\left(\sqrt{3}+\sqrt{2}+\sqrt{5}\right)\left(\sqrt{3}+\sqrt{2}-\sqrt{5}\right)}=\frac{\sqrt{3}+\sqrt{2}+\sqrt{5}}{\left(\sqrt{3}+\sqrt{2}\right)^2-\left(\sqrt{5}\right)^2}-\frac{\sqrt{3}+\sqrt{2}-\sqrt{5}}{\left(\sqrt{3}+\sqrt{2}\right)^2-\left(\sqrt{5}\right)^2}\)

a) \(\frac{1}{2+\sqrt{3}}+\frac{\sqrt{2}}{\sqrt{6}}-\frac{2}{3+\sqrt{3}}\)

\(=\frac{3\sqrt{3}+3}{\sqrt{3}.\left(2+\sqrt{3}\right)\left(3+\sqrt{3}\right)}+\frac{9+5\sqrt{3}}{\sqrt{3}.\left(2+\sqrt{3}\right)\left(3+\sqrt{3}\right)}-\frac{2.\left(2\sqrt{3}+3\right)}{\sqrt{3}.\left(2+\sqrt{3}\right)\left(3+\sqrt{3}\right)}\)

\(=\frac{3\sqrt{3}+3+9+5\sqrt{3}-2\left(2\sqrt{3}+3\right)}{\sqrt{3}.\left(2+\sqrt{3}\right)\left(3+\sqrt{3}\right)}\)

\(=\frac{-2\left(2\sqrt{3}+3\right)+8\sqrt{3}+3+9}{\sqrt{3}.\left(2+\sqrt{3}\right)\left(3+\sqrt{3}\right)}\)

\(=\frac{-2\left(2\sqrt{3}+3\right)+8\sqrt{3}+3+9}{9\sqrt{3}+15}\)

\(=\frac{4\sqrt{3}+6}{9\sqrt{3}+15}\)

\(=\frac{3-\sqrt{3}}{3}\)

b) \(\frac{\sqrt{5}-2}{5+2\sqrt{5}}-\frac{1}{2+\sqrt{5}}+\frac{1}{\sqrt{5}}\)

\(=\frac{\left(\sqrt{5}-2\right)\left(2\sqrt{5}+5\right)}{\sqrt{5}.\left(2+\sqrt{5}\right)\left(5+2\sqrt{5}\right)}-\frac{5\sqrt{5}+10}{\sqrt{5}.\left(2+\sqrt{5}\right)\left(5+2\sqrt{5}\right)}+\frac{20+9\sqrt{5}}{\sqrt{5}.\left(2+\sqrt{5}\right)\left(5+2\sqrt{5}\right)}\)

\(=\frac{\left(\sqrt{5}-2\right)\left(2\sqrt{5}+5\right)-\left(5\sqrt{5}+10\right)+20-9\sqrt{5}}{20\sqrt{5}+45}\)

\(=\frac{5\sqrt{5}+10}{20\sqrt{5}+45}\)

\(=\frac{5\left(\sqrt{5}+2\right)}{5\left(4\sqrt{5}+9\right)}\)

\(=\frac{\sqrt{5}+2}{4\sqrt{5}+9}\)

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

\(\frac{3}{\sqrt{5}-\sqrt{2}}+\frac{1}{\sqrt{2}-1}+\frac{4}{3-\sqrt{5}}\)

\(=\frac{\left(\sqrt{5}-\sqrt{2}\right)\left(\sqrt{5}+\sqrt{2}\right)}{\sqrt{5}-\sqrt{2}}+\frac{\left(\sqrt{2}-1\right)\left(\sqrt{2}+1\right)}{\sqrt{2}-1}\)

\(-\frac{\left(3-\sqrt{5}\right)\left(3+\sqrt{5}\right)}{3-\sqrt{5}}\)

\(=\sqrt{5}+\sqrt{2}+\sqrt{2}+1-3-\sqrt{5}\)

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

\(\frac{1}{2+\sqrt{3}}+\frac{\sqrt{2}}{\sqrt{6}}-\frac{4}{3+\sqrt{3}}\)

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

\(=2-\sqrt{3}+\frac{1}{\sqrt{3}}-\frac{4}{3+\sqrt{3}}\)