1.次の数を簡略化しなさい。
(1)\(\sqrt{20}\)
\(=2\sqrt{5}\)
(2)\(\sqrt{44}\)
\(=2\sqrt{11}\)
(3)\(\sqrt{75}\)
\(=5\sqrt{3}\)
(4)\(\sqrt{48}\)
\(=4\sqrt{3}\)
(5)\(\sqrt{72}\)
\(=6\sqrt{2}\)
(6)\(\sqrt{\frac{7}{100}}\)
\(=\frac{\sqrt{7}}{10}\)
(7)\(\sqrt{45}\)
\(=3\sqrt{5}\)
(8)\(\sqrt{54}\)
\(=3\sqrt{6}\)
(9)\(\sqrt{128}\)
\(=8\sqrt{2}\)
2.次の数の分母を有理化しなさい。
(1)\(\frac{2}{\sqrt{5}}\)
\(=\frac{2\sqrt{5}}{\sqrt{5}\sqrt{5}}\)
\(=\frac{2\sqrt{5}}{5}\)
(2)\(\frac{\sqrt{8}}{\sqrt{7}}\)
\(=\frac{\sqrt{8}\sqrt{7}}{\sqrt{7}\sqrt{7}}\)
\(=\frac{\sqrt{56}}{7}\)
\(=\frac{2\sqrt{14}}{7}\)
(3)\(\frac{\sqrt{3}}{2\sqrt{10}}\)
\(=\frac{\sqrt{3}\sqrt{10}}{2\sqrt{10}\sqrt{10}}\)
\(=\frac{\sqrt{30}}{20}\)
(4)\(\frac{4}{\sqrt{3}}\)
\(=\frac{4\sqrt{3}}{\sqrt{3}\sqrt{3}}\)
\(=\frac{4\sqrt{3}}{3}\)
(5)\(-\frac{\sqrt{7}}{\sqrt{6}}\)
\(=-\frac{\sqrt{7}\sqrt{6}}{\sqrt{6}\sqrt{6}}\)
\(=-\frac{\sqrt{42}}{6}\)
(6)\(\frac{5\sqrt{2}}{\sqrt{10}}\)
\(=\frac{5\sqrt{2}\sqrt{10}}{\sqrt{10}\sqrt{10}}\)
\(=\frac{5\sqrt{20}}{10}\)
\(=\frac{10\sqrt{5}}{10}\)
\(=\sqrt{5}\)
(7)\(\frac{5}{\sqrt{3}}\)
\(=\frac{5\sqrt{3}}{\sqrt{3}\sqrt{3}}\)
\(=\frac{5\sqrt{3}}{3}\)
(8)\(\frac{7}{\sqrt{7}}\)
\(=\frac{7\sqrt{7}}{\sqrt{7}\sqrt{7}}\)
\(=\frac{7\sqrt{7}}{7}\)
\(=\sqrt{7}\)
(9)\(\frac{2\sqrt{3}}{\sqrt{2}}\)
\(=\frac{2\sqrt{3}\sqrt{2}}{\sqrt{2}\sqrt{2}}\)
\(=\frac{2\sqrt{6}}{2}\)
\(=\sqrt{6}\)
3.次の計算をしなさい。
(1)\(\sqrt{5}\sqrt{6}\)
\(=\sqrt{30}\)
(2)\(\sqrt{3}\sqrt{7}\)
\(=\sqrt{21}\)
(3)\(\sqrt{2}\sqrt{13}\)
\(=\sqrt{26}\)
(4)\(\sqrt{11}\sqrt{5}\)
\(=\sqrt{55}\)
(5)\(\sqrt{2}\sqrt{5}\sqrt{7}\)
\(=\sqrt{70}\)
(6)\(\sqrt{6}\sqrt{11}\sqrt{7}\)
\(=\sqrt{462}\)
(7)\(\sqrt{10}÷\sqrt{6}\)
\(=\frac{\sqrt{5}}{\sqrt{3}}\)
\(=\frac{\sqrt{15}}{3}\)
(8)\(\sqrt{15}÷\sqrt{35}\)
\(=\frac{\sqrt{3}}{\sqrt{7}}\)
\(=\frac{\sqrt{21}}{7}\)
(9)\(\sqrt{42}÷\sqrt{22}\)
\(=\frac{\sqrt{21}}{\sqrt{11}}\)
\(=\frac{\sqrt{231}}{11}\)
(10)\(\sqrt{26}÷\sqrt{20}\)
\(=\frac{\sqrt{13}}{\sqrt{10}}\)
\(=\frac{\sqrt{130}}{10}\)
(11)\(\sqrt{14}÷\sqrt{7}\)
\(=\sqrt{2}\)
(12)\(\sqrt{12}÷\sqrt{3}\)
\(=\sqrt{4}\)
\(=2\)
(13)\(\sqrt{8}\sqrt{75}\)
\(=2\sqrt{2}×5\sqrt{3}\)
\(=10\sqrt{6}\)
(14)\(\sqrt{45}\sqrt{18}\)
\(=3\sqrt{5}×3\sqrt{2}\)
\(=9\sqrt{10}\)
(15)\(\sqrt{108}\sqrt{28}\)
\(=6\sqrt{3}×2\sqrt{7}\)
\(=12\sqrt{21}\)
(16)\(\sqrt{75}÷\sqrt{24}\)
\(=5\sqrt{3}÷2\sqrt{6}\)
\(=\frac{5}{2\sqrt{2}}\)
\(=\frac{5\sqrt{2}}{4}\)
(17)\(\sqrt{56}÷\sqrt{40}\)
\(=2\sqrt{14}÷2\sqrt{10}\)
\(=\frac{\sqrt{7}}{\sqrt{5}}\)
\(=\frac{\sqrt{35}}{5}\)
(18)\(\sqrt{32}÷\sqrt{50}\)
\(=4\sqrt{2}÷5\sqrt{2}\)
\(=\frac{4}{5}\)
(19)\(\sqrt{18}÷\sqrt{21}\)
\(=\frac{\sqrt{6}}{\sqrt{7}}\)
\(=\frac{\sqrt{42}}{7}\)
(20)\(3\sqrt{5}÷\sqrt{3}\)
\(=\frac{3\sqrt{15}}{3}\)
\(=\sqrt{15}\)
(21)\(2\sqrt{7}÷\sqrt{6}\)
\(=\frac{2\sqrt{42}}{6}\)
\(=\frac{\sqrt{42}}{3}\)
(22)\(\sqrt{24}\sqrt{27}\)
\(=2\sqrt{6}×3\sqrt{3}\)
\(=6\sqrt{18}\)
\(=18\sqrt{2}\)
(23)\(\sqrt{12}\sqrt{20}\sqrt{8}\)
\(=2\sqrt{3}×2\sqrt{5}×2\sqrt{2}\)
\(=8\sqrt{30}\)
(24)\(4\sqrt{6}÷\sqrt{12}\)
\(=\frac{4\sqrt{6}}{2\sqrt{3}}\)
\(=2\sqrt{2}\)
(25)\(\sqrt{54}÷\sqrt{6}\)
\(=\sqrt{9}\)
\(=3\)
(26)\(5\sqrt{6}+3\sqrt{6}\)
\(=8\sqrt{6}\)
(27)\(8\sqrt{10}+2\sqrt{10}\)
\(=10\sqrt{10}\)
(28)\(7\sqrt{3}-4\sqrt{3}\)
\(=3\sqrt{3}\)
(29)\(2\sqrt{7}-9\sqrt{7}\)
\(=-7\sqrt{7}\)
(30)\(3\sqrt{2}+4\sqrt{2}+\sqrt{2}\)
\(=8\sqrt{2}\)
(31)\(6\sqrt{5}-\sqrt{5}-2\sqrt{5}\)
\(=3\sqrt{5}\)
(32)\(\sqrt{48}+\sqrt{75}\)
\(=4\sqrt{3}+5\sqrt{3}\)
\(=9\sqrt{3}\)
(33)\(\sqrt{45}-\sqrt{20}\)
\(=3\sqrt{5}-2\sqrt{5}\)
\(=\sqrt{5}\)
(34)\(\sqrt{28}+\sqrt{63}-\sqrt{7}\)
\(=2\sqrt{7}+3\sqrt{7}-\sqrt{7}\)
\(=4\sqrt{7}\)
(35)\(\sqrt{24}+\sqrt{6}-\sqrt{96}\)
\(=2\sqrt{6}+\sqrt{6}-4\sqrt{6}\)
\(=-\sqrt{6}\)
(36)\(\sqrt{27}+\sqrt{7}+\sqrt{3}-\sqrt{28}\)
\(=3\sqrt{3}+\sqrt{7}+\sqrt{3}-2\sqrt{7}\)
\(=4\sqrt{3}-\sqrt{7}\)
(37)\(\sqrt{32}-3\sqrt{6}-\sqrt{18}+4\sqrt{6}\)
\(=4\sqrt{2}-3\sqrt{6}-3\sqrt{2}+4\sqrt{6}\)
\(=\sqrt{2}+\sqrt{6}\)
(38)\(\sqrt{45}-\sqrt{8}+\sqrt{20}-\sqrt{50}\)
\(=3\sqrt{5}-2\sqrt{2}+2\sqrt{5}-5\sqrt{2}\)
\(=5\sqrt{5}-5\sqrt{2}\)
(39)\(\sqrt{75}-\sqrt{54}+\sqrt{96}-\sqrt{108}\)
\(=5\sqrt{3}-3\sqrt{6}+4\sqrt{6}-6\sqrt{3}\)
\(=\sqrt{6}-\sqrt{3}\)
(40)\(\sqrt{8}-\frac{2}{\sqrt{2}}\)
\(=2\sqrt{2}-\frac{2\sqrt{2}}{2}\)
\(=2\sqrt{2}-\sqrt{2}\)
\(=\sqrt{2}\)
(41)\(\sqrt{24}+\frac{2\sqrt{2}}{\sqrt{3}}\)
\(=2\sqrt{6}+\frac{2\sqrt{6}}{3}\)
\(=\frac{6\sqrt{6}+2\sqrt{6}}{3}\)
\(=\frac{8\sqrt{6}}{3}\)
(42)\(\frac{\sqrt{5}}{\sqrt{2}}-\frac{\sqrt{2}}{\sqrt{5}}+\sqrt{10}\)
\(=\frac{\sqrt{10}}{2}-\frac{\sqrt{10}}{5}+\sqrt{10}\)
\(=\frac{5\sqrt{10}-2\sqrt{10}+10\sqrt{10}}{10}\)
\(=\frac{13\sqrt{10}}{10}\)
(43)\(\sqrt{7}\sqrt{35}\)
\(=\sqrt{245}\)
\(=7\sqrt{5}\)
(44)\(\sqrt{24}\sqrt{80}\)
\(=2\sqrt{6}×4\sqrt{5}\)
\(=8\sqrt{30}\)
(45)\(\sqrt{8}\sqrt{50}\sqrt{27}\)
\(=2\sqrt{2}×5\sqrt{2}×3\sqrt{3}\)
\(=60\sqrt{3}\)
(46)\(\sqrt{30}÷\sqrt{12}\)
\(=\sqrt{30}÷2\sqrt{3}\)
\(=\frac{\sqrt{10}}{2}\)
(47)\(\sqrt{48}÷\sqrt{18}\)
\(=4\sqrt{3}÷3\sqrt{2}\)
\(=\frac{4\sqrt{6}}{6}\)
\(=\frac{2\sqrt{6}}{3}\)
(48)\(\sqrt{6}\sqrt{20}÷\sqrt{15}\)
\(=\frac{\sqrt{6}\sqrt{20}}{\sqrt{15}}\)
\(=\sqrt{8}\)
\(=2\sqrt{2}\)
(49)\(5\sqrt{5}-7\sqrt{5}\)
\(=-2\sqrt{5}\)
(50)\(\sqrt{3}-\sqrt{12}+\sqrt{27}\)
\(=\sqrt{3}-2\sqrt{3}+3\sqrt{3}\)
\(=2\sqrt{3}\)
(51)\(2\sqrt{3}-\sqrt{45}+\sqrt{27}-\sqrt{80}\)
\(=2\sqrt{3}-3\sqrt{5}+3\sqrt{3}-4\sqrt{5}\)
\(=5\sqrt{3}-7\sqrt{5}\)
(52)\(\frac{1}{\sqrt{5}}+\frac{1}{3\sqrt{3}}+\sqrt{3}\)
\(=\frac{\sqrt{5}}{5}+\frac{\sqrt{3}}{9}+\sqrt{3}\)
\(=\frac{9\sqrt{5}+5\sqrt{3}+45\sqrt{3}}{45}\)
\(=\frac{9\sqrt{5}+50\sqrt{3}}{45}\)
(53)\(\sqrt{7}(\sqrt{2}+\sqrt{5})\)
\(=\sqrt{14}+\sqrt{35}\)
(54)\(\sqrt{2}(\sqrt{3}-2\sqrt{6})\)
\(=\sqrt{6}-2\sqrt{12}\)
\(=\sqrt{6}-4\sqrt{3}\)
(55)\(\sqrt{8}(\sqrt{12}+\sqrt{6})\)
\(=2\sqrt{2}(2\sqrt{3}+\sqrt{6})\)
\(=4\sqrt{6}+2\sqrt{12}\)
\(=4\sqrt{6}+4\sqrt{3}\)
(56)\(3\sqrt{3}(\sqrt{50}+\sqrt{75})\)
\(=3\sqrt{3}(5\sqrt{2}+5\sqrt{3})\)
\(=15\sqrt{6}+15\sqrt{9}\)
\(=15\sqrt{6}+45\)
(57)\((\sqrt{7}+3)(\sqrt{7}+2)\)
\(=7+5\sqrt{7}+6\)
\(=13+5\sqrt{7}\)
(58)\((\sqrt{2}-1)^2\)
\(=2-2\sqrt{2}+1\)
\(=3-2\sqrt{2}\)
(59)\((2\sqrt{5}-3)(2\sqrt{5}+3)\)
\(=20-9\)
\(=11\)
(60)\((\sqrt{3}+1)^2-2(\sqrt{3}+1)\)
\(=(\sqrt{3}+1)(\sqrt{3}+1-2)\)
\(=(\sqrt{3}+1)(\sqrt{3}-1)\)
\(=3-1\)
\(=2\)
(61)\(\sqrt{3}(\sqrt{6}-\sqrt{12})\)
\(=\sqrt{18}-\sqrt{36}\)
\(=3\sqrt{2}-6\)
(62)\((\sqrt{48}+\sqrt{27})÷2\sqrt{3}\)
\(=(4\sqrt{3}+3\sqrt{3})÷2\sqrt{3}\)
\(=(7\sqrt{3})÷2\sqrt{3}\)
\(=\frac{7}{2}\)
(63)\((2+\sqrt{7})(2-\sqrt{7})\)
\(=4-7\)
\(=-3\)
(64)\((2\sqrt{5}+1)^2\)
\(=20+4\sqrt{5}+1\)
\(=21+4\sqrt{5}\)
(65)\((\sqrt{3}-1)(\sqrt{3}+8)\)
\(=3+7\sqrt{3}-8\)
\(=-5+7\sqrt{3}\)
(66)\((2+\sqrt{6})(-3+\sqrt{6})\)
\(=6-\sqrt{6}-6\)
\(=-\sqrt{6}\)
(67)\((\sqrt{3}+\sqrt{5})^2\)
\(=3+2\sqrt{15}+5\)
\(=8+2\sqrt{15}\)
(68)\((\sqrt{7}+\sqrt{2})(\sqrt{7}-\sqrt{2})\)
\(=7-2\)
\(=5\)
(69)\((3-\sqrt{2}+\sqrt{5})^2\)
\(=(3-\sqrt{2})^2+2\sqrt{5}(3-\sqrt{2})+5\)
\(=9-6\sqrt{2}+2+6\sqrt{5}-2\sqrt{10}+5\)
\(=16-6\sqrt{2}+6\sqrt{5}-2\sqrt{10}\)
(70)\((2-\sqrt{6})^2-4(2-\sqrt{6})+4\)
\(=(2-\sqrt{6}-2)^2\)
\(=6\)
4.\(\sqrt{3}=1.732,\sqrt{30}=5.477\)とするとき、次の値を求めなさい。
(1)\(\sqrt{3000}\)
\(=10\sqrt{30}\)
\(=10×5.477\)
\(=54.77\)
(2)\(\sqrt{30000}\)
\(=100\sqrt{3}\)
\(=100×1.732\)
\(=173.2\)
(3)\(\sqrt{0.3}\)
\(=\frac{\sqrt{30}}{10}\)
\(=\frac{5.477}{10}\)
\(=0.5477\)
5.\(\sqrt{5.2}=2.280,\sqrt{52}=7.211\)とするとき、次の値を求めなさい。
(1)\(\sqrt{520}\)
\(=10\sqrt{5.2}\)
\(=10×2.280\)
\(=22.8\)
(2)\(\sqrt{520000}\)
\(=100\sqrt{52}\)
\(=100×7.211\)
\(=721.1\)
(3)\(\sqrt{0.0052}\)
\(=\frac{\sqrt{52}}{100}\)
\(=\frac{7.211}{100}\)
\(=0.07211\)
6.\(x=\sqrt{3}+1,y=\sqrt{3}-1\)とするとき、次の値を求めなさい。
(1)\(x^2+2xy+y^2\)
\(=(x+y)^2\)
\(=(\sqrt{3}+1+\sqrt{3}-1)^2\)
\(=(2\sqrt{3})^2\)
\(=12\)
(2)\(x^2-y^2\)
\(=(x+y)(x-y)\)
\(=(\sqrt{3}+1+\sqrt{3}-1)(\sqrt{3}+1-\sqrt{3}+1)\)
\(=(2\sqrt{3})×2\)
\(=4\sqrt{3}\)
7.\(x=\sqrt{3}+\sqrt{2},y=\sqrt{3}-\sqrt{2}\)とするとき、次の値を求めなさい。
(1)\(3x^2-3y^2\)
\(=3(x^2-y^2)\)
\(=3(x+y)(x-y)\)
\(=3(\sqrt{3}+\sqrt{2}+\sqrt{3}-\sqrt{2})\)
\(\ \ \ ×(\sqrt{3}+\sqrt{2}-\sqrt{3}+\sqrt{2})\)
\(=3(2\sqrt{3})(2\sqrt{2})\)
\(=12\sqrt{6}\)
(2)\(2x^2-4xy+2y^2\)
\(=2(x^2-2xy+y^2)\)
\(=2(x-y)^2\)
\(=2(\sqrt{3}+\sqrt{2}-\sqrt{3}+\sqrt{2})^2\)
\(=2(2\sqrt{2})^2\)
\(=16\)