{"id":33,"date":"2004-06-25T15:37:11","date_gmt":"2004-06-25T15:37:11","guid":{"rendered":"http:\/\/www.sixthform.info\/maths\/?p=33"},"modified":"2004-06-25T15:37:11","modified_gmt":"2004-06-25T15:37:11","slug":"powers","status":"publish","type":"post","link":"https:\/\/www.sixthform.info\/maths\/?p=33","title":{"rendered":"Powers"},"content":{"rendered":"<p>I was asked recently why <img src='\/maths\/latexrender\/pictures\/ccc35de62fd016c8d36b21cde39396fb.gif' title='2^0=1' alt='2^0=1' align=absmiddle>. Remember that if <i>n<\/i> is a positive whole number then <img src='\/maths\/latexrender\/pictures\/4160ba789a392fbb287c2e28a573731f.gif' title='2^n= \\underset{n \\text{ times}}{\\underbrace{2 \\times 2 \\times 2\\times \\dots \\times 2}}' alt='2^n= \\underset{n \\text{ times}}{\\underbrace{2 \\times 2 \\times 2\\times \\dots \\times 2}}' align=absmiddle>. Clearly you can&#8217;t multiply 2 by itself 0 times \ud83d\ude15 <\/p>\n<p>The key, when extending properties of the number system, is to use definitions that work for every number. So, for example<\/p>\n<ul><img src='\/maths\/latexrender\/pictures\/719c2d69a0265125462f55a3ea631ce2.gif' title='\\dfrac{2^5}{2^2}=\\dfrac{2 \\times 2 \\times 2 \\; \\times \\! \\not{2} \\; \\times \\! \\not{2}}{\\not{2}\\; \\times \\! \\not{2}}=2^3=2^{5-2}' alt='\\dfrac{2^5}{2^2}=\\dfrac{2 \\times 2 \\times 2 \\; \\times \\! \\not{2} \\; \\times \\! \\not{2}}{\\not{2}\\; \\times \\! \\not{2}}=2^3=2^{5-2}' align=absmiddle><\/ul>\n<p>which gives you the rule that <\/p>\n<ul>to divide powers you subtract the <i>indices<\/i> (the small superscripted numbers)<\/ul>\n<p>This leads to <\/p>\n<ul><img src='\/maths\/latexrender\/pictures\/0d53b3e77f53429438512fa5b9322d38.gif' title='1=\\dfrac{2^3}{2^3}=2^{3-3}=2^0' alt='1=\\dfrac{2^3}{2^3}=2^{3-3}=2^0' align=absmiddle><\/ul>\n<p> Similarly, <img src='\/maths\/latexrender\/pictures\/92a3a74f1feee8b1b94b04f88f86cc20.gif' title='a^0=1' alt='a^0=1' align=absmiddle> for any positive real number.<\/p>\n<p>What about zero powers of non-negative powers? <img src='\/maths\/latexrender\/pictures\/601341530f30ffce8390632db2aa2e4f.gif' title='0^0' alt='0^0' align=absmiddle> is a controversial case I have mentioned on <a href=\"http:\/\/www.sixthform.info\/maths\/index.php?m=20040229\" target=\"_blank\">29 February (Q2.)<\/a> and see <a href=\"http:\/\/mathforum.org\/dr.math\/faq\/faq.0.to.0.power.html\" target=\"_blank\">Dr Math FAQ<\/a> for more on this.<\/p>\n<p>And if the number is negative? Great care is needed in this case. For example, using only real numbers, <img src='\/maths\/latexrender\/pictures\/0539aa78ce6258a809e97e738b337fa8.gif' title='(-1)^{1\/3}=-1' alt='(-1)^{1\/3}=-1' align=absmiddle> but <img src='\/maths\/latexrender\/pictures\/8769a57432b585e123f43a238db6f623.gif' title='(-1)^{1\/2}' alt='(-1)^{1\/2}' align=absmiddle> is not a real number. The problem arises because the general definition of a power is given by <img src='\/maths\/latexrender\/pictures\/190ae633d7d7928ffb2fea7d90f8f83e.gif' title='a^x=e^{x\\ln a}' alt='a^x=e^{x\\ln a}' align=absmiddle> and <img src='\/maths\/latexrender\/pictures\/bd3e28c3af8f38399b11cf6cee12dbfd.gif' title='\\ln a' alt='\\ln a' align=absmiddle> is undefined if <i>a<\/i> is negative or 0. Using complex numbers (which helps with <img src='\/maths\/latexrender\/pictures\/8769a57432b585e123f43a238db6f623.gif' title='(-1)^{1\/2}' alt='(-1)^{1\/2}' align=absmiddle>) just makes things more complex \ud83d\ude15 &#8211; see <a href=\"http:\/\/mathforum.org\/dr.math\/faq\/faq.0.to.0.power.html\" target=\"_blank\">Log of Complex Number<\/a> <\/p>\n","protected":false},"excerpt":{"rendered":"<p>I was asked recently why . Remember that if n is a positive whole number then . Clearly you can&#8217;t multiply 2 by itself 0 times \ud83d\ude15 The key, when extending properties of the number system, is to use definitions that work for every number. So, for example which gives you the rule that to [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-33","post","type-post","status-publish","format-standard","hentry","category-articles"],"_links":{"self":[{"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/posts\/33","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=33"}],"version-history":[{"count":0,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/posts\/33\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=33"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=33"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=33"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}