第2章 2.8

(1)

$\frac{2}{\sqrt{6}} = \frac{2\sqrt{6}}{6} = \frac{\sqrt{6}}{3}$

(2)

$\frac{1}{\sqrt{7}-\sqrt{3}} = \frac{\left(\sqrt{7}+\sqrt{3}\right)}{\left(\sqrt{7}-\sqrt{3}\right)\left(\sqrt{7}+\sqrt{3}\right)} = \frac{\sqrt{7}+\sqrt{3}}{7-3} = \frac{\sqrt{7}+\sqrt{3}}{4}$

(3)

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

(4)

$\frac{4-\sqrt{2}}{3+\sqrt{2}} = \frac{\left(4-\sqrt{2}\right)\left(3-\sqrt{2}\right)}{\left(3+\sqrt{2}\right)\left(3-\sqrt{2}\right)} = \frac{12-7\sqrt{2}+2}{9-2} = \frac{14-7\sqrt{2}}{7} = 2-\sqrt{2}$

(5)

$\frac{\sqrt{6}-\sqrt{2}}{\sqrt{6}+\sqrt{2}} = \frac{\left(\sqrt{6}-\sqrt{2}\right)\left(\sqrt{6}-\sqrt{2}\right)}{\left(\sqrt{6}+\sqrt{2}\right)\left(\sqrt{6}-\sqrt{2}\right)} = \frac{6-2\sqrt{12}+2}{6-2} = \frac{8-4\sqrt{3}}{4} = 2-\sqrt{3}$

(6)

$\frac{2\sqrt{7}+\sqrt{3}}{\sqrt{7}+2\sqrt{3}} = \frac{\left(2\sqrt{7}+\sqrt{3}\right)\left(\sqrt{7}-2\sqrt{3}\right)}{\left(\sqrt{7}+2\sqrt{3}\right)\left(\sqrt{7}-2\sqrt{3}\right)} = \frac{14-3\sqrt{21}-6}{7-12} = \frac{8-3\sqrt{21}}{-5} = \frac{3\sqrt{21}-8}{5}$