{"id":983,"date":"2020-06-27T13:59:12","date_gmt":"2020-06-27T05:59:12","guid":{"rendered":"https:\/\/study-together.com\/edu\/?page_id=983"},"modified":"2020-11-02T23:16:54","modified_gmt":"2020-11-02T15:16:54","slug":"laws-of-integral-indices","status":"publish","type":"page","link":"https:\/\/www.study-together.com\/edu\/secondary\/form1_maths\/laws-of-integral-indices\/","title":{"rendered":"3. \u6307\u6578\u53ca\u9032\u5236 (Laws of integral indices)"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-page\" data-elementor-id=\"983\" class=\"elementor elementor-983\">\n\t\t\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-dc80c4f elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"dc80c4f\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-66 elementor-top-column elementor-element elementor-element-d9ef4ef\" data-id=\"d9ef4ef\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-c3e2df5 elementor-widget elementor-widget-text-editor\" data-id=\"c3e2df5\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<h2>3.1\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u6307\u6578\u8a18\u6578\u6cd5 (Index Notation)<\/h2><div>\u76f8\u4fe1\u5927\u5bb6\u90fd\u77e5\u9053&#8221;999\u500b3\u52a0\u57cb&#8221;\u5605\u6578\u5f0f\u4fc2\u53ef\u4ee5\u5229\u7528\u4e58\u6cd5\u5beb\u6210&#8221;\\( 999 \\times 3\\)&#8221;\u3002<\/div><p>\u800c\u6240\u8b02\u5605\u300c\u6307\u6578\u8a18\u6578\u6cd5\u300d\u5176\u5be6\u4fc2\u7528\u569f\u7c21\u5316&#8221;999\u500b5\u4e58\u57cb&#8221;\u5605\u6578\u5f0f\u3002<\/p><p style=\"text-align: left; padding-left: 40px;\">\\(\\begin{align}\\underbrace{5\\times 5\\times 5\\times &#8230;.\\times 5} &amp; = 5^{999} \\\\<br \/>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 999\u500b5\u4e58\u57cb \\quad \\quad &amp; \\end{align}\\)<\/p><p>\u4e00\u822c\u569f\u8b1b\uff0c<\/p><p style=\"text-align: left; padding-left: 40px;\">\\(\\begin{align}\\underbrace{a\\times a\\times a\\times &#8230;.\\times a} &amp; = a^{n} &amp; \\\\<br \/>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 n\u500ba\u4e58\u57cb \\quad \\quad &amp; \\end{align}\\)<\/p><ul><li>\u55ba\\(a^{n}\\)\u5165\u9762\uff0c<em>\\(a\\)<\/em>\u7a31\u70ba\u300c\u5e95 (base)\u300d\uff0c\\(n\\)\u7a31\u70ba\u300c\u6307\u6578 (index)\u300d\u3002<\/li><li>\\(a^{n}\\)\u53ef\u4ee5\u8b80\u6210 \u300ca\u5605n\u6b21\u65b9\u300d \u6216 \u300c<em>a<\/em>\u7684<em>n<\/em>\u6b21\u51aa\u300d\uff1b\u82f1\u6587\u8b80\u6210 <em>n<\/em>-th power of <em>a<\/em> \u6216 a to the power of n\u3002<\/li><\/ul>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-3ba0eb7 elementor-widget elementor-widget-text-editor\" data-id=\"3ba0eb7\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<h2>3.2\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u6307\u6578\u5b9a\u5f8b (Laws of Indices)<\/h2><p>\u6307\u6578\u5b9a\u5f8b\u57fa\u672c\u4e0a\u4fc2\u6307\u4e0b\u9762\u5e7e\u689d\u6046\u7b49\u5f0f\uff1a<\/p><p>\\(\\begin{align}<br \/>1.\\quad &amp; a^{m} \\times a^{n} = a^{m + n} \\\\<br \/>2. \\quad &amp; a^{m} \\div a^{n} = a^{m &#8211; n} \\\\<br \/>3. \\quad &amp; (a^{m} )^{n} = a^{m \\times n} \\\\<br \/>4. \\quad &amp; (ab)^{n} = a^{n}b^{n} \\\\<br \/>5. \\quad &amp; (\\frac{a}{b})^{n} = \\frac{a^{n}}{b^{n}} \\\\<br \/>\\end{align}\\)<\/p><p>\u540c\u5b78\u6700\u7dca\u8981\u4fc2\u8b58\u5f97\u5229\u7528\u4e0a\u9762\u5e7e\u689d\u516c\u5f0f\u569f\u5316\u7c21\u6709\u6307\u6578\u5605\u6578\u5f0f\u5c31\u53ef\u4ee5\u4e86\u3002<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-44f731c elementor-widget elementor-widget-text-editor\" data-id=\"44f731c\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<h2>3.3\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u9032\u5236<\/h2><p>\u300c\u9032\u5236\u300d\u53c8\u53eb\u300c\u9032\u4f4d\u5236\u300d\uff0c\u5176\u5be6\u53ea\u4fc2\u4e00\u7a2e\u8a18\u6578\u65b9\u5f0f\u3002\u900f\u904e\u9032\u5236\uff0c\u6211\u5730\u53ef\u4ee5\u5229\u7528\u6709\u9650\u5605\u6578\u5b57\u6216\u7b26\u865f\u569f\u4ee3\u8868\u6240\u6709\u6578\u503c\u3002<\/p><ul><li>\u53ef\u7528\u5605\u6578\u5b57\u6216\u7b26\u865f(\u6216\u5143\u865f)\u5605\u6578\u76ee\u7a31\u70ba\u9032\u5236\u5605\u57fa\u6578\u6216\u5e95\u6578\u3002<\/li><li>\u57fa\u6578\u4fc2 n\uff0c\u5c31\u53eb n \u9032\u4f4d\u5236\u3002<\/li><li>\u55f0\uff24\u6578\u5b57\u7b26\u865f\u5c31\u53eb\u6578\u78bc\u3002<\/li><\/ul><p>\u5927\u5bb6\u6700\u5e38\u7528\u5605\u7576\u7136\u4fc2\u5341\u9032\u4f4d\uff0c\u7576\u4e2d\u5229\u7528 10 \u500b\u963f\u62c9\u4f2f\u6578\u5b57 0\u81f39 \u569f\u8a18\u6578\u3002\u9664\u6b64\u4e4b\u5916\uff0c\u5176\u5be6\u65e5\u5e38\u751f\u6d3b\u4e2d\u6211\u5730\u90fd\u7d93\u5e38\u63a5\u89f8\u5230\u5514\u540c\u5605\u9032\u5236\u3002\u4f8b\u5982\uff1a<\/p><ul><li>\u4e8c\u9032\u5236\uff1a\u5ee3\u6cdb\u7528\u55ba\u6578\u78bc\u8a2d\u5099\u4e2d<\/li><li>\u5341\u4e8c\u9032\u5236\uff1a\u6642\u8fb0\u3001\u6708\u4efd\u3001\u6578\u91cf\u55ae\u4f4d\uff08\u5982\u4e00\u6253\u86cb\uff09<\/li><li>\u5341\u516d\u9032\u5236\uff1a\u5ee3\u6cdb\u7528\u55ba\u96fb\u8166\u7a0b\u5f0f\u3001\u8a08\u6578\u6a5f<\/li><li>\u516d\u5341\u9032\u5236\uff1a\u79d2\u3001\u5206<\/li><\/ul><h3>3.3.1\u00a0\u00a0\u00a0\u00a0 \u6578\u78bc (Digits)<\/h3><p>\u4e0b\u9762\u5217\u51fa\u5497\u5e7e\u7a2e\u9032\u5236\u5605\u6578\u78bc\u540c\u5be6\u969b\u4f8b\u5b50\uff1a<\/p><ul><li>\u5341\u9032\u5236 (denary system)\uff1a\u963f\u62c9\u4f2f\u6578\u5b57 0\u81f39<br \/>e.g. 35 \uff08\u56e0\u70ba\u5341\u9032\u5236\u4fc2\u6700\u7576\u7528\u5605\uff0c\u6240\u4ee5\u901a\u5e38\u5514\u5beb\u500b\u5e95\u657810)<\/li><li>\u4e8c\u9032\u5236 (binary system)\uff1a0, 1<br \/>e.g.\u00a0 1101<sub>2<\/sub><\/li><li>\u5341\u516d\u9032\u5236 (hexadecimal system)\uff1a0 \u2013 9, A, B, C, D, E, F<br \/>e.g.\u00a0 A34F<sub>16<\/sub><\/li><\/ul><h3>3.3.2\u00a0\u00a0\u00a0\u00a0 \u4f4d\u503c (Place Value)<\/h3><p>\u7121\u8ad6\u4fc2\u54a9\u9032\u5236\uff0c\u6700\u53f3\u908a\u5605&#8221;\u6578\u78bc&#8221;\u90fd\u4fc2\u6700\u7d30\uff0c\u4ee3\u8868\u500b\u4f4d\u3002\u4f46\u518d\u5411\u5de6\u7747\uff0c\u6bcf\u500b\u6578\u78bc\u5605\u503c\u90fd\u5514\u540c\u3002<\/p><table><tbody><tr><td width=\"88\">\u5341\u9032\u5236<\/td><td width=\"85\">\u842c(10<sup>4<\/sup>)<\/td><td width=\"88\">\u5343(10<sup>3<\/sup>)<\/td><td width=\"88\">\u767e(10<sup>2<\/sup>)<\/td><td width=\"88\">\u5341(10)<\/td><td width=\"88\">\u500b(1)<\/td><\/tr><tr><td width=\"88\">\u4e8c\u9032\u5236<\/td><td width=\"85\">2<sup>4<\/sup><\/td><td width=\"88\">2<sup>3<\/sup><\/td><td width=\"88\">2<sup>2<\/sup><\/td><td width=\"88\">2<\/td><td width=\"88\">1<\/td><\/tr><tr><td width=\"88\">\u5341\u516d\u9032\u5236<\/td><td width=\"85\">16<sup>4<\/sup><\/td><td width=\"88\">16<sup>3<\/sup><\/td><td width=\"88\">16<sup>2<\/sup><\/td><td width=\"88\">16<\/td><td width=\"88\">1<\/td><\/tr><\/tbody><\/table><p>\u7576\u6211\u5730\u53bb\u7406\u89e34235<sub>10<\/sub>\u5230\u5e95\u6709\u5e7e\u5927\u5605\u6642\u5019\uff0c<\/p><ul><li style=\"list-style-type: none;\"><ul><li>&#8220;5&#8221;\u4fc2\u4ee3\u8868\u4e94\u500b1<\/li><li>&#8220;3&#8221;\u4fc2\u4ee3\u8868\u4e09\u500b10<\/li><li>&#8220;2&#8221;\u4fc2\u4ee3\u8868\u5169\u500b100\u00a0\u00a0 *\u7559\u610f \\(100 = 10^{2}\\)<\/li><li>&#8220;4&#8221;\u4fc2\u4ee3\u8868\u56db\u500b1000\u00a0 *\u7559\u610f \\(1000 = 10^{3}\\)<\/li><\/ul><\/li><\/ul><p>\u6240\u4ee5\u6211\u5730\u5148\u53eb4235\u505a&#8221;4\u5343\u53472\u767e3\u53415&#8243;\uff0c\u4ea6\u53ef\u7406\u89e3\u6210<\/p><p>\\(4235 = 5 + (3 \\times 10^{1}) + (2 \\times 10^{2}) + (4 \\times 10^{3}) \\)<\/p><p>\u4f8b1\uff1a\u00a0\u00a0\u00a0\u00a0\u00a0 \u8a08\u7b97\u51fa101<sub>2<\/sub>\u7684\u503c\u3002<\/p><p>\\(\\begin{align}<br \/>101_{2} &amp; =\u00a0 1 + (0 \\times 2^{1}) + (1 \\times 2^{2}) \\\\<br \/>&amp; = 1 + 0 + (1 \\times 4) \\\\<br \/>&amp; = 5<br \/>\\end{align}\\)<\/p><p>\u4f8b2\uff1a\u00a0\u00a0\u00a0\u00a0\u00a0 \u8a08\u7b97\u51faAC<sub>16<\/sub>\u7684\u503c\u3002<\/p><p>\\(\\begin{align}<br \/>AC_{16} &amp; =\u00a0 12 + (10 \\times 16^{1}) \\\\<br \/>&amp; = 12 + 160 \\\\<br \/>&amp; = 172<br \/>\\end{align}\\)<\/p><h3>3.3.3\u00a0\u00a0\u00a0\u00a0 \u9032\u5236\u9593\u4e4b\u8f49\u63db (Conversion between numeral systems)<\/h3><h3>3.3.3.1\u00a0 \u975e\u5341\u9032\u5236\u8f49\u63db\u70ba\u5341\u9032\u5236<\/h3><p>\u5176\u5be6\u300c\u975e\u5341\u9032\u5236\u8f49\u63db\u70ba\u5341\u9032\u5236\u300d\u5373\u4fc2\u8b1b\u89e3\u4f4d\u503c\u5165\u9762\u5605\u4f8b\u5b50\uff0c\u6bd4\u500b\u5514\u4fc2\u5341\u9032\u5236\u5605\u6578\u4f60\uff0c\u53eb\u4f60\u8a08\u8fd4\u500b\u503c\u51fa\u569f\u3002<\/p><h3>3.3.3.2\u00a0 \u5341\u9032\u5236\u8f49\u63db\u70ba\u975e\u5341\u9032\u5236<\/h3><p>\u8981\u5c07\u4e00\u500b\u5341\u9032\u5236\u5605\u6578\u7528\u975e\u5341\u9032\u5236\u5605\u5beb\u6cd5\u5beb\u51fa\u569f\uff0c\u6211\u5730\u8981\u4e0d\u65b7\u9664\u4ee5\u76ee\u6a19\u9032\u5236\u57fa\u5e95\uff0c\u76f4\u5230\u5546\u6578\u70ba 0\uff0c\u4e4b\u5f8c\u53cd\u5411 \u53d6\u51fa\u300e\u9918\u6578\u300f\u3002<\/p><p>\u4e0d\u904e\u5176\u5be6\u5481\u5beb\uff0c\u6211\u8ad7\u5514\u4fc2\u597d\u591a\u4eba\u660e\u3002\u6700\u7c21\u55ae\u5605\u65b9\u6cd5\u4fc2\u7747\u4f8b\u5b50\u3002<\/p><p>\u4f8b\uff1a\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u628a87<sub>10<\/sub>\u63db\u70ba\u4e8c\u9032\u5236<\/p><p>\u89e3\uff1a\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/p><p><img fetchpriority=\"high\" decoding=\"async\" class=\"alignnone size-medium wp-image-1021\" src=\"https:\/\/study-together.com\/edu\/wp-content\/uploads\/2020\/06\/f1Maths_dec-to-binary-ex1-300x274.png\" alt=\"\u4ee5\u4e8c\u9032\u5236\u5beb\u51fa87\" width=\"300\" height=\"274\" srcset=\"https:\/\/www.study-together.com\/edu\/wp-content\/uploads\/2020\/06\/f1Maths_dec-to-binary-ex1-300x274.png 300w, https:\/\/www.study-together.com\/edu\/wp-content\/uploads\/2020\/06\/f1Maths_dec-to-binary-ex1-768x702.png 768w, https:\/\/www.study-together.com\/edu\/wp-content\/uploads\/2020\/06\/f1Maths_dec-to-binary-ex1.png 925w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p><p>\u6240\u4ee5\u00a0 87<sub>10<\/sub>\u00a0 =\u00a0 1010111<sub>2<\/sub><\/p><p>\u4f8b\uff1a\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u628a766<sub>10<\/sub>\u63db\u70ba16\u9032\u5236<\/p><p>\u89e3\uff1a<\/p><p><img decoding=\"async\" class=\"alignnone size-medium wp-image-1022\" src=\"https:\/\/study-together.com\/edu\/wp-content\/uploads\/2020\/06\/f1Maths_dec-10_hexa_ex1-300x175.jpg\" alt=\"766\u8f49\u63db\u621016\u9032\u5236\" width=\"300\" height=\"175\" srcset=\"https:\/\/www.study-together.com\/edu\/wp-content\/uploads\/2020\/06\/f1Maths_dec-10_hexa_ex1-300x175.jpg 300w, https:\/\/www.study-together.com\/edu\/wp-content\/uploads\/2020\/06\/f1Maths_dec-10_hexa_ex1.jpg 308w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p><p>\u6240\u4ee5\u00a0 766<sub>10<\/sub>\u00a0 =\u00a0 2FE<sub>16<\/sub><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<section class=\"elementor-section elementor-inner-section elementor-element elementor-element-a7fbc50 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"a7fbc50\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-50 elementor-inner-column elementor-element elementor-element-ba7e55d\" data-id=\"ba7e55d\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-4e3f3eb elementor-widget elementor-widget-text-editor\" data-id=\"4e3f3eb\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<a href=\"https:\/\/www.study-together.com\/edu\/secondary\/form1_maths\/algebraic-expressions-equations\/\"><i class=\"far fa-arrow-alt-circle-left\" aria-hidden=\"true\"><\/i> \u4ee3\u6578\u5f0f\u548c\u65b9\u7a0b\u5f0f<\/a>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-50 elementor-inner-column elementor-element elementor-element-8e96c02\" data-id=\"8e96c02\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-bcfa50f elementor-widget elementor-widget-text-editor\" data-id=\"bcfa50f\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p style=\"text-align: right;\"><a href=\"https:\/\/www.study-together.com\/edu\/secondary\/form1_maths\/polynomials\/\">\u591a\u9805\u5f0f <i class=\"far fa-arrow-alt-circle-right\" aria-hidden=\"true\"><\/i><\/a><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-3f58c70\" data-id=\"3f58c70\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-8921088 elementor-widget elementor-widget-sidebar\" data-id=\"8921088\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"sidebar.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"sidebar-box widget-oceanwp-custom-menu custom-menu-widget clr\"><style type=\"text\/css\">.ocean_custom_menu-2 > ul > li > a, .custom-menu-widget .ocean_custom_menu-2 .dropdown-menu .sub-menu li a.menu-link{font-size:16px;line-height:17px;}.custom-menu-widget .ocean_custom_menu-2.oceanwp-custom-menu > ul.click-menu .open-this{font-size:16px;}<\/style><h4 class=\"widget-title\">\u4e2d\u4e00\u6578\u5b78\u3010\u76ee\u9304\u3011<\/h4><div class=\"oceanwp-custom-menu clr ocean_custom_menu-2 left 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href=\"https:\/\/www.study-together.com\/edu\/secondary\/form1_maths\/estimation-in-measurements\/\" class=\"menu-link\">10. \u91cf\u5ea6\u65b9\u9762\u7684\u4f30\u8a08 (Estimation in measurements)<\/a><\/li>\n<li  id=\"menu-item-1136\" class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-1136\"><a href=\"https:\/\/www.study-together.com\/edu\/secondary\/form1_maths\/area-and-volume\/\" class=\"menu-link\">11. \u9762\u7a4d\u53ca\u9ad4\u7a4d (Area and Volume)<\/a><\/li>\n<li  id=\"menu-item-1150\" class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-1150\"><a href=\"https:\/\/www.study-together.com\/edu\/secondary\/form1_maths\/introduction-to-coordinates\/\" class=\"menu-link\">12.   \u5750\u6a19\u7c21\u4ecb (Introduction to Coordinates)<\/a><\/li>\n<li  id=\"menu-item-1164\" class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-1164\"><a 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