{"id":18182,"date":"2022-04-07T18:00:59","date_gmt":"2022-04-07T18:00:59","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/?post_type=chapter&#038;p=18182"},"modified":"2022-04-28T23:14:32","modified_gmt":"2022-04-28T23:14:32","slug":"cr-3-introduction-to-algebra-simplifying-expressions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/chapter\/cr-3-introduction-to-algebra-simplifying-expressions\/","title":{"raw":"CR.3: Introduction to Algebra, Simplifying Expressions","rendered":"CR.3: Introduction to Algebra, Simplifying Expressions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Use variables to represent unknown quantities in algebraic expressions<\/li>\r\n \t<li>Identify the variables and constants in an algebraic expression<\/li>\r\n \t<li>Use words and symbols to represent algebraic operations on variables and constants<\/li>\r\n \t<li>Simplify expressions with exponents containing integer bases and variable bases<\/li>\r\n \t<li>Evaluate an expression for a given value<\/li>\r\n \t<li>Identify the variables and constants in a term<\/li>\r\n \t<li>Identify the coefficient of a variable term<\/li>\r\n \t<li>Identify and combine like terms in an expression<\/li>\r\n \t<li>Apply the distributive property to simplify an algebraic expression<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2 data-type=\"title\">Use Variables and Algebraic Symbols<\/h2>\r\nGreg and Alex have the same birthday, but they were born in different years. This year Greg is [latex]20[\/latex] years old and Alex is [latex]23[\/latex], so Alex is [latex]3[\/latex] years older than Greg. When Greg was [latex]12[\/latex], Alex was [latex]15[\/latex]. When Greg is [latex]35[\/latex], Alex will be [latex]38[\/latex]. No matter what Greg\u2019s age is, Alex\u2019s age will always be [latex]3[\/latex] years more, right?\r\n\r\nIn the language of algebra, we say that Greg\u2019s age and Alex\u2019s age are variable and the three is a constant. The ages change, or vary, so age is a variable. The [latex]3[\/latex] years between them always stays the same, so the age difference is the constant.\r\n\r\nIn algebra, letters of the alphabet are used to represent variables. Suppose we call Greg\u2019s age [latex]g[\/latex]. Then we could use [latex]g+3[\/latex] to represent Alex\u2019s age. See the table below.\r\n<table style=\"width: 40%;\" summary=\"This table has five rows and two columns. The first row is a header row and is labeled \">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th data-align=\"center\">Greg\u2019s age<\/th>\r\n<th data-align=\"center\">Alex\u2019s age<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-align=\"center\">[latex]12[\/latex]<\/td>\r\n<td data-align=\"center\">[latex]15[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"center\">[latex]20[\/latex]<\/td>\r\n<td data-align=\"center\">[latex]23[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"center\">[latex]35[\/latex]<\/td>\r\n<td data-align=\"center\">[latex]38[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"center\">[latex]g[\/latex]<\/td>\r\n<td data-align=\"center\">[latex]g+3[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nLetters are used to represent variables. Letters often used for variables are [latex]x,y,a,b,\\text{ and }c[\/latex].\r\n<div class=\"textbox shaded\">\r\n<h3>Variables and Constants<\/h3>\r\nA variable is a letter that represents a number or quantity whose value may change.\r\nA constant is a number whose value always stays the same.\r\n\r\n<\/div>\r\nTo write algebraically, we need some symbols as well as numbers and variables. There are several types of symbols we will be using. In Whole Numbers, we introduced the symbols for the four basic arithmetic operations: addition, subtraction, multiplication, and division. We will summarize them here, along with words we use for the operations and the result.\r\n<table class=\"unnumbered\" style=\"width: 40%;\" summary=\"This table has five rows and four columns. The first row is a header row. Each column is labeled accordingly: the first is labeled \" data-label=\"\">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th data-align=\"left\"><strong>Operation<\/strong><\/th>\r\n<th data-align=\"left\"><strong>Notation<\/strong><\/th>\r\n<th data-align=\"left\"><strong>Say:<\/strong><\/th>\r\n<th data-align=\"left\"><strong>The result is\u2026<\/strong><\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">Addition<\/td>\r\n<td data-align=\"left\">[latex]a+b[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]a\\text{ plus }b[\/latex]<\/td>\r\n<td data-align=\"left\">the sum of [latex]a[\/latex] and [latex]b[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">Subtraction<\/td>\r\n<td data-align=\"left\">[latex]a-b[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]a\\text{ minus }b[\/latex]<\/td>\r\n<td data-align=\"left\">the difference of [latex]a[\/latex] and [latex]b[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">Multiplication<\/td>\r\n<td data-align=\"left\">[latex]a\\cdot b,\\left(a\\right)\\left(b\\right),\\left(a\\right)b,a\\left(b\\right)[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]a\\text{ times }b[\/latex]<\/td>\r\n<td data-align=\"left\">The product of [latex]a[\/latex] and [latex]b[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">Division<\/td>\r\n<td data-align=\"left\">[latex]a\\div b,a\/b,\\frac{a}{b},b\\overline{)a}[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]a[\/latex] divided by [latex]b[\/latex]<\/td>\r\n<td data-align=\"left\">The quotient of [latex]a[\/latex] and [latex]b[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nIn algebra, the cross symbol, [latex]\\times [\/latex], is not used to show multiplication because that symbol may cause confusion. Does [latex]3xy[\/latex] mean [latex]3\\times y[\/latex] (three times [latex]y[\/latex] ) or [latex]3\\cdot x\\cdot y[\/latex] (three times [latex]x\\text{ times }y[\/latex] )? To make it clear, use \u2022 or parentheses for multiplication.\r\nWe perform these operations on two numbers. When translating from symbolic form to words, or from words to symbolic form, pay attention to the words <em data-effect=\"italics\">of<\/em> or <em data-effect=\"italics\">and<\/em> to help you find the numbers.\r\n<ul id=\"fs-id1969800\" data-labeled-item=\"true\">\r\n \t<li>The <em data-effect=\"italics\">sum\u00a0<\/em><strong><em data-effect=\"italics\">of<\/em><\/strong> [latex]5[\/latex] <strong><em data-effect=\"italics\">and<\/em><\/strong> [latex]3[\/latex] means add [latex]5[\/latex] plus [latex]3[\/latex], which we write as [latex]5+3[\/latex].<\/li>\r\n \t<li>The <em data-effect=\"italics\">difference\u00a0<\/em><strong><em data-effect=\"italics\">of<\/em><\/strong> [latex]9[\/latex] <strong><em data-effect=\"italics\">and<\/em><\/strong> [latex]2[\/latex] means subtract [latex]9[\/latex] minus [latex]2[\/latex], which we write as [latex]9 - 2[\/latex].<\/li>\r\n \t<li>The <em data-effect=\"italics\">product\u00a0<\/em><strong><em data-effect=\"italics\">of<\/em><\/strong> [latex]4[\/latex] <strong><em data-effect=\"italics\">and<\/em><\/strong> [latex]8[\/latex] means multiply [latex]4[\/latex] times [latex]8[\/latex], which we can write as [latex]4\\cdot 8[\/latex].<\/li>\r\n \t<li>The <em data-effect=\"italics\">quotient\u00a0<\/em><strong><em data-effect=\"italics\">of<\/em><\/strong> [latex]20[\/latex] <strong><em data-effect=\"italics\">and<\/em><\/strong> [latex]5[\/latex] means divide [latex]20[\/latex] by [latex]5[\/latex], which we can write as [latex]20\\div 5[\/latex].<\/li>\r\n<\/ul>\r\n<div class=\"textbox exercises\">\r\n<h3>Exercises<\/h3>\r\nTranslate from algebra to words:\r\n<ol>\r\n \t<li>[latex]12+14[\/latex]<\/li>\r\n \t<li>[latex]\\left(30\\right)\\left(5\\right)[\/latex]<\/li>\r\n \t<li>[latex]64\\div 8[\/latex]<\/li>\r\n \t<li>[latex]x-y[\/latex]<\/li>\r\n<\/ol>\r\nSolution:\r\n<table class=\"unnumbered unstyled\" style=\"width: 40%;\" summary=\".\" data-label=\"\">\r\n<tbody>\r\n<tr style=\"height: 15.7812px;\">\r\n<td style=\"height: 15.7812px;\">1.<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\" data-align=\"center\">[latex]12+14[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\" data-align=\"center\">[latex]12[\/latex] plus [latex]14[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\" data-align=\"center\">the sum of twelve and fourteen<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table style=\"width: 40%;\">\r\n<tbody>\r\n<tr>\r\n<td>2.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"center\">[latex]\\left(30\\right)\\left(5\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"center\">[latex]30[\/latex] times [latex]5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"center\">the product of thirty and five<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table class=\"unnumbered unstyled\" style=\"width: 40%;\" summary=\".\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td>3.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"center\">[latex]64\\div 8[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"center\">[latex]64[\/latex] divided by [latex]8[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"center\">the quotient of sixty-four and eight<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table class=\"unnumbered unstyled\" style=\"width: 40%;\" summary=\".\" data-label=\"\">\r\n<tbody>\r\n<tr style=\"height: 15.5625px;\">\r\n<td style=\"height: 15.5625px;\">4.<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\" data-align=\"center\">[latex]x-y[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\" data-align=\"center\">[latex]x[\/latex] minus [latex]y[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\" data-align=\"center\">the difference of [latex]x[\/latex] and [latex]y[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>TRY\u00a0IT<\/h3>\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=144651&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"330\"><\/iframe>\r\n\r\n<iframe id=\"mom2\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=144652&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"250\"><\/iframe>\r\n\r\n<\/div>\r\n<h2>Exponents<\/h2>\r\nRemember that an exponent indicates repeated multiplication of the same quantity. For example, [latex]{2}^{4}[\/latex] means to multiply four factors of [latex]2[\/latex], so [latex]{2}^{4}[\/latex] means [latex]2\\cdot 2\\cdot 2\\cdot 2[\/latex]. This format is known as exponential notation.\r\n<div class=\"textbox shaded\">\r\n<h3>Exponential Notation<\/h3>\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224353\/CNX_BMath_Figure_10_02_013_img.png\" alt=\"On the left side, a raised to the m is shown. The m is labeled in blue as an exponent. The a is labeled in red as the base. On the right, it says a to the m means multiply m factors of a. Below this, it says a to the m equals a times a times a times a, with m factors written below in blue.\" \/>\r\nThis is read [latex]a[\/latex] to the [latex]{m}^{\\mathrm{th}}[\/latex] power.\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nIn the expression [latex]{a}^{m}[\/latex], the exponent tells us how many times we use the base [latex]a[\/latex] as a factor.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224355\/CNX_BMath_Figure_10_02_014_img.png\" alt=\"On the left side, 7 to the 3rd power is shown. Below is 7 times 7 times 7, with 3 factors written below. On the right side, parentheses negative 8 to the 5th power is shown. Below is negative 8 times negative 8 times negative 8 times negative 8 times negative 8, with 5 factors written below.\" \/>\r\nBefore we begin working with variable expressions containing exponents, let\u2019s simplify a few expressions involving only numbers.\r\n\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify:\r\n\r\n1. [latex]{5}^{3}[\/latex]\r\n2. [latex]{9}^{1}[\/latex]\r\n\r\nSolution\r\n<table id=\"eip-id1168469452397\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>1.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]{5}^{3}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply [latex]3[\/latex] factors of [latex]5[\/latex].<\/td>\r\n<td>[latex]5\\cdot 5\\cdot 5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]125[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168046009892\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>2.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]{9}^{1}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply [latex]1[\/latex] factor of [latex]9[\/latex].<\/td>\r\n<td>[latex]9[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146094[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify:\r\n\r\n1. [latex]{\\left(-3\\right)}^{4}[\/latex]\r\n2. [latex]{-3}^{4}[\/latex]\r\n[reveal-answer q=\"152453\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"152453\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168468562526\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>1.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]{\\left(-3\\right)}^{4}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply four factors of [latex]\u22123[\/latex].<\/td>\r\n<td>[latex]\\left(-3\\right)\\left(-3\\right)\\left(-3\\right)\\left(-3\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]81[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168048408997\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>2.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]{-3}^{4}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply two factors.<\/td>\r\n<td>[latex]-\\left(3\\cdot 3\\cdot 3\\cdot 3\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]-81[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNotice the similarities and differences in parts 1 and 2. Why are the answers different? In part 1 the parentheses tell us to raise the [latex](\u22123)[\/latex] to the [latex]4[\/latex]<sup>th<\/sup> power. In part 2 we raise only the [latex]3[\/latex] to the [latex]4[\/latex]<sup>th<\/sup> power and then find the opposite.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146097[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nRewrite the following without exponents:\u00a0[latex]-8a^{5}b[\/latex].\r\n\r\n[reveal-answer q=\"152460\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"152460\"]\r\n\r\nSolution\r\n\r\nThe expression [latex]a^{5}[\/latex] means you are multiplying [latex]a[\/latex] five times, so it is [latex]{a}\\cdot{a}\\cdot{a}\\cdot{a}\\cdot{a}[\/latex].\u00a0 Therefore, your answer is [latex]-8\\cdot{a}\\cdot{a}\\cdot{a}\\cdot{a}\\cdot{a}\\cdot{b}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try it<\/h3>\r\nRewrite the following without exponents:\u00a0[latex]11r^{4}s^{3}[\/latex].\r\n\r\n[reveal-answer q=\"152461\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"152461\"]\r\n\r\n[latex]11\\cdot{r}\\cdot{r}\\cdot{r}\\cdot{r}\\cdot{s}\\cdot{s}\\cdot{s}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>\u00a0Evaluate Algebraic Expressions<\/h2>\r\nTo evaluate an expression, we substitute the given number for the variable in the expression and then simplify the expression using the order of operations.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nEvaluate [latex]x+7[\/latex] when\r\n<ol>\r\n \t<li>[latex]x=3[\/latex]<\/li>\r\n \t<li>[latex]x=12[\/latex]<\/li>\r\n<\/ol>\r\nSolution:\r\n\r\n1. To evaluate, substitute [latex]3[\/latex] for [latex]x[\/latex] in the expression, and then simplify.\r\n<table id=\"eip-id1166566546426\" class=\"unnumbered unstyled\" summary=\"The image shows the given expression x plus 7. Substitute 3 for x. The expression becomes 3 plus x which is 10.\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]x+7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute.<\/td>\r\n<td>[latex]\\color{red}{3}+7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add.<\/td>\r\n<td>[latex]10[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWhen [latex]x=3[\/latex], the expression [latex]x+7[\/latex] has a value of [latex]10[\/latex].\r\n2. To evaluate, substitute [latex]12[\/latex] for [latex]x[\/latex] in the expression, and then simplify.\r\n<table id=\"eip-id1166566410105\" class=\"unnumbered unstyled\" summary=\"The image shows the given expression x plus 7, substitute 12 for x. The expression becomes 12 plus x which is 19.\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]x+7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute.<\/td>\r\n<td>[latex]\\color{red}{12}+7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add.<\/td>\r\n<td>[latex]19[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWhen [latex]x=12[\/latex], the expression [latex]x+7[\/latex] has a value of [latex]19[\/latex].\r\n\r\nNotice that we got different results for parts 1 and 2 even though we started with the same expression. This is because the values used for [latex]x[\/latex] were different. When we evaluate an expression, the value varies depending on the value used for the variable.\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]144878[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nEvaluate [latex]9x - 2,[\/latex] when\r\n<ol>\r\n \t<li>[latex]x=5[\/latex]<\/li>\r\n \t<li>[latex]x=1[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"711463\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"711463\"]\r\n\r\nSolution\r\nRemember [latex]ab[\/latex] means [latex]a[\/latex] times [latex]b[\/latex], so [latex]9x[\/latex] means [latex]9[\/latex] times [latex]x[\/latex].\r\n1. To evaluate the expression when [latex]x=5[\/latex], we substitute [latex]5[\/latex] for [latex]x[\/latex], and then simplify.\r\n<table id=\"eip-id1168469462966\" class=\"unnumbered unstyled\" summary=\"The image shows the given expression nine x minus 2. Substitute 5 for x. The expression becomes 9 times 5 minus 2. Multiply first. Nine times 5 is 45 and the expression is now 45 minus 2. Subtract to get 43.\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]9x-2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute [latex]\\color{red}{5}[\/latex] for x.<\/td>\r\n<td>[latex]9\\cdot\\color{red}{5}-2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]45-2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract.<\/td>\r\n<td>[latex]43[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n2. To evaluate the expression when [latex]x=1[\/latex], we substitute [latex]1[\/latex] for [latex]x[\/latex], and then simplify.\r\n<table id=\"eip-id1168468440939\" class=\"unnumbered unstyled\" summary=\"The image shows the given expression nine x minus 2. Substitute 1 for x. The expression becomes 9 times 1 minus 2. Multiply first. Nine times 1 is 9 and the expression is now 9 minus 2. Subtract to get 7.\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]9x-2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute [latex]\\color{red}{1}[\/latex] for x.<\/td>\r\n<td>[latex]9(\\color{red}{1})-2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]9-2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract.<\/td>\r\n<td>[latex]7[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNotice that in part 1 that we wrote [latex]9\\cdot 5[\/latex] and in part 2 we wrote [latex]9\\left(1\\right)[\/latex]. Both the dot and the parentheses tell us to multiply.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nEvaluate [latex]\\frac{n}{d},[\/latex] when [latex]n=35[\/latex] and [latex]d=7[\/latex].\r\n\r\n[reveal-answer q=\"711470\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"711470\"]\r\n\r\nSolution\r\nTo evaluate the expression when [latex]n=35[\/latex] and [latex]d=7[\/latex], we substitute [latex]35[\/latex] for [latex]n[\/latex], and\u00a0[latex]7[\/latex] for [latex]d[\/latex] then simplify.\r\n<table id=\"eip-id1168469462966\" class=\"unnumbered unstyled\" summary=\"The image shows the given expression nine x minus 2. Substitute 5 for x. The expression becomes 9 times 5 minus 2. Multiply first. Nine times 5 is 45 and the expression is now 45 minus 2. Subtract to get 43.\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]\\frac{n}{d}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute [latex]\\color{red}{35}[\/latex] for n and\u00a0[latex]\\color{blue}{7}[\/latex] for d<\/td>\r\n<td>[latex]\\frac{\\color{red}{35}}{\\color{blue}{7}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Divide.<\/td>\r\n<td>[latex]\\color{red}{35}\/\\color{blue}{7}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]5[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]141843[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nEvaluate [latex]{x}^{2}[\/latex] when [latex]x=10[\/latex].\r\n[reveal-answer q=\"729694\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"729694\"]\r\n\r\nSolution\r\nWe substitute [latex]10[\/latex] for [latex]x[\/latex], and then simplify the expression.\r\n<table id=\"eip-id1168468538199\" class=\"unnumbered unstyled\" summary=\"The image shows the given expression x squared. Substitute 10 for x. The expression becomes 10 squared. By the definition of exponents, 10 squared is 10 times 10. Multiply to get 100.\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]x^2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute [latex]\\color{red}{10}[\/latex] for x.<\/td>\r\n<td>[latex]{\\color{red}{10}}^{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the definition of exponent.<\/td>\r\n<td>[latex]10\\cdot 10[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]100[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWhen [latex]x=10[\/latex], the expression [latex]{x}^{2}[\/latex] has a value of [latex]100[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]144879[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\n[latex]\\text{Evaluate }{2}^{x}\\text{ when }x=5[\/latex].\r\n[reveal-answer q=\"920379\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"920379\"]\r\n\r\nSolution\r\nIn this expression, the variable is an exponent.\r\n<table id=\"eip-id1168469574741\" class=\"unnumbered unstyled\" summary=\"The image shows the given expression 2 to the power of x. Substitute 5 for x. The expression becomes 2 to the fifth power. By the definition of exponents, 2 to the fifth power is 2 times 2 times 2 times 2 times 2, or 5 factors of 2. Multiply from left to right to get 32.\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]2^x[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute [latex]\\color{red}{5}[\/latex] for x.<\/td>\r\n<td>[latex]{2}^{\\color{red}{5}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the definition of exponent.<\/td>\r\n<td>[latex]2\\cdot2\\cdot2\\cdot2\\cdot2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]32[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWhen [latex]x=5[\/latex], the expression [latex]{2}^{x}[\/latex] has a value of [latex]32[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]144882[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\n[latex]\\text{Evaluate }3x+4y - 6\\text{ when }x=10\\text{ and }y=2[\/latex].\r\n[reveal-answer q=\"769566\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"769566\"]\r\n\r\nSolution\r\n\r\n&nbsp;\r\n\r\nThis expression contains two variables, so we must make two substitutions.\r\n<table id=\"eip-id1168467158036\" class=\"unnumbered unstyled\" summary=\"The image shows the given expression three x plus four y minus 6. Substitute 10 for x and 2 for y. The expression becomes 3 times 10 plus 4 times 2 minus 6. Perform multiplication from left to right. Three times 10 is 30 and 4 times 2 is 8. The expression becomes 30 plus 8 minus 6. Add and subtract from left to right. Thirty plus 8 is 38. Thirty-eight minus 6 is 32.\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]3x+4y-6[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute [latex]\\color{red}{10}[\/latex] for x and [latex]\\color{blue}{2}[\/latex] for y.<\/td>\r\n<td>[latex]3(\\color{red}{10})+4(\\color{blue}{2})-6[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]30+8-6[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add and subtract left to right.<\/td>\r\n<td>[latex]32[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWhen [latex]x=10[\/latex] and [latex]y=2[\/latex], the expression [latex]3x+4y - 6[\/latex] has a value of [latex]32[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>TRY\u00a0IT<\/h3>\r\n[ohm_question]144884[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\n[latex]\\text{Evaluate }2{x}^{2}+3x+8\\text{ when }x=4[\/latex].\r\n[reveal-answer q=\"971697\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"971697\"]\r\n\r\nSolution\r\nWe need to be careful when an expression has a variable with an exponent. In this expression, [latex]2{x}^{2}[\/latex] means [latex]2\\cdot x\\cdot x[\/latex] and is different from the expression [latex]{\\left(2x\\right)}^{2}[\/latex], which means [latex]2x\\cdot 2x[\/latex].\r\n<table id=\"eip-id1168466011069\" class=\"unnumbered unstyled\" summary=\"The image shows the given expression two x squared plus three x plus 8. Substitute 4 for each x. The expression becomes 2 times 4 squared plus 3 times 4 plus 8. Simplify exponents first. Four squared is 16 so the expression becomes 2 times 16 plus 3 times 4 plus 8. Next perform multiplication from left to right. Two times 16 is 32 and 3 times 4 is 12. The expression becomes 32 plus 12 plus 8. Add from left to right. Thirty-two plus 12 is 44. Forty-four plus 8 is 52.\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]2x^2+3x+8[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute [latex]\\color{red}{4}[\/latex] for each x.<\/td>\r\n<td>[latex]2{(\\color{red}{4})}^{2}+3(\\color{red}{4})+8[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify [latex]{4}^{2}[\/latex] .<\/td>\r\n<td>[latex]2(16)+3(4)+8[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]32+12+8[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add.<\/td>\r\n<td>[latex]52[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]144886[\/ohm_question]\r\n\r\n<\/div>\r\nIn the video below we show more examples of how to substitute a value for variable in an expression, then evaluate the expression.\r\n\r\nhttps:\/\/youtu.be\/dkFIVfJTG9E\r\n<h2 data-type=\"title\">Identify Terms, Coefficients, and Like Terms<\/h2>\r\nAlgebraic expressions are made up of <em data-effect=\"italics\">terms<\/em>. A term is a constant or the product of a constant and one or more variables. Some examples of terms are [latex]7,y,5{x}^{2},9a,\\text{and }13xy[\/latex].\r\n\r\nThe constant that multiplies the variable(s) in a term is called the coefficient. We can think of the coefficient as the number <em data-effect=\"italics\">in front of<\/em> the variable. The coefficient of the term [latex]3x[\/latex] is [latex]3[\/latex]. When we write [latex]x[\/latex], the coefficient is [latex]1[\/latex], since [latex]x=1\\cdot x[\/latex]. The table below gives the coefficients for each of the terms in the left column.\r\n<table id=\"fs-id2266631\" summary=\"This table has five rows and two columns. The first row is a header row and it labels each column. The first column is labeled \">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th data-align=\"center\">Term<\/th>\r\n<th data-align=\"center\">Coefficient<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-align=\"center\">[latex]7[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]7[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"center\">[latex]9a[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]9[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"center\">[latex]y[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"center\">[latex]5{x}^{2}[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]5[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nAn algebraic expression may consist of one or more terms added or subtracted. In this chapter, we will only work with terms that are added together. The table below gives some examples of algebraic expressions with various numbers of terms. Notice that we include the operation before a term with it.\r\n<table id=\"fs-id1596496\" summary=\"This table has six rows and two columns. The first row is a header row and it labels each column. The first column is labeled \">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th data-align=\"center\">Expression<\/th>\r\n<th data-align=\"center\">Terms<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]7[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]7[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]y[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]y[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]x+7[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]x,7[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]2x+7y+4[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]2x,7y,4[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]3{x}^{2}+4{x}^{2}+5y+3[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]3{x}^{2},4{x}^{2},5y,3[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div><\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nIdentify each term in the expression [latex]9b+15{x}^{2}+a+6[\/latex]. Then identify the coefficient of each term.\r\n\r\nSolution:\r\nThe expression has four terms. They are [latex]9b,15{x}^{2},a[\/latex], and [latex]6[\/latex].\r\n<ul>\r\n \t<li>The coefficient of [latex]9b[\/latex] is [latex]9[\/latex].<\/li>\r\n \t<li>The coefficient of [latex]15{x}^{2}[\/latex] is [latex]15[\/latex].<\/li>\r\n \t<li>Remember that if no number is written before a variable, the coefficient is [latex]1[\/latex]. So the coefficient of [latex]a[\/latex] is [latex]1[\/latex].<\/li>\r\n \t<li>The coefficient of a constant is the constant, so the coefficient of [latex]6[\/latex] is [latex]6[\/latex].<\/li>\r\n<\/ul>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]144899[\/ohm_question]\r\n\r\n<\/div>\r\nSome terms share common traits. Look at the following terms. Which ones seem to have traits in common?\r\n\r\n[latex]5x,7,{n}^{2},4,3x,9{n}^{2}[\/latex]\r\nWhich of these terms are like terms?\r\n<ul id=\"fs-id1627987\" data-bullet-style=\"bullet\">\r\n \t<li>The terms [latex]7[\/latex] and [latex]4[\/latex] are both constant terms.<\/li>\r\n \t<li>The terms [latex]5x[\/latex] and [latex]3x[\/latex] are both terms with [latex]x[\/latex].<\/li>\r\n \t<li>The terms [latex]{n}^{2}[\/latex] and [latex]9{n}^{2}[\/latex] both have [latex]{n}^{2}[\/latex].<\/li>\r\n<\/ul>\r\nTerms are called like terms if they have the same variables and exponents. All constant terms are also like terms. So among the terms [latex]5x,7,{n}^{2},4,3x,9{n}^{2}[\/latex],\r\n<ul>\r\n \t<li>[latex]7[\/latex] and [latex]4[\/latex] are like terms.<\/li>\r\n \t<li>[latex]5x[\/latex] and [latex]3x[\/latex] are like terms.<\/li>\r\n \t<li>[latex]{n}^{2}[\/latex] and [latex]9{n}^{2}[\/latex] are like terms.<\/li>\r\n<\/ul>\r\n<div class=\"textbox shaded\">\r\n<h3>Like Terms<\/h3>\r\nTerms that are either constants or have the same variables with the same exponents are like terms.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nIdentify the like terms:\r\n<ol>\r\n \t<li>[latex]{y}^{3},7{x}^{2},14,23,4{y}^{3},9x,5{x}^{2}[\/latex]<\/li>\r\n \t<li>[latex]4{x}^{2}+2x+5{x}^{2}+6x+40x+8xy[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"169480\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"169480\"]\r\n\r\nSolution:\r\n1. [latex]{y}^{3},7{x}^{2},14,23,4{y}^{3},9x,5{x}^{2}[\/latex]\r\nLook at the variables and exponents. The expression contains [latex]{y}^{3},{x}^{2},x[\/latex], and constants.\r\nThe terms [latex]{y}^{3}[\/latex] and [latex]4{y}^{3}[\/latex] are like terms because they both have [latex]{y}^{3}[\/latex].\r\nThe terms [latex]7{x}^{2}[\/latex] and [latex]5{x}^{2}[\/latex] are like terms because they both have [latex]{x}^{2}[\/latex].\r\nThe terms [latex]14[\/latex] and [latex]23[\/latex] are like terms because they are both constants.\r\nThe term [latex]9x[\/latex] does not have any like terms in this list since no other terms have the variable [latex]x[\/latex] raised to the power of [latex]1[\/latex].\r\n2. [latex]4{x}^{2}+2x+5{x}^{2}+6x+40x+8xy[\/latex]\r\nLook at the variables and exponents. The expression contains the terms [latex]4{x}^{2},2x,5{x}^{2},6x,40x,\\text{and}8xy[\/latex]\r\nThe terms [latex]4{x}^{2}[\/latex] and [latex]5{x}^{2}[\/latex] are like terms because they both have [latex]{x}^{2}[\/latex].\r\nThe terms [latex]2x,6x,\\text{and}40x[\/latex] are like terms because they all have [latex]x[\/latex].\r\nThe term [latex]8xy[\/latex] has no like terms in the given expression because no other terms contain the two variables [latex]xy[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146540[\/ohm_question]\r\n\r\n<\/div>\r\n<h2 data-type=\"title\">Simplify Expressions by Combining Like Terms<\/h2>\r\nWe can simplify an expression by combining the like terms. What do you think [latex]3x+6x[\/latex] would simplify to? If you thought [latex]9x[\/latex], you would be right!\r\n\r\nWe can see why this works by writing both terms as addition problems.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215839\/CNX_BMath_Figure_02_02_001_img.png\" alt=\"The image shows the expression 3 x plus 6 x. The 3 x represents x plus x plus x. The 6 x represents x plus x plus x plus x plus x plus x. The expression 3 x plus 6 x becomes x plus x plus x plus x plus x plus x plus x plus x plus x. This simplifies to a total of 9 x's or the term 9 x.\" data-media-type=\"image\/png\" \/>\r\nAdd the coefficients and keep the same variable. It doesn\u2019t matter what [latex]x[\/latex] is. If you have [latex]3[\/latex] of something and add [latex]6[\/latex] more of the same thing, the result is [latex]9[\/latex] of them. For example, [latex]3[\/latex] oranges plus [latex]6[\/latex] oranges is [latex]9[\/latex] oranges. We will discuss the mathematical properties behind this later.\r\n\r\nThe expression [latex]3x+6x[\/latex] has only two terms. When an expression contains more terms, it may be helpful to rearrange the terms so that like terms are together. The Commutative Property of Addition says that we can change the order of addends without changing the sum. So we could rearrange the following expression before combining like terms.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215840\/CNX_BMath_Figure_02_02_015_img.png\" alt=\"The image shows the expression 3 x plus 4 y plus 2 x plus 6 y. The position of the middle terms, 4 y and 2 x, can be switched so that the expression becomes 3 x plus 2 x plus 4 y plus 6 y. Now the terms containing x are together and the terms containing y are together.\" data-media-type=\"image\/png\" \/>\r\nNow it is easier to see the like terms to be combined.\r\n<div class=\"textbox shaded\">\r\n<h3>Combine like terms<\/h3>\r\n<ol id=\"eip-id1168466010921\" class=\"stepwise\" data-number-style=\"arabic\">\r\n \t<li>Identify like terms.<\/li>\r\n \t<li>Rearrange the expression so like terms are together.<\/li>\r\n \t<li>Add the coefficients of the like terms.<\/li>\r\n<\/ol>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify the expression: [latex]3x+7+4x+5[\/latex].\r\n[reveal-answer q=\"668173\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"668173\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168468738000\" class=\"unnumbered unstyled\" summary=\"The image shows the expression 3 x plus 7 plus 4 x plus 5. Three x and 4 x are like terms as are 7 and 5. The middle terms, 7 and 4 x, can be rearranged so that the like terms are together. The expressions becomes 3 x plus 4 x plus 7 plus 5. Now the like terms can be combined by adding the coefficients of the like terms. Three x plus 4 x is 7 x and 7 plus 5 is 12. The expression becomes 7 x plus 12.\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]3x+7+4x+5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Identify the like terms.<\/td>\r\n<td>[latex]\\color{red}{3x}+\\color{blue}{7}+\\color{red}{4x}+\\color{blue}{5}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Rearrange the expression, so the like terms are together.<\/td>\r\n<td>[latex]\\color{red}{3x}+\\color{red}{4x}+\\color{blue}{7}+\\color{blue}{5}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add the coefficients of the like terms.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215843\/CNX_BMath_Figure_02_02_022_img-04.png\" alt=\".\" data-media-type=\"image\/png\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The original expression is simplified to...<\/td>\r\n<td>[latex]7x+12[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]144900[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify the expression: [latex]8x+7{x}^{2}-{x}^{2}-+4x[\/latex].\r\n[reveal-answer q=\"94190\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"94190\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168467195542\" class=\"unnumbered unstyled\" summary=\"The image shows the expression 7 x squared plus 8 x plus x squared plus 4 x. Seven x squared and x squared are like terms as are 8 x and 4 x. The middle terms, 8 x and x squared, can be rearranged so that the like terms are together. The expressions becomes 7 x squared plus x squared plus 8 x plus 4 x. Now the like terms can be combined by adding the coefficients of the like terms. 7 x squared plus x squared is 8 x squared and 8 x plus 4 x is 12 x. The expression becomes 8 x squared plus 12 x.\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]8x+7{x}^{2}-{x}^{2}-+4x[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Identify the like terms.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Rearrange the expression so like terms are together.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add the coefficients of the like terms.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThese are not like terms and cannot be combined. So [latex]8{x}^{2}+12x[\/latex] is in simplest form.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]144905[\/ohm_question]\r\n\r\n<\/div>\r\nIn the following video, we present more examples of how to combine like terms given an algebraic expression.\r\n\r\nhttps:\/\/youtu.be\/KMUCQ_Pwt7o\r\n<h2>Simplify Expressions Using the Distributive Property<\/h2>\r\n<div class=\"textbox shaded\">\r\n<h3>Distributive Property<\/h3>\r\nIf [latex]a,b,c[\/latex] are real numbers, then\r\n\r\n[latex]a\\left(b+c\\right)=ab+ac[\/latex]\r\n\r\n<\/div>\r\nIn algebra, we use the Distributive Property to remove parentheses as we simplify expressions. For example, if we are asked to simplify the expression [latex]3\\left(x+4\\right)[\/latex], the order of operations says to work in the parentheses first. But we cannot add [latex]x[\/latex] and [latex]4[\/latex], since they are not like terms. So we use the Distributive Property, as shown in the next example.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]3\\left(x+4\\right)[\/latex]\r\n\r\nSolution:\r\n<table id=\"eip-id1168468605528\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>[latex]3\\left(x+4\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Distribute.<\/td>\r\n<td>[latex]3\\cdot x+3\\cdot 4[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]3x+12[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\nSome students find it helpful to draw in arrows to remind them how to use the Distributive Property. Then the first step in the previous example would look like this:\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24222343\/CNX_BMath_Figure_07_03_003_img.png\" alt=\"The image shows the expression x plus 4 in parentheses with the number 3 outside the parentheses on the left. There are two arrows pointing from the top of the three. One arrow points to the top of the x. The other arrow points to the top of the 4.\" \/>\r\n\r\n[latex]3\\cdot x+3\\cdot 4[\/latex]\r\n\r\nNow you try.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146473[\/ohm_question]\r\n\r\n<\/div>\r\nIn our next example, there is a coefficient on the variable y. When you use the distributive property, you multiply the two numbers together, just like simplifying any product. You will also see another example where the expression in parentheses is subtraction, rather than addition. \u00a0You will need to be careful to change the sign of your product.\r\n\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]6\\left(5y+1\\right)[\/latex].\r\n\r\n[reveal-answer q=\"645849\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"645849\"]\r\n\r\n&nbsp;\r\n\r\nSolution:\r\n<table id=\"eip-id1168466125869\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The image shows the expression 5 y plus 1 in parentheses with the number 6 outside the parentheses on the left. Simplify by distributing the 6 through the parentheses to get the expression 6 times 5 y plus 6 times 1. Simplify further by multiplying 6 times 5 y to get 30 y and 6 times 1 to get 6. The expression simplifies to 30 y plus 6.\">\r\n<tbody>\r\n<tr>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24222344\/CNX_BMath_Figure_07_03_025_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Distribute.<\/td>\r\n<td>[latex]6\\cdot 5y+6\\cdot 1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]30y+6[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\nSimplify: [latex]2\\left(x - 3\\right)[\/latex]\r\n[reveal-answer q=\"877652\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"877652\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168466277015\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The image shows the expression x minus 3 in parentheses with the number 2 outside the parentheses on the left. Simplify by distributing the 2 through the parentheses to get the expression 2 times x minus 2 times 3. Simplify further by multiplying 2 times x to get 2 x and 2 times 3 to get 6. The expression simplifies to 2 x minus 6.\">\r\n<tbody>\r\n<tr>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24222347\/CNX_BMath_Figure_07_03_026_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Distribute.<\/td>\r\n<td>[latex]2\\cdot x--2\\cdot 3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]2x--6[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNow you try.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146474[\/ohm_question]\r\n\r\n[ohm_question]146475[\/ohm_question]\r\n\r\n<\/div>\r\nThe distributive property can be used to simplify expressions that look slightly different from [latex]a\\left(b+c\\right)[\/latex]. Here are two other forms.\r\n<div class=\"textbox shaded\">\r\n<h3>different Forms of the Distributive Property<\/h3>\r\nIf [latex]a,b,c[\/latex] are real numbers, then\r\n\r\n[latex]a\\left(b+c\\right)=ab+ac[\/latex]\r\nOther forms\r\n\r\n[latex]a\\left(b-c\\right)=ab-ac[\/latex]\r\n[latex]\\left(b+c\\right)a=ba+ca[\/latex]\r\n\r\n<\/div>\r\nIn teh following video we show more examples of using the distributive property.\r\n\r\nhttps:\/\/youtu.be\/Nt8V5cEvAz8","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Use variables to represent unknown quantities in algebraic expressions<\/li>\n<li>Identify the variables and constants in an algebraic expression<\/li>\n<li>Use words and symbols to represent algebraic operations on variables and constants<\/li>\n<li>Simplify expressions with exponents containing integer bases and variable bases<\/li>\n<li>Evaluate an expression for a given value<\/li>\n<li>Identify the variables and constants in a term<\/li>\n<li>Identify the coefficient of a variable term<\/li>\n<li>Identify and combine like terms in an expression<\/li>\n<li>Apply the distributive property to simplify an algebraic expression<\/li>\n<\/ul>\n<\/div>\n<h2 data-type=\"title\">Use Variables and Algebraic Symbols<\/h2>\n<p>Greg and Alex have the same birthday, but they were born in different years. This year Greg is [latex]20[\/latex] years old and Alex is [latex]23[\/latex], so Alex is [latex]3[\/latex] years older than Greg. When Greg was [latex]12[\/latex], Alex was [latex]15[\/latex]. When Greg is [latex]35[\/latex], Alex will be [latex]38[\/latex]. No matter what Greg\u2019s age is, Alex\u2019s age will always be [latex]3[\/latex] years more, right?<\/p>\n<p>In the language of algebra, we say that Greg\u2019s age and Alex\u2019s age are variable and the three is a constant. The ages change, or vary, so age is a variable. The [latex]3[\/latex] years between them always stays the same, so the age difference is the constant.<\/p>\n<p>In algebra, letters of the alphabet are used to represent variables. Suppose we call Greg\u2019s age [latex]g[\/latex]. Then we could use [latex]g+3[\/latex] to represent Alex\u2019s age. See the table below.<\/p>\n<table style=\"width: 40%;\" summary=\"This table has five rows and two columns. The first row is a header row and is labeled\">\n<thead>\n<tr valign=\"top\">\n<th data-align=\"center\">Greg\u2019s age<\/th>\n<th data-align=\"center\">Alex\u2019s age<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td data-align=\"center\">[latex]12[\/latex]<\/td>\n<td data-align=\"center\">[latex]15[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"center\">[latex]20[\/latex]<\/td>\n<td data-align=\"center\">[latex]23[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"center\">[latex]35[\/latex]<\/td>\n<td data-align=\"center\">[latex]38[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"center\">[latex]g[\/latex]<\/td>\n<td data-align=\"center\">[latex]g+3[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Letters are used to represent variables. Letters often used for variables are [latex]x,y,a,b,\\text{ and }c[\/latex].<\/p>\n<div class=\"textbox shaded\">\n<h3>Variables and Constants<\/h3>\n<p>A variable is a letter that represents a number or quantity whose value may change.<br \/>\nA constant is a number whose value always stays the same.<\/p>\n<\/div>\n<p>To write algebraically, we need some symbols as well as numbers and variables. There are several types of symbols we will be using. In Whole Numbers, we introduced the symbols for the four basic arithmetic operations: addition, subtraction, multiplication, and division. We will summarize them here, along with words we use for the operations and the result.<\/p>\n<table class=\"unnumbered\" style=\"width: 40%;\" summary=\"This table has five rows and four columns. The first row is a header row. Each column is labeled accordingly: the first is labeled\" data-label=\"\">\n<thead>\n<tr valign=\"top\">\n<th data-align=\"left\"><strong>Operation<\/strong><\/th>\n<th data-align=\"left\"><strong>Notation<\/strong><\/th>\n<th data-align=\"left\"><strong>Say:<\/strong><\/th>\n<th data-align=\"left\"><strong>The result is\u2026<\/strong><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td data-align=\"left\">Addition<\/td>\n<td data-align=\"left\">[latex]a+b[\/latex]<\/td>\n<td data-align=\"left\">[latex]a\\text{ plus }b[\/latex]<\/td>\n<td data-align=\"left\">the sum of [latex]a[\/latex] and [latex]b[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">Subtraction<\/td>\n<td data-align=\"left\">[latex]a-b[\/latex]<\/td>\n<td data-align=\"left\">[latex]a\\text{ minus }b[\/latex]<\/td>\n<td data-align=\"left\">the difference of [latex]a[\/latex] and [latex]b[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">Multiplication<\/td>\n<td data-align=\"left\">[latex]a\\cdot b,\\left(a\\right)\\left(b\\right),\\left(a\\right)b,a\\left(b\\right)[\/latex]<\/td>\n<td data-align=\"left\">[latex]a\\text{ times }b[\/latex]<\/td>\n<td data-align=\"left\">The product of [latex]a[\/latex] and [latex]b[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">Division<\/td>\n<td data-align=\"left\">[latex]a\\div b,a\/b,\\frac{a}{b},b\\overline{)a}[\/latex]<\/td>\n<td data-align=\"left\">[latex]a[\/latex] divided by [latex]b[\/latex]<\/td>\n<td data-align=\"left\">The quotient of [latex]a[\/latex] and [latex]b[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>In algebra, the cross symbol, [latex]\\times[\/latex], is not used to show multiplication because that symbol may cause confusion. Does [latex]3xy[\/latex] mean [latex]3\\times y[\/latex] (three times [latex]y[\/latex] ) or [latex]3\\cdot x\\cdot y[\/latex] (three times [latex]x\\text{ times }y[\/latex] )? To make it clear, use \u2022 or parentheses for multiplication.<br \/>\nWe perform these operations on two numbers. When translating from symbolic form to words, or from words to symbolic form, pay attention to the words <em data-effect=\"italics\">of<\/em> or <em data-effect=\"italics\">and<\/em> to help you find the numbers.<\/p>\n<ul id=\"fs-id1969800\" data-labeled-item=\"true\">\n<li>The <em data-effect=\"italics\">sum\u00a0<\/em><strong><em data-effect=\"italics\">of<\/em><\/strong> [latex]5[\/latex] <strong><em data-effect=\"italics\">and<\/em><\/strong> [latex]3[\/latex] means add [latex]5[\/latex] plus [latex]3[\/latex], which we write as [latex]5+3[\/latex].<\/li>\n<li>The <em data-effect=\"italics\">difference\u00a0<\/em><strong><em data-effect=\"italics\">of<\/em><\/strong> [latex]9[\/latex] <strong><em data-effect=\"italics\">and<\/em><\/strong> [latex]2[\/latex] means subtract [latex]9[\/latex] minus [latex]2[\/latex], which we write as [latex]9 - 2[\/latex].<\/li>\n<li>The <em data-effect=\"italics\">product\u00a0<\/em><strong><em data-effect=\"italics\">of<\/em><\/strong> [latex]4[\/latex] <strong><em data-effect=\"italics\">and<\/em><\/strong> [latex]8[\/latex] means multiply [latex]4[\/latex] times [latex]8[\/latex], which we can write as [latex]4\\cdot 8[\/latex].<\/li>\n<li>The <em data-effect=\"italics\">quotient\u00a0<\/em><strong><em data-effect=\"italics\">of<\/em><\/strong> [latex]20[\/latex] <strong><em data-effect=\"italics\">and<\/em><\/strong> [latex]5[\/latex] means divide [latex]20[\/latex] by [latex]5[\/latex], which we can write as [latex]20\\div 5[\/latex].<\/li>\n<\/ul>\n<div class=\"textbox exercises\">\n<h3>Exercises<\/h3>\n<p>Translate from algebra to words:<\/p>\n<ol>\n<li>[latex]12+14[\/latex]<\/li>\n<li>[latex]\\left(30\\right)\\left(5\\right)[\/latex]<\/li>\n<li>[latex]64\\div 8[\/latex]<\/li>\n<li>[latex]x-y[\/latex]<\/li>\n<\/ol>\n<p>Solution:<\/p>\n<table class=\"unnumbered unstyled\" style=\"width: 40%;\" summary=\".\" data-label=\"\">\n<tbody>\n<tr style=\"height: 15.7812px;\">\n<td style=\"height: 15.7812px;\">1.<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\" data-align=\"center\">[latex]12+14[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\" data-align=\"center\">[latex]12[\/latex] plus [latex]14[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\" data-align=\"center\">the sum of twelve and fourteen<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table style=\"width: 40%;\">\n<tbody>\n<tr>\n<td>2.<\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">[latex]\\left(30\\right)\\left(5\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">[latex]30[\/latex] times [latex]5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">the product of thirty and five<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table class=\"unnumbered unstyled\" style=\"width: 40%;\" summary=\".\" data-label=\"\">\n<tbody>\n<tr>\n<td>3.<\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">[latex]64\\div 8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">[latex]64[\/latex] divided by [latex]8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">the quotient of sixty-four and eight<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table class=\"unnumbered unstyled\" style=\"width: 40%;\" summary=\".\" data-label=\"\">\n<tbody>\n<tr style=\"height: 15.5625px;\">\n<td style=\"height: 15.5625px;\">4.<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\" data-align=\"center\">[latex]x-y[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\" data-align=\"center\">[latex]x[\/latex] minus [latex]y[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\" data-align=\"center\">the difference of [latex]x[\/latex] and [latex]y[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>TRY\u00a0IT<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=144651&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"330\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"mom2\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=144652&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<h2>Exponents<\/h2>\n<p>Remember that an exponent indicates repeated multiplication of the same quantity. For example, [latex]{2}^{4}[\/latex] means to multiply four factors of [latex]2[\/latex], so [latex]{2}^{4}[\/latex] means [latex]2\\cdot 2\\cdot 2\\cdot 2[\/latex]. This format is known as exponential notation.<\/p>\n<div class=\"textbox shaded\">\n<h3>Exponential Notation<\/h3>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224353\/CNX_BMath_Figure_10_02_013_img.png\" alt=\"On the left side, a raised to the m is shown. The m is labeled in blue as an exponent. The a is labeled in red as the base. On the right, it says a to the m means multiply m factors of a. Below this, it says a to the m equals a times a times a times a, with m factors written below in blue.\" \/><br \/>\nThis is read [latex]a[\/latex] to the [latex]{m}^{\\mathrm{th}}[\/latex] power.<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>In the expression [latex]{a}^{m}[\/latex], the exponent tells us how many times we use the base [latex]a[\/latex] as a factor.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224355\/CNX_BMath_Figure_10_02_014_img.png\" alt=\"On the left side, 7 to the 3rd power is shown. Below is 7 times 7 times 7, with 3 factors written below. On the right side, parentheses negative 8 to the 5th power is shown. Below is negative 8 times negative 8 times negative 8 times negative 8 times negative 8, with 5 factors written below.\" \/><br \/>\nBefore we begin working with variable expressions containing exponents, let\u2019s simplify a few expressions involving only numbers.<\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify:<\/p>\n<p>1. [latex]{5}^{3}[\/latex]<br \/>\n2. [latex]{9}^{1}[\/latex]<\/p>\n<p>Solution<\/p>\n<table id=\"eip-id1168469452397\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td>1.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]{5}^{3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply [latex]3[\/latex] factors of [latex]5[\/latex].<\/td>\n<td>[latex]5\\cdot 5\\cdot 5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]125[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168046009892\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td>2.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]{9}^{1}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply [latex]1[\/latex] factor of [latex]9[\/latex].<\/td>\n<td>[latex]9[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146094\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146094&theme=oea&iframe_resize_id=ohm146094&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify:<\/p>\n<p>1. [latex]{\\left(-3\\right)}^{4}[\/latex]<br \/>\n2. [latex]{-3}^{4}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q152453\">Show Solution<\/span><\/p>\n<div id=\"q152453\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168468562526\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td>1.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]{\\left(-3\\right)}^{4}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply four factors of [latex]\u22123[\/latex].<\/td>\n<td>[latex]\\left(-3\\right)\\left(-3\\right)\\left(-3\\right)\\left(-3\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]81[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168048408997\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td>2.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]{-3}^{4}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply two factors.<\/td>\n<td>[latex]-\\left(3\\cdot 3\\cdot 3\\cdot 3\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]-81[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Notice the similarities and differences in parts 1 and 2. Why are the answers different? In part 1 the parentheses tell us to raise the [latex](\u22123)[\/latex] to the [latex]4[\/latex]<sup>th<\/sup> power. In part 2 we raise only the [latex]3[\/latex] to the [latex]4[\/latex]<sup>th<\/sup> power and then find the opposite.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146097\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146097&theme=oea&iframe_resize_id=ohm146097&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Rewrite the following without exponents:\u00a0[latex]-8a^{5}b[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q152460\">Show Solution<\/span><\/p>\n<div id=\"q152460\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<p>The expression [latex]a^{5}[\/latex] means you are multiplying [latex]a[\/latex] five times, so it is [latex]{a}\\cdot{a}\\cdot{a}\\cdot{a}\\cdot{a}[\/latex].\u00a0 Therefore, your answer is [latex]-8\\cdot{a}\\cdot{a}\\cdot{a}\\cdot{a}\\cdot{a}\\cdot{b}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try it<\/h3>\n<p>Rewrite the following without exponents:\u00a0[latex]11r^{4}s^{3}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q152461\">Show Solution<\/span><\/p>\n<div id=\"q152461\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]11\\cdot{r}\\cdot{r}\\cdot{r}\\cdot{r}\\cdot{s}\\cdot{s}\\cdot{s}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>\u00a0Evaluate Algebraic Expressions<\/h2>\n<p>To evaluate an expression, we substitute the given number for the variable in the expression and then simplify the expression using the order of operations.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Evaluate [latex]x+7[\/latex] when<\/p>\n<ol>\n<li>[latex]x=3[\/latex]<\/li>\n<li>[latex]x=12[\/latex]<\/li>\n<\/ol>\n<p>Solution:<\/p>\n<p>1. To evaluate, substitute [latex]3[\/latex] for [latex]x[\/latex] in the expression, and then simplify.<\/p>\n<table id=\"eip-id1166566546426\" class=\"unnumbered unstyled\" summary=\"The image shows the given expression x plus 7. Substitute 3 for x. The expression becomes 3 plus x which is 10.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]x+7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute.<\/td>\n<td>[latex]\\color{red}{3}+7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add.<\/td>\n<td>[latex]10[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>When [latex]x=3[\/latex], the expression [latex]x+7[\/latex] has a value of [latex]10[\/latex].<br \/>\n2. To evaluate, substitute [latex]12[\/latex] for [latex]x[\/latex] in the expression, and then simplify.<\/p>\n<table id=\"eip-id1166566410105\" class=\"unnumbered unstyled\" summary=\"The image shows the given expression x plus 7, substitute 12 for x. The expression becomes 12 plus x which is 19.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]x+7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute.<\/td>\n<td>[latex]\\color{red}{12}+7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add.<\/td>\n<td>[latex]19[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>When [latex]x=12[\/latex], the expression [latex]x+7[\/latex] has a value of [latex]19[\/latex].<\/p>\n<p>Notice that we got different results for parts 1 and 2 even though we started with the same expression. This is because the values used for [latex]x[\/latex] were different. When we evaluate an expression, the value varies depending on the value used for the variable.<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm144878\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144878&theme=oea&iframe_resize_id=ohm144878&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Evaluate [latex]9x - 2,[\/latex] when<\/p>\n<ol>\n<li>[latex]x=5[\/latex]<\/li>\n<li>[latex]x=1[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q711463\">Show Solution<\/span><\/p>\n<div id=\"q711463\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<br \/>\nRemember [latex]ab[\/latex] means [latex]a[\/latex] times [latex]b[\/latex], so [latex]9x[\/latex] means [latex]9[\/latex] times [latex]x[\/latex].<br \/>\n1. To evaluate the expression when [latex]x=5[\/latex], we substitute [latex]5[\/latex] for [latex]x[\/latex], and then simplify.<\/p>\n<table id=\"eip-id1168469462966\" class=\"unnumbered unstyled\" summary=\"The image shows the given expression nine x minus 2. Substitute 5 for x. The expression becomes 9 times 5 minus 2. Multiply first. Nine times 5 is 45 and the expression is now 45 minus 2. Subtract to get 43.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]9x-2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute [latex]\\color{red}{5}[\/latex] for x.<\/td>\n<td>[latex]9\\cdot\\color{red}{5}-2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]45-2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtract.<\/td>\n<td>[latex]43[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>2. To evaluate the expression when [latex]x=1[\/latex], we substitute [latex]1[\/latex] for [latex]x[\/latex], and then simplify.<\/p>\n<table id=\"eip-id1168468440939\" class=\"unnumbered unstyled\" summary=\"The image shows the given expression nine x minus 2. Substitute 1 for x. The expression becomes 9 times 1 minus 2. Multiply first. Nine times 1 is 9 and the expression is now 9 minus 2. Subtract to get 7.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]9x-2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute [latex]\\color{red}{1}[\/latex] for x.<\/td>\n<td>[latex]9(\\color{red}{1})-2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]9-2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtract.<\/td>\n<td>[latex]7[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Notice that in part 1 that we wrote [latex]9\\cdot 5[\/latex] and in part 2 we wrote [latex]9\\left(1\\right)[\/latex]. Both the dot and the parentheses tell us to multiply.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Evaluate [latex]\\frac{n}{d},[\/latex] when [latex]n=35[\/latex] and [latex]d=7[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q711470\">Show Solution<\/span><\/p>\n<div id=\"q711470\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<br \/>\nTo evaluate the expression when [latex]n=35[\/latex] and [latex]d=7[\/latex], we substitute [latex]35[\/latex] for [latex]n[\/latex], and\u00a0[latex]7[\/latex] for [latex]d[\/latex] then simplify.<\/p>\n<table id=\"eip-id1168469462966\" class=\"unnumbered unstyled\" summary=\"The image shows the given expression nine x minus 2. Substitute 5 for x. The expression becomes 9 times 5 minus 2. Multiply first. Nine times 5 is 45 and the expression is now 45 minus 2. Subtract to get 43.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]\\frac{n}{d}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute [latex]\\color{red}{35}[\/latex] for n and\u00a0[latex]\\color{blue}{7}[\/latex] for d<\/td>\n<td>[latex]\\frac{\\color{red}{35}}{\\color{blue}{7}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Divide.<\/td>\n<td>[latex]\\color{red}{35}\/\\color{blue}{7}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]5[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm141843\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=141843&theme=oea&iframe_resize_id=ohm141843&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Evaluate [latex]{x}^{2}[\/latex] when [latex]x=10[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q729694\">Show Solution<\/span><\/p>\n<div id=\"q729694\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<br \/>\nWe substitute [latex]10[\/latex] for [latex]x[\/latex], and then simplify the expression.<\/p>\n<table id=\"eip-id1168468538199\" class=\"unnumbered unstyled\" summary=\"The image shows the given expression x squared. Substitute 10 for x. The expression becomes 10 squared. By the definition of exponents, 10 squared is 10 times 10. Multiply to get 100.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]x^2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute [latex]\\color{red}{10}[\/latex] for x.<\/td>\n<td>[latex]{\\color{red}{10}}^{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the definition of exponent.<\/td>\n<td>[latex]10\\cdot 10[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]100[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>When [latex]x=10[\/latex], the expression [latex]{x}^{2}[\/latex] has a value of [latex]100[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm144879\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144879&theme=oea&iframe_resize_id=ohm144879&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>[latex]\\text{Evaluate }{2}^{x}\\text{ when }x=5[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q920379\">Show Solution<\/span><\/p>\n<div id=\"q920379\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<br \/>\nIn this expression, the variable is an exponent.<\/p>\n<table id=\"eip-id1168469574741\" class=\"unnumbered unstyled\" summary=\"The image shows the given expression 2 to the power of x. Substitute 5 for x. The expression becomes 2 to the fifth power. By the definition of exponents, 2 to the fifth power is 2 times 2 times 2 times 2 times 2, or 5 factors of 2. Multiply from left to right to get 32.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]2^x[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute [latex]\\color{red}{5}[\/latex] for x.<\/td>\n<td>[latex]{2}^{\\color{red}{5}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the definition of exponent.<\/td>\n<td>[latex]2\\cdot2\\cdot2\\cdot2\\cdot2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]32[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>When [latex]x=5[\/latex], the expression [latex]{2}^{x}[\/latex] has a value of [latex]32[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm144882\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144882&theme=oea&iframe_resize_id=ohm144882&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>[latex]\\text{Evaluate }3x+4y - 6\\text{ when }x=10\\text{ and }y=2[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q769566\">Show Solution<\/span><\/p>\n<div id=\"q769566\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<p>&nbsp;<\/p>\n<p>This expression contains two variables, so we must make two substitutions.<\/p>\n<table id=\"eip-id1168467158036\" class=\"unnumbered unstyled\" summary=\"The image shows the given expression three x plus four y minus 6. Substitute 10 for x and 2 for y. The expression becomes 3 times 10 plus 4 times 2 minus 6. Perform multiplication from left to right. Three times 10 is 30 and 4 times 2 is 8. The expression becomes 30 plus 8 minus 6. Add and subtract from left to right. Thirty plus 8 is 38. Thirty-eight minus 6 is 32.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]3x+4y-6[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute [latex]\\color{red}{10}[\/latex] for x and [latex]\\color{blue}{2}[\/latex] for y.<\/td>\n<td>[latex]3(\\color{red}{10})+4(\\color{blue}{2})-6[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]30+8-6[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add and subtract left to right.<\/td>\n<td>[latex]32[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>When [latex]x=10[\/latex] and [latex]y=2[\/latex], the expression [latex]3x+4y - 6[\/latex] has a value of [latex]32[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>TRY\u00a0IT<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm144884\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144884&theme=oea&iframe_resize_id=ohm144884&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>[latex]\\text{Evaluate }2{x}^{2}+3x+8\\text{ when }x=4[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q971697\">Show Solution<\/span><\/p>\n<div id=\"q971697\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<br \/>\nWe need to be careful when an expression has a variable with an exponent. In this expression, [latex]2{x}^{2}[\/latex] means [latex]2\\cdot x\\cdot x[\/latex] and is different from the expression [latex]{\\left(2x\\right)}^{2}[\/latex], which means [latex]2x\\cdot 2x[\/latex].<\/p>\n<table id=\"eip-id1168466011069\" class=\"unnumbered unstyled\" summary=\"The image shows the given expression two x squared plus three x plus 8. Substitute 4 for each x. The expression becomes 2 times 4 squared plus 3 times 4 plus 8. Simplify exponents first. Four squared is 16 so the expression becomes 2 times 16 plus 3 times 4 plus 8. Next perform multiplication from left to right. Two times 16 is 32 and 3 times 4 is 12. The expression becomes 32 plus 12 plus 8. Add from left to right. Thirty-two plus 12 is 44. Forty-four plus 8 is 52.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]2x^2+3x+8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute [latex]\\color{red}{4}[\/latex] for each x.<\/td>\n<td>[latex]2{(\\color{red}{4})}^{2}+3(\\color{red}{4})+8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify [latex]{4}^{2}[\/latex] .<\/td>\n<td>[latex]2(16)+3(4)+8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]32+12+8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add.<\/td>\n<td>[latex]52[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm144886\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144886&theme=oea&iframe_resize_id=ohm144886&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the video below we show more examples of how to substitute a value for variable in an expression, then evaluate the expression.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex: Substitute and Evaluate  Expressions x^2+3, (x+3)^2, x^2+2x+3\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/dkFIVfJTG9E?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2 data-type=\"title\">Identify Terms, Coefficients, and Like Terms<\/h2>\n<p>Algebraic expressions are made up of <em data-effect=\"italics\">terms<\/em>. A term is a constant or the product of a constant and one or more variables. Some examples of terms are [latex]7,y,5{x}^{2},9a,\\text{and }13xy[\/latex].<\/p>\n<p>The constant that multiplies the variable(s) in a term is called the coefficient. We can think of the coefficient as the number <em data-effect=\"italics\">in front of<\/em> the variable. The coefficient of the term [latex]3x[\/latex] is [latex]3[\/latex]. When we write [latex]x[\/latex], the coefficient is [latex]1[\/latex], since [latex]x=1\\cdot x[\/latex]. The table below gives the coefficients for each of the terms in the left column.<\/p>\n<table id=\"fs-id2266631\" summary=\"This table has five rows and two columns. The first row is a header row and it labels each column. The first column is labeled\">\n<thead>\n<tr valign=\"top\">\n<th data-align=\"center\">Term<\/th>\n<th data-align=\"center\">Coefficient<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td data-align=\"center\">[latex]7[\/latex]<\/td>\n<td data-align=\"left\">[latex]7[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"center\">[latex]9a[\/latex]<\/td>\n<td data-align=\"left\">[latex]9[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"center\">[latex]y[\/latex]<\/td>\n<td data-align=\"left\">[latex]1[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"center\">[latex]5{x}^{2}[\/latex]<\/td>\n<td data-align=\"left\">[latex]5[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>An algebraic expression may consist of one or more terms added or subtracted. In this chapter, we will only work with terms that are added together. The table below gives some examples of algebraic expressions with various numbers of terms. Notice that we include the operation before a term with it.<\/p>\n<table id=\"fs-id1596496\" summary=\"This table has six rows and two columns. The first row is a header row and it labels each column. The first column is labeled\">\n<thead>\n<tr valign=\"top\">\n<th data-align=\"center\">Expression<\/th>\n<th data-align=\"center\">Terms<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]7[\/latex]<\/td>\n<td data-align=\"left\">[latex]7[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]y[\/latex]<\/td>\n<td data-align=\"left\">[latex]y[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]x+7[\/latex]<\/td>\n<td data-align=\"left\">[latex]x,7[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]2x+7y+4[\/latex]<\/td>\n<td data-align=\"left\">[latex]2x,7y,4[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]3{x}^{2}+4{x}^{2}+5y+3[\/latex]<\/td>\n<td data-align=\"left\">[latex]3{x}^{2},4{x}^{2},5y,3[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div><\/div>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Identify each term in the expression [latex]9b+15{x}^{2}+a+6[\/latex]. Then identify the coefficient of each term.<\/p>\n<p>Solution:<br \/>\nThe expression has four terms. They are [latex]9b,15{x}^{2},a[\/latex], and [latex]6[\/latex].<\/p>\n<ul>\n<li>The coefficient of [latex]9b[\/latex] is [latex]9[\/latex].<\/li>\n<li>The coefficient of [latex]15{x}^{2}[\/latex] is [latex]15[\/latex].<\/li>\n<li>Remember that if no number is written before a variable, the coefficient is [latex]1[\/latex]. So the coefficient of [latex]a[\/latex] is [latex]1[\/latex].<\/li>\n<li>The coefficient of a constant is the constant, so the coefficient of [latex]6[\/latex] is [latex]6[\/latex].<\/li>\n<\/ul>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm144899\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144899&theme=oea&iframe_resize_id=ohm144899&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Some terms share common traits. Look at the following terms. Which ones seem to have traits in common?<\/p>\n<p>[latex]5x,7,{n}^{2},4,3x,9{n}^{2}[\/latex]<br \/>\nWhich of these terms are like terms?<\/p>\n<ul id=\"fs-id1627987\" data-bullet-style=\"bullet\">\n<li>The terms [latex]7[\/latex] and [latex]4[\/latex] are both constant terms.<\/li>\n<li>The terms [latex]5x[\/latex] and [latex]3x[\/latex] are both terms with [latex]x[\/latex].<\/li>\n<li>The terms [latex]{n}^{2}[\/latex] and [latex]9{n}^{2}[\/latex] both have [latex]{n}^{2}[\/latex].<\/li>\n<\/ul>\n<p>Terms are called like terms if they have the same variables and exponents. All constant terms are also like terms. So among the terms [latex]5x,7,{n}^{2},4,3x,9{n}^{2}[\/latex],<\/p>\n<ul>\n<li>[latex]7[\/latex] and [latex]4[\/latex] are like terms.<\/li>\n<li>[latex]5x[\/latex] and [latex]3x[\/latex] are like terms.<\/li>\n<li>[latex]{n}^{2}[\/latex] and [latex]9{n}^{2}[\/latex] are like terms.<\/li>\n<\/ul>\n<div class=\"textbox shaded\">\n<h3>Like Terms<\/h3>\n<p>Terms that are either constants or have the same variables with the same exponents are like terms.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Identify the like terms:<\/p>\n<ol>\n<li>[latex]{y}^{3},7{x}^{2},14,23,4{y}^{3},9x,5{x}^{2}[\/latex]<\/li>\n<li>[latex]4{x}^{2}+2x+5{x}^{2}+6x+40x+8xy[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q169480\">Show Solution<\/span><\/p>\n<div id=\"q169480\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<br \/>\n1. [latex]{y}^{3},7{x}^{2},14,23,4{y}^{3},9x,5{x}^{2}[\/latex]<br \/>\nLook at the variables and exponents. The expression contains [latex]{y}^{3},{x}^{2},x[\/latex], and constants.<br \/>\nThe terms [latex]{y}^{3}[\/latex] and [latex]4{y}^{3}[\/latex] are like terms because they both have [latex]{y}^{3}[\/latex].<br \/>\nThe terms [latex]7{x}^{2}[\/latex] and [latex]5{x}^{2}[\/latex] are like terms because they both have [latex]{x}^{2}[\/latex].<br \/>\nThe terms [latex]14[\/latex] and [latex]23[\/latex] are like terms because they are both constants.<br \/>\nThe term [latex]9x[\/latex] does not have any like terms in this list since no other terms have the variable [latex]x[\/latex] raised to the power of [latex]1[\/latex].<br \/>\n2. [latex]4{x}^{2}+2x+5{x}^{2}+6x+40x+8xy[\/latex]<br \/>\nLook at the variables and exponents. The expression contains the terms [latex]4{x}^{2},2x,5{x}^{2},6x,40x,\\text{and}8xy[\/latex]<br \/>\nThe terms [latex]4{x}^{2}[\/latex] and [latex]5{x}^{2}[\/latex] are like terms because they both have [latex]{x}^{2}[\/latex].<br \/>\nThe terms [latex]2x,6x,\\text{and}40x[\/latex] are like terms because they all have [latex]x[\/latex].<br \/>\nThe term [latex]8xy[\/latex] has no like terms in the given expression because no other terms contain the two variables [latex]xy[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146540\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146540&theme=oea&iframe_resize_id=ohm146540&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2 data-type=\"title\">Simplify Expressions by Combining Like Terms<\/h2>\n<p>We can simplify an expression by combining the like terms. What do you think [latex]3x+6x[\/latex] would simplify to? If you thought [latex]9x[\/latex], you would be right!<\/p>\n<p>We can see why this works by writing both terms as addition problems.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215839\/CNX_BMath_Figure_02_02_001_img.png\" alt=\"The image shows the expression 3 x plus 6 x. The 3 x represents x plus x plus x. The 6 x represents x plus x plus x plus x plus x plus x. The expression 3 x plus 6 x becomes x plus x plus x plus x plus x plus x plus x plus x plus x. This simplifies to a total of 9 x's or the term 9 x.\" data-media-type=\"image\/png\" \/><br \/>\nAdd the coefficients and keep the same variable. It doesn\u2019t matter what [latex]x[\/latex] is. If you have [latex]3[\/latex] of something and add [latex]6[\/latex] more of the same thing, the result is [latex]9[\/latex] of them. For example, [latex]3[\/latex] oranges plus [latex]6[\/latex] oranges is [latex]9[\/latex] oranges. We will discuss the mathematical properties behind this later.<\/p>\n<p>The expression [latex]3x+6x[\/latex] has only two terms. When an expression contains more terms, it may be helpful to rearrange the terms so that like terms are together. The Commutative Property of Addition says that we can change the order of addends without changing the sum. So we could rearrange the following expression before combining like terms.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215840\/CNX_BMath_Figure_02_02_015_img.png\" alt=\"The image shows the expression 3 x plus 4 y plus 2 x plus 6 y. The position of the middle terms, 4 y and 2 x, can be switched so that the expression becomes 3 x plus 2 x plus 4 y plus 6 y. Now the terms containing x are together and the terms containing y are together.\" data-media-type=\"image\/png\" \/><br \/>\nNow it is easier to see the like terms to be combined.<\/p>\n<div class=\"textbox shaded\">\n<h3>Combine like terms<\/h3>\n<ol id=\"eip-id1168466010921\" class=\"stepwise\" data-number-style=\"arabic\">\n<li>Identify like terms.<\/li>\n<li>Rearrange the expression so like terms are together.<\/li>\n<li>Add the coefficients of the like terms.<\/li>\n<\/ol>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify the expression: [latex]3x+7+4x+5[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q668173\">Show Solution<\/span><\/p>\n<div id=\"q668173\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168468738000\" class=\"unnumbered unstyled\" summary=\"The image shows the expression 3 x plus 7 plus 4 x plus 5. Three x and 4 x are like terms as are 7 and 5. The middle terms, 7 and 4 x, can be rearranged so that the like terms are together. The expressions becomes 3 x plus 4 x plus 7 plus 5. Now the like terms can be combined by adding the coefficients of the like terms. Three x plus 4 x is 7 x and 7 plus 5 is 12. The expression becomes 7 x plus 12.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]3x+7+4x+5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Identify the like terms.<\/td>\n<td>[latex]\\color{red}{3x}+\\color{blue}{7}+\\color{red}{4x}+\\color{blue}{5}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Rearrange the expression, so the like terms are together.<\/td>\n<td>[latex]\\color{red}{3x}+\\color{red}{4x}+\\color{blue}{7}+\\color{blue}{5}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add the coefficients of the like terms.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215843\/CNX_BMath_Figure_02_02_022_img-04.png\" alt=\".\" data-media-type=\"image\/png\" \/><\/td>\n<\/tr>\n<tr>\n<td>The original expression is simplified to&#8230;<\/td>\n<td>[latex]7x+12[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm144900\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144900&theme=oea&iframe_resize_id=ohm144900&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify the expression: [latex]8x+7{x}^{2}-{x}^{2}-+4x[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q94190\">Show Solution<\/span><\/p>\n<div id=\"q94190\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168467195542\" class=\"unnumbered unstyled\" summary=\"The image shows the expression 7 x squared plus 8 x plus x squared plus 4 x. Seven x squared and x squared are like terms as are 8 x and 4 x. The middle terms, 8 x and x squared, can be rearranged so that the like terms are together. The expressions becomes 7 x squared plus x squared plus 8 x plus 4 x. Now the like terms can be combined by adding the coefficients of the like terms. 7 x squared plus x squared is 8 x squared and 8 x plus 4 x is 12 x. The expression becomes 8 x squared plus 12 x.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]8x+7{x}^{2}-{x}^{2}-+4x[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Identify the like terms.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Rearrange the expression so like terms are together.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Add the coefficients of the like terms.<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>These are not like terms and cannot be combined. So [latex]8{x}^{2}+12x[\/latex] is in simplest form.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm144905\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144905&theme=oea&iframe_resize_id=ohm144905&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the following video, we present more examples of how to combine like terms given an algebraic expression.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Simplify Expressions by Combining Like Terms (No Negatives)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/KMUCQ_Pwt7o?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Simplify Expressions Using the Distributive Property<\/h2>\n<div class=\"textbox shaded\">\n<h3>Distributive Property<\/h3>\n<p>If [latex]a,b,c[\/latex] are real numbers, then<\/p>\n<p>[latex]a\\left(b+c\\right)=ab+ac[\/latex]<\/p>\n<\/div>\n<p>In algebra, we use the Distributive Property to remove parentheses as we simplify expressions. For example, if we are asked to simplify the expression [latex]3\\left(x+4\\right)[\/latex], the order of operations says to work in the parentheses first. But we cannot add [latex]x[\/latex] and [latex]4[\/latex], since they are not like terms. So we use the Distributive Property, as shown in the next example.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify: [latex]3\\left(x+4\\right)[\/latex]<\/p>\n<p>Solution:<\/p>\n<table id=\"eip-id1168468605528\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\n<tbody>\n<tr>\n<td>[latex]3\\left(x+4\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Distribute.<\/td>\n<td>[latex]3\\cdot x+3\\cdot 4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]3x+12[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>Some students find it helpful to draw in arrows to remind them how to use the Distributive Property. Then the first step in the previous example would look like this:<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24222343\/CNX_BMath_Figure_07_03_003_img.png\" alt=\"The image shows the expression x plus 4 in parentheses with the number 3 outside the parentheses on the left. There are two arrows pointing from the top of the three. One arrow points to the top of the x. The other arrow points to the top of the 4.\" \/><\/p>\n<p>[latex]3\\cdot x+3\\cdot 4[\/latex]<\/p>\n<p>Now you try.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146473\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146473&theme=oea&iframe_resize_id=ohm146473&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In our next example, there is a coefficient on the variable y. When you use the distributive property, you multiply the two numbers together, just like simplifying any product. You will also see another example where the expression in parentheses is subtraction, rather than addition. \u00a0You will need to be careful to change the sign of your product.<\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify: [latex]6\\left(5y+1\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q645849\">Show Solution<\/span><\/p>\n<div id=\"q645849\" class=\"hidden-answer\" style=\"display: none\">\n<p>&nbsp;<\/p>\n<p>Solution:<\/p>\n<table id=\"eip-id1168466125869\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The image shows the expression 5 y plus 1 in parentheses with the number 6 outside the parentheses on the left. Simplify by distributing the 6 through the parentheses to get the expression 6 times 5 y plus 6 times 1. Simplify further by multiplying 6 times 5 y to get 30 y and 6 times 1 to get 6. The expression simplifies to 30 y plus 6.\">\n<tbody>\n<tr>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24222344\/CNX_BMath_Figure_07_03_025_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Distribute.<\/td>\n<td>[latex]6\\cdot 5y+6\\cdot 1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]30y+6[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<p>Simplify: [latex]2\\left(x - 3\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q877652\">Show Solution<\/span><\/p>\n<div id=\"q877652\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168466277015\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The image shows the expression x minus 3 in parentheses with the number 2 outside the parentheses on the left. Simplify by distributing the 2 through the parentheses to get the expression 2 times x minus 2 times 3. Simplify further by multiplying 2 times x to get 2 x and 2 times 3 to get 6. The expression simplifies to 2 x minus 6.\">\n<tbody>\n<tr>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24222347\/CNX_BMath_Figure_07_03_026_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Distribute.<\/td>\n<td>[latex]2\\cdot x--2\\cdot 3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]2x--6[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>Now you try.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146474\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146474&theme=oea&iframe_resize_id=ohm146474&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm146475\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146475&theme=oea&iframe_resize_id=ohm146475&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>The distributive property can be used to simplify expressions that look slightly different from [latex]a\\left(b+c\\right)[\/latex]. Here are two other forms.<\/p>\n<div class=\"textbox shaded\">\n<h3>different Forms of the Distributive Property<\/h3>\n<p>If [latex]a,b,c[\/latex] are real numbers, then<\/p>\n<p>[latex]a\\left(b+c\\right)=ab+ac[\/latex]<br \/>\nOther forms<\/p>\n<p>[latex]a\\left(b-c\\right)=ab-ac[\/latex]<br \/>\n[latex]\\left(b+c\\right)a=ba+ca[\/latex]<\/p>\n<\/div>\n<p>In teh following video we show more examples of using the distributive property.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Ex 1:  The Distributive Property\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/Nt8V5cEvAz8?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n","protected":false},"author":264444,"menu_order":3,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-18182","chapter","type-chapter","status-publish","hentry"],"part":18142,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/18182","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/users\/264444"}],"version-history":[{"count":31,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/18182\/revisions"}],"predecessor-version":[{"id":18698,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/18182\/revisions\/18698"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/parts\/18142"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/18182\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/media?parent=18182"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapter-type?post=18182"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/contributor?post=18182"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/license?post=18182"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}