{"id":85,"date":"2024-10-16T18:28:57","date_gmt":"2024-10-16T18:28:57","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/?post_type=chapter&#038;p=85"},"modified":"2024-10-18T21:25:52","modified_gmt":"2024-10-18T21:25:52","slug":"radicals-and-rational-exponents-fresh-take","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/chapter\/radicals-and-rational-exponents-fresh-take\/","title":{"raw":"Radicals and Rational Exponents: Fresh Take","rendered":"Radicals and Rational Exponents: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n \t<li>Calculate square roots and apply them in basic operations like addition, subtraction, and rationalizing the denominators.<\/li>\r\n \t<li>Use the product and quotient rules to simplify expressions that include square roots.<\/li>\r\n \t<li>Understand how to use rational exponents in expressions and recognize their connection to roots for simplifying calculations.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Radical Expressions<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Definition of Square Root:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">A square root of [latex]a[\/latex] is any number [latex]x[\/latex] such that [latex]x^2 = a[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">For positive [latex]a[\/latex], there are always two square roots: positive and negative<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Principal Square Root:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Denoted by [latex]\\sqrt{a}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">The non-negative square root of a number<\/li>\r\n \t<li class=\"whitespace-normal break-words\">What calculators provide when computing square roots<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Radical Expression:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\sqrt{a}[\/latex] is called a radical expression<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\sqrt{}[\/latex] symbol is the radical<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]a[\/latex] under the radical is the radicand<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Evaluate each expression.\r\n<ol>\r\n \t<li>[latex]\\sqrt{225}[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt{\\sqrt{81}}[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt{25 - 9}[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt{36}+\\sqrt{121}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"98241\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"98241\"]\r\n<ol>\r\n \t<li>[latex]15[\/latex]<\/li>\r\n \t<li>[latex]3[\/latex]<\/li>\r\n \t<li>[latex]4[\/latex]<\/li>\r\n \t<li>[latex]17[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">In the following video you will see more examples of how to simplify radical expressions with variables.<iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/q7LqsKPoAKo?si=_5otc54ahycpXrOJ\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/section>\r\n<h2>Simplifying Square Roots and Expressing Them in Lowest Terms<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Simplifying Square Roots:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Rewrite as a rational number times the square root of a number with no perfect square factors<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Product Rule for Square Roots:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">For non-negative real numbers [latex]a[\/latex] and [latex]b[\/latex]: [latex]\\sqrt{a \\times b} = \\sqrt{a} \\times \\sqrt{b}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Perfect Squares:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Integers that are squares of other integers (e.g., 16, 25, 36)<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Simplification Process:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Factor the radicand into perfect square and non-perfect square parts<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Apply the product rule<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Simplify<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Variables in Square Roots:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\sqrt{x^2} = |x|[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Even powers can be simplified, odd powers leave one factor under the radical<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Using the Quotient Rule to Simplify Square Roots<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Quotient Rule for Square Roots:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">For non-negative real numbers [latex]a[\/latex] and [latex]b[\/latex], where [latex]b \\neq 0[\/latex]: [latex]\\sqrt{\\frac{a}{b}} = \\frac{\\sqrt{a}}{\\sqrt{b}}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Application of the Quotient Rule:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Separate the square root of a fraction into the quotient of two square roots<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Simplify numerator and denominator separately<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Simplification Process:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Apply the quotient rule<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Simplify each radical expression<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Combine results<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Simplify [latex]\\dfrac{\\sqrt{9{a}^{5}{b}^{14}}}{\\sqrt{3{a}^{4}{b}^{5}}}[\/latex].[reveal-answer q=\"157179\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"157179\"][latex]{b}^{4}\\sqrt{3ab}[\/latex][\/hidden-answer]<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">In the following video you will see more examples of how to simplify radical expressions with variables.<iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/q7LqsKPoAKo?si=4c0cKyIhqrrDJdPZ\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/section>\r\n<h2 data-type=\"title\">Adding and Subtracting Square Roots<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Adding and Subtracting Square Roots:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Only possible with identical radicands<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Combine coefficients of like terms<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Simplifying Radical Expressions:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Simplify each term before combining<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Factor out perfect squares from radicands<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Rationalizing Denominators:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Remove radicals from the denominator<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Use the identity property of multiplication<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Methods for Rationalizing:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Single term denominator: Multiply by [latex]\\sqrt{c}[\/latex] for [latex]b\\sqrt{c}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Complex denominator: Use the conjugate for [latex]a + b\\sqrt{c}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Conjugates:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">For [latex]a + \\sqrt{b}[\/latex], the conjugate is [latex]a - \\sqrt{b}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Product of a term and its conjugate: [latex](a + \\sqrt{b})(a - \\sqrt{b}) = a^2 - b[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Add [latex]\\sqrt{5}+6\\sqrt{20}[\/latex].[reveal-answer q=\"21382\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"21382\"][latex]13\\sqrt{5}[\/latex][\/hidden-answer]<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch this video to see more examples of adding roots.\r\n<iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/S3fGUeALy7E?si=zk3DllilIKwCgP2m\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/section><section class=\"textbox example\" aria-label=\"Example\">Subtract [latex]3\\sqrt{80x}-4\\sqrt{45x}[\/latex].[reveal-answer q=\"236912\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"236912\"][latex]0[\/latex][\/hidden-answer]<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">In the next video, we show more examples of how to subtract radicals.\u00a0<\/section>\r\n<h3>Rationalizing Denominators<\/h3>\r\n<section class=\"textbox example\" aria-label=\"Example\">Write [latex]\\dfrac{12\\sqrt{3}}{\\sqrt{2}}[\/latex] in simplest form.[reveal-answer q=\"497322\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"497322\"][latex]6\\sqrt{6}[\/latex][\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Write [latex]\\dfrac{7}{2+\\sqrt{3}}[\/latex] in simplest form.[reveal-answer q=\"132932\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"132932\"][latex]14 - 7\\sqrt{3}[\/latex][\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/vINRIRgeKqU?si=0YL870F94WAcVnmJ\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/section>\r\n<h2>Rational Roots<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea<\/strong>\r\n\r\n&nbsp;\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">nth Roots:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\sqrt[n]{a}[\/latex] is a number that, when raised to the nth power, equals [latex]a[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]n[\/latex] is called the index of the radical<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Principal nth Root:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Denoted as [latex]\\sqrt[n]{a}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Has the same sign as [latex]a[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">When raised to the nth power, equals [latex]a[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Rational Exponents:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Alternative notation for radical expressions<\/li>\r\n \t<li class=\"whitespace-normal break-words\">General form: [latex]a^{\\frac{m}{n}} = (\\sqrt[n]{a})^m = \\sqrt[n]{a^m}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Properties of Rational Exponents:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Follow the same rules as integer exponents<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Can simplify expressions by combining like bases<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Conversion between Radicals and Rational Exponents:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\sqrt[n]{x} = x^{\\frac{1}{n}}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\sqrt[n]{x^m} = x^{\\frac{m}{n}}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Simplify.\r\n<ol>\r\n \t<li>[latex]\\sqrt[3]{-216}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{3\\sqrt[4]{80}}{\\sqrt[4]{5}}[\/latex]<\/li>\r\n \t<li>[latex]6\\sqrt[3]{9,000}+7\\sqrt[3]{576}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"15987\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"15987\"]\r\n<ol>\r\n \t<li>[latex]-6[\/latex]<\/li>\r\n \t<li>[latex]6[\/latex]<\/li>\r\n \t<li>[latex]88\\sqrt[3]{9}[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Write [latex]{9}^{\\frac{5}{2}}[\/latex] as a radical. Simplify.[reveal-answer q=\"937831\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"937831\"][latex]{\\left(\\sqrt{9}\\right)}^{5}={3}^{5}=243[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Write [latex]x\\sqrt{{\\left(5y\\right)}^{9}}[\/latex] using a rational exponent.[reveal-answer q=\"522860\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"522860\"][latex]x{\\left(5y\\right)}^{\\frac{9}{2}}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Simplify the following:\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]{\\left(8x\\right)}^{\\frac{1}{3}}\\left(14{x}^{\\frac{6}{5}}\\right)[\/latex]<\/li>\r\n \t<li>[latex]\\large{\\frac{\\sqrt{y}}{y^\\frac{2}{5}}}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"95703\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"95703\"]\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]28{x}^{\\frac{23}{15}}[\/latex]<\/li>\r\n \t<li>\r\n<p style=\"padding-left: 30px;\">[latex]y^{\\frac{1}{2}}-y^{\\frac{2}{5}}[\/latex]<\/p>\r\n<p style=\"padding-left: 30px;\">[latex]y^{\\frac{5}{10}}-y^{\\frac{4}{10}}[\/latex]<\/p>\r\n<p style=\"padding-left: 30px;\">[latex]y^{\\frac{1}{10}}[\/latex] or [latex]\\sqrt[10]{y}[\/latex]<\/p>\r\n<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch this video to see more examples of how to write a radical with a fractional exponent.\r\n<iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/L5Z_3RrrVjA?si=CwBb5XBGEkeVeqBt\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Calculate square roots and apply them in basic operations like addition, subtraction, and rationalizing the denominators.<\/li>\n<li>Use the product and quotient rules to simplify expressions that include square roots.<\/li>\n<li>Understand how to use rational exponents in expressions and recognize their connection to roots for simplifying calculations.<\/li>\n<\/ul>\n<\/section>\n<h2>Radical Expressions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Definition of Square Root:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">A square root of [latex]a[\/latex] is any number [latex]x[\/latex] such that [latex]x^2 = a[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">For positive [latex]a[\/latex], there are always two square roots: positive and negative<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Principal Square Root:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Denoted by [latex]\\sqrt{a}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">The non-negative square root of a number<\/li>\n<li class=\"whitespace-normal break-words\">What calculators provide when computing square roots<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Radical Expression:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]\\sqrt{a}[\/latex] is called a radical expression<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\sqrt{}[\/latex] symbol is the radical<\/li>\n<li class=\"whitespace-normal break-words\">[latex]a[\/latex] under the radical is the radicand<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Evaluate each expression.<\/p>\n<ol>\n<li>[latex]\\sqrt{225}[\/latex]<\/li>\n<li>[latex]\\sqrt{\\sqrt{81}}[\/latex]<\/li>\n<li>[latex]\\sqrt{25 - 9}[\/latex]<\/li>\n<li>[latex]\\sqrt{36}+\\sqrt{121}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q98241\">Show Solution<\/button><\/p>\n<div id=\"q98241\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]15[\/latex]<\/li>\n<li>[latex]3[\/latex]<\/li>\n<li>[latex]4[\/latex]<\/li>\n<li>[latex]17[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">In the following video you will see more examples of how to simplify radical expressions with variables.<iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/q7LqsKPoAKo?si=_5otc54ahycpXrOJ\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/section>\n<h2>Simplifying Square Roots and Expressing Them in Lowest Terms<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Simplifying Square Roots:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Rewrite as a rational number times the square root of a number with no perfect square factors<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Product Rule for Square Roots:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">For non-negative real numbers [latex]a[\/latex] and [latex]b[\/latex]: [latex]\\sqrt{a \\times b} = \\sqrt{a} \\times \\sqrt{b}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Perfect Squares:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Integers that are squares of other integers (e.g., 16, 25, 36)<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Simplification Process:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Factor the radicand into perfect square and non-perfect square parts<\/li>\n<li class=\"whitespace-normal break-words\">Apply the product rule<\/li>\n<li class=\"whitespace-normal break-words\">Simplify<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Variables in Square Roots:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]\\sqrt{x^2} = |x|[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Even powers can be simplified, odd powers leave one factor under the radical<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<h2>Using the Quotient Rule to Simplify Square Roots<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Quotient Rule for Square Roots:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">For non-negative real numbers [latex]a[\/latex] and [latex]b[\/latex], where [latex]b \\neq 0[\/latex]: [latex]\\sqrt{\\frac{a}{b}} = \\frac{\\sqrt{a}}{\\sqrt{b}}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Application of the Quotient Rule:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Separate the square root of a fraction into the quotient of two square roots<\/li>\n<li class=\"whitespace-normal break-words\">Simplify numerator and denominator separately<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Simplification Process:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Apply the quotient rule<\/li>\n<li class=\"whitespace-normal break-words\">Simplify each radical expression<\/li>\n<li class=\"whitespace-normal break-words\">Combine results<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Simplify [latex]\\dfrac{\\sqrt{9{a}^{5}{b}^{14}}}{\\sqrt{3{a}^{4}{b}^{5}}}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q157179\">Show Solution<\/button><\/p>\n<div id=\"q157179\" class=\"hidden-answer\" style=\"display: none\">[latex]{b}^{4}\\sqrt{3ab}[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">In the following video you will see more examples of how to simplify radical expressions with variables.<iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/q7LqsKPoAKo?si=4c0cKyIhqrrDJdPZ\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/section>\n<h2 data-type=\"title\">Adding and Subtracting Square Roots<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Adding and Subtracting Square Roots:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Only possible with identical radicands<\/li>\n<li class=\"whitespace-normal break-words\">Combine coefficients of like terms<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Simplifying Radical Expressions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Simplify each term before combining<\/li>\n<li class=\"whitespace-normal break-words\">Factor out perfect squares from radicands<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Rationalizing Denominators:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Remove radicals from the denominator<\/li>\n<li class=\"whitespace-normal break-words\">Use the identity property of multiplication<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Methods for Rationalizing:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Single term denominator: Multiply by [latex]\\sqrt{c}[\/latex] for [latex]b\\sqrt{c}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Complex denominator: Use the conjugate for [latex]a + b\\sqrt{c}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Conjugates:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">For [latex]a + \\sqrt{b}[\/latex], the conjugate is [latex]a - \\sqrt{b}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Product of a term and its conjugate: [latex](a + \\sqrt{b})(a - \\sqrt{b}) = a^2 - b[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Add [latex]\\sqrt{5}+6\\sqrt{20}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q21382\">Show Solution<\/button><\/p>\n<div id=\"q21382\" class=\"hidden-answer\" style=\"display: none\">[latex]13\\sqrt{5}[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch this video to see more examples of adding roots.<br \/>\n<iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/S3fGUeALy7E?si=zk3DllilIKwCgP2m\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Subtract [latex]3\\sqrt{80x}-4\\sqrt{45x}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q236912\">Show Solution<\/button><\/p>\n<div id=\"q236912\" class=\"hidden-answer\" style=\"display: none\">[latex]0[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">In the next video, we show more examples of how to subtract radicals.\u00a0<\/section>\n<h3>Rationalizing Denominators<\/h3>\n<section class=\"textbox example\" aria-label=\"Example\">Write [latex]\\dfrac{12\\sqrt{3}}{\\sqrt{2}}[\/latex] in simplest form.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q497322\">Show Solution<\/button><\/p>\n<div id=\"q497322\" class=\"hidden-answer\" style=\"display: none\">[latex]6\\sqrt{6}[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Write [latex]\\dfrac{7}{2+\\sqrt{3}}[\/latex] in simplest form.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q132932\">Show Solution<\/button><\/p>\n<div id=\"q132932\" class=\"hidden-answer\" style=\"display: none\">[latex]14 - 7\\sqrt{3}[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/vINRIRgeKqU?si=0YL870F94WAcVnmJ\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/section>\n<h2>Rational Roots<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<\/strong><\/p>\n<p>&nbsp;<\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">nth Roots:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]\\sqrt[n]{a}[\/latex] is a number that, when raised to the nth power, equals [latex]a[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]n[\/latex] is called the index of the radical<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Principal nth Root:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Denoted as [latex]\\sqrt[n]{a}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Has the same sign as [latex]a[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">When raised to the nth power, equals [latex]a[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Rational Exponents:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Alternative notation for radical expressions<\/li>\n<li class=\"whitespace-normal break-words\">General form: [latex]a^{\\frac{m}{n}} = (\\sqrt[n]{a})^m = \\sqrt[n]{a^m}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Properties of Rational Exponents:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Follow the same rules as integer exponents<\/li>\n<li class=\"whitespace-normal break-words\">Can simplify expressions by combining like bases<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Conversion between Radicals and Rational Exponents:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]\\sqrt[n]{x} = x^{\\frac{1}{n}}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\sqrt[n]{x^m} = x^{\\frac{m}{n}}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Simplify.<\/p>\n<ol>\n<li>[latex]\\sqrt[3]{-216}[\/latex]<\/li>\n<li>[latex]\\dfrac{3\\sqrt[4]{80}}{\\sqrt[4]{5}}[\/latex]<\/li>\n<li>[latex]6\\sqrt[3]{9,000}+7\\sqrt[3]{576}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q15987\">Show Solution<\/button><\/p>\n<div id=\"q15987\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]-6[\/latex]<\/li>\n<li>[latex]6[\/latex]<\/li>\n<li>[latex]88\\sqrt[3]{9}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Write [latex]{9}^{\\frac{5}{2}}[\/latex] as a radical. Simplify.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q937831\">Show Solution<\/button><\/p>\n<div id=\"q937831\" class=\"hidden-answer\" style=\"display: none\">[latex]{\\left(\\sqrt{9}\\right)}^{5}={3}^{5}=243[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Write [latex]x\\sqrt{{\\left(5y\\right)}^{9}}[\/latex] using a rational exponent.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q522860\">Show Solution<\/button><\/p>\n<div id=\"q522860\" class=\"hidden-answer\" style=\"display: none\">[latex]x{\\left(5y\\right)}^{\\frac{9}{2}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Simplify the following:<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>[latex]{\\left(8x\\right)}^{\\frac{1}{3}}\\left(14{x}^{\\frac{6}{5}}\\right)[\/latex]<\/li>\n<li>[latex]\\large{\\frac{\\sqrt{y}}{y^\\frac{2}{5}}}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q95703\">Show Solution<\/button><\/p>\n<div id=\"q95703\" class=\"hidden-answer\" style=\"display: none\">\n<ol style=\"list-style-type: lower-alpha;\">\n<li>[latex]28{x}^{\\frac{23}{15}}[\/latex]<\/li>\n<li>\n<p style=\"padding-left: 30px;\">[latex]y^{\\frac{1}{2}}-y^{\\frac{2}{5}}[\/latex]<\/p>\n<p style=\"padding-left: 30px;\">[latex]y^{\\frac{5}{10}}-y^{\\frac{4}{10}}[\/latex]<\/p>\n<p style=\"padding-left: 30px;\">[latex]y^{\\frac{1}{10}}[\/latex] or [latex]\\sqrt[10]{y}[\/latex]<\/p>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch this video to see more examples of how to write a radical with a fractional exponent.<br \/>\n<iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/L5Z_3RrrVjA?si=CwBb5XBGEkeVeqBt\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/section>\n","protected":false},"author":15,"menu_order":28,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":28,"module-header":"- Select Header -","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\/collegealgebrademo\/wp-json\/pressbooks\/v2\/chapters\/85"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":1,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/pressbooks\/v2\/chapters\/85\/revisions"}],"predecessor-version":[{"id":94,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/pressbooks\/v2\/chapters\/85\/revisions\/94"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/pressbooks\/v2\/parts\/28"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/pressbooks\/v2\/chapters\/85\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/wp\/v2\/media?parent=85"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/pressbooks\/v2\/chapter-type?post=85"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/wp\/v2\/contributor?post=85"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/wp-json\/wp\/v2\/license?post=85"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}