{"id":1704,"date":"2016-06-24T23:03:50","date_gmt":"2016-06-24T23:03:50","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/?post_type=chapter&#038;p=1704"},"modified":"2019-07-24T20:53:52","modified_gmt":"2019-07-24T20:53:52","slug":"read-evaluate-algebraic-expressions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/chapter\/read-evaluate-algebraic-expressions\/","title":{"raw":"Evaluate Algebraic Expressions","rendered":"Evaluate Algebraic Expressions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Define and identify constants in an algebraic expression<\/li>\r\n \t<li>Evaluate algebraic expressions for different values<\/li>\r\n<\/ul>\r\n<\/div>\r\nIn mathematics, we may see expressions such as [latex]x+5,\\dfrac{4}{3}\\normalsize\\pi {r}^{3}[\/latex], or [latex]\\sqrt{2{m}^{3}{n}^{2}}[\/latex]. In the expression [latex]x+5[\/latex], [latex]5[\/latex] is called a <strong>constant<\/strong> because it does not vary and <em>x<\/em> is called a <strong>variable<\/strong> because it does. (In naming the variable, ignore any exponents or radicals containing the variable.) An <strong>algebraic expression<\/strong> is a collection of constants and variables joined together by the algebraic operations of addition, subtraction, multiplication, and division.\r\n\r\nThe examples in this section include exponents. Recall that an exponent is shorthand for writing repeated multiplication of the same number. When variables have exponents, it means repeated multiplication of the same variable. The base of an exponent is the number or variable being multiplied, and the exponent tells us how many times to multiply.\r\n\r\nFor example,\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{ll}\\left(-3\\right)^{5}=\\left(-3\\right)\\cdot\\left(-3\\right)\\cdot\\left(-3\\right)\\cdot\\left(-3\\right)\\cdot\\left(-3\\right)\\,\\,\\, &amp; 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) &amp; \\left(yz\\right)^{3}=\\left(yz\\right)\\cdot\\left(yz\\right)\\cdot\\left(yz\\right)\\end{array}[\/latex]<\/div>\r\nIn each case, the exponent tells us how many factors of the base to use regardless of whether the base consists of constants or variables.\r\nIn the following example, we will practice identifying constants and variables in mathematical expressions.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nList 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]\\dfrac{4}{3}\\normalsize\\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=\"308507\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"308507\"]\r\n<table summary=\"A table with four rows and three columns. The first entry of the first row reads: expression, 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>\r\n<th style=\"text-align: left;\">Expression<\/th>\r\n<th style=\"text-align: left;\">Constants<\/th>\r\n<th style=\"text-align: left;\">Variables<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>1.\u00a0 [latex] x + 5 [\/latex]<\/td>\r\n<td>[latex]5[\/latex]<\/td>\r\n<td>[latex]x[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2.\u00a0 [latex]\\dfrac{4}{3}\\normalsize\\pi {r}^{3}[\/latex]<\/td>\r\n<td>[latex]\\dfrac{4}{3}\\normalsize,\\pi [\/latex]<\/td>\r\n<td>[latex]r[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>3.\u00a0 [latex]\\sqrt{2{m}^{3}{n}^{2}}[\/latex]<\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]m,n[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nAny 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. Replace each variable in the expression with the given value then simplify the resulting expression using the order of operations. If the algebraic expression contains more than one variable, replace each variable with its assigned value and simplify the expression as before. In the next example we show how to substitute various types of numbers into a mathematical expression.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nEvaluate the expression [latex]2x - 7[\/latex] for each value for [latex]x[\/latex]<em>.<\/em>\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]x=0[\/latex]<\/li>\r\n \t<li>[latex]x=1[\/latex]<\/li>\r\n \t<li>[latex]x=\\dfrac{1}{2}[\/latex]<\/li>\r\n \t<li>[latex]x=-4[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"664833\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"664833\"]\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>Substitute 0 for [latex]x[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{ll}2x-7 \\hfill&amp; = 2\\left(0\\right)-7 \\\\ \\hfill&amp; =0-7 \\\\ \\hfill&amp; =-7\\end{array}[\/latex]<\/div><\/li>\r\n \t<li>Substitute 1 for [latex]x[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{ll}2x-7 \\hfill&amp; = 2\\left(1\\right)-7 \\\\ \\hfill&amp; =2-7 \\\\ \\hfill&amp; =-5\\end{array}[\/latex]<\/div><\/li>\r\n \t<li>Substitute [latex]\\frac{1}{2}[\/latex] for [latex]x[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{ll}2x-7 \\hfill&amp; = 2\\left(\\frac{1}{2}\\right)-7 \\\\ \\hfill&amp; =1-7 \\\\ \\hfill&amp; =-6\\end{array}[\/latex]<\/div><\/li>\r\n \t<li>Substitute [latex]-4[\/latex] for [latex]x[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{ll}2x-7 \\hfill&amp; = 2\\left(-4\\right)-7 \\\\ \\hfill&amp; =-8-7 \\\\ \\hfill&amp; =-15\\end{array}[\/latex]<\/div><\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNow we will show more examples of evaluating a variety of mathematical expressions for various values.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nEvaluate each expression for the given values.\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]x+5[\/latex] for [latex]x=-5[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{t}{2t - 1}[\/latex] for [latex]t=10[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{4}{3}\\normalsize\\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=\"208163\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"208163\"]\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>Substitute [latex]-5[\/latex] for [latex]x[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{ll}x+5\\hfill&amp;=\\left(-5\\right)+5 \\\\ \\hfill&amp;=0\\end{array}[\/latex]<\/div><\/li>\r\n \t<li>Substitute 10 for [latex]t[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{ll}\\frac{t}{2t-1}\\hfill&amp; =\\frac{\\left(10\\right)}{2\\left(10\\right)-1} \\\\ \\hfill&amp; =\\frac{10}{20-1} \\\\ \\hfill&amp; =\\frac{10}{19}\\end{array}[\/latex]<\/div><\/li>\r\n \t<li>Substitute 5 for [latex]r[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{ll}\\frac{4}{3}\\pi r^{3} \\hfill&amp; =\\frac{4}{3}\\pi\\left(5\\right)^{3} \\\\ \\hfill&amp; =\\frac{4}{3}\\pi\\left(125\\right) \\\\ \\hfill&amp; =\\frac{500}{3}\\pi\\end{array}[\/latex]<\/div><\/li>\r\n \t<li>Substitute 11 for [latex]a[\/latex] and \u20138 for [latex]b[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{ll}a+ab+b \\hfill&amp; =\\left(11\\right)+\\left(11\\right)\\left(-8\\right)+\\left(-8\\right) \\\\ \\hfill&amp; =11-88-8 \\\\ \\hfill&amp; =-85\\end{array}[\/latex]<\/div><\/li>\r\n \t<li>Substitute 2 for [latex]m[\/latex] and 3 for [latex]n[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{ll}\\sqrt{2m^{3}n^{2}} \\hfill&amp; =\\sqrt{2\\left(2\\right)^{3}\\left(3\\right)^{2}} \\\\ \\hfill&amp; =\\sqrt{2\\left(8\\right)\\left(9\\right)} \\\\ \\hfill&amp; =\\sqrt{144} \\\\ \\hfill&amp; =12\\end{array}[\/latex]<\/div><\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video we present more examples of evaluating a variety of expressions for given values.\r\n\r\nhttps:\/\/youtu.be\/MkRdwV4n91g","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Define and identify constants in an algebraic expression<\/li>\n<li>Evaluate algebraic expressions for different values<\/li>\n<\/ul>\n<\/div>\n<p>In mathematics, we may see expressions such as [latex]x+5,\\dfrac{4}{3}\\normalsize\\pi {r}^{3}[\/latex], or [latex]\\sqrt{2{m}^{3}{n}^{2}}[\/latex]. In the expression [latex]x+5[\/latex], [latex]5[\/latex] is called a <strong>constant<\/strong> because it does not vary and <em>x<\/em> is called a <strong>variable<\/strong> because it does. (In naming the variable, ignore any exponents or radicals containing the variable.) An <strong>algebraic expression<\/strong> is a collection of constants and variables joined together by the algebraic operations of addition, subtraction, multiplication, and division.<\/p>\n<p>The examples in this section include exponents. Recall that an exponent is shorthand for writing repeated multiplication of the same number. When variables have exponents, it means repeated multiplication of the same variable. The base of an exponent is the number or variable being multiplied, and the exponent tells us how many times to multiply.<\/p>\n<p>For example,<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{ll}\\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)\\end{array}[\/latex]<\/div>\n<p>In each case, the exponent tells us how many factors of the base to use regardless of whether the base consists of constants or variables.<br \/>\nIn the following example, we will practice identifying constants and variables in mathematical expressions.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>List the constants and variables for each algebraic expression.<\/p>\n<ol>\n<li>[latex]x+5[\/latex]<\/li>\n<li>[latex]\\dfrac{4}{3}\\normalsize\\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\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q308507\">Show Solution<\/span><\/p>\n<div id=\"q308507\" class=\"hidden-answer\" style=\"display: none\">\n<table summary=\"A table with four rows and three columns. The first entry of the first row reads: expression, 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>\n<th style=\"text-align: left;\">Expression<\/th>\n<th style=\"text-align: left;\">Constants<\/th>\n<th style=\"text-align: left;\">Variables<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>1.\u00a0 [latex]x + 5[\/latex]<\/td>\n<td>[latex]5[\/latex]<\/td>\n<td>[latex]x[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>2.\u00a0 [latex]\\dfrac{4}{3}\\normalsize\\pi {r}^{3}[\/latex]<\/td>\n<td>[latex]\\dfrac{4}{3}\\normalsize,\\pi[\/latex]<\/td>\n<td>[latex]r[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>3.\u00a0 [latex]\\sqrt{2{m}^{3}{n}^{2}}[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]m,n[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\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. Replace each variable in the expression with the given value then simplify the resulting expression using the order of operations. If the algebraic expression contains more than one variable, replace each variable with its assigned value and simplify the expression as before. In the next example we show how to substitute various types of numbers into a mathematical expression.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Evaluate the expression [latex]2x - 7[\/latex] for each value for [latex]x[\/latex]<em>.<\/em><\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>[latex]x=0[\/latex]<\/li>\n<li>[latex]x=1[\/latex]<\/li>\n<li>[latex]x=\\dfrac{1}{2}[\/latex]<\/li>\n<li>[latex]x=-4[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q664833\">Show Solution<\/span><\/p>\n<div id=\"q664833\" class=\"hidden-answer\" style=\"display: none\">\n<ol style=\"list-style-type: lower-alpha;\">\n<li>Substitute 0 for [latex]x[\/latex].\n<div style=\"text-align: center;\">[latex]\\begin{array}{ll}2x-7 \\hfill& = 2\\left(0\\right)-7 \\\\ \\hfill& =0-7 \\\\ \\hfill& =-7\\end{array}[\/latex]<\/div>\n<\/li>\n<li>Substitute 1 for [latex]x[\/latex].\n<div style=\"text-align: center;\">[latex]\\begin{array}{ll}2x-7 \\hfill& = 2\\left(1\\right)-7 \\\\ \\hfill& =2-7 \\\\ \\hfill& =-5\\end{array}[\/latex]<\/div>\n<\/li>\n<li>Substitute [latex]\\frac{1}{2}[\/latex] for [latex]x[\/latex].\n<div style=\"text-align: center;\">[latex]\\begin{array}{ll}2x-7 \\hfill& = 2\\left(\\frac{1}{2}\\right)-7 \\\\ \\hfill& =1-7 \\\\ \\hfill& =-6\\end{array}[\/latex]<\/div>\n<\/li>\n<li>Substitute [latex]-4[\/latex] for [latex]x[\/latex].\n<div style=\"text-align: center;\">[latex]\\begin{array}{ll}2x-7 \\hfill& = 2\\left(-4\\right)-7 \\\\ \\hfill& =-8-7 \\\\ \\hfill& =-15\\end{array}[\/latex]<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>Now we will show more examples of evaluating a variety of mathematical expressions for various values.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Evaluate each expression for the given values.<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>[latex]x+5[\/latex] for [latex]x=-5[\/latex]<\/li>\n<li>[latex]\\dfrac{t}{2t - 1}[\/latex] for [latex]t=10[\/latex]<\/li>\n<li>[latex]\\dfrac{4}{3}\\normalsize\\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\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q208163\">Show Solution<\/span><\/p>\n<div id=\"q208163\" class=\"hidden-answer\" style=\"display: none\">\n<ol style=\"list-style-type: lower-alpha;\">\n<li>Substitute [latex]-5[\/latex] for [latex]x[\/latex].\n<div style=\"text-align: center;\">[latex]\\begin{array}{ll}x+5\\hfill&=\\left(-5\\right)+5 \\\\ \\hfill&=0\\end{array}[\/latex]<\/div>\n<\/li>\n<li>Substitute 10 for [latex]t[\/latex].\n<div style=\"text-align: center;\">[latex]\\begin{array}{ll}\\frac{t}{2t-1}\\hfill& =\\frac{\\left(10\\right)}{2\\left(10\\right)-1} \\\\ \\hfill& =\\frac{10}{20-1} \\\\ \\hfill& =\\frac{10}{19}\\end{array}[\/latex]<\/div>\n<\/li>\n<li>Substitute 5 for [latex]r[\/latex].\n<div style=\"text-align: center;\">[latex]\\begin{array}{ll}\\frac{4}{3}\\pi r^{3} \\hfill& =\\frac{4}{3}\\pi\\left(5\\right)^{3} \\\\ \\hfill& =\\frac{4}{3}\\pi\\left(125\\right) \\\\ \\hfill& =\\frac{500}{3}\\pi\\end{array}[\/latex]<\/div>\n<\/li>\n<li>Substitute 11 for [latex]a[\/latex] and \u20138 for [latex]b[\/latex].\n<div style=\"text-align: center;\">[latex]\\begin{array}{ll}a+ab+b \\hfill& =\\left(11\\right)+\\left(11\\right)\\left(-8\\right)+\\left(-8\\right) \\\\ \\hfill& =11-88-8 \\\\ \\hfill& =-85\\end{array}[\/latex]<\/div>\n<\/li>\n<li>Substitute 2 for [latex]m[\/latex] and 3 for [latex]n[\/latex].\n<div style=\"text-align: center;\">[latex]\\begin{array}{ll}\\sqrt{2m^{3}n^{2}} \\hfill& =\\sqrt{2\\left(2\\right)^{3}\\left(3\\right)^{2}} \\\\ \\hfill& =\\sqrt{2\\left(8\\right)\\left(9\\right)} \\\\ \\hfill& =\\sqrt{144} \\\\ \\hfill& =12\\end{array}[\/latex]<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video we present more examples of evaluating a variety of expressions for given values.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Evaluate Various Algebraic Expressions\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/MkRdwV4n91g?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-1704\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Evaluate Various Algebraic Expressions. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/MkRdwV4n91g\">https:\/\/youtu.be\/MkRdwV4n91g<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>College Algebra: Evaluating Algebraic Expressions. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t 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