{"id":1555,"date":"2024-05-28T18:12:35","date_gmt":"2024-05-28T18:12:35","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=1555"},"modified":"2024-12-04T18:29:16","modified_gmt":"2024-12-04T18:29:16","slug":"module-6-background-youll-need-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/module-6-background-youll-need-1\/","title":{"raw":"Algebraic Operations on Functions: Background You'll Need 1","rendered":"Algebraic Operations on Functions: Background You&#8217;ll Need 1"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n \t<li>Simplify and calculate an algebraic equation<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Algebraic Expressions<\/h2>\r\nAn <strong>algebraic expression<\/strong> is a collection of constants and variables joined together by the algebraic operations of addition, subtraction, multiplication, and division. For example, [latex]3x + 2y - 7[\/latex] is an algebraic expression that contains two variables [latex]x[\/latex] and [latex]y[\/latex] and three constants [latex]3[\/latex], [latex]2[\/latex], and [latex]7[\/latex].\r\n\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>constant, variable, algebraic expression<\/h3>\r\n<ul>\r\n \t<li>A <strong>constant<\/strong> is a fixed value or a number that does not change in a particular context.<\/li>\r\n \t<li>A <strong>variable<\/strong> is a symbol that represents a value or quantity that can change or vary in a given situation or context.<\/li>\r\n \t<li>An <strong>algebraic expression<\/strong> is a mathematical phrase or combination of numbers, variables, and arithmetic operations such as addition, subtraction, multiplication, and division.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/section>We have already seen some real number examples of exponential notation, a shorthand method of writing products of the same factor. When variables are used, the constants and variables are treated the same way.\r\n<div style=\"text-align: center;\">[latex]\\begin{align}&amp;\\left(-3\\right)^{5}=\\left(-3\\right)\\cdot\\left(-3\\right)\\cdot\\left(-3\\right)\\cdot\\left(-3\\right)\\cdot\\left(-3\\right) &amp;&amp; x^{5}=x\\cdot x\\cdot x\\cdot x\\cdot x \\\\ &amp;\\left(2\\cdot7\\right)^{3}=\\left(2\\cdot7\\right)\\cdot\\left(2\\cdot7\\right)\\cdot\\left(2\\cdot7\\right) &amp;&amp; \\left(yz\\right)^{3}=\\left(yz\\right)\\cdot\\left(yz\\right)\\cdot\\left(yz\\right)\\\\ \\text{ }\\end{align}[\/latex]<\/div>\r\nIn each case, the exponent tells us how many factors of the base to use, whether the base consists of constants or variables. When naming the variable, ignore any exponents or radicals containing the variable.\r\n\r\n<section class=\"textbox example\">List the constants and variables for each algebraic expression.\r\n<ol>\r\n \t<li>[latex]x + 5[\/latex]<\/li>\r\n \t<li>[latex]\\frac{4}{3}\\pi {r}^{3}[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt{2{m}^{3}{n}^{2}}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"790423\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"790423\"]\r\n<table style=\"height: 64px;\" summary=\"A table with four rows and three columns. The first entry of the first row is empty, but the second entry reads: Constants, and the third reads: Variables. The first entry of the second row reads: x plus five. The second column entry reads: five. The third column entry reads: x. The first entry of the third row reads: four-thirds pi times r cubed. The second column entry reads: four-thirds, pi. The third column entry reads: r. The first entry of the fourth row reads: the square root of two times m cubed times n squared. The second column entry reads: two. The third column entry reads: m, n.\">\r\n<thead>\r\n<tr style=\"height: 16px;\">\r\n<th style=\"height: 16px; width: 247.703px;\"><\/th>\r\n<th style=\"height: 16px; width: 179.398px;\">Constants<\/th>\r\n<th style=\"height: 16px; width: 112.898px;\">Variables<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr style=\"height: 16px;\">\r\n<td style=\"height: 16px; width: 248.203px;\">1. [latex]x + 5[\/latex]<\/td>\r\n<td style=\"height: 16px; width: 180.398px;\">[latex]5[\/latex]<\/td>\r\n<td style=\"height: 16px; width: 113.398px;\">[latex]x[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16px;\">\r\n<td style=\"height: 16px; width: 248.203px;\">2. [latex]\\frac{4}{3}\\pi {r}^{3}[\/latex]<\/td>\r\n<td style=\"height: 16px; width: 180.398px;\">[latex]\\frac{4}{3},\\pi [\/latex]<\/td>\r\n<td style=\"height: 16px; width: 113.398px;\">[latex]r[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16px;\">\r\n<td style=\"height: 16px; width: 248.203px;\">3. [latex]\\sqrt{2{m}^{3}{n}^{2}}[\/latex]<\/td>\r\n<td style=\"height: 16px; width: 180.398px;\">[latex]2[\/latex]<\/td>\r\n<td style=\"height: 16px; width: 113.398px;\">[latex]m,n[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]6606[\/ohm2_question]<\/section>Any variable in an algebraic expression may take on or be assigned different values. When that happens, the value of the algebraic expression changes. To evaluate an algebraic expression means to determine the value of the expression for a given value of each variable in the expression.\r\n\r\n<section class=\"textbox questionHelp\"><strong>How To: Evaluate Algebraic Expressions\r\n<\/strong>[latex]\\\\[\/latex]<strong>\r\n<\/strong>Use the following steps to evaluate an algebraic expression:\r\n<ol>\r\n \t<li>Replace each variable in the expression with the given value<\/li>\r\n \t<li>Simplify the resulting expression using the order of operations<\/li>\r\n<\/ol>\r\nNote: If the algebraic expression contains more than one variable, replace each variable with its assigned value and simplify the expression as before.\r\n\r\n<\/section><section class=\"textbox recall\"><strong>Be Careful when simplifying fractions! Why does the fraction [latex]\\dfrac{(25)}{3(25)-1}[\/latex] not simplify to [latex]\\dfrac{\\cancel{(25)}}{3\\cancel{(25)}-1}=\\dfrac{1}{3-1}=\\dfrac{1}{2}[\/latex]?<\/strong>\r\n\r\n<hr \/>\r\n\r\nUsing the inverse property of multiplication, we are permitted to \"cancel out\" common factors in the numerator and denominator such that [latex]\\dfrac{a}{a}=1[\/latex].But be careful! We have no rule that allows us to cancel numbers in the top and bottom of a fractions that are contained in sums or differences. You'll see this idea reappear frequently throughout the course.\r\n\r\n<\/section><section class=\"textbox example\">Evaluate each expression for the given values.\r\n<ol>\r\n \t<li>[latex]x+5[\/latex] for [latex]x=-5[\/latex]<\/li>\r\n \t<li>[latex]\\frac{t}{2t - 1}[\/latex] for [latex]t=10[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{4}{3}\\pi {r}^{3}[\/latex] for [latex]r=5[\/latex]<\/li>\r\n \t<li>[latex]a+ab+b[\/latex] for [latex]a=11,b=-8[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt{2{m}^{3}{n}^{2}}[\/latex] for [latex]m=2,n=3[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"182854\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"182854\"]\r\n<ol>\r\n \t<li>Substitute [latex]-5[\/latex] for [latex]x[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{align}x+5 &amp;=\\left(-5\\right)+5 \\\\ &amp;=0\\end{align}[\/latex]<\/div><\/li>\r\n \t<li>Substitute [latex]10[\/latex] for [latex]t[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{align}\\frac{t}{2t-1} &amp; =\\frac{\\left(10\\right)}{2\\left(10\\right)-1} \\\\ &amp; =\\frac{10}{20-1} \\\\ &amp; =\\frac{10}{19}\\end{align}[\/latex]<\/div><\/li>\r\n \t<li>Substitute [latex]5[\/latex] for [latex]r[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{align}\\frac{4}{3}\\pi r^{3} &amp; =\\frac{4}{3}\\pi\\left(5\\right)^{3} \\\\ &amp; =\\frac{4}{3}\\pi\\left(125\\right) \\\\ &amp; =\\frac{500}{3}\\pi\\end{align}[\/latex]<\/div><\/li>\r\n \t<li>Substitute [latex]11[\/latex] for [latex]a[\/latex] and \u20138 for [latex]b[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{align}a+ab+b &amp; =\\left(11\\right)+\\left(11\\right)\\left(-8\\right)+\\left(-8\\right) \\\\ &amp; =11-8-8 \\\\ &amp; =-85\\end{align}[\/latex]<\/div><\/li>\r\n \t<li>Substitute [latex]2[\/latex] for [latex]m[\/latex] and 3 for [latex]n[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{align}\\sqrt{2m^{3}n^{2}} &amp; =\\sqrt{2\\left(2\\right)^{3}\\left(3\\right)^{2}} \\\\ &amp; =\\sqrt{2\\left(8\\right)\\left(9\\right)} \\\\ &amp; =\\sqrt{144} \\\\ &amp; =12\\end{align}[\/latex]<\/div><\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]6621[\/ohm2_question]<\/section><section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]6622[\/ohm2_question]<\/section><section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]6753[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Simplify and calculate an algebraic equation<\/li>\n<\/ul>\n<\/section>\n<h2>Algebraic Expressions<\/h2>\n<p>An <strong>algebraic expression<\/strong> is a collection of constants and variables joined together by the algebraic operations of addition, subtraction, multiplication, and division. For example, [latex]3x + 2y - 7[\/latex] is an algebraic expression that contains two variables [latex]x[\/latex] and [latex]y[\/latex] and three constants [latex]3[\/latex], [latex]2[\/latex], and [latex]7[\/latex].<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>constant, variable, algebraic expression<\/h3>\n<ul>\n<li>A <strong>constant<\/strong> is a fixed value or a number that does not change in a particular context.<\/li>\n<li>A <strong>variable<\/strong> is a symbol that represents a value or quantity that can change or vary in a given situation or context.<\/li>\n<li>An <strong>algebraic expression<\/strong> is a mathematical phrase or combination of numbers, variables, and arithmetic operations such as addition, subtraction, multiplication, and division.<\/li>\n<\/ul>\n<\/div>\n<\/section>\n<p>We have already seen some real number examples of exponential notation, a shorthand method of writing products of the same factor. When variables are used, the constants and variables are treated the same way.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{align}&\\left(-3\\right)^{5}=\\left(-3\\right)\\cdot\\left(-3\\right)\\cdot\\left(-3\\right)\\cdot\\left(-3\\right)\\cdot\\left(-3\\right) && x^{5}=x\\cdot x\\cdot x\\cdot x\\cdot x \\\\ &\\left(2\\cdot7\\right)^{3}=\\left(2\\cdot7\\right)\\cdot\\left(2\\cdot7\\right)\\cdot\\left(2\\cdot7\\right) && \\left(yz\\right)^{3}=\\left(yz\\right)\\cdot\\left(yz\\right)\\cdot\\left(yz\\right)\\\\ \\text{ }\\end{align}[\/latex]<\/div>\n<p>In each case, the exponent tells us how many factors of the base to use, whether the base consists of constants or variables. When naming the variable, ignore any exponents or radicals containing the variable.<\/p>\n<section class=\"textbox example\">List the constants and variables for each algebraic expression.<\/p>\n<ol>\n<li>[latex]x + 5[\/latex]<\/li>\n<li>[latex]\\frac{4}{3}\\pi {r}^{3}[\/latex]<\/li>\n<li>[latex]\\sqrt{2{m}^{3}{n}^{2}}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q790423\">Show Solution<\/button><\/p>\n<div id=\"q790423\" class=\"hidden-answer\" style=\"display: none\">\n<table style=\"height: 64px;\" summary=\"A table with four rows and three columns. The first entry of the first row is empty, but the second entry reads: Constants, and the third reads: Variables. The first entry of the second row reads: x plus five. The second column entry reads: five. The third column entry reads: x. The first entry of the third row reads: four-thirds pi times r cubed. The second column entry reads: four-thirds, pi. The third column entry reads: r. The first entry of the fourth row reads: the square root of two times m cubed times n squared. The second column entry reads: two. The third column entry reads: m, n.\">\n<thead>\n<tr style=\"height: 16px;\">\n<th style=\"height: 16px; width: 247.703px;\"><\/th>\n<th style=\"height: 16px; width: 179.398px;\">Constants<\/th>\n<th style=\"height: 16px; width: 112.898px;\">Variables<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr style=\"height: 16px;\">\n<td style=\"height: 16px; width: 248.203px;\">1. [latex]x + 5[\/latex]<\/td>\n<td style=\"height: 16px; width: 180.398px;\">[latex]5[\/latex]<\/td>\n<td style=\"height: 16px; width: 113.398px;\">[latex]x[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 16px;\">\n<td style=\"height: 16px; width: 248.203px;\">2. [latex]\\frac{4}{3}\\pi {r}^{3}[\/latex]<\/td>\n<td style=\"height: 16px; width: 180.398px;\">[latex]\\frac{4}{3},\\pi[\/latex]<\/td>\n<td style=\"height: 16px; width: 113.398px;\">[latex]r[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 16px;\">\n<td style=\"height: 16px; width: 248.203px;\">3. [latex]\\sqrt{2{m}^{3}{n}^{2}}[\/latex]<\/td>\n<td style=\"height: 16px; width: 180.398px;\">[latex]2[\/latex]<\/td>\n<td style=\"height: 16px; width: 113.398px;\">[latex]m,n[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm6606\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=6606&theme=lumen&iframe_resize_id=ohm6606&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<p>Any variable in an algebraic expression may take on or be assigned different values. When that happens, the value of the algebraic expression changes. To evaluate an algebraic expression means to determine the value of the expression for a given value of each variable in the expression.<\/p>\n<section class=\"textbox questionHelp\"><strong>How To: Evaluate Algebraic Expressions<br \/>\n<\/strong>[latex]\\\\[\/latex]<strong><br \/>\n<\/strong>Use the following steps to evaluate an algebraic expression:<\/p>\n<ol>\n<li>Replace each variable in the expression with the given value<\/li>\n<li>Simplify the resulting expression using the order of operations<\/li>\n<\/ol>\n<p>Note: If the algebraic expression contains more than one variable, replace each variable with its assigned value and simplify the expression as before.<\/p>\n<\/section>\n<section class=\"textbox recall\"><strong>Be Careful when simplifying fractions! Why does the fraction [latex]\\dfrac{(25)}{3(25)-1}[\/latex] not simplify to [latex]\\dfrac{\\cancel{(25)}}{3\\cancel{(25)}-1}=\\dfrac{1}{3-1}=\\dfrac{1}{2}[\/latex]?<\/strong><\/p>\n<hr \/>\n<p>Using the inverse property of multiplication, we are permitted to &#8220;cancel out&#8221; common factors in the numerator and denominator such that [latex]\\dfrac{a}{a}=1[\/latex].But be careful! We have no rule that allows us to cancel numbers in the top and bottom of a fractions that are contained in sums or differences. You&#8217;ll see this idea reappear frequently throughout the course.<\/p>\n<\/section>\n<section class=\"textbox example\">Evaluate each expression for the given values.<\/p>\n<ol>\n<li>[latex]x+5[\/latex] for [latex]x=-5[\/latex]<\/li>\n<li>[latex]\\frac{t}{2t - 1}[\/latex] for [latex]t=10[\/latex]<\/li>\n<li>[latex]\\dfrac{4}{3}\\pi {r}^{3}[\/latex] for [latex]r=5[\/latex]<\/li>\n<li>[latex]a+ab+b[\/latex] for [latex]a=11,b=-8[\/latex]<\/li>\n<li>[latex]\\sqrt{2{m}^{3}{n}^{2}}[\/latex] for [latex]m=2,n=3[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q182854\">Show Solution<\/button><\/p>\n<div id=\"q182854\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>Substitute [latex]-5[\/latex] for [latex]x[\/latex].\n<div style=\"text-align: center;\">[latex]\\begin{align}x+5 &=\\left(-5\\right)+5 \\\\ &=0\\end{align}[\/latex]<\/div>\n<\/li>\n<li>Substitute [latex]10[\/latex] for [latex]t[\/latex].\n<div style=\"text-align: center;\">[latex]\\begin{align}\\frac{t}{2t-1} & =\\frac{\\left(10\\right)}{2\\left(10\\right)-1} \\\\ & =\\frac{10}{20-1} \\\\ & =\\frac{10}{19}\\end{align}[\/latex]<\/div>\n<\/li>\n<li>Substitute [latex]5[\/latex] for [latex]r[\/latex].\n<div style=\"text-align: center;\">[latex]\\begin{align}\\frac{4}{3}\\pi r^{3} & =\\frac{4}{3}\\pi\\left(5\\right)^{3} \\\\ & =\\frac{4}{3}\\pi\\left(125\\right) \\\\ & =\\frac{500}{3}\\pi\\end{align}[\/latex]<\/div>\n<\/li>\n<li>Substitute [latex]11[\/latex] for [latex]a[\/latex] and \u20138 for [latex]b[\/latex].\n<div style=\"text-align: center;\">[latex]\\begin{align}a+ab+b & =\\left(11\\right)+\\left(11\\right)\\left(-8\\right)+\\left(-8\\right) \\\\ & =11-8-8 \\\\ & =-85\\end{align}[\/latex]<\/div>\n<\/li>\n<li>Substitute [latex]2[\/latex] for [latex]m[\/latex] and 3 for [latex]n[\/latex].\n<div style=\"text-align: center;\">[latex]\\begin{align}\\sqrt{2m^{3}n^{2}} & =\\sqrt{2\\left(2\\right)^{3}\\left(3\\right)^{2}} \\\\ & =\\sqrt{2\\left(8\\right)\\left(9\\right)} \\\\ & =\\sqrt{144} \\\\ & =12\\end{align}[\/latex]<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm6621\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=6621&theme=lumen&iframe_resize_id=ohm6621&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm6622\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=6622&theme=lumen&iframe_resize_id=ohm6622&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm6753\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=6753&theme=lumen&iframe_resize_id=ohm6753&source=tnh\" width=\"100%\" 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