{"id":2621,"date":"2025-08-13T18:15:48","date_gmt":"2025-08-13T18:15:48","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=2621"},"modified":"2026-03-16T16:06:29","modified_gmt":"2026-03-16T16:06:29","slug":"exponential-and-logarithmic-functions-background-youll-need-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/exponential-and-logarithmic-functions-background-youll-need-3\/","title":{"raw":"Exponential and Logarithmic Functions: Background You'll Need 3","rendered":"Exponential and Logarithmic Functions: Background You&#8217;ll Need 3"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li><span data-sheets-root=\"1\">Rewrite expressions negative exponents and rational exponents<\/span><\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>negative exponents<\/h3>\r\nA <strong data-start=\"518\" data-end=\"539\">negative exponent<\/strong> tells us to take the reciprocal of the base and make the exponent positive.\r\n\r\n&nbsp;\r\n<p style=\"text-align: center;\">[latex]a^{-n}=\\frac{1}{a^n}[\/latex] where [latex]a\\ne 0[\/latex]<\/p>\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]311407 [\/ohm_question]<\/section><section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>rational exponents<\/h3>\r\n<p data-start=\"1029\" data-end=\"1088\">A <strong data-start=\"1031\" data-end=\"1052\">rational exponent<\/strong> is another way of writing a root.<\/p>\r\n\r\n<ul data-start=\"1089\" data-end=\"1199\">\r\n \t<li data-start=\"1089\" data-end=\"1144\">\r\n<p data-start=\"1091\" data-end=\"1144\">The denominator of the exponent gives the root.<\/p>\r\n<\/li>\r\n \t<li data-start=\"1145\" data-end=\"1199\">\r\n<p data-start=\"1147\" data-end=\"1199\">The numerator of the exponent gives the power.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<p style=\"text-align: center;\">[latex]a^\\frac{m}{n}=\\sqrt[n]{a^m}=(\\sqrt[n]{a})^m)[\/latex]<\/p>\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Rewrite [latex]\\sqrt{27}[\/latex] as an exponent.\r\n\r\n\r\n[reveal-answer q=\"162586\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"162586\"]First rewrite the root as an exponent: [latex](27)^\\frac{1}{2}[\/latex] but we can also rewrite the base [latex]27[\/latex]: [latex](3^3)^\\frac{1}{2}=(3)^\\frac{3}{2}[\/latex][\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]311409[\/ohm_question]<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li><span data-sheets-root=\"1\">Rewrite expressions negative exponents and rational exponents<\/span><\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>negative exponents<\/h3>\n<p>A <strong data-start=\"518\" data-end=\"539\">negative exponent<\/strong> tells us to take the reciprocal of the base and make the exponent positive.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\">[latex]a^{-n}=\\frac{1}{a^n}[\/latex] where [latex]a\\ne 0[\/latex]<\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm311407\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=311407&theme=lumen&iframe_resize_id=ohm311407&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>rational exponents<\/h3>\n<p data-start=\"1029\" data-end=\"1088\">A <strong data-start=\"1031\" data-end=\"1052\">rational exponent<\/strong> is another way of writing a root.<\/p>\n<ul data-start=\"1089\" data-end=\"1199\">\n<li data-start=\"1089\" data-end=\"1144\">\n<p data-start=\"1091\" data-end=\"1144\">The denominator of the exponent gives the root.<\/p>\n<\/li>\n<li data-start=\"1145\" data-end=\"1199\">\n<p data-start=\"1147\" data-end=\"1199\">The numerator of the exponent gives the power.<\/p>\n<\/li>\n<\/ul>\n<p style=\"text-align: center;\">[latex]a^\\frac{m}{n}=\\sqrt[n]{a^m}=(\\sqrt[n]{a})^m)[\/latex]<\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Rewrite [latex]\\sqrt{27}[\/latex] as an exponent.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q162586\">Show Answer<\/button><\/p>\n<div id=\"q162586\" class=\"hidden-answer\" style=\"display: none\">First rewrite the root as an exponent: [latex](27)^\\frac{1}{2}[\/latex] but we can also rewrite the base [latex]27[\/latex]: [latex](3^3)^\\frac{1}{2}=(3)^\\frac{3}{2}[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm311409\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=311409&theme=lumen&iframe_resize_id=ohm311409&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":67,"menu_order":4,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":105,"module-header":"background_you_need","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2621"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/67"}],"version-history":[{"count":8,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2621\/revisions"}],"predecessor-version":[{"id":5870,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2621\/revisions\/5870"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/105"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2621\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=2621"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=2621"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=2621"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=2621"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}