{"id":792,"date":"2025-06-20T17:13:07","date_gmt":"2025-06-20T17:13:07","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/?post_type=chapter&#038;p=792"},"modified":"2025-08-29T14:19:30","modified_gmt":"2025-08-29T14:19:30","slug":"introduction-to-differential-equations-background-youll-need-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/introduction-to-differential-equations-background-youll-need-3\/","title":{"raw":"Introduction to Differential Equations: Background You'll Need 3","rendered":"Introduction to Differential Equations: Background You&#8217;ll Need 3"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n \t<li><span data-sheets-root=\"1\">Apply the properties of exponents to simplify exponential expressions<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Exponential Functions<\/h2>\r\nAny function of the form [latex]f(x)=b^x[\/latex], where [latex]b&gt;0, \\, b \\ne 1[\/latex], is an <strong>exponential function<\/strong> with base [latex]b[\/latex] and exponent [latex]x[\/latex]. Exponential functions have constant bases and variable exponents.\r\n\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>exponential function<\/h3>\r\nFor any real number [latex]x[\/latex], an exponential function is a function with the form\r\n<p style=\"text-align: center;\">[latex]f(x)=ab^x[\/latex]<\/p>\r\nwhere,\r\n<ul>\r\n \t<li>[latex]a[\/latex] is a non-zero real number called the initial value and<\/li>\r\n \t<li>[latex]b[\/latex] is any positive real number ([latex]b&gt;0[\/latex]) such that [latex]b\u22601[\/latex].<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/section>To evaluate an exponential function with the form [latex]f(x)=b^x[\/latex], we simply substitute [latex]x[\/latex] with the given value, and calculate the resulting power.\r\n\r\n<section class=\"textbox example\">Let [latex]f(x)=2^x[\/latex]. What is [latex]f(3)[\/latex]?<center>[latex]\\begin{array}{rcl} f(x) &amp; = &amp; 2^x \\\\ f(3) &amp; = &amp; 2^3 &amp; \\quad \\text{Substitute } x = 3. \\\\ &amp; = &amp; 8 &amp; \\quad \\text{Evaluate the power.} \\end{array} [\/latex]<\/center>\u00a0<\/section>To evaluate an exponential function with a form other than the basic form, it is important to follow the order of operations.\r\n\r\n<section class=\"textbox example\">Let [latex]f(x)=30(2)^x[\/latex]. What is [latex]f(3)[\/latex]?<center>[latex]\\begin{array}{rcll} f(x) &amp; = &amp; 30(2)^x &amp; \\\\ f(3) &amp; = &amp; 30(2)^3 &amp; \\quad \\text{Substitute } x = 3. \\\\ &amp; = &amp; 30(8) &amp; \\quad \\text{Simplify the power first.} \\\\ &amp; = &amp; 240 &amp; \\quad \\text{Multiply.} \\end{array} [\/latex]<\/center>Note that if the order of operations were not followed, the result would be incorrect:<center>[latex]f(3)=30(2)^3\u226060^3=216,000[\/latex]<\/center><\/section><section class=\"textbox questionHelp\"><b>How To: Evaluating Exponential Functions<\/b>\r\n<ol>\r\n \t<li>Given an exponential function, identify [latex]a[\/latex], [latex]b[\/latex], and the value of [latex]x[\/latex] you're being asked to substitute into the function.<\/li>\r\n \t<li>Replace the variable [latex]x[\/latex] in the function with the given number.<\/li>\r\n \t<li>Compute the value of [latex]b^x[\/latex]. This means raising the base [latex]b[\/latex] to the power of [latex]x[\/latex].<\/li>\r\n \t<li>If there is a coefficient [latex]a[\/latex] in front of the base, multiply the result of [latex]b^x[\/latex] by [latex]a[\/latex]. If [latex]a[\/latex] is [latex]1[\/latex], this step does not change the value.<\/li>\r\n \t<li>Simplify the expression if necessary. This could involve performing any additional multiplication or addition\/subtraction if the function has more terms.<\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox example\">Let [latex]f(x)=5(3)^x+1[\/latex]. Evaluate [latex]f(2)[\/latex] without using a calculator.[reveal-answer q=\"586760\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"586760\"]Follow the order of operations. Be sure to pay attention to the parentheses.<center>[latex]\\begin{array}{rcll} f(x) &amp; = &amp; 5(3)^{x+1} &amp; \\\\ f(2) &amp; = &amp; 5(3)^{2+1} &amp; \\quad \\text{Substitute } x = 2. \\\\ &amp; = &amp; 5(3)^3 &amp; \\quad \\text{Add the exponents.} \\\\ &amp; = &amp; 5(27) &amp; \\quad \\text{Simplify the power.} \\\\ &amp; = &amp; 135 &amp; \\quad \\text{Multiply.} \\end{array} [\/latex]<\/center>[\/hidden-answer]<\/section><section class=\"textbox tryIt\">[ohm_question hide_question_numbers=1]284250[\/ohm_question]<\/section><section class=\"textbox example\">\r\n<p id=\"fs-id1170572169653\">Suppose a particular population of bacteria is known to double in size every [latex]4[\/latex] hours. If a culture starts with [latex]1000[\/latex] bacteria, the number of bacteria after [latex]4[\/latex] hours is [latex]n(4)=1000\u00b72[\/latex]. The number of bacteria after [latex]8[\/latex] hours is [latex]n(8)=n(4)\u00b72=1000\u00b72^2[\/latex].<\/p>\r\nIn general, the number of bacteria after [latex]4m[\/latex] hours is [latex]n(4m)=1000\u00b72^m[\/latex]. Letting [latex]t=4m[\/latex], we see that the number of bacteria after [latex]t[\/latex] hours is [latex]n(t)=1000\u00b72^{t\/4}[\/latex].\r\n\r\nFind the number of bacteria after [latex]6[\/latex] hours, [latex]10[\/latex] hours, and [latex]24[\/latex] hours.\r\n\r\n[reveal-answer q=\"fs-id1170572550969\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572550969\"]\r\n<p id=\"fs-id1170572550969\">The number of bacteria after [latex]6[\/latex] hours is given by [latex]n(6)=1000\u00b72^{6\/4} \\approx 2828[\/latex] bacteria.<\/p>\r\nThe number of bacteria after [latex]10[\/latex] hours is given by [latex]n(10)=1000\u00b72^{10\/4} \\approx 5657[\/latex] bacteria.\r\n\r\nThe number of bacteria after [latex]24[\/latex] hours is given by [latex]n(24)=1000\u00b72^{24\/4}=1000\u00b72^6=64,000[\/latex] bacteria.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>\r\n<p id=\"fs-id1170572481226\">The Laws of Exponents are fundamental rules that govern the operations involving powers. These rules are essential for simplifying expressions and are foundational for higher-level math.<\/p>\r\n\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>laws of exponents<\/h3>\r\n<ol id=\"fs-id1170572481268\">\r\n \t<li>The <strong>Product of Powers<\/strong> rule states that when you multiply two exponents with the same base, you can add the exponents.<center>[latex]b^x\u00b7b^y=b^{x+y}[\/latex]<\/center><\/li>\r\n \t<li>The <strong>Quotient of Powers<\/strong> rule tells us that when dividing exponents with the same base, we subtract the exponents.<center>[latex]\\large\\frac{b^x}{b^y} \\normalsize = b^{x-y}[\/latex]<\/center><\/li>\r\n \t<li>The <strong>Power of a Power<\/strong> rule shows that when taking an exponent to another exponent, we multiply the exponents.<center>[latex](b^x)^y=b^{xy}[\/latex]<\/center><\/li>\r\n \t<li>The <strong>Power of a Product<\/strong> rule lets us know that when raising a product to an exponent, each factor in the product is raised to the exponent.<center>[latex](ab)^x=a^x b^x[\/latex]<\/center><\/li>\r\n \t<li>The <strong>Power of a Quotient<\/strong> rule indicates that when a quotient is raised to an exponent, both the numerator and the denominator are raised to the exponent.<center>[latex]\\dfrac{a^x}{b^x} =\\left(\\dfrac{a}{b}\\right)^x[\/latex]<\/center><\/li>\r\n<\/ol>\r\n<em>Note: This is true for any constants [latex]a&gt;0, \\, b&gt;0[\/latex], and for all [latex]x[\/latex] and [latex]y[\/latex]<\/em>\r\n\r\n<\/section><section class=\"textbox example\">\r\n<p id=\"fs-id1170572440102\">Use the laws of exponents to simplify each of the following expressions.<\/p>\r\n\r\n<ol id=\"fs-id1170572440106\" style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]\\large \\frac{(2x^{2\/3})^3}{(4x^{-1\/3})^2}[\/latex]<\/li>\r\n \t<li>[latex]\\large \\frac{(x^3 y^{-1})^2}{(xy^2)^{-2}}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"fs-id1170572453127\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572453127\"]\r\n<ol id=\"fs-id1170572453127\" style=\"list-style-type: lower-alpha;\">\r\n \t<li>We can simplify as follows:\r\n<div id=\"fs-id1170570966957\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\large \\frac{(2x^{2\/3})^3}{(4x^{-1\/3})^2} \\normalsize = \\large \\frac{2^3(x^{2\/3})^3}{4^2(x^{-1\/3})^2} \\normalsize = \\large \\frac{8x^2}{16x^{-2\/3}} \\normalsize = \\large \\frac{x^2x^{2\/3}}{2} \\normalsize = \\large \\frac{x^{8\/3}}{2}[\/latex]<\/div><\/li>\r\n \t<li>We can simplify as follows:\r\n<div id=\"fs-id1170573582280\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\large \\frac{(x^3y^{-1})^2}{(xy^2)^{-2}} \\normalsize = \\large \\frac{(x^3)^2(y^{-1})^2}{x^{-2}(y^2)^{-2}} \\normalsize = \\large \\frac{x^6y^{-2}}{x^{-2}y^{-4}} \\normalsize = x^6x^2y^{-2}y^4 = x^8y^2[\/latex]<\/div><\/li>\r\n<\/ol>\r\nWatch the following video to see the worked solution to this example.\r\n\r\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/tOkk_pSFpzk?controls=0&amp;start=212&amp;end=380&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/center>For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.You can view the transcript for this video using <a href=\"https:\/\/oerfiles.s3-us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/ExponentialAndLogarithmicFunctions212to380_transcript.txt\" target=\"_blank\" rel=\"noopener\">this link<\/a> (opens in new window).\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox proTip\">When you encounter a negative exponent on a term in the denominator of a fraction, you can transform it into a positive exponent by moving the term to the numerator.<center>[latex]\\frac{1}{a^-n}=a^{n}[\/latex]<\/center>Using this rule can significantly simplify expressions involving exponents.<\/section><section class=\"textbox tryIt\">[ohm_question hide_question_numbers=1]123515[\/ohm_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li><span data-sheets-root=\"1\">Apply the properties of exponents to simplify exponential expressions<\/span><\/li>\n<\/ul>\n<\/section>\n<h2>Exponential Functions<\/h2>\n<p>Any function of the form [latex]f(x)=b^x[\/latex], where [latex]b>0, \\, b \\ne 1[\/latex], is an <strong>exponential function<\/strong> with base [latex]b[\/latex] and exponent [latex]x[\/latex]. Exponential functions have constant bases and variable exponents.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>exponential function<\/h3>\n<p>For any real number [latex]x[\/latex], an exponential function is a function with the form<\/p>\n<p style=\"text-align: center;\">[latex]f(x)=ab^x[\/latex]<\/p>\n<p>where,<\/p>\n<ul>\n<li>[latex]a[\/latex] is a non-zero real number called the initial value and<\/li>\n<li>[latex]b[\/latex] is any positive real number ([latex]b>0[\/latex]) such that [latex]b\u22601[\/latex].<\/li>\n<\/ul>\n<\/div>\n<\/section>\n<p>To evaluate an exponential function with the form [latex]f(x)=b^x[\/latex], we simply substitute [latex]x[\/latex] with the given value, and calculate the resulting power.<\/p>\n<section class=\"textbox example\">Let [latex]f(x)=2^x[\/latex]. What is [latex]f(3)[\/latex]?<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{rcl} f(x) & = & 2^x \\\\ f(3) & = & 2^3 & \\quad \\text{Substitute } x = 3. \\\\ & = & 8 & \\quad \\text{Evaluate the power.} \\end{array}[\/latex]<\/div>\n<p>\u00a0<\/section>\n<p>To evaluate an exponential function with a form other than the basic form, it is important to follow the order of operations.<\/p>\n<section class=\"textbox example\">Let [latex]f(x)=30(2)^x[\/latex]. What is [latex]f(3)[\/latex]?<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{rcll} f(x) & = & 30(2)^x & \\\\ f(3) & = & 30(2)^3 & \\quad \\text{Substitute } x = 3. \\\\ & = & 30(8) & \\quad \\text{Simplify the power first.} \\\\ & = & 240 & \\quad \\text{Multiply.} \\end{array}[\/latex]<\/div>\n<p>Note that if the order of operations were not followed, the result would be incorrect:<\/p>\n<div style=\"text-align: center;\">[latex]f(3)=30(2)^3\u226060^3=216,000[\/latex]<\/div>\n<\/section>\n<section class=\"textbox questionHelp\"><b>How To: Evaluating Exponential Functions<\/b><\/p>\n<ol>\n<li>Given an exponential function, identify [latex]a[\/latex], [latex]b[\/latex], and the value of [latex]x[\/latex] you&#8217;re being asked to substitute into the function.<\/li>\n<li>Replace the variable [latex]x[\/latex] in the function with the given number.<\/li>\n<li>Compute the value of [latex]b^x[\/latex]. This means raising the base [latex]b[\/latex] to the power of [latex]x[\/latex].<\/li>\n<li>If there is a coefficient [latex]a[\/latex] in front of the base, multiply the result of [latex]b^x[\/latex] by [latex]a[\/latex]. If [latex]a[\/latex] is [latex]1[\/latex], this step does not change the value.<\/li>\n<li>Simplify the expression if necessary. This could involve performing any additional multiplication or addition\/subtraction if the function has more terms.<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\">Let [latex]f(x)=5(3)^x+1[\/latex]. Evaluate [latex]f(2)[\/latex] without using a calculator.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q586760\">Show Answer<\/button><\/p>\n<div id=\"q586760\" class=\"hidden-answer\" style=\"display: none\">Follow the order of operations. Be sure to pay attention to the parentheses.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{rcll} f(x) & = & 5(3)^{x+1} & \\\\ f(2) & = & 5(3)^{2+1} & \\quad \\text{Substitute } x = 2. \\\\ & = & 5(3)^3 & \\quad \\text{Add the exponents.} \\\\ & = & 5(27) & \\quad \\text{Simplify the power.} \\\\ & = & 135 & \\quad \\text{Multiply.} \\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm284250\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=284250&theme=lumen&iframe_resize_id=ohm284250&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1170572169653\">Suppose a particular population of bacteria is known to double in size every [latex]4[\/latex] hours. If a culture starts with [latex]1000[\/latex] bacteria, the number of bacteria after [latex]4[\/latex] hours is [latex]n(4)=1000\u00b72[\/latex]. The number of bacteria after [latex]8[\/latex] hours is [latex]n(8)=n(4)\u00b72=1000\u00b72^2[\/latex].<\/p>\n<p>In general, the number of bacteria after [latex]4m[\/latex] hours is [latex]n(4m)=1000\u00b72^m[\/latex]. Letting [latex]t=4m[\/latex], we see that the number of bacteria after [latex]t[\/latex] hours is [latex]n(t)=1000\u00b72^{t\/4}[\/latex].<\/p>\n<p>Find the number of bacteria after [latex]6[\/latex] hours, [latex]10[\/latex] hours, and [latex]24[\/latex] hours.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170572550969\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170572550969\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572550969\">The number of bacteria after [latex]6[\/latex] hours is given by [latex]n(6)=1000\u00b72^{6\/4} \\approx 2828[\/latex] bacteria.<\/p>\n<p>The number of bacteria after [latex]10[\/latex] hours is given by [latex]n(10)=1000\u00b72^{10\/4} \\approx 5657[\/latex] bacteria.<\/p>\n<p>The number of bacteria after [latex]24[\/latex] hours is given by [latex]n(24)=1000\u00b72^{24\/4}=1000\u00b72^6=64,000[\/latex] bacteria.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<p id=\"fs-id1170572481226\">The Laws of Exponents are fundamental rules that govern the operations involving powers. These rules are essential for simplifying expressions and are foundational for higher-level math.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>laws of exponents<\/h3>\n<ol id=\"fs-id1170572481268\">\n<li>The <strong>Product of Powers<\/strong> rule states that when you multiply two exponents with the same base, you can add the exponents.\n<div style=\"text-align: center;\">[latex]b^x\u00b7b^y=b^{x+y}[\/latex]<\/div>\n<\/li>\n<li>The <strong>Quotient of Powers<\/strong> rule tells us that when dividing exponents with the same base, we subtract the exponents.\n<div style=\"text-align: center;\">[latex]\\large\\frac{b^x}{b^y} \\normalsize = b^{x-y}[\/latex]<\/div>\n<\/li>\n<li>The <strong>Power of a Power<\/strong> rule shows that when taking an exponent to another exponent, we multiply the exponents.\n<div style=\"text-align: center;\">[latex](b^x)^y=b^{xy}[\/latex]<\/div>\n<\/li>\n<li>The <strong>Power of a Product<\/strong> rule lets us know that when raising a product to an exponent, each factor in the product is raised to the exponent.\n<div style=\"text-align: center;\">[latex](ab)^x=a^x b^x[\/latex]<\/div>\n<\/li>\n<li>The <strong>Power of a Quotient<\/strong> rule indicates that when a quotient is raised to an exponent, both the numerator and the denominator are raised to the exponent.\n<div style=\"text-align: center;\">[latex]\\dfrac{a^x}{b^x} =\\left(\\dfrac{a}{b}\\right)^x[\/latex]<\/div>\n<\/li>\n<\/ol>\n<p><em>Note: This is true for any constants [latex]a>0, \\, b>0[\/latex], and for all [latex]x[\/latex] and [latex]y[\/latex]<\/em><\/p>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1170572440102\">Use the laws of exponents to simplify each of the following expressions.<\/p>\n<ol id=\"fs-id1170572440106\" style=\"list-style-type: lower-alpha;\">\n<li>[latex]\\large \\frac{(2x^{2\/3})^3}{(4x^{-1\/3})^2}[\/latex]<\/li>\n<li>[latex]\\large \\frac{(x^3 y^{-1})^2}{(xy^2)^{-2}}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170572453127\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170572453127\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1170572453127\" style=\"list-style-type: lower-alpha;\">\n<li>We can simplify as follows:\n<div id=\"fs-id1170570966957\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\large \\frac{(2x^{2\/3})^3}{(4x^{-1\/3})^2} \\normalsize = \\large \\frac{2^3(x^{2\/3})^3}{4^2(x^{-1\/3})^2} \\normalsize = \\large \\frac{8x^2}{16x^{-2\/3}} \\normalsize = \\large \\frac{x^2x^{2\/3}}{2} \\normalsize = \\large \\frac{x^{8\/3}}{2}[\/latex]<\/div>\n<\/li>\n<li>We can simplify as follows:\n<div id=\"fs-id1170573582280\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\large \\frac{(x^3y^{-1})^2}{(xy^2)^{-2}} \\normalsize = \\large \\frac{(x^3)^2(y^{-1})^2}{x^{-2}(y^2)^{-2}} \\normalsize = \\large \\frac{x^6y^{-2}}{x^{-2}y^{-4}} \\normalsize = x^6x^2y^{-2}y^4 = x^8y^2[\/latex]<\/div>\n<\/li>\n<\/ol>\n<p>Watch the following video to see the worked solution to this example.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/tOkk_pSFpzk?controls=0&amp;start=212&amp;end=380&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/div>\n<p>For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.You can view the transcript for this video using <a href=\"https:\/\/oerfiles.s3-us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/ExponentialAndLogarithmicFunctions212to380_transcript.txt\" target=\"_blank\" rel=\"noopener\">this link<\/a> (opens in new window).<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox proTip\">When you encounter a negative exponent on a term in the denominator of a fraction, you can transform it into a positive exponent by moving the term to the numerator.<\/p>\n<div style=\"text-align: center;\">[latex]\\frac{1}{a^-n}=a^{n}[\/latex]<\/div>\n<p>Using this rule can significantly simplify expressions involving exponents.<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm123515\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=123515&theme=lumen&iframe_resize_id=ohm123515&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":15,"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":669,"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\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/792"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":6,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/792\/revisions"}],"predecessor-version":[{"id":2073,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/792\/revisions\/2073"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/669"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/792\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=792"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=792"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=792"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=792"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}