{"id":1014,"date":"2024-04-10T15:01:46","date_gmt":"2024-04-10T15:01:46","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=1014"},"modified":"2024-08-05T12:24:36","modified_gmt":"2024-08-05T12:24:36","slug":"more-basic-functions-and-graphs-background-youll-need-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/more-basic-functions-and-graphs-background-youll-need-2\/","title":{"raw":"More Basic Functions and Graphs: Background You\u2019ll Need 2","rendered":"More Basic Functions and Graphs: Background You\u2019ll Need 2"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Convert between radical and rational exponent notations<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Radical and Rational Exponent Notations<\/h2>\r\n<p><strong>Radical equations<\/strong> are equations that contain variables in the <strong>radicand<\/strong> (the expression under a radical symbol), such as<\/p>\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{ccc} \\sqrt{3y+18}=x &amp; \\\\ \\sqrt{x+3}=y-3 &amp; \\\\ \\sqrt{x+5}-\\sqrt{y - 3}=2\\end{array}[\/latex]<\/div>\r\n<p>Radical equations are manipulated by eliminating each radical, one at a time until you have solved for the indicated variable.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>radical equation<\/h3>\r\n<p>An equation containing terms with a variable in the radicand is called a <strong>radical equation<\/strong>.<\/p>\r\n<\/section>\r\n<p><strong>Rational exponents<\/strong> are exponents that are fractions, where the numerator is a power and the denominator is a root. For example, [latex]{16}^{\\frac{1}{2}}[\/latex] is another way of writing [latex]\\sqrt{16}[\/latex] and [latex]{8}^{\\frac{2}{3}}[\/latex] is another way of writing [latex]\\left(\\sqrt[3]{8}\\right)^2[\/latex].<\/p>\r\n<p>We can solve equations in which a variable is raised to a rational exponent by raising both sides of the equation to the reciprocal of the exponent. The reason we raise the equation to the reciprocal of the exponent is because we want to eliminate the exponent on the variable term, and a number multiplied by its reciprocal equals [latex]1[\/latex].<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>rational exponent<\/h3>\r\n<p>A <strong>rational exponent<\/strong> indicates a power in the numerator and a root in the denominator. There are multiple ways of writing an expression, a variable, or a number with a rational exponent:<\/p>\r\n<center>[latex]{a}^{\\frac{m}{n}}={\\left({a}^{\\frac{1}{n}}\\right)}^{m}={\\left({a}^{m}\\right)}^{\\frac{1}{n}}=\\sqrt[n]{{a}^{m}}={\\left(\\sqrt[n]{a}\\right)}^{m}[\/latex]<\/center><\/section>\r\n<p>To convert a radical to an exponent notation, remember that the [latex]n[\/latex]th root of a number can be written as a power with a fractional exponent. The denominator of the fraction is the root's index ([latex]n[\/latex]), and the numerator is the power to which the radicand is raised.<\/p>\r\n<section class=\"textbox example\">The square root of [latex]a[\/latex], written as [latex]\\sqrt{a}[\/latex], can be expressed as [latex]a^{\\frac{1}{2}}[\/latex].<br \/>\r\nThe cube root of [latex]a[\/latex], written as\u00a0 [latex]\\sqrt[3]{a}[\/latex]\u200b, can be expressed as [latex]a^{\\frac{1}{3}}[\/latex].<\/section>\r\n<p>Conversely, to convert an expression from exponent notation to radical notation, use the denominator of the exponent's fraction as the index of the radical, and the numerator as the power inside the radical.<\/p>\r\n<section class=\"textbox example\">[latex]a^{\\frac{3}{2}}[\/latex] can be written as [latex]\\sqrt[2]{a^3}[\/latex]\u200b or [latex]\\sqrt[3]{a^3}[\/latex]\u200b.<br \/>\r\n[latex]a^{\\frac{2}{5}}[\/latex] can be written as [latex]\\sqrt[5]{a^2}[\/latex]\u200b.<\/section>\r\n<section class=\"textbox example\">Convert the fifth root of [latex]x^3[\/latex], written as [latex]\\sqrt[5]{a^3}[\/latex]\u200b, to exponent notation.<br \/>\r\n<br \/>\r\n[reveal-answer q=\"298131\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"298131\"]<br \/>\r\n<ol>\r\n\t<li>Identify the index of the root, which is [latex]5[\/latex] in this case.<\/li>\r\n\t<li>Write the radicand (the number under the radical) with an exponent that represents the power it is raised to, which is 3 here.<\/li>\r\n\t<li>Combine the root and the power into a single exponent using the rule [latex]{a}^{\\frac{m}{n}}=\\sqrt[n]{{a}^{m}}[\/latex]\u200b<\/li>\r\n\t<li>Apply the rule to get [latex]x^{\\frac{3}{5}}[\/latex].<\/li>\r\n<\/ol>\r\n\r\nThe exponent notation for [latex]\\sqrt[5]{a^3}[\/latex]\u200b is [latex]x^{\\frac{3}{5}}[\/latex]. [\/hidden-answer]<\/section>\r\n<section class=\"textbox example\">Convert [latex]y^{\\frac{4}{3}}[\/latex] into radical notation.<br \/>\r\n<br \/>\r\n[reveal-answer q=\"433508\"]Show Answer[\/reveal-answer] [hidden-answer a=\"433508\"]\r\n\r\n<ol>\r\n\t<li>The denominator of the fraction in the exponent ([latex]3[\/latex]) is the index of the radical.<\/li>\r\n\t<li>The numerator ([latex]4[\/latex]) will be the power to which the radicand is raised inside the radical.<\/li>\r\n\t<li>Write the radical with the index and raise the radicand to the power of the numerator.<\/li>\r\n\t<li>The conversion is then written as\u00a0[latex]\\sqrt[3]{y^4}[\/latex].<\/li>\r\n<\/ol>\r\n\r\nThe radical notation for [latex]y^{\\frac{4}{3}}[\/latex] is [latex]\\sqrt[3]{y^4}[\/latex]. [\/hidden-answer]<\/section>\r\n<section class=\"textbox tryIt\">\r\n<p>[ohm_question hide_question_numbers=1]284086-287002[\/ohm_question]<\/p>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Convert between radical and rational exponent notations<\/li>\n<\/ul>\n<\/section>\n<h2>Radical and Rational Exponent Notations<\/h2>\n<p><strong>Radical equations<\/strong> are equations that contain variables in the <strong>radicand<\/strong> (the expression under a radical symbol), such as<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{ccc} \\sqrt{3y+18}=x & \\\\ \\sqrt{x+3}=y-3 & \\\\ \\sqrt{x+5}-\\sqrt{y - 3}=2\\end{array}[\/latex]<\/div>\n<p>Radical equations are manipulated by eliminating each radical, one at a time until you have solved for the indicated variable.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>radical equation<\/h3>\n<p>An equation containing terms with a variable in the radicand is called a <strong>radical equation<\/strong>.<\/p>\n<\/section>\n<p><strong>Rational exponents<\/strong> are exponents that are fractions, where the numerator is a power and the denominator is a root. For example, [latex]{16}^{\\frac{1}{2}}[\/latex] is another way of writing [latex]\\sqrt{16}[\/latex] and [latex]{8}^{\\frac{2}{3}}[\/latex] is another way of writing [latex]\\left(\\sqrt[3]{8}\\right)^2[\/latex].<\/p>\n<p>We can solve equations in which a variable is raised to a rational exponent by raising both sides of the equation to the reciprocal of the exponent. The reason we raise the equation to the reciprocal of the exponent is because we want to eliminate the exponent on the variable term, and a number multiplied by its reciprocal equals [latex]1[\/latex].<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>rational exponent<\/h3>\n<p>A <strong>rational exponent<\/strong> indicates a power in the numerator and a root in the denominator. There are multiple ways of writing an expression, a variable, or a number with a rational exponent:<\/p>\n<div style=\"text-align: center;\">[latex]{a}^{\\frac{m}{n}}={\\left({a}^{\\frac{1}{n}}\\right)}^{m}={\\left({a}^{m}\\right)}^{\\frac{1}{n}}=\\sqrt[n]{{a}^{m}}={\\left(\\sqrt[n]{a}\\right)}^{m}[\/latex]<\/div>\n<\/section>\n<p>To convert a radical to an exponent notation, remember that the [latex]n[\/latex]th root of a number can be written as a power with a fractional exponent. The denominator of the fraction is the root&#8217;s index ([latex]n[\/latex]), and the numerator is the power to which the radicand is raised.<\/p>\n<section class=\"textbox example\">The square root of [latex]a[\/latex], written as [latex]\\sqrt{a}[\/latex], can be expressed as [latex]a^{\\frac{1}{2}}[\/latex].<br \/>\nThe cube root of [latex]a[\/latex], written as\u00a0 [latex]\\sqrt[3]{a}[\/latex]\u200b, can be expressed as [latex]a^{\\frac{1}{3}}[\/latex].<\/section>\n<p>Conversely, to convert an expression from exponent notation to radical notation, use the denominator of the exponent&#8217;s fraction as the index of the radical, and the numerator as the power inside the radical.<\/p>\n<section class=\"textbox example\">[latex]a^{\\frac{3}{2}}[\/latex] can be written as [latex]\\sqrt[2]{a^3}[\/latex]\u200b or [latex]\\sqrt[3]{a^3}[\/latex]\u200b.<br \/>\n[latex]a^{\\frac{2}{5}}[\/latex] can be written as [latex]\\sqrt[5]{a^2}[\/latex]\u200b.<\/section>\n<section class=\"textbox example\">Convert the fifth root of [latex]x^3[\/latex], written as [latex]\\sqrt[5]{a^3}[\/latex]\u200b, to exponent notation.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q298131\">Show Answer<\/button><\/p>\n<div id=\"q298131\" class=\"hidden-answer\" style=\"display: none\"><\/p>\n<ol>\n<li>Identify the index of the root, which is [latex]5[\/latex] in this case.<\/li>\n<li>Write the radicand (the number under the radical) with an exponent that represents the power it is raised to, which is 3 here.<\/li>\n<li>Combine the root and the power into a single exponent using the rule [latex]{a}^{\\frac{m}{n}}=\\sqrt[n]{{a}^{m}}[\/latex]\u200b<\/li>\n<li>Apply the rule to get [latex]x^{\\frac{3}{5}}[\/latex].<\/li>\n<\/ol>\n<p>The exponent notation for [latex]\\sqrt[5]{a^3}[\/latex]\u200b is [latex]x^{\\frac{3}{5}}[\/latex]. <\/p><\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">Convert [latex]y^{\\frac{4}{3}}[\/latex] into radical notation.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q433508\">Show Answer<\/button> <\/p>\n<div id=\"q433508\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>The denominator of the fraction in the exponent ([latex]3[\/latex]) is the index of the radical.<\/li>\n<li>The numerator ([latex]4[\/latex]) will be the power to which the radicand is raised inside the radical.<\/li>\n<li>Write the radical with the index and raise the radicand to the power of the numerator.<\/li>\n<li>The conversion is then written as\u00a0[latex]\\sqrt[3]{y^4}[\/latex].<\/li>\n<\/ol>\n<p>The radical notation for [latex]y^{\\frac{4}{3}}[\/latex] is [latex]\\sqrt[3]{y^4}[\/latex]. <\/p><\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm284086\" class=\"resizable\" 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