{"id":1220,"date":"2024-05-07T01:42:28","date_gmt":"2024-05-07T01:42:28","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=1220"},"modified":"2025-08-13T15:38:17","modified_gmt":"2025-08-13T15:38:17","slug":"non-linear-equations-and-inequalities-background-youll-need-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/non-linear-equations-and-inequalities-background-youll-need-1\/","title":{"raw":"Non-Linear Equations: Background You'll Need 1","rendered":"Non-Linear Equations: Background You&#8217;ll Need 1"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n \t<li><span data-sheets-root=\"1\" data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Simplify and rewrite rational exponents.&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:7041,&quot;3&quot;:{&quot;1&quot;:0},&quot;10&quot;:0,&quot;11&quot;:4,&quot;12&quot;:0,&quot;14&quot;:{&quot;1&quot;:2,&quot;2&quot;:0},&quot;15&quot;:&quot;Calibri&quot;}\">Simplify and rewrite rational exponents.<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Rational Exponents<\/h2>\r\nRadical expressions can also be written without using the radical symbol. We can use rational (fractional) exponents. The index must be a positive integer. If the index [latex]n[\/latex] is even, then [latex]x[\/latex] cannot be negative.\r\n<table class=\"center\" style=\"border-collapse: collapse; width: 57.2575%;\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 27.0409%; text-align: center;\">Radical Form<\/td>\r\n<td style=\"width: 24.4404%; text-align: center;\">Exponent Form<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 27.0409%; text-align: center;\"><span style=\"font-size: 14pt;\">[latex]\\sqrt{x}[\/latex]<\/span><\/td>\r\n<td style=\"width: 24.4404%; text-align: center;\"><span style=\"font-size: 14pt;\">[latex]x^{\\frac{1}{2}}[\/latex]<\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 27.0409%; text-align: center;\"><span style=\"font-size: 14pt;\">[latex]\\sqrt[3]{x}[\/latex]<\/span><\/td>\r\n<td style=\"width: 24.4404%; text-align: center;\"><span style=\"font-size: 14pt;\">[latex]x^{\\frac{1}{3}}[\/latex]<\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 27.0409%; text-align: center;\"><span style=\"font-size: 14pt;\">[latex]\\sqrt[4]{x}[\/latex]<\/span><\/td>\r\n<td style=\"width: 24.4404%; text-align: center;\"><span style=\"font-size: 14pt;\">[latex]x^{\\frac{1}{4}}[\/latex]<\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 27.0409%; text-align: center;\"><span style=\"font-size: 14pt;\">...<\/span><\/td>\r\n<td style=\"width: 24.4404%; text-align: center;\"><span style=\"font-size: 14pt;\">...<\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 27.0409%; text-align: center;\"><span style=\"font-size: 14pt;\">[latex]\\sqrt[n]{x}[\/latex]<\/span><\/td>\r\n<td style=\"width: 24.4404%; text-align: center;\"><span style=\"font-size: 14pt;\">[latex]x^{\\frac{1}{n}}[\/latex]<\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">We can also have rational exponents with numerators other than 1.<\/section><section class=\"textbox keyTakeaway\">\r\n<h3>rational exponents<\/h3>\r\nRational exponents are another way to express principal [latex]\\text{n}^{\\text{th}}[\/latex] roots.\r\n\r\n&nbsp;\r\n\r\nThe general form for converting between a radical expression with a radical symbol and one with a rational exponent is\r\n<div style=\"text-align: center;\">[latex]\\begin{align}{a}^{\\frac{m}{n}}={\\left(\\sqrt[n]{a}\\right)}^{m}=\\sqrt[n]{{a}^{m}}\\end{align}[\/latex]<\/div>\r\n\r\n[caption id=\"attachment_741\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-741 size-medium\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/04\/25175536\/Screen-Shot-2016-07-29-at-3.56.45-PM-300x179.png\" alt=\"\" width=\"300\" height=\"179\" \/> Converting principal root to rational exponent[\/caption]\r\n\r\n<\/section><section class=\"textbox example\">Write [latex]{9}^{\\frac{5}{2}}[\/latex] as a radical and then simplify.[reveal-answer q=\"55790\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"55790\"] [latex]\\begin{align*} \\text{Convert exponent to radical} &amp;: &amp; 9^{5\/2} &amp;= \\left(9^{1\/2}\\right)^5 &amp; \\\\ \\text{Simplify the square root} &amp;: &amp; &amp;= (\\sqrt{9})^5 &amp; \\\\ \\text{Calculate \\( \\sqrt{9} \\)} &amp;: &amp; &amp;= 3^5 &amp; \\\\ \\text{Compute the power} &amp;: &amp; &amp;= 243 &amp; \\end{align*}[\/latex][\/hidden-answer]<\/section><section class=\"textbox recall\" aria-label=\"Recall\">When a base has a negative exponent, you can rewrite it as the reciprocal of the base with a positive exponent. In mathematical terms:<center>[latex] a^{-x} = \\frac{1}{a^x} [\/latex]<\/center>This rule helps simplify expressions where negative exponents are present by turning them into fractions with positive exponents.<\/section><section class=\"textbox example\">\r\n<div class=\"flex-shrink-0 flex flex-col relative items-end\">\r\n<div class=\"pt-0.5\">\r\n<div class=\"gizmo-shadow-stroke flex h-6 w-6 items-center justify-center overflow-hidden rounded-full\">\r\n<div class=\"relative flex\">Write [latex]\\dfrac{4}{\\sqrt[7]{{a}^{2}}}[\/latex] using a rational exponent.<strong>\r\n<\/strong>[reveal-answer q=\"796457\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"796457\"]<span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">[latex]\\begin{align*} \\text{Start with:} &amp; \\quad \\frac{4}{\\sqrt[7]{a^2}} &amp; \\text{Original expression.} \\\\ \\text{Rewrite the radical:} &amp; \\quad \\frac{4}{a^{\\frac{2}{7}}} &amp; \\text{Express the 7th root of } a^2 \\text{ as } a^{2\/7}. \\\\ \\text{Apply the negative exponent rule:} &amp; \\quad 4a^{-\\frac{2}{7}} &amp; \\text{Transform } \\frac{1}{a^{\\frac{2}{7}}} \\text{ to } a^{-\\frac{2}{7}} \\text{ using the rule } \\frac{1}{a^x} = a^{-x}. \\end{align*}[\/latex]<\/span>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Simplify:\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]5\\left(2{x}^{\\frac{3}{4}}\\right)\\left(3{x}^{\\frac{1}{5}}\\right)[\/latex]<\/li>\r\n \t<li>[latex]{\\left(\\dfrac{16}{9}\\right)}^{-\\frac{1}{2}}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"803060\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"803060\"]\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]\\begin{align}5\\left(2{x}^{\\frac{3}{4}}\\right)\\left(3{x}^{\\frac{1}{5}}\\right) &amp; = 30{x}^{\\frac{3}{4}}{x}^{\\frac{1}{5}}&amp;&amp; \\text{Multiply the coefficients}. \\\\ &amp; = 30{x}^{\\frac{3}{4}+\\frac{1}{5}}&amp;&amp; \\text{Use properties of exponents}. \\\\ &amp; = 30{x}^{\\frac{19}{20}}&amp;&amp; \\text{Simplify}. \\end{align}[\/latex]<\/li>\r\n \t<li>[latex]\\begin{align}{\\left(\\dfrac{16}{9}\\right)}^{-\\frac{1}{2}} &amp; = {\\left(\\frac{9}{16}\\right)}^{\\frac{1}{2}}&amp;&amp; \\text{Use definition of negative exponents}. \\\\ &amp; = \\sqrt{\\frac{9}{16}}&amp;&amp; \\text{Rewrite as a radical}. \\\\ &amp; = \\frac{\\sqrt{9}}{\\sqrt{16}}&amp;&amp; \\text{Use the quotient rule}. \\\\ &amp; = \\frac{3}{4}&amp;&amp; \\text{Simplify}. \\end{align}[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]18994[\/ohm2_question]<\/section><section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]18995[\/ohm2_question]<\/section><section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]18996[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li><span data-sheets-root=\"1\" data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Simplify and rewrite rational exponents.&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:7041,&quot;3&quot;:{&quot;1&quot;:0},&quot;10&quot;:0,&quot;11&quot;:4,&quot;12&quot;:0,&quot;14&quot;:{&quot;1&quot;:2,&quot;2&quot;:0},&quot;15&quot;:&quot;Calibri&quot;}\">Simplify and rewrite rational exponents.<\/span><\/li>\n<\/ul>\n<\/section>\n<h2>Rational Exponents<\/h2>\n<p>Radical expressions can also be written without using the radical symbol. We can use rational (fractional) exponents. The index must be a positive integer. If the index [latex]n[\/latex] is even, then [latex]x[\/latex] cannot be negative.<\/p>\n<table class=\"center\" style=\"border-collapse: collapse; width: 57.2575%;\">\n<tbody>\n<tr>\n<td style=\"width: 27.0409%; text-align: center;\">Radical Form<\/td>\n<td style=\"width: 24.4404%; text-align: center;\">Exponent Form<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 27.0409%; text-align: center;\"><span style=\"font-size: 14pt;\">[latex]\\sqrt{x}[\/latex]<\/span><\/td>\n<td style=\"width: 24.4404%; text-align: center;\"><span style=\"font-size: 14pt;\">[latex]x^{\\frac{1}{2}}[\/latex]<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 27.0409%; text-align: center;\"><span style=\"font-size: 14pt;\">[latex]\\sqrt[3]{x}[\/latex]<\/span><\/td>\n<td style=\"width: 24.4404%; text-align: center;\"><span style=\"font-size: 14pt;\">[latex]x^{\\frac{1}{3}}[\/latex]<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 27.0409%; text-align: center;\"><span style=\"font-size: 14pt;\">[latex]\\sqrt[4]{x}[\/latex]<\/span><\/td>\n<td style=\"width: 24.4404%; text-align: center;\"><span style=\"font-size: 14pt;\">[latex]x^{\\frac{1}{4}}[\/latex]<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 27.0409%; text-align: center;\"><span style=\"font-size: 14pt;\">&#8230;<\/span><\/td>\n<td style=\"width: 24.4404%; text-align: center;\"><span style=\"font-size: 14pt;\">&#8230;<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 27.0409%; text-align: center;\"><span style=\"font-size: 14pt;\">[latex]\\sqrt[n]{x}[\/latex]<\/span><\/td>\n<td style=\"width: 24.4404%; text-align: center;\"><span style=\"font-size: 14pt;\">[latex]x^{\\frac{1}{n}}[\/latex]<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">We can also have rational exponents with numerators other than 1.<\/section>\n<section class=\"textbox keyTakeaway\">\n<h3>rational exponents<\/h3>\n<p>Rational exponents are another way to express principal [latex]\\text{n}^{\\text{th}}[\/latex] roots.<\/p>\n<p>&nbsp;<\/p>\n<p>The general form for converting between a radical expression with a radical symbol and one with a rational exponent is<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{align}{a}^{\\frac{m}{n}}={\\left(\\sqrt[n]{a}\\right)}^{m}=\\sqrt[n]{{a}^{m}}\\end{align}[\/latex]<\/div>\n<figure id=\"attachment_741\" aria-describedby=\"caption-attachment-741\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-741 size-medium\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/04\/25175536\/Screen-Shot-2016-07-29-at-3.56.45-PM-300x179.png\" alt=\"\" width=\"300\" height=\"179\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/04\/25175536\/Screen-Shot-2016-07-29-at-3.56.45-PM-300x179.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/04\/25175536\/Screen-Shot-2016-07-29-at-3.56.45-PM-65x39.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/04\/25175536\/Screen-Shot-2016-07-29-at-3.56.45-PM-225x134.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/04\/25175536\/Screen-Shot-2016-07-29-at-3.56.45-PM-350x208.png 350w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/04\/25175536\/Screen-Shot-2016-07-29-at-3.56.45-PM.png 356w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-741\" class=\"wp-caption-text\">Converting principal root to rational exponent<\/figcaption><\/figure>\n<\/section>\n<section class=\"textbox example\">Write [latex]{9}^{\\frac{5}{2}}[\/latex] as a radical and then simplify.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q55790\">Show Answer<\/button><\/p>\n<div id=\"q55790\" class=\"hidden-answer\" style=\"display: none\"> [latex]\\begin{align*} \\text{Convert exponent to radical} &: & 9^{5\/2} &= \\left(9^{1\/2}\\right)^5 & \\\\ \\text{Simplify the square root} &: & &= (\\sqrt{9})^5 & \\\\ \\text{Calculate \\( \\sqrt{9} \\)} &: & &= 3^5 & \\\\ \\text{Compute the power} &: & &= 243 & \\end{align*}[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox recall\" aria-label=\"Recall\">When a base has a negative exponent, you can rewrite it as the reciprocal of the base with a positive exponent. In mathematical terms:<\/p>\n<div style=\"text-align: center;\">[latex]a^{-x} = \\frac{1}{a^x}[\/latex]<\/div>\n<p>This rule helps simplify expressions where negative exponents are present by turning them into fractions with positive exponents.<\/section>\n<section class=\"textbox example\">\n<div class=\"flex-shrink-0 flex flex-col relative items-end\">\n<div class=\"pt-0.5\">\n<div class=\"gizmo-shadow-stroke flex h-6 w-6 items-center justify-center overflow-hidden rounded-full\">\n<div class=\"relative flex\">Write [latex]\\dfrac{4}{\\sqrt[7]{{a}^{2}}}[\/latex] using a rational exponent.<strong><br \/>\n<\/strong><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q796457\">Show Answer<\/button><\/p>\n<div id=\"q796457\" class=\"hidden-answer\" style=\"display: none\"><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">[latex]\\begin{align*} \\text{Start with:} & \\quad \\frac{4}{\\sqrt[7]{a^2}} & \\text{Original expression.} \\\\ \\text{Rewrite the radical:} & \\quad \\frac{4}{a^{\\frac{2}{7}}} & \\text{Express the 7th root of } a^2 \\text{ as } a^{2\/7}. \\\\ \\text{Apply the negative exponent rule:} & \\quad 4a^{-\\frac{2}{7}} & \\text{Transform } \\frac{1}{a^{\\frac{2}{7}}} \\text{ to } a^{-\\frac{2}{7}} \\text{ using the rule } \\frac{1}{a^x} = a^{-x}. \\end{align*}[\/latex]<\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Simplify:<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>[latex]5\\left(2{x}^{\\frac{3}{4}}\\right)\\left(3{x}^{\\frac{1}{5}}\\right)[\/latex]<\/li>\n<li>[latex]{\\left(\\dfrac{16}{9}\\right)}^{-\\frac{1}{2}}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q803060\">Show Solution<\/button><\/p>\n<div id=\"q803060\" class=\"hidden-answer\" style=\"display: none\">\n<ol style=\"list-style-type: lower-alpha;\">\n<li>[latex]\\begin{align}5\\left(2{x}^{\\frac{3}{4}}\\right)\\left(3{x}^{\\frac{1}{5}}\\right) & = 30{x}^{\\frac{3}{4}}{x}^{\\frac{1}{5}}&& \\text{Multiply the coefficients}. \\\\ & = 30{x}^{\\frac{3}{4}+\\frac{1}{5}}&& \\text{Use properties of exponents}. \\\\ & = 30{x}^{\\frac{19}{20}}&& \\text{Simplify}. \\end{align}[\/latex]<\/li>\n<li>[latex]\\begin{align}{\\left(\\dfrac{16}{9}\\right)}^{-\\frac{1}{2}} & = {\\left(\\frac{9}{16}\\right)}^{\\frac{1}{2}}&& \\text{Use definition of negative exponents}. \\\\ & = \\sqrt{\\frac{9}{16}}&& \\text{Rewrite as a radical}. \\\\ & = \\frac{\\sqrt{9}}{\\sqrt{16}}&& \\text{Use the quotient rule}. \\\\ & = \\frac{3}{4}&& \\text{Simplify}. \\end{align}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm18994\" 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