{"id":425,"date":"2025-02-13T19:44:49","date_gmt":"2025-02-13T19:44:49","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/applications-of-integration-background-youll-need-3\/"},"modified":"2025-02-13T19:44:49","modified_gmt":"2025-02-13T19:44:49","slug":"applications-of-integration-background-youll-need-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/applications-of-integration-background-youll-need-3\/","title":{"raw":"Applications of Integration: Background You'll Need 3","rendered":"Applications of Integration: Background You&#8217;ll Need 3"},"content":{"raw":"\n<section class=\"textbox learningGoals\">\n<ul>\n\t<li>Work with expressions that include square roots and raising numbers to powers.<\/li>\n<\/ul>\n<\/section>\n<h2>Manipulating Expressions Involving Square Roots and Powers<\/h2>\n<p>Manipulating expressions involving square roots and powers is a fundamental skill necessary for understanding more advanced calculus topics, such as integration and differentiation. Understanding how to manipulate square roots and powers allows you to simplify expressions and solve equations effectively. Some key ways to simplify these expressions are:<\/p>\n<ul>\n\t<li>The power rule for exponents: [latex] (a^m)^n = a^{mn}[\/latex]<\/li>\n\t<li>The product rule for exponents: [latex]a^m \\cdot a^n = a^{m+n} [\/latex]<\/li>\n\t<li>The quotient rule for exponents: [latex] \\frac{a^m}{a^n} = a^{m-n}[\/latex]<\/li>\n\t<li>Simplifying square roots:[latex]\\sqrt{a} \\cdot \\sqrt{b} = \\sqrt{ab}[\/latex]<\/li>\n<\/ul>\n<h3>The Power Rule for Exponents<\/h3>\n<p>Another word for an exponent is power. You have likely seen or heard an example such as [latex]3^5[\/latex] can be described as [latex]3[\/latex] raised to the [latex]5[\/latex]th power. In this section, we will further expand our capabilities with exponents. We will learn what to do when a term with a power is raised to another power, what to do when two numbers or variables are multiplied and both are raised to an exponent, and what to do when numbers or variables that are divided are raised to a power. We will begin by raising powers to powers.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>the power rule for exponents<\/h3>\n<p>For any positive number [latex]x[\/latex] and integers [latex]a[\/latex] and [latex]b[\/latex]: [latex]\\left(x^{a}\\right)^{b}=x^{a\\cdot{b}}[\/latex].<\/p>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\">[ohm_question hide_question_numbers=1]287140[\/ohm_question]<\/section>\n<h3>Raise a Product to a Power<\/h3>\n<p>Raising a product to a power is a fundamental operation in algebra that demonstrates how exponents interact with multiplication. This operation is widely used across various mathematical disciplines, including geometry, where it might be used to calculate the volume of shapes, and in finance, where it can be used to calculate compounded interest over multiple periods.<\/p>\n<p>The rule simplifies the process of working with powers of products. Instead of multiplying the base numbers repeatedly, we apply the exponent to each factor individually. This is based on the distributive property of exponents over multiplication.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>a product raised to a power<\/h3>\n<p>For any nonzero numbers [latex]a[\/latex] and [latex]b[\/latex] and any integer [latex]x[\/latex], [latex]\\left(ab\\right)^{x}=a^{x}\\cdot{b^{x}}[\/latex].<\/p>\n<\/div>\n<\/section>\n<section class=\"textbox example\">Simplify the following:&nbsp;<center>[latex]\\left(2yz\\right)^{6}[\/latex]<\/center>[reveal-answer q=\"368657\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"368657\"]Apply the exponent to each number in the product.[latex]2^{6}y^{6}z^{6}[\/latex]<br>\n<center>Answer: [latex]\\left(2yz\\right)^{6}=64y^{6}z^{6}[\/latex]<\/center>[\/hidden-answer]<\/section>\n<p>If the variable has an exponent with it, use the Power Rule: multiply the exponents.<\/p>\n<section class=\"textbox example\">Simplify the following:<center>[latex]\\left(\u22127a^{4}b\\right)^{2}[\/latex]<\/center>[reveal-answer q=\"136794\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"136794\"]Apply the exponent [latex]2[\/latex]&nbsp;to each factor within the parentheses.[latex]\\left(\u22127\\right)^{2}\\left(a^{4}\\right)^{2}\\left(b\\right)^{2}[\/latex]Square the coefficient and use the Power Rule to square&nbsp;[latex]\\left(a^{4}\\right)^{2}[\/latex].\n\n<p style=\"text-align: center;\">[latex]49a^{4\\cdot2}b^{2}[\/latex]<\/p>\n<p>Simplify.<\/p>\n<p style=\"text-align: center;\">[latex]49a^{8}b^{2}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<center>Answer: [latex]\\left(-7a^{4}b\\right)^{2}=49a^{8}b^{2}[\/latex]<\/center>\n<p>[\/hidden-answer]<\/p>\n<\/section>\n<h3>The Product Rule for Exponents<\/h3>\n<p>The Product Rule for Exponents is one of the essential rules in algebra that simplifies the process of working with powers. This rule is pivotal when dealing with exponential expressions, particularly when multiplying them. In essence, it tells us that when we multiply two exponents with the same base, we can simply add the exponents to get the new power of the base.<\/p>\n<p>This rule is extremely useful in various mathematical and real-world applications, such as calculating compound interest, understanding scientific notation, or solving problems in physics and engineering. By using the Product Rule, we can manage and simplify complex expressions without the need for lengthy multiplication.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>the product rule for exponents<\/h3>\n<p>For any number [latex]x[\/latex] and any integers [latex]a[\/latex] and [latex]b[\/latex],&nbsp;[latex]\\left(x^{a}\\right)\\left(x^{b}\\right) = x^{a+b}[\/latex].<br>\n&nbsp;<br>\nTo multiply exponential terms with the same base, add the exponents.<\/p>\n<\/div>\n<\/section>\n<p><img class=\"wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659-300x265.png\" alt=\"Caution\" width=\"88\" height=\"78\">Caution! When you are reading mathematical rules, it is important to pay attention to the conditions on the rule. For example, when using the product rule, you may only apply it when the terms being multiplied have the same base and the exponents are integers. Conditions on mathematical rules are often given before the rule is stated, as in this example it says \"For any number [latex]x[\/latex] and any integers [latex]a[\/latex] and [latex]b[\/latex].\"<\/p>\n<section class=\"textbox example\">Simplify the following:\n\n<p style=\"text-align: center;\">[latex](a^{3})(a^{7})[\/latex]<\/p>\n<p>[reveal-answer q=\"356596\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"356596\"]The base of both exponents is [latex]a[\/latex], so the product rule applies.<\/p>\n<p style=\"text-align: center;\">[latex]\\left(a^{3}\\right)\\left(a^{7}\\right)[\/latex]<\/p>\n<p>Add the exponents with a common base.<\/p>\n<p style=\"text-align: center;\">[latex]a^{3+7}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<center>Answer: [latex]\\left(a^{3}\\right)\\left(a^{7}\\right) = a^{10}[\/latex]<\/center>\n<p>[\/hidden-answer]<\/p>\n<\/section>\n<p>When multiplying more complicated terms, multiply the coefficients and then multiply the variables.<\/p>\n<section class=\"textbox example\">Simplify the following:\n\n<p style=\"text-align: center;\">[latex]5a^{4}\\cdot7a^{6}[\/latex]<\/p>\n<p>[reveal-answer q=\"215459\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"215459\"]Multiply the coefficients.<\/p>\n<p style=\"text-align: center;\">[latex]35\\cdot{a}^{4}\\cdot{a}^{6}[\/latex]<\/p>\n<p>The base of both exponents is [latex]a[\/latex], so the product rule applies. Add the exponents.<\/p>\n<p style=\"text-align: center;\">[latex]35\\cdot{a}^{4+6}[\/latex]<\/p>\n<p>Add the exponents with a common base.<\/p>\n<p style=\"text-align: center;\">[latex]35\\cdot{a}^{10}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<center>Answer: [latex]5a^{4}\\cdot7a^{6}=35a^{10}[\/latex]<\/center>\n<p>[\/hidden-answer]<\/p>\n<\/section>\n<section class=\"textbox tryIt\">[ohm_question hide_question_numbers=1]287141[\/ohm_question]<\/section>\n<h3>The Quotient (Division) Rule for Exponents<\/h3>\n<p>The Quotient Rule for Exponents is as crucial as the Product Rule and serves as its counterpart for division. This rule assists in simplifying expressions where we have exponential terms with the same base being divided. It states that when you divide exponents with the same base, you can subtract the exponents.<\/p>\n<p>This rule has significant practical applications, especially in fields that involve calculations of rates of change, decay, or growth when they are decreasing, such as in the case of depreciation in finance or radioactive decay in physics.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>the quotient (division) rule for exponents<\/h3>\n<p>For any non-zero number [latex]x[\/latex] and any integers [latex]a[\/latex] and [latex]b[\/latex]:<\/p>\n<center>[latex] \\displaystyle \\frac{{{x}^{a}}}{{{x}^{b}}}={{x}^{a-b}}[\/latex]<\/center>\n<p>To divide exponential terms with the same base, subtract the exponents.<\/p>\n<\/div>\n<\/section>\n<section class=\"textbox example\">Evaluate the following:<center>[latex] \\displaystyle \\frac{{{4}^{9}}}{{{4}^{4}}}[\/latex]<\/center>[reveal-answer q=\"96156\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"96156\"]These two exponents have the same base, [latex]4[\/latex]. According to the Quotient Rule, you can subtract the power in the denominator from the power in the numerator.\n\n<p style=\"text-align: center;\">[latex] \\displaystyle {{4}^{9-4}}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<center>[latex] \\displaystyle \\frac{{{4}^{9}}}{{{4}^{4}}}=4^{5}[\/latex]<\/center>\n<p>[\/hidden-answer]<\/p>\n<\/section>\n<p>When dividing terms that also contain coefficients, divide the coefficients and then divide variable powers with the same base by subtracting the exponents.<\/p>\n<section class=\"textbox example\">Simplify the following:<center>[latex] \\displaystyle \\frac{12{{x}^{4}}}{2x}[\/latex]<\/center>[reveal-answer q=\"23604\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"23604\"]Separate into numerical and variable factors.\n\n<p style=\"text-align: center;\">[latex] \\displaystyle \\left( \\frac{12}{2} \\right)\\left( \\frac{{{x}^{4}}}{x} \\right)[\/latex]<\/p>\n<p>Since the bases of the exponents are the same, you can apply the Quotient Rule. Divide the coefficients and subtract the exponents of matching variables.<\/p>\n<p style=\"text-align: center;\">[latex] \\displaystyle 6\\left( {{x}^{4-1}} \\right)[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<center>[latex] \\displaystyle \\frac{12{{x}^{4}}}{2x}[\/latex]=[latex] \\displaystyle 6{{x}^{3}}[\/latex]<\/center>\n<p>[\/hidden-answer]<\/p>\n<\/section>\n<section class=\"textbox tryIt\">[ohm_question hide_question_numbers=1]287143[\/ohm_question]<\/section>\n<h3>Simplifying Square Roots and Expressing Them in Lowest Terms<\/h3>\n<p>To <strong>simplify a square root<\/strong> means that we rewrite the square root as a rational number times the square root of a number that has no perfect square factors. The act of changing a square root into such a form is simplifying the square root.<\/p>\n<section class=\"textbox recall\">\n<p>The number inside the square root symbol is referred to as the radicand. So in the expression [latex]\\sqrt{a}[\/latex] the number [latex]a[\/latex] is referred to as the radicand.<\/p>\n<\/section>\n<p>Before discussing how to simplify a square root, we need to introduce a rule about square roots.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>the product rule for square roots<\/h3>\n<p>The square root of a product of numbers equals the product of the square roots of those number.<\/p>\n<p>Given that [latex]a[\/latex] and [latex]b[\/latex] are nonnegative real numbers,<\/p>\n<center>[latex]\\sqrt{a \\times {b}}=\\sqrt{a} \\times \\sqrt{b}[\/latex]<\/center><\/div>\n<\/section>\n<p>Using this formula, we can factor an integer inside a square root into a perfect square times another integer. Then the square root can be applied to the perfect square, leaving an integer times the square root of another integer. If the number remaining under the square root has no perfect square factors, then we\u2019ve simplified the square root into its lowest terms.<\/p>\n<section class=\"textbox proTip\">\n<p>A perfect square is an integer that can be expressed as the square of another integer. For example, [latex]16[\/latex], [latex]25[\/latex], and [latex]36[\/latex] are perfect squares because they are [latex]4^2[\/latex], [latex]5^2[\/latex], and [latex]6^2[\/latex], respectively.<\/p>\n<\/section>\n<section class=\"textbox questionHelp\">\n<p><strong>How to: To simplify a square root the lowest terms when [latex]n[\/latex] is an integer<\/strong><\/p>\n<ul>\n\t<li><strong>Step 1:<\/strong> Determine the largest perfect square factor of [latex]n[\/latex], which we denote [latex]a^2[\/latex].<\/li>\n\t<li><strong>Step 2:<\/strong> Factor [latex]n[\/latex] into [latex]a^2\u00d7b[\/latex].<\/li>\n\t<li><strong>Step 3:<\/strong> Apply [latex]\\sqrt{a^2 \\times b} =\\sqrt{a^2} \\times \\sqrt{b}[\/latex].<\/li>\n\t<li><strong>Step 4:<\/strong> Write [latex]\\sqrt{n}[\/latex] in its simplified form, [latex]a\\sqrt{b}[\/latex].<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox example\">\n<p>Simplify [latex]\\sqrt{180}[\/latex] and express in lowest terms.&nbsp;<\/p>\n<p>[reveal-answer q=\"214538\"]Show Solution[\/reveal-answer] [hidden-answer a=\"214538\"]<\/p>\n<p>Begin by finding the largest perfect square that is a factor of [latex]180[\/latex]. We can do this by writing out the factor pairs of [latex]180[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]1 \\times 180, \\enspace 2 \\times 90, \\enspace 3 \\times 60, \\enspace 4 \\times 45, \\enspace 5 \\times 36, \\enspace 6 \\times 30, \\enspace 9 \\times 20, \\enspace 10 \\times 18, \\enspace 12 \\times 15[\/latex]<\/p>\n<p>Looking at the list of factors, the perfect squares are [latex]4[\/latex], [latex]9[\/latex], and [latex]36[\/latex]. The largest is [latex]36[\/latex], so we factor the into [latex]36\u00d75=6^2\u00d75[\/latex]. In the formula, [latex]a=6[\/latex] and [latex]b=5[\/latex].<\/p>\n<p>Apply [latex]\\sqrt{a^2 \\times b}=\\sqrt{a^2} \\times \\sqrt{b}[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt{6^2 \\times 5}=\\sqrt{6^2} \\times \\sqrt{5}[\/latex]<\/p>\n<p>The simplified form of [latex]\\sqrt{180}[\/latex] is [latex]6\\sqrt{5}[\/latex].&nbsp;<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/section>\n<section class=\"textbox example\">\n<p>Simplify [latex]\\sqrt{330}[\/latex] and express in lowest terms.<\/p>\n<p>[reveal-answer q=\"214558\"]Show Solution[\/reveal-answer] [hidden-answer a=\"214558\"]<\/p>\n<p>Begin by finding the largest perfect square that is a factor of [latex]330[\/latex]. We can do this by writing out the factor pairs of [latex]330[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]1 \\times 330, \\enspace 2 \\times 165, \\enspace 3 \\times 110, \\enspace 5 \\times 66, \\enspace 6 \\times 55, \\enspace 10 \\times 33, \\enspace 11 \\times 30, \\enspace 15 \\times 22[\/latex]<\/p>\n<p>Looking at the list of factors, there are no perfect squares other than [latex]1[\/latex], which means [latex]\\sqrt{330}[\/latex] is already expressed in lowest terms.<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/section>\n<section class=\"textbox tryIt\">\n<p>[ohm_question hide_question_numbers=1]287142[\/ohm_question]<\/p>\n<\/section>\n","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Work with expressions that include square roots and raising numbers to powers.<\/li>\n<\/ul>\n<\/section>\n<h2>Manipulating Expressions Involving Square Roots and Powers<\/h2>\n<p>Manipulating expressions involving square roots and powers is a fundamental skill necessary for understanding more advanced calculus topics, such as integration and differentiation. Understanding how to manipulate square roots and powers allows you to simplify expressions and solve equations effectively. Some key ways to simplify these expressions are:<\/p>\n<ul>\n<li>The power rule for exponents: [latex](a^m)^n = a^{mn}[\/latex]<\/li>\n<li>The product rule for exponents: [latex]a^m \\cdot a^n = a^{m+n}[\/latex]<\/li>\n<li>The quotient rule for exponents: [latex]\\frac{a^m}{a^n} = a^{m-n}[\/latex]<\/li>\n<li>Simplifying square roots:[latex]\\sqrt{a} \\cdot \\sqrt{b} = \\sqrt{ab}[\/latex]<\/li>\n<\/ul>\n<h3>The Power Rule for Exponents<\/h3>\n<p>Another word for an exponent is power. You have likely seen or heard an example such as [latex]3^5[\/latex] can be described as [latex]3[\/latex] raised to the [latex]5[\/latex]th power. In this section, we will further expand our capabilities with exponents. We will learn what to do when a term with a power is raised to another power, what to do when two numbers or variables are multiplied and both are raised to an exponent, and what to do when numbers or variables that are divided are raised to a power. We will begin by raising powers to powers.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>the power rule for exponents<\/h3>\n<p>For any positive number [latex]x[\/latex] and integers [latex]a[\/latex] and [latex]b[\/latex]: [latex]\\left(x^{a}\\right)^{b}=x^{a\\cdot{b}}[\/latex].<\/p>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm287140\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=287140&theme=lumen&iframe_resize_id=ohm287140&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h3>Raise a Product to a Power<\/h3>\n<p>Raising a product to a power is a fundamental operation in algebra that demonstrates how exponents interact with multiplication. This operation is widely used across various mathematical disciplines, including geometry, where it might be used to calculate the volume of shapes, and in finance, where it can be used to calculate compounded interest over multiple periods.<\/p>\n<p>The rule simplifies the process of working with powers of products. Instead of multiplying the base numbers repeatedly, we apply the exponent to each factor individually. This is based on the distributive property of exponents over multiplication.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>a product raised to a power<\/h3>\n<p>For any nonzero numbers [latex]a[\/latex] and [latex]b[\/latex] and any integer [latex]x[\/latex], [latex]\\left(ab\\right)^{x}=a^{x}\\cdot{b^{x}}[\/latex].<\/p>\n<\/div>\n<\/section>\n<section class=\"textbox example\">Simplify the following:&nbsp;<\/p>\n<div style=\"text-align: center;\">[latex]\\left(2yz\\right)^{6}[\/latex]<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q368657\">Show Solution<\/button><\/p>\n<div id=\"q368657\" class=\"hidden-answer\" style=\"display: none\">Apply the exponent to each number in the product.[latex]2^{6}y^{6}z^{6}[\/latex]<\/p>\n<div style=\"text-align: center;\">Answer: [latex]\\left(2yz\\right)^{6}=64y^{6}z^{6}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/section>\n<p>If the variable has an exponent with it, use the Power Rule: multiply the exponents.<\/p>\n<section class=\"textbox example\">Simplify the following:<\/p>\n<div style=\"text-align: center;\">[latex]\\left(\u22127a^{4}b\\right)^{2}[\/latex]<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q136794\">Show Solution<\/button><\/p>\n<div id=\"q136794\" class=\"hidden-answer\" style=\"display: none\">Apply the exponent [latex]2[\/latex]&nbsp;to each factor within the parentheses.[latex]\\left(\u22127\\right)^{2}\\left(a^{4}\\right)^{2}\\left(b\\right)^{2}[\/latex]Square the coefficient and use the Power Rule to square&nbsp;[latex]\\left(a^{4}\\right)^{2}[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]49a^{4\\cdot2}b^{2}[\/latex]<\/p>\n<p>Simplify.<\/p>\n<p style=\"text-align: center;\">[latex]49a^{8}b^{2}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<div style=\"text-align: center;\">Answer: [latex]\\left(-7a^{4}b\\right)^{2}=49a^{8}b^{2}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/section>\n<h3>The Product Rule for Exponents<\/h3>\n<p>The Product Rule for Exponents is one of the essential rules in algebra that simplifies the process of working with powers. This rule is pivotal when dealing with exponential expressions, particularly when multiplying them. In essence, it tells us that when we multiply two exponents with the same base, we can simply add the exponents to get the new power of the base.<\/p>\n<p>This rule is extremely useful in various mathematical and real-world applications, such as calculating compound interest, understanding scientific notation, or solving problems in physics and engineering. By using the Product Rule, we can manage and simplify complex expressions without the need for lengthy multiplication.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>the product rule for exponents<\/h3>\n<p>For any number [latex]x[\/latex] and any integers [latex]a[\/latex] and [latex]b[\/latex],&nbsp;[latex]\\left(x^{a}\\right)\\left(x^{b}\\right) = x^{a+b}[\/latex].<br \/>\n&nbsp;<br \/>\nTo multiply exponential terms with the same base, add the exponents.<\/p>\n<\/div>\n<\/section>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659-300x265.png\" alt=\"Caution\" width=\"88\" height=\"78\" \/>Caution! When you are reading mathematical rules, it is important to pay attention to the conditions on the rule. For example, when using the product rule, you may only apply it when the terms being multiplied have the same base and the exponents are integers. Conditions on mathematical rules are often given before the rule is stated, as in this example it says &#8220;For any number [latex]x[\/latex] and any integers [latex]a[\/latex] and [latex]b[\/latex].&#8221;<\/p>\n<section class=\"textbox example\">Simplify the following:<\/p>\n<p style=\"text-align: center;\">[latex](a^{3})(a^{7})[\/latex]<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q356596\">Show Solution<\/button><\/p>\n<div id=\"q356596\" class=\"hidden-answer\" style=\"display: none\">The base of both exponents is [latex]a[\/latex], so the product rule applies.<\/p>\n<p style=\"text-align: center;\">[latex]\\left(a^{3}\\right)\\left(a^{7}\\right)[\/latex]<\/p>\n<p>Add the exponents with a common base.<\/p>\n<p style=\"text-align: center;\">[latex]a^{3+7}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<div style=\"text-align: center;\">Answer: [latex]\\left(a^{3}\\right)\\left(a^{7}\\right) = a^{10}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/section>\n<p>When multiplying more complicated terms, multiply the coefficients and then multiply the variables.<\/p>\n<section class=\"textbox example\">Simplify the following:<\/p>\n<p style=\"text-align: center;\">[latex]5a^{4}\\cdot7a^{6}[\/latex]<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q215459\">Show Solution<\/button><\/p>\n<div id=\"q215459\" class=\"hidden-answer\" style=\"display: none\">Multiply the coefficients.<\/p>\n<p style=\"text-align: center;\">[latex]35\\cdot{a}^{4}\\cdot{a}^{6}[\/latex]<\/p>\n<p>The base of both exponents is [latex]a[\/latex], so the product rule applies. Add the exponents.<\/p>\n<p style=\"text-align: center;\">[latex]35\\cdot{a}^{4+6}[\/latex]<\/p>\n<p>Add the exponents with a common base.<\/p>\n<p style=\"text-align: center;\">[latex]35\\cdot{a}^{10}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<div style=\"text-align: center;\">Answer: [latex]5a^{4}\\cdot7a^{6}=35a^{10}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm287141\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=287141&theme=lumen&iframe_resize_id=ohm287141&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h3>The Quotient (Division) Rule for Exponents<\/h3>\n<p>The Quotient Rule for Exponents is as crucial as the Product Rule and serves as its counterpart for division. This rule assists in simplifying expressions where we have exponential terms with the same base being divided. It states that when you divide exponents with the same base, you can subtract the exponents.<\/p>\n<p>This rule has significant practical applications, especially in fields that involve calculations of rates of change, decay, or growth when they are decreasing, such as in the case of depreciation in finance or radioactive decay in physics.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>the quotient (division) rule for exponents<\/h3>\n<p>For any non-zero number [latex]x[\/latex] and any integers [latex]a[\/latex] and [latex]b[\/latex]:<\/p>\n<div style=\"text-align: center;\">[latex]\\displaystyle \\frac{{{x}^{a}}}{{{x}^{b}}}={{x}^{a-b}}[\/latex]<\/div>\n<p>To divide exponential terms with the same base, subtract the exponents.<\/p>\n<\/div>\n<\/section>\n<section class=\"textbox example\">Evaluate the following:<\/p>\n<div style=\"text-align: center;\">[latex]\\displaystyle \\frac{{{4}^{9}}}{{{4}^{4}}}[\/latex]<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q96156\">Show Solution<\/button><\/p>\n<div id=\"q96156\" class=\"hidden-answer\" style=\"display: none\">These two exponents have the same base, [latex]4[\/latex]. According to the Quotient Rule, you can subtract the power in the denominator from the power in the numerator.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle {{4}^{9-4}}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<div style=\"text-align: center;\">[latex]\\displaystyle \\frac{{{4}^{9}}}{{{4}^{4}}}=4^{5}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/section>\n<p>When dividing terms that also contain coefficients, divide the coefficients and then divide variable powers with the same base by subtracting the exponents.<\/p>\n<section class=\"textbox example\">Simplify the following:<\/p>\n<div style=\"text-align: center;\">[latex]\\displaystyle \\frac{12{{x}^{4}}}{2x}[\/latex]<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q23604\">Show Solution<\/button><\/p>\n<div id=\"q23604\" class=\"hidden-answer\" style=\"display: none\">Separate into numerical and variable factors.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\left( \\frac{12}{2} \\right)\\left( \\frac{{{x}^{4}}}{x} \\right)[\/latex]<\/p>\n<p>Since the bases of the exponents are the same, you can apply the Quotient Rule. Divide the coefficients and subtract the exponents of matching variables.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle 6\\left( {{x}^{4-1}} \\right)[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<div style=\"text-align: center;\">[latex]\\displaystyle \\frac{12{{x}^{4}}}{2x}[\/latex]=[latex]\\displaystyle 6{{x}^{3}}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm287143\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=287143&theme=lumen&iframe_resize_id=ohm287143&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h3>Simplifying Square Roots and Expressing Them in Lowest Terms<\/h3>\n<p>To <strong>simplify a square root<\/strong> means that we rewrite the square root as a rational number times the square root of a number that has no perfect square factors. The act of changing a square root into such a form is simplifying the square root.<\/p>\n<section class=\"textbox recall\">\n<p>The number inside the square root symbol is referred to as the radicand. So in the expression [latex]\\sqrt{a}[\/latex] the number [latex]a[\/latex] is referred to as the radicand.<\/p>\n<\/section>\n<p>Before discussing how to simplify a square root, we need to introduce a rule about square roots.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>the product rule for square roots<\/h3>\n<p>The square root of a product of numbers equals the product of the square roots of those number.<\/p>\n<p>Given that [latex]a[\/latex] and [latex]b[\/latex] are nonnegative real numbers,<\/p>\n<div style=\"text-align: center;\">[latex]\\sqrt{a \\times {b}}=\\sqrt{a} \\times \\sqrt{b}[\/latex]<\/div>\n<\/div>\n<\/section>\n<p>Using this formula, we can factor an integer inside a square root into a perfect square times another integer. Then the square root can be applied to the perfect square, leaving an integer times the square root of another integer. If the number remaining under the square root has no perfect square factors, then we\u2019ve simplified the square root into its lowest terms.<\/p>\n<section class=\"textbox proTip\">\n<p>A perfect square is an integer that can be expressed as the square of another integer. For example, [latex]16[\/latex], [latex]25[\/latex], and [latex]36[\/latex] are perfect squares because they are [latex]4^2[\/latex], [latex]5^2[\/latex], and [latex]6^2[\/latex], respectively.<\/p>\n<\/section>\n<section class=\"textbox questionHelp\">\n<p><strong>How to: To simplify a square root the lowest terms when [latex]n[\/latex] is an integer<\/strong><\/p>\n<ul>\n<li><strong>Step 1:<\/strong> Determine the largest perfect square factor of [latex]n[\/latex], which we denote [latex]a^2[\/latex].<\/li>\n<li><strong>Step 2:<\/strong> Factor [latex]n[\/latex] into [latex]a^2\u00d7b[\/latex].<\/li>\n<li><strong>Step 3:<\/strong> Apply [latex]\\sqrt{a^2 \\times b} =\\sqrt{a^2} \\times \\sqrt{b}[\/latex].<\/li>\n<li><strong>Step 4:<\/strong> Write [latex]\\sqrt{n}[\/latex] in its simplified form, [latex]a\\sqrt{b}[\/latex].<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox example\">\n<p>Simplify [latex]\\sqrt{180}[\/latex] and express in lowest terms.&nbsp;<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q214538\">Show Solution<\/button> <\/p>\n<div id=\"q214538\" class=\"hidden-answer\" style=\"display: none\">\n<p>Begin by finding the largest perfect square that is a factor of [latex]180[\/latex]. We can do this by writing out the factor pairs of [latex]180[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]1 \\times 180, \\enspace 2 \\times 90, \\enspace 3 \\times 60, \\enspace 4 \\times 45, \\enspace 5 \\times 36, \\enspace 6 \\times 30, \\enspace 9 \\times 20, \\enspace 10 \\times 18, \\enspace 12 \\times 15[\/latex]<\/p>\n<p>Looking at the list of factors, the perfect squares are [latex]4[\/latex], [latex]9[\/latex], and [latex]36[\/latex]. The largest is [latex]36[\/latex], so we factor the into [latex]36\u00d75=6^2\u00d75[\/latex]. In the formula, [latex]a=6[\/latex] and [latex]b=5[\/latex].<\/p>\n<p>Apply [latex]\\sqrt{a^2 \\times b}=\\sqrt{a^2} \\times \\sqrt{b}[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt{6^2 \\times 5}=\\sqrt{6^2} \\times \\sqrt{5}[\/latex]<\/p>\n<p>The simplified form of [latex]\\sqrt{180}[\/latex] is [latex]6\\sqrt{5}[\/latex].&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p>Simplify [latex]\\sqrt{330}[\/latex] and express in lowest terms.<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q214558\">Show Solution<\/button> <\/p>\n<div id=\"q214558\" class=\"hidden-answer\" style=\"display: none\">\n<p>Begin by finding the largest perfect square that is a factor of [latex]330[\/latex]. We can do this by writing out the factor pairs of [latex]330[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]1 \\times 330, \\enspace 2 \\times 165, \\enspace 3 \\times 110, \\enspace 5 \\times 66, \\enspace 6 \\times 55, \\enspace 10 \\times 33, \\enspace 11 \\times 30, \\enspace 15 \\times 22[\/latex]<\/p>\n<p>Looking at the list of factors, there are no perfect squares other than [latex]1[\/latex], which means [latex]\\sqrt{330}[\/latex] is already expressed in lowest terms.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm287142\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=287142&theme=lumen&iframe_resize_id=ohm287142&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n","protected":false},"author":6,"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":421,"module-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\/425"}],"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\/6"}],"version-history":[{"count":0,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/425\/revisions"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/421"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/425\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=425"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=425"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=425"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=425"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}