高専の数学解説

第2章 2.9

(1)

52−2=522−2=52−222=322\frac{5}{\sqrt{2}} - \sqrt{2} = \frac{5\sqrt{2}}{2} - \sqrt{2} = \frac{5\sqrt{2} - 2\sqrt{2}}{2} = \frac{3\sqrt{2}}{2}

(2)

327+572=6+3514=4114=411414\frac{3\sqrt{2}}{\sqrt{7}} + \frac{5\sqrt{7}}{\sqrt{2}} = \frac{6+35}{\sqrt{14}} = \frac{41}{\sqrt{14}} = \frac{41\sqrt{14}}{14}

(3)

12+3−12−3=2−3−2−3(2+3)(2−3)=−234−3=−23\frac{1}{2+\sqrt{3}} - \frac{1}{2-\sqrt{3}} = \frac{2-\sqrt{3}-2-\sqrt{3}}{(2+\sqrt{3})(2-\sqrt{3})} = \frac{-2\sqrt{3}}{4-3} = -2\sqrt{3}

(4)

13+5+15+7=3−53−5+5−75−7=−3−72=7−32\frac{1}{\sqrt{3}+\sqrt{5}} + \frac{1}{\sqrt{5}+\sqrt{7}} = \frac{\sqrt{3}-\sqrt{5}}{3-5} + \frac{\sqrt{5}-\sqrt{7}}{5-7} = -\frac{\sqrt{3}-\sqrt{7}}{2} = \frac{\sqrt{7}-\sqrt{3}}{2}

(5)

5−35+3+5+35−3=(5−3)2+(5+3)2(5+3)(5−3)=5−215+3+5+215+32=162=8\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}} + \frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}} = \frac{(\sqrt{5}-\sqrt{3})^2 + (\sqrt{5}+\sqrt{3})^2}{(\sqrt{5}+\sqrt{3})(\sqrt{5}-\sqrt{3})} = \frac{5-2\sqrt{15}+3+5+2\sqrt{15}+3}{2} = \frac{16}{2} = 8

解説: r31bn1z