【高校数学Ⅰ】1-2-2 平方根｜問題集
1.次の平方根を答えなさい。
(1)\(9\)
\(\pm3\)
(2)\(7\)
\(\pm\sqrt{7}\)
2.次の式を簡単にしなさい。
(1)\((\sqrt{3})^2\)
\(=3\)
(2)\(\sqrt{(-5)^2}\)
\(=5\)
(3)\((-\sqrt{5})^2\)
\(=5\)
(4)\(-\sqrt{3^2}\)
\(=-3\)
(5)\(\sqrt{12}\)
\(=2\sqrt{3}\)
(6)\(\sqrt{3}\sqrt{6}\)
\(=\sqrt{18}\)
\(=3\sqrt{2}\)
\(=3\sqrt{2}\)
(7)\(\displaystyle \frac{\sqrt{20}}{\sqrt{5}}\)
\(=\sqrt{4}\)
\(=2\)
\(=2\)
(8)\(\sqrt{0.08}\)
\(\displaystyle =\sqrt{\frac{8}{100}}\)
\(\displaystyle =\frac{2\sqrt{2}}{10}\)
\(\displaystyle =\frac{\sqrt{2}}{5}\)
\(\displaystyle =\frac{2\sqrt{2}}{10}\)
\(\displaystyle =\frac{\sqrt{2}}{5}\)
(9)\(\sqrt{27}\)
\(=3\sqrt{3}\)
(10)\(\sqrt{6}\sqrt{15}\)
\(=\sqrt{90}\)
\(=3\sqrt{10}\)
\(=3\sqrt{10}\)
(11)\(\displaystyle \frac{\sqrt{50}}{\sqrt{2}}\)
\(=\sqrt{25}\)
\(=5\)
\(=5\)
(12)\(\sqrt{0.12}\)
\(\displaystyle =\sqrt{\frac{12}{100}}\)
\(\displaystyle =\frac{2\sqrt{3}}{10}\)
\(\displaystyle =\frac{\sqrt{3}}{5}\)
\(\displaystyle =\frac{2\sqrt{3}}{10}\)
\(\displaystyle =\frac{\sqrt{3}}{5}\)
(13)\(-\sqrt{4}\)
\(=-2\)
(14)\(\displaystyle \sqrt{\frac{9}{4}}\)
\(\displaystyle =\frac{3}{2}\)
(15)\(\sqrt{121}\)
\(=11\)
(16)\(\displaystyle -\sqrt{\frac{1}{64}}\)
\(\displaystyle =-\frac{1}{8}\)
(17)\(\sqrt{5}\sqrt{7}\)
\(=\sqrt{35}\)
(18)\(\displaystyle \frac{\sqrt{8}}{\sqrt{12}}\)
\(\displaystyle =\frac{\sqrt{2}}{\sqrt{3}}\)
\(\displaystyle =\frac{\sqrt{6}}{3}\)
\(\displaystyle =\frac{\sqrt{6}}{3}\)
(19)\(\sqrt{48}\)
\(=4\sqrt{3}\)
(20)\(\sqrt{2}\sqrt{3}\)
\(=\sqrt{6}\)
(21)\(\displaystyle \frac{\sqrt{5}}{\sqrt{15}}\)
\(\displaystyle =\frac{\sqrt{1}}{\sqrt{3}}\)
\(\displaystyle =\frac{\sqrt{3}}{3}\)
\(\displaystyle =\frac{\sqrt{3}}{3}\)
(22)\(\sqrt{32}\)
\(=4\sqrt{2}\)
3.次の計算をしなさい。
(1)\(5\sqrt{3}-2\sqrt{3}+\sqrt{3}\)
\(=4\sqrt{3}\)
(2)\(\sqrt{2}+\sqrt{32}-\sqrt{72}\)
\(=\sqrt{2}+4\sqrt{2}-6\sqrt{2}\)
\(=-\sqrt{2}\)
\(=-\sqrt{2}\)
(3)\((5\sqrt{2}-3\sqrt{3})-(2\sqrt{2}+\sqrt{3})\)
\(=5\sqrt{2}-3\sqrt{3}-2\sqrt{2}-\sqrt{3}\)
\(=3\sqrt{2}-4\sqrt{3}\)
\(=3\sqrt{2}-4\sqrt{3}\)
(4)\((2\sqrt{5}+3\sqrt{6})-(\sqrt{96}-\sqrt{45})\)
\(=2\sqrt{5}+3\sqrt{6}-4\sqrt{6}+3\sqrt{5}\)
\(=5\sqrt{5}-\sqrt{6}\)
\(=5\sqrt{5}-\sqrt{6}\)
(5)\((4\sqrt{2}+3\sqrt{5})(2\sqrt{2}-\sqrt{5})\)
\(=16-4\sqrt{10}+6\sqrt{10}-15\)
\(=1+2\sqrt{10}\)
\(=1+2\sqrt{10}\)
(6)\((2\sqrt{3}-\sqrt{6})(\sqrt{3}+3\sqrt{6})\)
\(=6+6\sqrt{18}-\sqrt{18}-18\)
\(=-12+15\sqrt{2}\)
\(=-12+15\sqrt{2}\)
(7)\((\sqrt{3}+\sqrt{2})(\sqrt{3}-\sqrt{2})\)
\(=3-2\)
\(=1\)
\(=1\)
(8)\((3-\sqrt{5})(3+\sqrt{5})\)
\(=9-5\)
\(=4\)
\(=4\)
(9)\((\sqrt{3}+\sqrt{2})^2\)
\(=3+2\sqrt{6}+2\)
\(=5+2\sqrt{6}\)
\(=5+2\sqrt{6}\)
(10)\((2\sqrt{3}-\sqrt{2})^2\)
\(=12-4\sqrt{6}+2\)
\(=14-4\sqrt{6}\)
\(=14-4\sqrt{6}\)
(11)\(\sqrt{20}+\sqrt{80}\)
\(=2\sqrt{5}+4\sqrt{5}\)
\(=6\sqrt{5}\)
\(=6\sqrt{5}\)
(12)\((2+\sqrt{3})(2-\sqrt{3})\)
\(=4-3\)
\(=1\)
\(=1\)
(13)\((1+\sqrt{5})^2\)
\(=1+2\sqrt{5}+5\)
\(=6+2\sqrt{5}\)
\(=6+2\sqrt{5}\)
(14)\((\sqrt{2}+\sqrt{3})(2\sqrt{2}-3\sqrt{3})\)
\(=4-3\sqrt{6}+2\sqrt{6}-9\)
\(=-5-\sqrt{6}\)
\(=-5-\sqrt{6}\)
(15)\(\sqrt{54}+\sqrt{96}\)
\(=3\sqrt{6}+4\sqrt{6}\)
\(=7\sqrt{6}\)
\(=7\sqrt{6}\)
(16)\((3-\sqrt{6})(3+\sqrt{6})\)
\(=9-6\)
\(=3\)
\(=3\)
(17)\((2-\sqrt{2})^2\)
\(=4-4\sqrt{2}+2\)
\(=6-4\sqrt{2}\)
\(=6-4\sqrt{2}\)
(18)\((1+2\sqrt{3})(3-\sqrt{3})\)
\(=3-\sqrt{3}+6\sqrt{3}-6\)
\(=-3-5\sqrt{3}\)
\(=-3-5\sqrt{3}\)
(19)\(-2\sqrt{2}-3\sqrt{2}+4\sqrt{2}\)
\(=-\sqrt{2}\)
(20)\((4-2\sqrt{5})(4+2\sqrt{5})\)
\(=16-20\)
\(=-4\)
\(=-4\)
(21)\((3-2\sqrt{3})^2\)
\(=9-12\sqrt{3}+12\)
\(=21-12\sqrt{3}\)
\(=21-12\sqrt{3}\)
(22)\(2\sqrt{5}-3\sqrt{5}+4\sqrt{5}\)
\(=3\sqrt{5}\)
(23)\(\sqrt{7}+\sqrt{28}+\sqrt{63}\)
\(=\sqrt{7}+2\sqrt{7}+3\sqrt{7}\)
\(=6\sqrt{7}\)
\(=6\sqrt{7}\)
(24)\((3\sqrt{5}+2\sqrt{2})-(\sqrt{8}-\sqrt{5})\)
\(=3\sqrt{5}+2\sqrt{2}-2\sqrt{2}+\sqrt{5}\)
\(=4\sqrt{5}\)
\(=4\sqrt{5}\)
(25)\((4\sqrt{2}+\sqrt{5})(\sqrt{3}-\sqrt{5})\)
\(=4\sqrt{6}-4\sqrt{10}+\sqrt{15}-5\)
(26)\((5-\sqrt{6})(3+\sqrt{6})\)
\(=15+5\sqrt{6}-3\sqrt{6}-6\)
\(=9+2\sqrt{6}\)
\(=9+2\sqrt{6}\)
(27)\((\sqrt{2}+2\sqrt{3})^2\)
\(=2+4\sqrt{6}+12\)
\(=14+4\sqrt{6}\)
\(=14+4\sqrt{6}\)
(28)\((3\sqrt{2}-\sqrt{5})(\sqrt{2}-\sqrt{3})\)
\(=6-3\sqrt{6}-\sqrt{10}+\sqrt{15}\)
(29)\((\sqrt{5}+3\sqrt{2})^2\)
\(=5+6\sqrt{10}+18\)
\(=23+6\sqrt{10}\)
\(=23+6\sqrt{10}\)
(30)\((2\sqrt{2}+\sqrt{7})(2\sqrt{2}-\sqrt{7})\)
\(=8-7\)
\(=1\)
\(=1\)
4.次の式を有理化しなさい。
(1)\(\displaystyle \frac{4}{3\sqrt{8}}\)
\(\displaystyle =\frac{4}{6\sqrt{2}}\)
\(\displaystyle =\frac{4\sqrt{2}}{12}\)
\(\displaystyle =\frac{\sqrt{2}}{3}\)
\(\displaystyle =\frac{4\sqrt{2}}{12}\)
\(\displaystyle =\frac{\sqrt{2}}{3}\)
(2)\(\displaystyle \frac{1}{\sqrt{3}+\sqrt{2}}\)
\(\displaystyle =\frac{\sqrt{3}-\sqrt{2}}{(\sqrt{3}+\sqrt{2})(\sqrt{3}-\sqrt{2})}\)
\(\displaystyle =\frac{\sqrt{3}-\sqrt{2}}{3-2}\)
\(=\sqrt{3}-\sqrt{2}\)
\(\displaystyle =\frac{\sqrt{3}-\sqrt{2}}{3-2}\)
\(=\sqrt{3}-\sqrt{2}\)
(3)\(\displaystyle \frac{2\sqrt{3}}{\sqrt{6}-2}\)
\(\displaystyle =\frac{2\sqrt{3}(\sqrt{6}+2)}{(\sqrt{6}-2)(\sqrt{6}+2)}\)
\(\displaystyle =\frac{2\sqrt{18}+4\sqrt{3}}{6-4}\)
\(\displaystyle =\frac{6\sqrt{2}+4\sqrt{3}}{2}\)
\(=3\sqrt{2}+2\sqrt{3}\)
\(\displaystyle =\frac{2\sqrt{18}+4\sqrt{3}}{6-4}\)
\(\displaystyle =\frac{6\sqrt{2}+4\sqrt{3}}{2}\)
\(=3\sqrt{2}+2\sqrt{3}\)
(4)\(\displaystyle \frac{\sqrt{5}+\sqrt{2}}{\sqrt{5}-\sqrt{2}}\)
\(\displaystyle =\frac{(\sqrt{5}+\sqrt{2})^2}{(\sqrt{5}-\sqrt{2})(\sqrt{5}+\sqrt{2})}\)
\(\displaystyle =\frac{5+2\sqrt{10}+2}{5-2}\)
\(\displaystyle =\frac{7+2\sqrt{10}}{3}\)
\(\displaystyle =\frac{5+2\sqrt{10}+2}{5-2}\)
\(\displaystyle =\frac{7+2\sqrt{10}}{3}\)
(5)\(\displaystyle \frac{1}{\sqrt{2}}\)
\(\displaystyle =\frac{\sqrt{2}}{\sqrt{2}\sqrt{2}}\)
\(\displaystyle =\frac{\sqrt{2}}{2}\)
\(\displaystyle =\frac{\sqrt{2}}{2}\)
(6)\(\displaystyle \frac{2+\sqrt{6}}{\sqrt{3}}\)
\(\displaystyle =\frac{\sqrt{3}(2+\sqrt{6})}{\sqrt{3}\sqrt{3}}\)
\(\displaystyle =\frac{2\sqrt{3}+\sqrt{18}}{3}\)
\(\displaystyle =\frac{2\sqrt{3}+3\sqrt{2}}{3}\)
\(\displaystyle =\frac{2\sqrt{3}+\sqrt{18}}{3}\)
\(\displaystyle =\frac{2\sqrt{3}+3\sqrt{2}}{3}\)
(7)\(\displaystyle \frac{1}{2+\sqrt{3}}\)
\(\displaystyle =\frac{2-\sqrt{3}}{(2+\sqrt{3})(2-\sqrt{3})}\)
\(\displaystyle =\frac{2-\sqrt{3}}{4-3}\)
\(=2-\sqrt{3}\)
\(\displaystyle =\frac{2-\sqrt{3}}{4-3}\)
\(=2-\sqrt{3}\)
(8)\(\displaystyle \frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}\)
\(\displaystyle =\frac{(\sqrt{5}+\sqrt{3})^2}{(\sqrt{5}-\sqrt{3})(\sqrt{5}+\sqrt{3})}\)
\(\displaystyle =\frac{5+2\sqrt{15}+3}{5-3}\)
\(\displaystyle =\frac{8+2\sqrt{15}}{2}\)
\(=4+\sqrt{15}\)
\(\displaystyle =\frac{5+2\sqrt{15}+3}{5-3}\)
\(\displaystyle =\frac{8+2\sqrt{15}}{2}\)
\(=4+\sqrt{15}\)
(9)\(\displaystyle \frac{2}{\sqrt{3}}\)
\(\displaystyle =\frac{2\sqrt{3}}{\sqrt{3}\sqrt{3}}\)
\(\displaystyle =\frac{2\sqrt{3}}{3}\)
\(\displaystyle =\frac{2\sqrt{3}}{3}\)
(10)\(\displaystyle \frac{1-\sqrt{6}}{\sqrt{2}}\)
\(\displaystyle =\frac{\sqrt{2}(1-\sqrt{6})}{\sqrt{2}\sqrt{2}}\)
\(\displaystyle =\frac{\sqrt{2}-\sqrt{12}}{2}\)
\(\displaystyle =\frac{\sqrt{2}-2\sqrt{3}}{2}\)
\(\displaystyle =\frac{\sqrt{2}-\sqrt{12}}{2}\)
\(\displaystyle =\frac{\sqrt{2}-2\sqrt{3}}{2}\)
(11)\(\displaystyle \frac{1}{\sqrt{2}-\sqrt{5}}\)
\(\displaystyle =\frac{\sqrt{2}+\sqrt{5}}{(\sqrt{2}-\sqrt{5})(\sqrt{2}+\sqrt{5})}\)
\(\displaystyle =\frac{\sqrt{2}+\sqrt{5}}{2-5}\)
\(\displaystyle =-\frac{\sqrt{2}+\sqrt{5}}{3}\)
\(\displaystyle =\frac{\sqrt{2}+\sqrt{5}}{2-5}\)
\(\displaystyle =-\frac{\sqrt{2}+\sqrt{5}}{3}\)
(12)\(\displaystyle \frac{3-2\sqrt{2}}{3+2\sqrt{2}}\)
\(\displaystyle =\frac{(3-2\sqrt{2})^2}{(3+2\sqrt{2})(3-2\sqrt{2})}\)
\(\displaystyle =\frac{9-12\sqrt{2}+8}{9-8}\)
\(=17-12\sqrt{2}\)
\(\displaystyle =\frac{9-12\sqrt{2}+8}{9-8}\)
\(=17-12\sqrt{2}\)
(13)\(\displaystyle \frac{\sqrt{7}}{\sqrt{3}}\)
\(\displaystyle =\frac{\sqrt{7}\sqrt{3}}{\sqrt{3}\sqrt{3}}\)
\(\displaystyle =\frac{\sqrt{21}}{3}\)
\(\displaystyle =\frac{\sqrt{21}}{3}\)
(14)\(\displaystyle \frac{\sqrt{5}}{\sqrt{5}-\sqrt{3}}\)
\(\displaystyle =\frac{\sqrt{5}(\sqrt{5}+\sqrt{3})}{(\sqrt{5}-\sqrt{3})(\sqrt{5}+\sqrt{3})}\)
\(\displaystyle =\frac{5+\sqrt{15}}{5-3}\)
\(\displaystyle =\frac{5+\sqrt{15}}{2}\)
\(\displaystyle =\frac{5+\sqrt{15}}{5-3}\)
\(\displaystyle =\frac{5+\sqrt{15}}{2}\)
(15)\(\displaystyle \frac{2}{3\sqrt{3}}\)
\(\displaystyle =\frac{2\sqrt{3}}{3\sqrt{3}\sqrt{3}}\)
\(\displaystyle =\frac{2\sqrt{3}}{9}\)
\(\displaystyle =\frac{2\sqrt{3}}{9}\)
(16)\(\displaystyle \frac{\sqrt{7}+\sqrt{3}}{\sqrt{7}-\sqrt{3}}\)
\(\displaystyle =\frac{(\sqrt{7}+\sqrt{3})^2}{(\sqrt{7}-\sqrt{3})(\sqrt{7}+\sqrt{3})}\)
\(\displaystyle =\frac{7+2\sqrt{21}+3}{7-3}\)
\(\displaystyle =\frac{5+\sqrt{21}}{2}\)
\(\displaystyle =\frac{7+2\sqrt{21}+3}{7-3}\)
\(\displaystyle =\frac{5+\sqrt{21}}{2}\)
次の学習に進もう！