{"id":16780,"date":"2010-12-03T03:31:23","date_gmt":"2010-12-02T19:31:23","guid":{"rendered":"http:\/\/highscope.ch.ntu.edu.tw\/wordpress\/?p=16780"},"modified":"2021-10-06T16:30:23","modified_gmt":"2021-10-06T08:30:23","slug":"%e6%8c%87%e6%95%b8%e5%be%8b%ef%bc%88exponential-law%ef%bc%89","status":"publish","type":"post","link":"http:\/\/localhost\/%e6%8c%87%e6%95%b8%e5%be%8b%ef%bc%88exponential-law%ef%bc%89\/","title":{"rendered":"\u6307\u6578\u5f8b\uff08Exponential law\uff09"},"content":{"rendered":"<div class=\"pf-content\"><p><strong><span style=\"color: #ff6600;\">\u6307\u6578\u5f8b\uff08Exponential law\uff09<\/span><\/strong><br \/>\n<strong><span style=\"color: #008000;\">\u570b\u7acb\u5317\u9580\u8fb2\u5de5\u8077\u696d\u5b78\u6821\u6578\u5b78\u79d1\u674e\u5efa\u5b97\u8001\u5e2b\/\u570b\u7acb\u53f0\u7063\u5e2b\u7bc4\u5927\u5b78\u6578\u5b78\u7cfb\u6d2a\u842c\u751f\u6559\u6388\u8cac\u4efb\u7de8\u8f2f<\/span><\/strong><\/p>\n<p>\u4e00\u500b\u5be6\u6578 $$a$$ \u81ea\u4e58 $$n$$\u6b21\u5f8c\u7684\u4e58\u7a4d($$n$$ \u662f\u81ea\u7136\u6578)\uff0c\u8a72\u5982\u4f55\u7528\u7cbe\u7c21\u7684\u6578\u5b78\u7b26\u865f\u8a9e\u8a00\u4f86\u8868\u793a\u5462\uff1f\u4f60\u662f\u5426\u80fd\u60f3\u5f97\u5230\u50cf\u7b1b\u5361\u5152\uff08Ren<strong>\u00e9 <\/strong>Descartes\uff09\u5728\u5341\u4e03\u4e16\u7d00\u6642\u6240\u7528\u65b9\u6cd5\uff0c\u5373\u5728\u6578\u5b57\u6216\u6587\u5b57\u7684\u53f3\u4e0a\u65b9\u7528\u5c0f\u62c9\u4f2f\u6578\u5b57\u4f86\u8868\u793a\u6b21\u65b9\u7684\u6982\u5ff5\u5462\uff1f\u4e5f\u5c31\u8aaa\u5c07 $$a$$ \u81ea\u4e58 $$n$$ \u6b21\u6216\u8005\u8aaa\u6709 $$n$$ \u500b $$a$$ \u81ea\u4e58\uff0c\u4ee5 $$a^n$$ \u4f86\u8868\u793a\uff0c\u5176\u4e2d $$a$$ \u7a31\u70ba\u5e95\u6578\uff0c$$n$$ \u7a31\u70ba\u6307\u6578\u3002<!--more--><\/p>\n<p>\u90a3 $$a$$ \u81ea\u4e58 $$n$$ \u6b21\u518d\u4e58\u4ee5 $$a$$ \u81ea\u4e58 $$m$$ \u6b21\uff0c\u90a3 $$a$$ \u7e3d\u5171\u81ea\u4e58\u4e86\u591a\u5c11\u6b21\uff1f\u800c\u7528\u6578\u5b78\u8a9e\u8a00\u5c31\u662f $$a^n\\cdot{a}^m=~?$$ $$(a^{n+m})$$\uff0c\u800c\u6b64\u6642\u6307\u6578\u70ba\u81ea\u7136\u6578\uff0c\u5373\u70ba\u6307\u6578\u5f8b\u7684\u7b2c\u4e00\u500b\u57fa\u672c\u6027\u8cea\u3002<\/p>\n<p>\u90a3\u6307\u6578\u5f8b\u7684\u7b2c\u4e8c\u500b\u57fa\u672c\u6027\u8cea\u70ba\u4f55\u5462\uff1f\u5373 $$a$$ \u81ea\u4e58 $$n$$ \u6b21\u5f97\u4e00\u6578\u70ba $$a^n$$\uff0c\u518d\u5c07 $$a^n$$ \u81ea\u4e58 $$m$$ \u6b21\uff0c\u5373 $$(a^n)^m=a^{nm}$$\u3002<\/p>\n<p>\u800c\u6307\u6578\u5f8b\u7684\u6700\u5f8c\u4e00\u500b\u6027\u8cea\u70ba $$ab$$ \u81ea\u4e58 $$m$$ \u6b21\uff0c\u662f\u5426\u548c $$a$$ \u81ea\u4e58 $$m$$ \u6b21\u518d\u4e58\u4ee5 $$b$$ \u81ea\u4e58 $$m$$ \u6b21\u4e00\u6a23\u5462\uff1f\u800c\u6b64\u4e00\u6027\u8cea\u7684\u8868\u73fe\u5f0f\u5b50\u70ba $$(ab)^m=a^m\\cdot{b}^m$$\u3002\u9019\u4e09\u500b\u6027\u8cea\uff0c\u5373\u70ba\u570b\u4e2d\u6240\u5b78\u6307\u6578\u70ba\u81ea\u7136\u6578\u6642\u7684\u6307\u6578\u5f8b\u6027\u8cea\u3002<\/p>\n<p>\u5728\u9ad8\u4e2d\u6642\uff0c\u70ba\u4e86\u4ee5\u5f8c\u6307\u6578\u51fd\u6578\u767c\u5c55\u70ba\u57fa\u790e\uff0c\u5fc5\u9808\u5c07\u6307\u6578\u5f8b\u7684\u6027\u8cea\uff0c\u4f9d\u5e8f\u7531\u96f6\u6307\u6578\u3001\u8ca0\u6574\u6578\u6307\u6578\u3001\u6709\u7406\u6578\u6307\u6578\u5230\u5be6\u6578\u6307\u6578\u4f9d\u5e8f\u7684\u64f4\u5145\u3002\u5176\u4e2d\uff0c\u96f6\u6307\u6578\u5373\u70ba $$a^0~(a\\neq{0})$$\uff0c\u6211\u5011\u5c07\u5176\u5b9a\u7fa9\u70ba $$a^0=1$$\uff0c\u6b64\u6642\uff0c\u6307\u6578\u5f8b\u6027\u8cea\u4f9d\u7136\u53ef\u4ee5\u6210\u7acb\u3002<\/p>\n<p>\u800c\u8ca0\u6574\u6578\u6307\u6578\u5373\u70ba $$a^{-n}~(a\\neq{0},~n\\in{N})$$\uff0c\u6211\u5011\u5c07\u5176\u5b9a\u7fa9\u70ba $$a^{-n}=\\frac{1}{a^n}$$\uff0c\u800c\u9019\u6a23\u5b9a\u7fa9\u65b9\u5f0f\u5728\u6307\u6578\u70ba\u8ca0\u6574\u6578\u6642\uff0c\u4f9d\u7136\u53ef\u4ee5\u6eff\u8db3\u6307\u6578\u6027\u8cea\u3002<\/p>\n<p>\u800c\u5728\u64f4\u5145\u70ba\u6709\u7406\u6578\u6307\u6578\u6642\uff0c\u5fc5\u9808\u5c07\u5e95\u6578 $$a$$ \u9650\u5236\u70ba\u6b63\u6578\uff0c\u56e0\u70ba\u52d8\u6839\u5b9a\u7406\u544a\u8a34\u6211\u5011 $$x^n=a$$ \u6642\uff0c\u82e5 $$a&gt;0$$\uff0c\u6070\u53ea\u6709\u4e00\u500b\u6b63\u5be6\u6839 $$\\sqrt[n]{a}$$\uff0c\u82e5 $$a&lt;0$$\uff0c\u5247\u4e0d\u4e00\u5b9a\u6709\u5be6\u6839\u5b58\u5728\u3002\u56e0\u800c\u5728\u52d8\u6839\u5b9a\u7406\u548c\u81ea\u7136\u6578\u6307\u6578\u5f8b\u4e0b\uff0c\u53ef\u4ee5\u77e5\u9053 $$a^{\\frac{1}{n}}$$ \u662f $$x^n$$ \u7684\u4e00\u6b63\u5be6\u6839\uff0c\u6240\u4ee5\uff0c\u53ef\u4ee5\u5b9a\u7fa9 $$a^{\\frac{1}{a}}=\\sqrt[n]{a}$$\u3002<\/p>\n<p>\u800c\u66f4\u4e00\u822c\u6027\u6709\u7406\u6578\u6307\u6578 $$a^r$$\uff0c$$r=\\frac{m}{n}~(n,m\\in{N})$$\uff0c\u53ef\u4ee5\u5b9a\u7fa9\u70ba $$a^r=a^{\\frac{m}{n}}=(\\sqrt[n]{n})^m$$\u3002\u5728\u6b64\u4e00\u5b9a\u7fa9\u4e0b\uff0c\u4f9d\u7136\u53ef\u4ee5\u8b49\u660e\u6eff\u8db3\u6307\u6578\u5f8b\u7684\u6027\u8cea\u3002<\/p>\n<p>\u5728\u9ad8\u4e2d\u968e\u6bb5\u8981\u5c07\u6307\u6578\u5f8b\u6027\u8cea\u63a8\u5ee3\u81f3\u5be6\u6578\u6307\u6578\uff0c\u727d\u6d89\u5be6\u6578\u5b8c\u5099\u6027\uff0c\u800c\u9019\u53ea\u80fd\u5728\u5927\u5b78\u5fae\u7a4d\u5206\u4e0a\u624d\u80fd\u8b49\u660e\u3002\u4f46\u900f\u904e\u73fe\u4eca\u8a08\u7b97\u6a5f\u7684\u767c\u9054\uff0c\u9084\u662f\u53ef\u4ee5\u900f\u904e\u96fb\u8166\u904b\u7b97\u7684\u903c\u8fd1\uff0c\u53bb\u611f\u53d7\u5230\u6307\u6578\u70ba\u7121\u7406\u6578\u6642\uff0c\u6307\u6578\u5f8b\u6027\u8cea\u7684\u6210\u7acb\u3002<\/p>\n<p>\u88dc\u8db3\u4e86\u6700\u5f8c\u4e00\u584a\u7121\u7406\u6578\u6eff\u8db3\u6307\u6578\u5f8b\u6027\u8cea\uff0c\u4e5f\u5c31\u5c07\u6307\u6578\u5f8b\u6eff\u8db3\u6240\u6709\u5be6\u6578\u6027\u8cea\u90fd\u89e3\u6c7a\u4e86\uff0c\u800c\u6307\u6578\u5f8b\u6027\u8cea\u5c31\u53ef\u4ee5\u63cf\u8ff0\u5982\u4e0b\uff1a\u5373\u8a2d $$a&gt;0$$\uff0c$$b&gt;0$$ \u6642\uff0c$$r,s\\in{R}$$\uff0c\u5247<\/p>\n<ul>\n<li>$$a^r\\cdot{a}^s=a^{r+s}$$<\/li>\n<li>$$(a^r)^s=a^{rs}$$<\/li>\n<li>$$a^r\\cdot{b}^r=(ab)^r$$<\/li>\n<\/ul>\n<p><strong>\u53c3\u8003\u6587\u737b\uff1a<\/strong><\/p>\n<ol>\n<li>\u6bdb\u723e\uff08Eli Maor\uff09\u8457\uff08\u912d\u60df\u539a\u8b6f\uff09\uff0c\u300a\u6bdb\u8d77\u4f86\u8aaa\uff45\u300b\uff0c\u53f0\u5317\uff1a\u5929\u4e0b\u9060\u898b\u51fa\u7248\u793e\uff0c2001\u5e74\u3002<\/li>\n<li>\u9673\u4ec1\u653f\uff0c\u300a\u4e0d\u53ef\u601d\u8b70\u7684\uff45\u300b\uff0c\u5317\u4eac\uff1a\u79d1\u5b78\u51fa\u7248\u793e\uff0c2005\u5e74\u3002<\/li>\n<\/ol>\n<\/div>","protected":false},"excerpt":{"rendered":"<p>\u672c\u6587\u7c21\u55ae\u6558\u8ff0\u9ad8\u4e2d\u968e\u6bb5\u6307\u6578\u5f8b\u6027\u8cea\uff0c\u7531\u6307\u6578\u662f\u6b63\u6574\u6578\u3001\u96f6\u3001\u8ca0\u6574\u6578\u3001\u6709\u7406\u6578\u4f9d\u5e8f\u64f4\u5145\u90fd\u53ef\u4ee5\u6eff\u8db3\u6307\u6578\u6578\u57fa\u672c\u6027\u8cea\u3002 <\/p>\n","protected":false},"author":50,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[221,111],"tags":[304,1610],"class_list":["post-16780","post","type-post","status-publish","format-standard","hentry","category-math03-02","category-mathematics00","tag-304","tag-1610","loop-entry","cat-221","cat-111","no-thumbnail"],"views":15257,"_links":{"self":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/16780","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/users\/50"}],"replies":[{"embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/comments?post=16780"}],"version-history":[{"count":1,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/16780\/revisions"}],"predecessor-version":[{"id":89254,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/16780\/revisions\/89254"}],"wp:attachment":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/media?parent=16780"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/categories?post=16780"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/tags?post=16780"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}