{"id":4622,"date":"2020-04-21T00:19:10","date_gmt":"2020-04-21T00:19:10","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/mathforliberalartscorequisite\/chapter\/an-introduction-to-the-language-of-algebra\/"},"modified":"2025-10-29T21:17:17","modified_gmt":"2025-10-29T21:17:17","slug":"an-introduction-to-the-language-of-algebra","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/mathforliberalartscorequisite\/chapter\/an-introduction-to-the-language-of-algebra\/","title":{"raw":"Introduction to The Language of Algebra","rendered":"Introduction to The Language of Algebra"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Use Variables and Algebraic Symbols\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>Use inequality symbols to compare two quantities<\/li>\r\n \t<li>Translate between words and inequality notation<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Identifying Expressions and Equations\r\n<ul>\r\n \t<li>Identify and write mathematical expressions using words and symbols<\/li>\r\n \t<li>Identify and write mathematical equations using words and symbols<\/li>\r\n \t<li>Use the order of operations to simplify mathematical expressions<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Simplifying Expressions Using the Order of Operations\r\n<ul>\r\n \t<li>Use exponential notation<\/li>\r\n \t<li>Write an exponential expression in expanded form<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>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>Greg\u2019s age<\/th>\r\n<th>Alex\u2019s age<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>[latex]12[\/latex]<\/td>\r\n<td>[latex]15[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]20[\/latex]<\/td>\r\n<td>[latex]23[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]35[\/latex]<\/td>\r\n<td>[latex]38[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]g[\/latex]<\/td>\r\n<td>[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 \">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th><strong>Operation<\/strong><\/th>\r\n<th><strong>Notation<\/strong><\/th>\r\n<th><strong>Say:<\/strong><\/th>\r\n<th><strong>The result is\u2026<\/strong><\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>Addition<\/td>\r\n<td>[latex]a+b[\/latex]<\/td>\r\n<td>[latex]a\\text{ plus }b[\/latex]<\/td>\r\n<td>the sum of [latex]a[\/latex] and [latex]b[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>Subtraction<\/td>\r\n<td>[latex]a-b[\/latex]<\/td>\r\n<td>[latex]a\\text{ minus }b[\/latex]<\/td>\r\n<td>the difference of [latex]a[\/latex] and [latex]b[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>Multiplication<\/td>\r\n<td>[latex]a\\cdot b,\\left(a\\right)\\left(b\\right),\\left(a\\right)b,a\\left(b\\right)[\/latex]<\/td>\r\n<td>[latex]a\\text{ times }b[\/latex]<\/td>\r\n<td>The product of [latex]a[\/latex] and [latex]b[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>Division<\/td>\r\n<td>[latex]a\\div b,a\/b,\\frac{a}{b},b\\overline{)a}[\/latex]<\/td>\r\n<td>[latex]a[\/latex] divided by [latex]b[\/latex]<\/td>\r\n<td>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\n\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>of<\/em> or <em>and<\/em> to help you find the numbers.\r\n<ul id=\"fs-id1969800\">\r\n \t<li>The <em>sum\u00a0<\/em><strong><em>of<\/em><\/strong> [latex]5[\/latex] <strong><em>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>difference\u00a0<\/em><strong><em>of<\/em><\/strong> [latex]9[\/latex] <strong><em>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>product\u00a0<\/em><strong><em>of<\/em><\/strong> [latex]4[\/latex] <strong><em>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>quotient\u00a0<\/em><strong><em>of<\/em><\/strong> [latex]20[\/latex] <strong><em>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>Example<\/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=\".\">\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;\">[latex]12+14[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\">[latex]12[\/latex] plus [latex]14[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\">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>[latex]\\left(30\\right)\\left(5\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]30[\/latex] times [latex]5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>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=\".\">\r\n<tbody>\r\n<tr>\r\n<td>3.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]64\\div 8[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]64[\/latex] divided by [latex]8[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>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=\".\">\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;\">[latex]x-y[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\">[latex]x[\/latex] minus [latex]y[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\">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[ohm_question hide_question_numbers=1]144651[\/ohm_question]\r\n\r\n[ohm_question hide_question_numbers=1]144652[\/ohm_question]\r\n\r\n<\/div>\r\nWhen two quantities have the same value, we say they are equal and connect them with an <em>equal sign<\/em>.\r\n<div class=\"textbox shaded\">\r\n<h3>Equality Symbol<\/h3>\r\n[latex]a=b[\/latex] is read [latex]a[\/latex] is equal to [latex]b[\/latex]\r\nThe symbol [latex]=[\/latex] is called the equal sign.\r\n\r\n<\/div>\r\nAn inequality is used in algebra to compare two quantities that may have different values. The number line can help you understand inequalities. Remember that on the number line the numbers get larger as they go from left to right. So if we know that [latex]b[\/latex] is greater than [latex]a[\/latex], it means that [latex]b[\/latex] is to the right of [latex]a[\/latex] on the number line. We use the symbols [latex]\\text{&lt;}[\/latex] and [latex]\\text{&gt;}[\/latex] for inequalities.\r\n<p style=\"text-align: center;\">[latex]a\\lt{b}[\/latex] is read [latex]a[\/latex] is less than [latex]b[\/latex]\r\n[latex]a[\/latex] is to the left of [latex]b[\/latex] on the number line<\/p>\r\n<p style=\"text-align: center;\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215726\/CNX_BMath_Figure_02_01_001.png\" alt=\"The figure shows a horizontal number line that begins with the letter a on the left then the letter b to its right.\" \/>\r\n[latex]a\\gt{b}[\/latex] is read [latex]a[\/latex] is greater than [latex]b[\/latex]\r\n[latex]a[\/latex] is to the right of [latex]b[\/latex] on the number line<\/p>\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215727\/CNX_BMath_Figure_02_01_002.png\" alt=\"The figure shows a horizontal number line that begins with the letter b on the left then the letter a to its right.\" \/>\r\n\r\nThe expressions [latex]a\\lt{b}\\text{ and }a\\gt{b}[\/latex] can be read from left-to-right or right-to-left, though in English we usually read from left-to-right. In general,\r\n<p style=\"text-align: center;\">[latex]a\\lt{b}\\text{ is equivalent to }b\\gt{a}[\/latex].<\/p>\r\n<p style=\"text-align: center;\">For example, [latex]7&lt;11\\text{ is equivalent to }11&gt;7[\/latex].<\/p>\r\n<p style=\"text-align: center;\">[latex]a\\gt{b}\\text{ is equivalent to }b\\lt{a}[\/latex].<\/p>\r\n<p style=\"text-align: center;\">.For example, [latex]17\\gt{4}\\text{ is equivalent to }4\\lt{17}[\/latex]<\/p>\r\nWhen we write an inequality symbol with a line under it, such as [latex]a\\le{b}[\/latex], it means [latex]a\\lt{b}[\/latex] or [latex]a=b[\/latex]. We read this [latex]a[\/latex] is less than or equal to [latex]b[\/latex]. Also, if we put a slash through an equal sign, [latex]\\ne[\/latex], it means not equal.\r\n\r\nWe summarize the symbols of equality and inequality in the table below.\r\n<table style=\"width: 40%;\" summary=\"This table has seven 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>Algebraic Notation<\/th>\r\n<th>Say<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>[latex]a=b[\/latex]<\/td>\r\n<td>[latex]a[\/latex] is equal to [latex]b[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]a\\ne b[\/latex]<\/td>\r\n<td>[latex]a[\/latex] is not equal to [latex]b[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]a\\lt{b}[\/latex]<\/td>\r\n<td>[latex]a[\/latex] is less than [latex]b[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]a\\gt{b}[\/latex]<\/td>\r\n<td>[latex]a[\/latex] is greater than [latex]b[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]a\\le b[\/latex]<\/td>\r\n<td>[latex]a[\/latex] is less than or equal to [latex]b[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]a\\ge b[\/latex]<\/td>\r\n<td>[latex]a[\/latex] is greater than or equal to [latex]b[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"textbox shaded\">\r\n<h3 style=\"text-align: left;\">Symbols [latex]&amp;lt[\/latex] and [latex]&amp;gt[\/latex]<\/h3>\r\n<p style=\"text-align: left;\">The symbols [latex]&amp;lt[\/latex] and [latex]&amp;gt[\/latex] each have a smaller side and a larger side.<\/p>\r\n<p style=\"text-align: center;\">smaller side [latex]&amp;lt[\/latex] larger side<\/p>\r\n<p style=\"text-align: center;\">larger side [latex]&amp;gt[\/latex] smaller side<\/p>\r\nThe smaller side of the symbol faces the smaller number and the larger faces the larger number.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nTranslate from algebra to words:\r\n<ol>\r\n \t<li>[latex]20\\le 35[\/latex]<\/li>\r\n \t<li>[latex]11\\ne 15 - 3[\/latex]<\/li>\r\n \t<li>[latex]9\\gt 10\\div 2[\/latex]<\/li>\r\n \t<li>[latex]x+2\\lt 10[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"346424\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"346424\"]\r\n\r\nSolution:\r\n<table style=\"width: 40%;\">\r\n<tbody>\r\n<tr>\r\n<td>1.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]20\\le 35[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]20[\/latex] is less than or equal to [latex]35[\/latex]<\/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>[latex]11\\ne 15 - 3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]11[\/latex] is not equal to [latex]15[\/latex] minus [latex]3[\/latex]<\/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>3.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]9&gt;10\\div 2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]9[\/latex] is greater than [latex]10[\/latex] divided by [latex]2[\/latex]<\/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>4.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]x+2&lt;10[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]x[\/latex] plus [latex]2[\/latex] is less than [latex]10[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>TRY\u00a0IT<\/h3>\r\n[ohm_question hide_question_numbers=1]144653[\/ohm_question]\r\n\r\n[ohm_question hide_question_numbers=1]144654[\/ohm_question]\r\n\r\n[ohm_question hide_question_numbers=1]144655[\/ohm_question]\r\n\r\n<\/div>\r\n<span style=\"color: #000000;\">In the following video we show more examples of how to write inequalities as words.<\/span>\r\n\r\nhttps:\/\/youtu.be\/q2ciQBwkjbk\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nThe information in the table below compares the fuel economy in miles-per-gallon (mpg) of several cars. Write the appropriate symbol [latex]\\text{=},\\text{&lt;},\\text{ or }\\text{&gt;}[\/latex] in each expression to compare the fuel economy of the cars.\r\n\r\n(credit: modification of work by Bernard Goldbach, Wikimedia Commons)\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215729\/CNX_BMath_Figure_02_01_003.png\" alt=\"This table has two rows and six columns. The first column is a header column and it labels each row The first row is labeled \" \/>\r\n<ol>\r\n \t<li>MPG of Prius_____ MPG of Mini Cooper<\/li>\r\n \t<li>MPG of Versa_____ MPG of Fit<\/li>\r\n \t<li>MPG of Mini Cooper_____ MPG of Fit<\/li>\r\n \t<li>MPG of Corolla_____ MPG of Versa<\/li>\r\n \t<li>MPG of Corolla_____ MPG of Prius<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"826680\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"826680\"]\r\n\r\nSolution\r\n<table style=\"width: 40%;\">\r\n<tbody>\r\n<tr>\r\n<td>1.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>MPG of Prius____MPG of Mini Cooper<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Find the values in the chart.<\/td>\r\n<td>48____27<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Compare.<\/td>\r\n<td>48 &gt; 27<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>MPG of Prius &gt; MPG of Mini Cooper<\/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>MPG of Versa____MPG of Fit<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Find the values in the chart.<\/td>\r\n<td>26____27<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Compare.<\/td>\r\n<td>26 &lt; 27<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>MPG of Versa &lt; MPG of Fit<\/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>3.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>MPG of Mini Cooper____MPG of Fit<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Find the values in the chart.<\/td>\r\n<td>27____27<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Compare.<\/td>\r\n<td>27 = 27<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>MPG of Mini Cooper = MPG of Fit<\/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>4.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>MPG of Corolla____MPG of Versa<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Find the values in the chart.<\/td>\r\n<td>28____26<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Compare.<\/td>\r\n<td>28 &gt; 26<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>MPG of Corolla &gt; MPG of Versa<\/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>5.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>MPG of Corolla____MPG of Prius<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Find the values in the chart.<\/td>\r\n<td>28____48<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Compare.<\/td>\r\n<td>28 &lt; 48<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>MPG of Corolla &lt; MPG of Prius<\/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\u00a0IT<\/h3>\r\n[ohm_question hide_question_numbers=1]144729[\/ohm_question]\r\n\r\n<\/div>\r\nGrouping symbols in algebra are much like the commas, colons, and other punctuation marks in written language. They indicate which expressions are to be kept together and separate from other expressions. The table below lists three of the most commonly used grouping symbols in algebra.\r\n<table style=\"width: 40%;\" summary=\"This table has four rows and two columns. The first row spans both columns and is a header reading \">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th style=\"width: 93.6508%;\" colspan=\"2\"><strong>Common Grouping Symbols<\/strong><\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td style=\"width: 61.4286%;\">parentheses<\/td>\r\n<td style=\"width: 32.2222%;\">( )<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td style=\"width: 61.4286%;\">brackets<\/td>\r\n<td style=\"width: 32.2222%;\">[ ]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td style=\"width: 61.4286%;\">braces<\/td>\r\n<td style=\"width: 32.2222%;\">{ }<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nHere are some examples of expressions that include grouping symbols. We will simplify expressions like these later in this section.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{cc}8\\left(14 - 8\\right)21 - 3\\\\\\left[2+4\\left(9 - 8\\right)\\right]\\\\24\\div \\left\\{13 - 2\\left[1\\left(6 - 5\\right)+4\\right]\\right\\}\\end{array}[\/latex]<\/p>\r\n\r\n<h2>Identify Expressions and Equations<\/h2>\r\nWhat is the difference in English between a phrase and a sentence? A phrase expresses a single thought that is incomplete by itself, but a sentence makes a complete statement. \"Running very fast\" is a phrase, but \"The football player was running very fast\" is a sentence. A sentence has a subject and a verb.\r\n\r\nIn algebra, we have <em>expressions<\/em> and <em>equations<\/em>. An expression is like a phrase. Here are some examples of expressions and how they relate to word phrases:\r\n<table id=\"fs-id2472125\" class=\"unnumbered\" summary=\"This table has five rows and three 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>Expression<\/th>\r\n<th>Words<\/th>\r\n<th>Phrase<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>[latex]3+5[\/latex]<\/td>\r\n<td>[latex]3\\text{ plus }5[\/latex]<\/td>\r\n<td>the sum of three and five<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]n - 1[\/latex]<\/td>\r\n<td>[latex]n[\/latex] minus one<\/td>\r\n<td>the difference of [latex]n[\/latex] and one<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]6\\cdot 7[\/latex]<\/td>\r\n<td>[latex]6\\text{ times }7[\/latex]<\/td>\r\n<td>the product of six and seven<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]\\frac{x}{y}[\/latex]<\/td>\r\n<td>[latex]x[\/latex] divided by [latex]y[\/latex]<\/td>\r\n<td>the quotient of [latex]x[\/latex] and [latex]y[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n\r\nNotice that the phrases do not form a complete sentence because the phrase does not have a verb. An equation is two expressions linked with an equal sign. When you read the words the symbols represent in an equation, you have a complete sentence in English. The equal sign gives the verb. Here are some examples of equations:\r\n<table id=\"fs-id2658369\" class=\"unnumbered\" summary=\"This table has six rows and two columns. The first row is a header row labeling each column. The first column is labeled \">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th>Equation<\/th>\r\n<th>Sentence<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>[latex]3+5=8[\/latex]<\/td>\r\n<td>The sum of three and five is equal to eight.<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]n - 1=14[\/latex]<\/td>\r\n<td>[latex]n[\/latex] minus one equals fourteen.<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]6\\cdot 7=42[\/latex]<\/td>\r\n<td>The product of six and seven is equal to forty-two.<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]x=53[\/latex]<\/td>\r\n<td>[latex]x[\/latex] is equal to fifty-three.<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]y+9=2y - 3[\/latex]<\/td>\r\n<td>[latex]y[\/latex] plus nine is equal to two [latex]y[\/latex] minus three.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div><\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Expressions and Equations<\/h3>\r\nAn expression is a number, a variable, or a combination of numbers and variables and operation symbols.\r\nAn equation is made up of two expressions connected by an equal sign.\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nDetermine if each is an expression or an equation:\r\n<ol>\r\n \t<li>[latex]16 - 6=10[\/latex]<\/li>\r\n \t<li>[latex]4\\cdot 2+1[\/latex]<\/li>\r\n \t<li>[latex]x\\div 25[\/latex]<\/li>\r\n \t<li>[latex]y+8=40[\/latex]<\/li>\r\n<\/ol>\r\nSolution\r\n<table id=\"eip-id1166346957031\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>1. [latex]16 - 6=10[\/latex]<\/td>\r\n<td>This is an equation\u2014two expressions are connected with an equal sign.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2. [latex]4\\cdot 2+1[\/latex]<\/td>\r\n<td>This is an expression\u2014no equal sign.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>3. [latex]x\\div 25[\/latex]<\/td>\r\n<td>This is an expression\u2014no equal sign.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>4. [latex]y+8=40[\/latex]<\/td>\r\n<td>This is an equation\u2014two expressions are connected with an equal sign.<\/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 hide_question_numbers=1]144735[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Use the order of operations to simplify mathematical expressions<\/li>\r\n \t<li>Simplify mathematical expressions involving addition, subtraction, multiplication, division, and exponents<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Simplify Expressions Containing Exponents<\/h2>\r\nTo simplify a numerical expression means to do all the math possible. For example, to simplify [latex]4\\cdot 2+1[\/latex] we\u2019d first multiply [latex]4\\cdot 2[\/latex] to get [latex]8[\/latex] and then add the [latex]1[\/latex] to get [latex]9[\/latex]. A good habit to develop is to work down the page, writing each step of the process below the previous step. The example just described would look like this:\r\n<p style=\"text-align: center;\">[latex]4\\cdot 2+1[\/latex]\r\n[latex]8+1[\/latex]\r\n[latex]9[\/latex]<\/p>\r\n<p style=\"text-align: left;\">Suppose we have the expression [latex]2\\cdot 2\\cdot 2\\cdot 2\\cdot 2\\cdot 2\\cdot 2\\cdot 2\\cdot 2[\/latex]. We could write this more compactly using exponential notation. Exponential notation is used in algebra to represent a quantity multiplied by itself several times. We write [latex]2\\cdot 2\\cdot 2[\/latex] as [latex]{2}^{3}[\/latex] and [latex]2\\cdot 2\\cdot 2\\cdot 2\\cdot 2\\cdot 2\\cdot 2\\cdot 2\\cdot 2[\/latex] as [latex]{2}^{9}[\/latex]. In expressions such as [latex]{2}^{3}[\/latex], the [latex]2[\/latex] is called the base and the [latex]3[\/latex] is called the exponent. The exponent tells us how many factors of the base we have to multiply.<\/p>\r\n<p style=\"text-align: center;\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215731\/CNX_BMath_Figure_02_01_003_img.png\" alt=\"The image shows the number two with the number three, in superscript, to the right of the two. The number two is labeled as \" \/>\r\n[latex]\\text{means multiply three factors of 2}[\/latex]\r\nWe say [latex]{2}^{3}[\/latex] is in exponential notation and [latex]2\\cdot 2\\cdot 2[\/latex] is in expanded notation.<\/p>\r\n\r\n<div class=\"textbox shaded\">\r\n<h3>Exponential Notation<\/h3>\r\nFor any expression [latex]{a}^{n},a[\/latex] is a factor multiplied by itself [latex]n[\/latex] times if [latex]n[\/latex] is a positive integer.\r\n\r\n[latex]{a}^{n}\\text{ means multiply }n\\text{ factors of }a[\/latex]\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215732\/CNX_BMath_Figure_02_01_010_img.png\" alt=\"At the top of the image is the letter a with the letter n, in superscript, to the right of the a. The letter a is labeled as \" \/>\r\nThe expression [latex]{a}^{n}[\/latex] is read [latex]a[\/latex] to the [latex]{n}^{th}[\/latex] power.\r\n\r\n<\/div>\r\nFor powers of [latex]n=2[\/latex] and [latex]n=3[\/latex], we have special names.\r\n<p style=\"text-align: center;\">[latex]a^2[\/latex] is read as \"[latex]a[\/latex] squared\"<\/p>\r\n<p style=\"text-align: center;\">[latex]a^3[\/latex] is read as \"[latex]a[\/latex] cubed\"<\/p>\r\n&nbsp;\r\n\r\nThe table below lists some examples of expressions written in exponential notation.\r\n<table id=\"fs-id1830286\" 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>Exponential Notation<\/th>\r\n<th>In Words<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>[latex]{7}^{2}[\/latex]<\/td>\r\n<td>[latex]7[\/latex] to the second power, or [latex]7[\/latex] squared<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]{5}^{3}[\/latex]<\/td>\r\n<td>[latex]5[\/latex] to the third power, or [latex]5[\/latex] cubed<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]{9}^{4}[\/latex]<\/td>\r\n<td>[latex]9[\/latex] to the fourth power<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]{12}^{5}[\/latex]<\/td>\r\n<td>[latex]12[\/latex] to the fifth power<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nWrite each expression in exponential form:\r\n<ol>\r\n \t<li>[latex]16\\cdot 16\\cdot 16\\cdot 16\\cdot 16\\cdot 16\\cdot 16[\/latex]<\/li>\r\n \t<li>[latex]\\text{9}\\cdot \\text{9}\\cdot \\text{9}\\cdot \\text{9}\\cdot \\text{9}[\/latex]<\/li>\r\n \t<li>[latex]x\\cdot x\\cdot x\\cdot x[\/latex]<\/li>\r\n \t<li>[latex]a\\cdot a\\cdot a\\cdot a\\cdot a\\cdot a\\cdot a\\cdot a[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"95827\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"95827\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168469627334\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>1. The base [latex]16[\/latex] is a factor [latex]7[\/latex] times.<\/td>\r\n<td>[latex]{16}^{7}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2. The base [latex]9[\/latex] is a factor [latex]5[\/latex] times.<\/td>\r\n<td>[latex]{9}^{5}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>3. The base [latex]x[\/latex] is a factor [latex]4[\/latex] times.<\/td>\r\n<td>[latex]{x}^{4}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>4. The base [latex]a[\/latex] is a factor [latex]8[\/latex] times.<\/td>\r\n<td>[latex]{a}^{8}[\/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 hide_question_numbers=1]144737[\/ohm_question]\r\n\r\n<\/div>\r\nIn the video below we show more examples of how to write an expression of repeated multiplication in exponential form.\r\n\r\nhttps:\/\/youtu.be\/HkPGTmAmg_s\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nWrite each exponential expression in expanded form:\r\n<ol>\r\n \t<li>[latex]{8}^{6}[\/latex]<\/li>\r\n \t<li>[latex]{x}^{5}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"20595\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"20595\"]\r\n\r\nSolution\r\n1. The base is [latex]8[\/latex] and the exponent is [latex]6[\/latex], so [latex]{8}^{6}[\/latex] means [latex]8\\cdot 8\\cdot 8\\cdot 8\\cdot 8\\cdot 8[\/latex]\r\n2. The base is [latex]x[\/latex] and the exponent is [latex]5[\/latex], so [latex]{x}^{5}[\/latex] means [latex]x\\cdot x\\cdot x\\cdot x\\cdot x[\/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 hide_question_numbers=1]144744[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nTo simplify an exponential expression without using a calculator, we write it in expanded form and then multiply the factors.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]{3}^{4}[\/latex]\r\n\r\n[reveal-answer q=\"534998\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"534998\"]\r\n\r\nSolution\r\n<table id=\"eip-id1164752752096\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]{3}^{4}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Expand the expression.<\/td>\r\n<td>[latex]3\\cdot 3\\cdot 3\\cdot 3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply left to right.<\/td>\r\n<td>[latex]9\\cdot 3\\cdot 3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]27\\cdot 3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]81[\/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 hide_question_numbers=1]144745[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Simplify Expressions Using the Order of Operations<\/h2>\r\nWe\u2019ve introduced most of the symbols and notation used in algebra, but now we need to clarify the order of operations. Otherwise, expressions may have different meanings, and they may result in different values.\r\n\r\nFor example, consider the expression:\r\n<p style=\"text-align: center;\">[latex]4+3\\cdot 7[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{cccc}\\hfill \\text{Some students say it simplifies to 49.}\\hfill &amp; &amp; &amp; \\hfill \\text{Some students say it simplifies to 25.}\\hfill \\\\ \\begin{array}{ccc}&amp; &amp; \\hfill 4+3\\cdot 7\\hfill \\\\ \\text{Since }4+3\\text{ gives 7.}\\hfill &amp; &amp; \\hfill 7\\cdot 7\\hfill \\\\ \\text{And }7\\cdot 7\\text{ is 49.}\\hfill &amp; &amp; \\hfill 49\\hfill \\end{array}&amp; &amp; &amp; \\begin{array}{ccc}&amp; &amp; \\hfill 4+3\\cdot 7\\hfill \\\\ \\text{Since }3\\cdot 7\\text{ is 21.}\\hfill &amp; &amp; \\hfill 4+21\\hfill \\\\ \\text{And }21+4\\text{ makes 25.}\\hfill &amp; &amp; \\hfill 25\\hfill \\end{array}\\hfill \\end{array}[\/latex]<\/p>\r\nImagine the confusion that could result if every problem had several different correct answers. The same expression should give the same result. So mathematicians established some guidelines called the order of operations, which outlines the order in which parts of an expression must be simplified.\r\n<div class=\"textbox shaded\">\r\n<h3>Order of Operations<\/h3>\r\nWhen simplifying mathematical expressions perform the operations in the following order:\r\n1. <strong>P<\/strong>arentheses and other Grouping Symbols\r\n<ul id=\"fs-id1171104029952\">\r\n \t<li>Simplify all expressions inside the parentheses or other grouping symbols, working on the innermost parentheses first.<\/li>\r\n<\/ul>\r\n2. <strong>E<\/strong>xponents\r\n<ul id=\"fs-id1171104407077\">\r\n \t<li>Simplify all expressions with exponents.<\/li>\r\n<\/ul>\r\n3. <strong>M<\/strong>ultiplication and <strong>D<\/strong>ivision\r\n<ul id=\"fs-id1171103140103\">\r\n \t<li>Perform all multiplication and division in order from left to right. These operations have equal priority.<\/li>\r\n<\/ul>\r\n4. <strong>A<\/strong>ddition and <strong>S<\/strong>ubtraction\r\n<ul id=\"fs-id1171104002792\">\r\n \t<li>Perform all addition and subtraction in order from left to right. These operations have equal priority.<\/li>\r\n<\/ul>\r\n<\/div>\r\n&nbsp;\r\n\r\nStudents often ask, \"How will I remember the order?\" Here is a way to help you remember: Take the first letter of each key word and substitute the silly phrase. <strong>P<\/strong>lease <strong>E<\/strong>xcuse <strong>M<\/strong>y <strong>D<\/strong>ear <strong>A<\/strong>unt <strong>S<\/strong>ally.\r\n<table id=\"fs-id1786633\" class=\"unnumbered\" summary=\"This table has five rows and two columns. The first row spans both columns and is a header reading \">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th colspan=\"2\"><strong>Order of Operations<\/strong><\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td><strong>P<\/strong>lease<\/td>\r\n<td><strong>P<\/strong>arentheses<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td><strong>E<\/strong>xcuse<\/td>\r\n<td><strong>E<\/strong>xponents<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td><strong>M<\/strong>y <strong>D<\/strong>ear<\/td>\r\n<td><strong>M<\/strong>ultiplication and <strong>D<\/strong>ivision<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td><strong>A<\/strong>unt <strong>S<\/strong>ally<\/td>\r\n<td><strong>A<\/strong>ddition and <strong>S<\/strong>ubtraction<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nIt\u2019s good that \u2018<strong>M<\/strong>y <strong>D<\/strong>ear\u2019 goes together, as this reminds us that <strong>m<\/strong>ultiplication and <strong>d<\/strong>ivision have equal priority. We do not always do multiplication before division or always do division before multiplication. We do them in order from left to right.\r\nSimilarly, \u2018<strong>A<\/strong>unt <strong>S<\/strong>ally\u2019 goes together and so reminds us that <strong>a<\/strong>ddition and <strong>s<\/strong>ubtraction also have equal priority and we do them in order from left to right.\r\n\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify the expressions:\r\n<ol>\r\n \t<li>[latex]4+3\\cdot 7[\/latex]<\/li>\r\n \t<li>[latex]\\left(4+3\\right)\\cdot 7[\/latex]<\/li>\r\n<\/ol>\r\nSolution:\r\n<table id=\"eip-id1164750479370\" class=\"unnumbered unstyled\" summary=\"The table shows the expression four plus three times seven. On the next line it states are there any parentheses in the expression? No. The line below that states are there any exponents in the expression? No. The next line states is there any multiplication or division in the expression? Yes. The next line states Multiply first and is followed by the expression of four plus three times seven. The expression is now four plus twenty-one. The last operation is addition. Add four and twenty-one to get twenty-five.\">\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]4+3\\cdot 7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Are there any <strong>p<\/strong>arentheses? No.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Are there any <strong>e<\/strong>xponents? No.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Is there any <strong>m<\/strong>ultiplication or <strong>d<\/strong>ivision? Yes.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply first.<\/td>\r\n<td>[latex]4+\\color{red}{3\\cdot 7}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add.<\/td>\r\n<td>[latex]4+21[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]25[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1164754514704\" class=\"unnumbered unstyled\" summary=\"The image shows the expression four plus three in parentheses times seven. Are there any parentheses? Yes, simplify inside the parentheses by adding four and three to get seven. The expression is now seven times seven. Is there any multiplication or division? Yes, multiply seven by seven to get forty-nine.\">\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](4+3)\\cdot 7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Are there any <strong>p<\/strong>arentheses? Yes.<\/td>\r\n<td>[latex]\\color{red}{(4+3)}\\cdot 7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify inside the parentheses.<\/td>\r\n<td>[latex](7)7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Are there any <strong>e<\/strong>xponents? No.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Is there any <strong>m<\/strong>ultiplication or <strong>d<\/strong>ivision? Yes.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]49[\/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]144748[\/ohm_question]\r\n\r\n[ohm_question]144751[\/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<ol>\r\n \t<li>[latex]\\text{18}\\div \\text{9}\\cdot \\text{2}[\/latex]<\/li>\r\n \t<li>[latex]\\text{18}\\cdot \\text{9}\\div \\text{2}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"604459\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"604459\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1164754213884\" class=\"unnumbered unstyled\" summary=\"The table shows the expression eighteen divided by nine times two. Are there any parentheses? No. Are there any exponents? No. Is there any multiplication or division? Yes, there is both multiplication and division. Perform multiplication and division from left to right so, divide first. Divide eighteen by nine to get two. The expression is now two times two. The last operation is multiplication. Multiply two by two to get four.\">\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]18\\div 9\\cdot 2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Are there any <strong>p<\/strong>arentheses? No.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Are there any <strong>e<\/strong>xponents? No.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Is there any <strong>m<\/strong>ultiplication or <strong>d<\/strong>ivision? Yes.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply and divide from left to right. Divide.<\/td>\r\n<td>[latex]\\color{red}{2}\\cdot 2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1164752720001\" class=\"unnumbered unstyled\" summary=\"The table shows the expression eighteen times nine divided by two. Are there any parentheses? No. Are there any exponents? No. Is there any multiplication or division? Yes, there is both multiplication and division. Perform multiplication and division from left to right so, multiply first. Multiply eighteen by nine to get one hundred sixty-two. The expression is now one hundred sixty-two divided by two. The last operation is division. divide one hundred sixty-two by two to get eighty-one.\">\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]18\\cdot 9\\div 2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Are there any <strong>p<\/strong>arentheses? No.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Are there any <strong>e<\/strong>xponents? No.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Is there any <strong>m<\/strong>ultiplication or <strong>d<\/strong>ivision? Yes.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply and divide from left to right.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]\\color{red}{162}\\div 2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Divide.<\/td>\r\n<td>[latex]81[\/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]144756[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]18\\div 6+4\\left(5 - 2\\right)[\/latex].\r\n[reveal-answer q=\"841846\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"841846\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1164754213917\" class=\"unnumbered unstyled\" summary=\"The image shows the expression eighteen divided by six plus four, in parentheses, five minus two. Are there any parentheses? Yes, perform the subtraction inside the parentheses. Five minus two becomes three inside the parentheses. The expression is now eighteen divided by six plus four, in parentheses, three. Are there any exponents? No. Is there any multiplication or division? Yes, there is both multiplication and division. Divide first because multiplication and division are performed left to right. Divide eighteen by six to get three. The expression is now three plus four, in parentheses, three. Now multiply four by the three in parentheses to get twelve. The expression becomes three plus twelve. Add three and twelve to get fifteen.\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]18\\div 6+4(5-2)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Parentheses? Yes, subtract first.<\/td>\r\n<td>[latex]18\\div 6+4(\\color{red}{3})[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Exponents? No.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiplication or division? Yes.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Divide first because we multiply and divide left to right.<\/td>\r\n<td>[latex]\\color{red}{3}+4(3)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Any other multiplication or division? Yes.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]3+\\color{red}{12}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Any other multiplication or division? No.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Any addition or subtraction? Yes.<\/td>\r\n<td>[latex]15[\/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]144758[\/ohm_question]\r\n\r\n<\/div>\r\nIn the video below we show another example of how to use the order of operations to simplify a mathematical expression.\r\n\r\nhttps:\/\/youtu.be\/qFUvF5-w9o0\r\n\r\n&nbsp;\r\n\r\nWhen there are multiple grouping symbols, we simplify the innermost parentheses first and work outward.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\n[latex]\\text{Simplify: }5+{2}^{3}+3\\left[6 - 3\\left(4 - 2\\right)\\right][\/latex].\r\n[reveal-answer q=\"221697\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"221697\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1164754129434\" class=\"unnumbered unstyled\" summary=\"The image shows the expression five plus two cubed plus three, open bracket, six minus three, open parentheses, four minus two close parentheses, close bracket. Are there any parentheses (or other grouping symbols)? Yes, there are parentheses within brackets. Start with the innermost grouping, the parentheses. Inside the parentheses subtract two from four to get two. The expression becomes five plus two cubed plus three, open bracket, six minus three, in parentheses, two, close bracket. Within the brackets is six minus three, in parentheses, two. Multiply three by two since multiplication comes before subtraction. The expression becomes five plus two cubed plus three, in brackets, six minus six. Finish simplifying inside the brackets by performing the subtraction. Six minus six leaves zero. The expression is now five plus two cubed plus three, in brackets, zero. Are there exponents? Yes, two cubed is eight and the expression becomes five plus eight plus three, in brackets, zero. Is there any multiplication or division? Yes, multiply three by zero to get zero. The expression is now five plus eight plus zero. The remaining operations are both addition. Perform the addition from left to right. five plus eight is thirteen. Thirteen plus zero is thirteen.\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]5+{2}^{3}+ 3[6-3(4-2)][\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Are there any parentheses (or other grouping symbol)? Yes.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Focus on the parentheses that are inside the brackets.<\/td>\r\n<td>[latex]5+{2}^{3}+ 3[6-3\\color{red}{(4-2)}][\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract.<\/td>\r\n<td>[latex]5+{2}^{3}+3[6-\\color{red}{3(2)}][\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Continue inside the brackets and multiply.<\/td>\r\n<td>[latex]5+{2}^{3}+3[6-\\color{red}{6}][\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Continue inside the brackets and subtract.<\/td>\r\n<td>[latex]5+{2}^{3}+3[\\color{red}{0}][\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The expression inside the brackets requires no further simplification.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Are there any exponents? Yes.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify exponents.<\/td>\r\n<td>[latex]5+\\color{red}{{2}^{3}}+3[0][\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Is there any multiplication or division? Yes.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]5+8+\\color{red}{3[0]}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Is there any addition or subtraction? Yes.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add.<\/td>\r\n<td>[latex]\\color{red}{5+8}+0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add.<\/td>\r\n<td>[latex]\\color{red}{13+0}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]13[\/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]144759[\/ohm_question]\r\n\r\n<\/div>\r\nIn the video below we show another example of how to use the order of operations to simplify an expression that contains exponents and grouping symbols.\r\n\r\nhttps:\/\/youtu.be\/8b-rf2AW3Ac\r\n\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]{2}^{3}+{3}^{4}\\div 3-{5}^{2}[\/latex].\r\n[reveal-answer q=\"199030\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"199030\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1164754324162\" class=\"unnumbered unstyled\" summary=\"The image shows the expression two cubed plus three to the fourth divided by three minus five squared. Are there any parentheses? No. Are there any exponents? Yes, several. Simplify each exponent. Two cubed is eight, three to the fourth is eighty-one, and five squared is twenty-five. The expression becomes eight plus eighty-one divided by three minus twenty-five. Is there any multiplication or division? Yes, just division. Divide eighty-one by three to get twenty-seven. The expression is now eight plus twenty-seven minus five. There is both addition and subtraction left. Perform these from left to right. Eight plus twenty-seven is thirty-five. Now the expression is thirty-five minus twenty five which leaves ten.\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]{2}^{3}+{3}^{4}\\div 3-{5}^{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>If an expression has several exponents, they may be simplified in the same step.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify exponents.<\/td>\r\n<td>[latex]\\color{red}{{2}^{3}}+\\color{red}{{3}^{4}}\\div 3-\\color{red}{{5}^{2}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Divide.<\/td>\r\n<td>[latex]8+\\color{red}{81\\div 3}-25[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add.<\/td>\r\n<td>[latex]\\color{red}{8+27}-25[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract.<\/td>\r\n<td>[latex]\\color{red}{35-25}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]10[\/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]144762[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Use Variables and Algebraic Symbols\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>Use inequality symbols to compare two quantities<\/li>\n<li>Translate between words and inequality notation<\/li>\n<\/ul>\n<\/li>\n<li>Identifying Expressions and Equations\n<ul>\n<li>Identify and write mathematical expressions using words and symbols<\/li>\n<li>Identify and write mathematical equations using words and symbols<\/li>\n<li>Use the order of operations to simplify mathematical expressions<\/li>\n<\/ul>\n<\/li>\n<li>Simplifying Expressions Using the Order of Operations\n<ul>\n<li>Use exponential notation<\/li>\n<li>Write an exponential expression in expanded form<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<h2>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>Greg\u2019s age<\/th>\n<th>Alex\u2019s age<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>[latex]12[\/latex]<\/td>\n<td>[latex]15[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]20[\/latex]<\/td>\n<td>[latex]23[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]35[\/latex]<\/td>\n<td>[latex]38[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]g[\/latex]<\/td>\n<td>[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\">\n<thead>\n<tr valign=\"top\">\n<th><strong>Operation<\/strong><\/th>\n<th><strong>Notation<\/strong><\/th>\n<th><strong>Say:<\/strong><\/th>\n<th><strong>The result is\u2026<\/strong><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>Addition<\/td>\n<td>[latex]a+b[\/latex]<\/td>\n<td>[latex]a\\text{ plus }b[\/latex]<\/td>\n<td>the sum of [latex]a[\/latex] and [latex]b[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Subtraction<\/td>\n<td>[latex]a-b[\/latex]<\/td>\n<td>[latex]a\\text{ minus }b[\/latex]<\/td>\n<td>the difference of [latex]a[\/latex] and [latex]b[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Multiplication<\/td>\n<td>[latex]a\\cdot b,\\left(a\\right)\\left(b\\right),\\left(a\\right)b,a\\left(b\\right)[\/latex]<\/td>\n<td>[latex]a\\text{ times }b[\/latex]<\/td>\n<td>The product of [latex]a[\/latex] and [latex]b[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Division<\/td>\n<td>[latex]a\\div b,a\/b,\\frac{a}{b},b\\overline{)a}[\/latex]<\/td>\n<td>[latex]a[\/latex] divided by [latex]b[\/latex]<\/td>\n<td>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.<\/p>\n<p>We 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>of<\/em> or <em>and<\/em> to help you find the numbers.<\/p>\n<ul id=\"fs-id1969800\">\n<li>The <em>sum\u00a0<\/em><strong><em>of<\/em><\/strong> [latex]5[\/latex] <strong><em>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>difference\u00a0<\/em><strong><em>of<\/em><\/strong> [latex]9[\/latex] <strong><em>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>product\u00a0<\/em><strong><em>of<\/em><\/strong> [latex]4[\/latex] <strong><em>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>quotient\u00a0<\/em><strong><em>of<\/em><\/strong> [latex]20[\/latex] <strong><em>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>Example<\/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=\".\">\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;\">[latex]12+14[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\">[latex]12[\/latex] plus [latex]14[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\">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>[latex]\\left(30\\right)\\left(5\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]30[\/latex] times [latex]5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>the product of thirty and five<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table class=\"unnumbered unstyled\" style=\"width: 40%;\" summary=\".\">\n<tbody>\n<tr>\n<td>3.<\/td>\n<\/tr>\n<tr>\n<td>[latex]64\\div 8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]64[\/latex] divided by [latex]8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>the quotient of sixty-four and eight<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table class=\"unnumbered unstyled\" style=\"width: 40%;\" summary=\".\">\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;\">[latex]x-y[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\">[latex]x[\/latex] minus [latex]y[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\">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=\"ohm144651\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144651&theme=oea&iframe_resize_id=ohm144651\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm144652\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144652&theme=oea&iframe_resize_id=ohm144652\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>When two quantities have the same value, we say they are equal and connect them with an <em>equal sign<\/em>.<\/p>\n<div class=\"textbox shaded\">\n<h3>Equality Symbol<\/h3>\n<p>[latex]a=b[\/latex] is read [latex]a[\/latex] is equal to [latex]b[\/latex]<br \/>\nThe symbol [latex]=[\/latex] is called the equal sign.<\/p>\n<\/div>\n<p>An inequality is used in algebra to compare two quantities that may have different values. The number line can help you understand inequalities. Remember that on the number line the numbers get larger as they go from left to right. So if we know that [latex]b[\/latex] is greater than [latex]a[\/latex], it means that [latex]b[\/latex] is to the right of [latex]a[\/latex] on the number line. We use the symbols [latex]\\text{<}[\/latex] and [latex]\\text{>}[\/latex] for inequalities.<\/p>\n<p style=\"text-align: center;\">[latex]a\\lt{b}[\/latex] is read [latex]a[\/latex] is less than [latex]b[\/latex]<br \/>\n[latex]a[\/latex] is to the left of [latex]b[\/latex] on the number line<\/p>\n<p style=\"text-align: center;\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215726\/CNX_BMath_Figure_02_01_001.png\" alt=\"The figure shows a horizontal number line that begins with the letter a on the left then the letter b to its right.\" \/><br \/>\n[latex]a\\gt{b}[\/latex] is read [latex]a[\/latex] is greater than [latex]b[\/latex]<br \/>\n[latex]a[\/latex] is to the right of [latex]b[\/latex] on the number line<\/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\/24215727\/CNX_BMath_Figure_02_01_002.png\" alt=\"The figure shows a horizontal number line that begins with the letter b on the left then the letter a to its right.\" \/><\/p>\n<p>The expressions [latex]a\\lt{b}\\text{ and }a\\gt{b}[\/latex] can be read from left-to-right or right-to-left, though in English we usually read from left-to-right. In general,<\/p>\n<p style=\"text-align: center;\">[latex]a\\lt{b}\\text{ is equivalent to }b\\gt{a}[\/latex].<\/p>\n<p style=\"text-align: center;\">For example, [latex]7<11\\text{ is equivalent to }11>7[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]a\\gt{b}\\text{ is equivalent to }b\\lt{a}[\/latex].<\/p>\n<p style=\"text-align: center;\">.For example, [latex]17\\gt{4}\\text{ is equivalent to }4\\lt{17}[\/latex]<\/p>\n<p>When we write an inequality symbol with a line under it, such as [latex]a\\le{b}[\/latex], it means [latex]a\\lt{b}[\/latex] or [latex]a=b[\/latex]. We read this [latex]a[\/latex] is less than or equal to [latex]b[\/latex]. Also, if we put a slash through an equal sign, [latex]\\ne[\/latex], it means not equal.<\/p>\n<p>We summarize the symbols of equality and inequality in the table below.<\/p>\n<table style=\"width: 40%;\" summary=\"This table has seven 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>Algebraic Notation<\/th>\n<th>Say<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>[latex]a=b[\/latex]<\/td>\n<td>[latex]a[\/latex] is equal to [latex]b[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]a\\ne b[\/latex]<\/td>\n<td>[latex]a[\/latex] is not equal to [latex]b[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]a\\lt{b}[\/latex]<\/td>\n<td>[latex]a[\/latex] is less than [latex]b[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]a\\gt{b}[\/latex]<\/td>\n<td>[latex]a[\/latex] is greater than [latex]b[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]a\\le b[\/latex]<\/td>\n<td>[latex]a[\/latex] is less than or equal to [latex]b[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]a\\ge b[\/latex]<\/td>\n<td>[latex]a[\/latex] is greater than or equal to [latex]b[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"textbox shaded\">\n<h3 style=\"text-align: left;\">Symbols [latex]&lt[\/latex] and [latex]&gt[\/latex]<\/h3>\n<p style=\"text-align: left;\">The symbols [latex]&lt[\/latex] and [latex]&gt[\/latex] each have a smaller side and a larger side.<\/p>\n<p style=\"text-align: center;\">smaller side [latex]&lt[\/latex] larger side<\/p>\n<p style=\"text-align: center;\">larger side [latex]&gt[\/latex] smaller side<\/p>\n<p>The smaller side of the symbol faces the smaller number and the larger faces the larger number.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Translate from algebra to words:<\/p>\n<ol>\n<li>[latex]20\\le 35[\/latex]<\/li>\n<li>[latex]11\\ne 15 - 3[\/latex]<\/li>\n<li>[latex]9\\gt 10\\div 2[\/latex]<\/li>\n<li>[latex]x+2\\lt 10[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q346424\">Show Solution<\/span><\/p>\n<div id=\"q346424\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table style=\"width: 40%;\">\n<tbody>\n<tr>\n<td>1.<\/td>\n<\/tr>\n<tr>\n<td>[latex]20\\le 35[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]20[\/latex] is less than or equal to [latex]35[\/latex]<\/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>[latex]11\\ne 15 - 3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]11[\/latex] is not equal to [latex]15[\/latex] minus [latex]3[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table style=\"width: 40%;\">\n<tbody>\n<tr>\n<td>3.<\/td>\n<\/tr>\n<tr>\n<td>[latex]9>10\\div 2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]9[\/latex] is greater than [latex]10[\/latex] divided by [latex]2[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table style=\"width: 40%;\">\n<tbody>\n<tr>\n<td>4.<\/td>\n<\/tr>\n<tr>\n<td>[latex]x+2<10[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]x[\/latex] plus [latex]2[\/latex] is less than [latex]10[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>TRY\u00a0IT<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm144653\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144653&theme=oea&iframe_resize_id=ohm144653\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm144654\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144654&theme=oea&iframe_resize_id=ohm144654\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm144655\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144655&theme=oea&iframe_resize_id=ohm144655\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p><span style=\"color: #000000;\">In the following video we show more examples of how to write inequalities as words.<\/span><\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Write Inequalities as Words\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/q2ciQBwkjbk?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>The information in the table below compares the fuel economy in miles-per-gallon (mpg) of several cars. Write the appropriate symbol [latex]\\text{=},\\text{<},\\text{ or }\\text{>}[\/latex] in each expression to compare the fuel economy of the cars.<\/p>\n<p>(credit: modification of work by Bernard Goldbach, Wikimedia Commons)<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215729\/CNX_BMath_Figure_02_01_003.png\" alt=\"This table has two rows and six columns. The first column is a header column and it labels each row The first row is labeled\" \/><\/p>\n<ol>\n<li>MPG of Prius_____ MPG of Mini Cooper<\/li>\n<li>MPG of Versa_____ MPG of Fit<\/li>\n<li>MPG of Mini Cooper_____ MPG of Fit<\/li>\n<li>MPG of Corolla_____ MPG of Versa<\/li>\n<li>MPG of Corolla_____ MPG of Prius<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q826680\">Show Solution<\/span><\/p>\n<div id=\"q826680\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table style=\"width: 40%;\">\n<tbody>\n<tr>\n<td>1.<\/td>\n<\/tr>\n<tr>\n<td>MPG of Prius____MPG of Mini Cooper<\/td>\n<\/tr>\n<tr>\n<td>Find the values in the chart.<\/td>\n<td>48____27<\/td>\n<\/tr>\n<tr>\n<td>Compare.<\/td>\n<td>48 &gt; 27<\/td>\n<\/tr>\n<tr>\n<td>MPG of Prius &gt; MPG of Mini Cooper<\/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>MPG of Versa____MPG of Fit<\/td>\n<\/tr>\n<tr>\n<td>Find the values in the chart.<\/td>\n<td>26____27<\/td>\n<\/tr>\n<tr>\n<td>Compare.<\/td>\n<td>26 &lt; 27<\/td>\n<\/tr>\n<tr>\n<td>MPG of Versa &lt; MPG of Fit<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table style=\"width: 40%;\">\n<tbody>\n<tr>\n<td>3.<\/td>\n<\/tr>\n<tr>\n<td>MPG of Mini Cooper____MPG of Fit<\/td>\n<\/tr>\n<tr>\n<td>Find the values in the chart.<\/td>\n<td>27____27<\/td>\n<\/tr>\n<tr>\n<td>Compare.<\/td>\n<td>27 = 27<\/td>\n<\/tr>\n<tr>\n<td>MPG of Mini Cooper = MPG of Fit<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table style=\"width: 40%;\">\n<tbody>\n<tr>\n<td>4.<\/td>\n<\/tr>\n<tr>\n<td>MPG of Corolla____MPG of Versa<\/td>\n<\/tr>\n<tr>\n<td>Find the values in the chart.<\/td>\n<td>28____26<\/td>\n<\/tr>\n<tr>\n<td>Compare.<\/td>\n<td>28 &gt; 26<\/td>\n<\/tr>\n<tr>\n<td>MPG of Corolla &gt; MPG of Versa<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table style=\"width: 40%;\">\n<tbody>\n<tr>\n<td>5.<\/td>\n<\/tr>\n<tr>\n<td>MPG of Corolla____MPG of Prius<\/td>\n<\/tr>\n<tr>\n<td>Find the values in the chart.<\/td>\n<td>28____48<\/td>\n<\/tr>\n<tr>\n<td>Compare.<\/td>\n<td>28 &lt; 48<\/td>\n<\/tr>\n<tr>\n<td>MPG of Corolla &lt; MPG of Prius<\/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\u00a0IT<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm144729\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144729&theme=oea&iframe_resize_id=ohm144729\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Grouping symbols in algebra are much like the commas, colons, and other punctuation marks in written language. They indicate which expressions are to be kept together and separate from other expressions. The table below lists three of the most commonly used grouping symbols in algebra.<\/p>\n<table style=\"width: 40%;\" summary=\"This table has four rows and two columns. The first row spans both columns and is a header reading\">\n<thead>\n<tr valign=\"top\">\n<th style=\"width: 93.6508%;\" colspan=\"2\"><strong>Common Grouping Symbols<\/strong><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td style=\"width: 61.4286%;\">parentheses<\/td>\n<td style=\"width: 32.2222%;\">( )<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td style=\"width: 61.4286%;\">brackets<\/td>\n<td style=\"width: 32.2222%;\">[ ]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td style=\"width: 61.4286%;\">braces<\/td>\n<td style=\"width: 32.2222%;\">{ }<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Here are some examples of expressions that include grouping symbols. We will simplify expressions like these later in this section.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{cc}8\\left(14 - 8\\right)21 - 3\\\\\\left[2+4\\left(9 - 8\\right)\\right]\\\\24\\div \\left\\{13 - 2\\left[1\\left(6 - 5\\right)+4\\right]\\right\\}\\end{array}[\/latex]<\/p>\n<h2>Identify Expressions and Equations<\/h2>\n<p>What is the difference in English between a phrase and a sentence? A phrase expresses a single thought that is incomplete by itself, but a sentence makes a complete statement. &#8220;Running very fast&#8221; is a phrase, but &#8220;The football player was running very fast&#8221; is a sentence. A sentence has a subject and a verb.<\/p>\n<p>In algebra, we have <em>expressions<\/em> and <em>equations<\/em>. An expression is like a phrase. Here are some examples of expressions and how they relate to word phrases:<\/p>\n<table id=\"fs-id2472125\" class=\"unnumbered\" summary=\"This table has five rows and three 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>Expression<\/th>\n<th>Words<\/th>\n<th>Phrase<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>[latex]3+5[\/latex]<\/td>\n<td>[latex]3\\text{ plus }5[\/latex]<\/td>\n<td>the sum of three and five<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]n - 1[\/latex]<\/td>\n<td>[latex]n[\/latex] minus one<\/td>\n<td>the difference of [latex]n[\/latex] and one<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]6\\cdot 7[\/latex]<\/td>\n<td>[latex]6\\text{ times }7[\/latex]<\/td>\n<td>the product of six and seven<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]\\frac{x}{y}[\/latex]<\/td>\n<td>[latex]x[\/latex] divided by [latex]y[\/latex]<\/td>\n<td>the quotient of [latex]x[\/latex] and [latex]y[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p>Notice that the phrases do not form a complete sentence because the phrase does not have a verb. An equation is two expressions linked with an equal sign. When you read the words the symbols represent in an equation, you have a complete sentence in English. The equal sign gives the verb. Here are some examples of equations:<\/p>\n<table id=\"fs-id2658369\" class=\"unnumbered\" summary=\"This table has six rows and two columns. The first row is a header row labeling each column. The first column is labeled\">\n<thead>\n<tr valign=\"top\">\n<th>Equation<\/th>\n<th>Sentence<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>[latex]3+5=8[\/latex]<\/td>\n<td>The sum of three and five is equal to eight.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]n - 1=14[\/latex]<\/td>\n<td>[latex]n[\/latex] minus one equals fourteen.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]6\\cdot 7=42[\/latex]<\/td>\n<td>The product of six and seven is equal to forty-two.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]x=53[\/latex]<\/td>\n<td>[latex]x[\/latex] is equal to fifty-three.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]y+9=2y - 3[\/latex]<\/td>\n<td>[latex]y[\/latex] plus nine is equal to two [latex]y[\/latex] minus three.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div><\/div>\n<div class=\"textbox shaded\">\n<h3>Expressions and Equations<\/h3>\n<p>An expression is a number, a variable, or a combination of numbers and variables and operation symbols.<br \/>\nAn equation is made up of two expressions connected by an equal sign.<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Determine if each is an expression or an equation:<\/p>\n<ol>\n<li>[latex]16 - 6=10[\/latex]<\/li>\n<li>[latex]4\\cdot 2+1[\/latex]<\/li>\n<li>[latex]x\\div 25[\/latex]<\/li>\n<li>[latex]y+8=40[\/latex]<\/li>\n<\/ol>\n<p>Solution<\/p>\n<table id=\"eip-id1166346957031\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\".\">\n<tbody>\n<tr>\n<td>1. [latex]16 - 6=10[\/latex]<\/td>\n<td>This is an equation\u2014two expressions are connected with an equal sign.<\/td>\n<\/tr>\n<tr>\n<td>2. [latex]4\\cdot 2+1[\/latex]<\/td>\n<td>This is an expression\u2014no equal sign.<\/td>\n<\/tr>\n<tr>\n<td>3. [latex]x\\div 25[\/latex]<\/td>\n<td>This is an expression\u2014no equal sign.<\/td>\n<\/tr>\n<tr>\n<td>4. [latex]y+8=40[\/latex]<\/td>\n<td>This is an equation\u2014two expressions are connected with an equal sign.<\/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=\"ohm144735\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144735&theme=oea&iframe_resize_id=ohm144735\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Use the order of operations to simplify mathematical expressions<\/li>\n<li>Simplify mathematical expressions involving addition, subtraction, multiplication, division, and exponents<\/li>\n<\/ul>\n<\/div>\n<h2>Simplify Expressions Containing Exponents<\/h2>\n<p>To simplify a numerical expression means to do all the math possible. For example, to simplify [latex]4\\cdot 2+1[\/latex] we\u2019d first multiply [latex]4\\cdot 2[\/latex] to get [latex]8[\/latex] and then add the [latex]1[\/latex] to get [latex]9[\/latex]. A good habit to develop is to work down the page, writing each step of the process below the previous step. The example just described would look like this:<\/p>\n<p style=\"text-align: center;\">[latex]4\\cdot 2+1[\/latex]<br \/>\n[latex]8+1[\/latex]<br \/>\n[latex]9[\/latex]<\/p>\n<p style=\"text-align: left;\">Suppose we have the expression [latex]2\\cdot 2\\cdot 2\\cdot 2\\cdot 2\\cdot 2\\cdot 2\\cdot 2\\cdot 2[\/latex]. We could write this more compactly using exponential notation. Exponential notation is used in algebra to represent a quantity multiplied by itself several times. We write [latex]2\\cdot 2\\cdot 2[\/latex] as [latex]{2}^{3}[\/latex] and [latex]2\\cdot 2\\cdot 2\\cdot 2\\cdot 2\\cdot 2\\cdot 2\\cdot 2\\cdot 2[\/latex] as [latex]{2}^{9}[\/latex]. In expressions such as [latex]{2}^{3}[\/latex], the [latex]2[\/latex] is called the base and the [latex]3[\/latex] is called the exponent. The exponent tells us how many factors of the base we have to multiply.<\/p>\n<p style=\"text-align: center;\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215731\/CNX_BMath_Figure_02_01_003_img.png\" alt=\"The image shows the number two with the number three, in superscript, to the right of the two. The number two is labeled as\" \/><br \/>\n[latex]\\text{means multiply three factors of 2}[\/latex]<br \/>\nWe say [latex]{2}^{3}[\/latex] is in exponential notation and [latex]2\\cdot 2\\cdot 2[\/latex] is in expanded notation.<\/p>\n<div class=\"textbox shaded\">\n<h3>Exponential Notation<\/h3>\n<p>For any expression [latex]{a}^{n},a[\/latex] is a factor multiplied by itself [latex]n[\/latex] times if [latex]n[\/latex] is a positive integer.<\/p>\n<p>[latex]{a}^{n}\\text{ means multiply }n\\text{ factors of }a[\/latex]<\/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\/24215732\/CNX_BMath_Figure_02_01_010_img.png\" alt=\"At the top of the image is the letter a with the letter n, in superscript, to the right of the a. The letter a is labeled as\" \/><br \/>\nThe expression [latex]{a}^{n}[\/latex] is read [latex]a[\/latex] to the [latex]{n}^{th}[\/latex] power.<\/p>\n<\/div>\n<p>For powers of [latex]n=2[\/latex] and [latex]n=3[\/latex], we have special names.<\/p>\n<p style=\"text-align: center;\">[latex]a^2[\/latex] is read as &#8220;[latex]a[\/latex] squared&#8221;<\/p>\n<p style=\"text-align: center;\">[latex]a^3[\/latex] is read as &#8220;[latex]a[\/latex] cubed&#8221;<\/p>\n<p>&nbsp;<\/p>\n<p>The table below lists some examples of expressions written in exponential notation.<\/p>\n<table id=\"fs-id1830286\" 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>Exponential Notation<\/th>\n<th>In Words<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>[latex]{7}^{2}[\/latex]<\/td>\n<td>[latex]7[\/latex] to the second power, or [latex]7[\/latex] squared<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]{5}^{3}[\/latex]<\/td>\n<td>[latex]5[\/latex] to the third power, or [latex]5[\/latex] cubed<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]{9}^{4}[\/latex]<\/td>\n<td>[latex]9[\/latex] to the fourth power<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]{12}^{5}[\/latex]<\/td>\n<td>[latex]12[\/latex] to the fifth power<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Write each expression in exponential form:<\/p>\n<ol>\n<li>[latex]16\\cdot 16\\cdot 16\\cdot 16\\cdot 16\\cdot 16\\cdot 16[\/latex]<\/li>\n<li>[latex]\\text{9}\\cdot \\text{9}\\cdot \\text{9}\\cdot \\text{9}\\cdot \\text{9}[\/latex]<\/li>\n<li>[latex]x\\cdot x\\cdot x\\cdot x[\/latex]<\/li>\n<li>[latex]a\\cdot a\\cdot a\\cdot a\\cdot a\\cdot a\\cdot a\\cdot a[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q95827\">Show Solution<\/span><\/p>\n<div id=\"q95827\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168469627334\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\".\">\n<tbody>\n<tr>\n<td>1. The base [latex]16[\/latex] is a factor [latex]7[\/latex] times.<\/td>\n<td>[latex]{16}^{7}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>2. The base [latex]9[\/latex] is a factor [latex]5[\/latex] times.<\/td>\n<td>[latex]{9}^{5}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>3. The base [latex]x[\/latex] is a factor [latex]4[\/latex] times.<\/td>\n<td>[latex]{x}^{4}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>4. The base [latex]a[\/latex] is a factor [latex]8[\/latex] times.<\/td>\n<td>[latex]{a}^{8}[\/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=\"ohm144737\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144737&theme=oea&iframe_resize_id=ohm144737\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the video below we show more examples of how to write an expression of repeated multiplication in exponential form.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Example:  Write Repeated Multiplication in Exponential Form\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/HkPGTmAmg_s?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Write each exponential expression in expanded form:<\/p>\n<ol>\n<li>[latex]{8}^{6}[\/latex]<\/li>\n<li>[latex]{x}^{5}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q20595\">Show Solution<\/span><\/p>\n<div id=\"q20595\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<br \/>\n1. The base is [latex]8[\/latex] and the exponent is [latex]6[\/latex], so [latex]{8}^{6}[\/latex] means [latex]8\\cdot 8\\cdot 8\\cdot 8\\cdot 8\\cdot 8[\/latex]<br \/>\n2. The base is [latex]x[\/latex] and the exponent is [latex]5[\/latex], so [latex]{x}^{5}[\/latex] means [latex]x\\cdot x\\cdot x\\cdot x\\cdot x[\/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=\"ohm144744\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144744&theme=oea&iframe_resize_id=ohm144744\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>To simplify an exponential expression without using a calculator, we write it in expanded form and then multiply the factors.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify: [latex]{3}^{4}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q534998\">Show Solution<\/span><\/p>\n<div id=\"q534998\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1164752752096\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]{3}^{4}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Expand the expression.<\/td>\n<td>[latex]3\\cdot 3\\cdot 3\\cdot 3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply left to right.<\/td>\n<td>[latex]9\\cdot 3\\cdot 3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]27\\cdot 3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]81[\/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=\"ohm144745\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144745&theme=oea&iframe_resize_id=ohm144745\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Simplify Expressions Using the Order of Operations<\/h2>\n<p>We\u2019ve introduced most of the symbols and notation used in algebra, but now we need to clarify the order of operations. Otherwise, expressions may have different meanings, and they may result in different values.<\/p>\n<p>For example, consider the expression:<\/p>\n<p style=\"text-align: center;\">[latex]4+3\\cdot 7[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{cccc}\\hfill \\text{Some students say it simplifies to 49.}\\hfill & & & \\hfill \\text{Some students say it simplifies to 25.}\\hfill \\\\ \\begin{array}{ccc}& & \\hfill 4+3\\cdot 7\\hfill \\\\ \\text{Since }4+3\\text{ gives 7.}\\hfill & & \\hfill 7\\cdot 7\\hfill \\\\ \\text{And }7\\cdot 7\\text{ is 49.}\\hfill & & \\hfill 49\\hfill \\end{array}& & & \\begin{array}{ccc}& & \\hfill 4+3\\cdot 7\\hfill \\\\ \\text{Since }3\\cdot 7\\text{ is 21.}\\hfill & & \\hfill 4+21\\hfill \\\\ \\text{And }21+4\\text{ makes 25.}\\hfill & & \\hfill 25\\hfill \\end{array}\\hfill \\end{array}[\/latex]<\/p>\n<p>Imagine the confusion that could result if every problem had several different correct answers. The same expression should give the same result. So mathematicians established some guidelines called the order of operations, which outlines the order in which parts of an expression must be simplified.<\/p>\n<div class=\"textbox shaded\">\n<h3>Order of Operations<\/h3>\n<p>When simplifying mathematical expressions perform the operations in the following order:<br \/>\n1. <strong>P<\/strong>arentheses and other Grouping Symbols<\/p>\n<ul id=\"fs-id1171104029952\">\n<li>Simplify all expressions inside the parentheses or other grouping symbols, working on the innermost parentheses first.<\/li>\n<\/ul>\n<p>2. <strong>E<\/strong>xponents<\/p>\n<ul id=\"fs-id1171104407077\">\n<li>Simplify all expressions with exponents.<\/li>\n<\/ul>\n<p>3. <strong>M<\/strong>ultiplication and <strong>D<\/strong>ivision<\/p>\n<ul id=\"fs-id1171103140103\">\n<li>Perform all multiplication and division in order from left to right. These operations have equal priority.<\/li>\n<\/ul>\n<p>4. <strong>A<\/strong>ddition and <strong>S<\/strong>ubtraction<\/p>\n<ul id=\"fs-id1171104002792\">\n<li>Perform all addition and subtraction in order from left to right. These operations have equal priority.<\/li>\n<\/ul>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Students often ask, &#8220;How will I remember the order?&#8221; Here is a way to help you remember: Take the first letter of each key word and substitute the silly phrase. <strong>P<\/strong>lease <strong>E<\/strong>xcuse <strong>M<\/strong>y <strong>D<\/strong>ear <strong>A<\/strong>unt <strong>S<\/strong>ally.<\/p>\n<table id=\"fs-id1786633\" class=\"unnumbered\" summary=\"This table has five rows and two columns. The first row spans both columns and is a header reading\">\n<thead>\n<tr valign=\"top\">\n<th colspan=\"2\"><strong>Order of Operations<\/strong><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td><strong>P<\/strong>lease<\/td>\n<td><strong>P<\/strong>arentheses<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td><strong>E<\/strong>xcuse<\/td>\n<td><strong>E<\/strong>xponents<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td><strong>M<\/strong>y <strong>D<\/strong>ear<\/td>\n<td><strong>M<\/strong>ultiplication and <strong>D<\/strong>ivision<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td><strong>A<\/strong>unt <strong>S<\/strong>ally<\/td>\n<td><strong>A<\/strong>ddition and <strong>S<\/strong>ubtraction<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>It\u2019s good that \u2018<strong>M<\/strong>y <strong>D<\/strong>ear\u2019 goes together, as this reminds us that <strong>m<\/strong>ultiplication and <strong>d<\/strong>ivision have equal priority. We do not always do multiplication before division or always do division before multiplication. We do them in order from left to right.<br \/>\nSimilarly, \u2018<strong>A<\/strong>unt <strong>S<\/strong>ally\u2019 goes together and so reminds us that <strong>a<\/strong>ddition and <strong>s<\/strong>ubtraction also have equal priority and we do them in order from left to right.<\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify the expressions:<\/p>\n<ol>\n<li>[latex]4+3\\cdot 7[\/latex]<\/li>\n<li>[latex]\\left(4+3\\right)\\cdot 7[\/latex]<\/li>\n<\/ol>\n<p>Solution:<\/p>\n<table id=\"eip-id1164750479370\" class=\"unnumbered unstyled\" summary=\"The table shows the expression four plus three times seven. On the next line it states are there any parentheses in the expression? No. The line below that states are there any exponents in the expression? No. The next line states is there any multiplication or division in the expression? Yes. The next line states Multiply first and is followed by the expression of four plus three times seven. The expression is now four plus twenty-one. The last operation is addition. Add four and twenty-one to get twenty-five.\">\n<tbody>\n<tr>\n<td>1.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]4+3\\cdot 7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Are there any <strong>p<\/strong>arentheses? No.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Are there any <strong>e<\/strong>xponents? No.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Is there any <strong>m<\/strong>ultiplication or <strong>d<\/strong>ivision? Yes.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Multiply first.<\/td>\n<td>[latex]4+\\color{red}{3\\cdot 7}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add.<\/td>\n<td>[latex]4+21[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]25[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1164754514704\" class=\"unnumbered unstyled\" summary=\"The image shows the expression four plus three in parentheses times seven. Are there any parentheses? Yes, simplify inside the parentheses by adding four and three to get seven. The expression is now seven times seven. Is there any multiplication or division? Yes, multiply seven by seven to get forty-nine.\">\n<tbody>\n<tr>\n<td>2.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex](4+3)\\cdot 7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Are there any <strong>p<\/strong>arentheses? Yes.<\/td>\n<td>[latex]\\color{red}{(4+3)}\\cdot 7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify inside the parentheses.<\/td>\n<td>[latex](7)7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Are there any <strong>e<\/strong>xponents? No.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Is there any <strong>m<\/strong>ultiplication or <strong>d<\/strong>ivision? Yes.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]49[\/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=\"ohm144748\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144748&theme=oea&iframe_resize_id=ohm144748&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm144751\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144751&theme=oea&iframe_resize_id=ohm144751&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<ol>\n<li>[latex]\\text{18}\\div \\text{9}\\cdot \\text{2}[\/latex]<\/li>\n<li>[latex]\\text{18}\\cdot \\text{9}\\div \\text{2}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q604459\">Show Solution<\/span><\/p>\n<div id=\"q604459\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1164754213884\" class=\"unnumbered unstyled\" summary=\"The table shows the expression eighteen divided by nine times two. Are there any parentheses? No. Are there any exponents? No. Is there any multiplication or division? Yes, there is both multiplication and division. Perform multiplication and division from left to right so, divide first. Divide eighteen by nine to get two. The expression is now two times two. The last operation is multiplication. Multiply two by two to get four.\">\n<tbody>\n<tr>\n<td>1.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]18\\div 9\\cdot 2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Are there any <strong>p<\/strong>arentheses? No.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Are there any <strong>e<\/strong>xponents? No.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Is there any <strong>m<\/strong>ultiplication or <strong>d<\/strong>ivision? Yes.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Multiply and divide from left to right. Divide.<\/td>\n<td>[latex]\\color{red}{2}\\cdot 2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]4[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1164752720001\" class=\"unnumbered unstyled\" summary=\"The table shows the expression eighteen times nine divided by two. Are there any parentheses? No. Are there any exponents? No. Is there any multiplication or division? Yes, there is both multiplication and division. Perform multiplication and division from left to right so, multiply first. Multiply eighteen by nine to get one hundred sixty-two. The expression is now one hundred sixty-two divided by two. The last operation is division. divide one hundred sixty-two by two to get eighty-one.\">\n<tbody>\n<tr>\n<td>2.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]18\\cdot 9\\div 2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Are there any <strong>p<\/strong>arentheses? No.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Are there any <strong>e<\/strong>xponents? No.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Is there any <strong>m<\/strong>ultiplication or <strong>d<\/strong>ivision? Yes.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Multiply and divide from left to right.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]\\color{red}{162}\\div 2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Divide.<\/td>\n<td>[latex]81[\/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=\"ohm144756\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144756&theme=oea&iframe_resize_id=ohm144756&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: [latex]18\\div 6+4\\left(5 - 2\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q841846\">Show Solution<\/span><\/p>\n<div id=\"q841846\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1164754213917\" class=\"unnumbered unstyled\" summary=\"The image shows the expression eighteen divided by six plus four, in parentheses, five minus two. Are there any parentheses? Yes, perform the subtraction inside the parentheses. Five minus two becomes three inside the parentheses. The expression is now eighteen divided by six plus four, in parentheses, three. Are there any exponents? No. Is there any multiplication or division? Yes, there is both multiplication and division. Divide first because multiplication and division are performed left to right. Divide eighteen by six to get three. The expression is now three plus four, in parentheses, three. Now multiply four by the three in parentheses to get twelve. The expression becomes three plus twelve. Add three and twelve to get fifteen.\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]18\\div 6+4(5-2)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Parentheses? Yes, subtract first.<\/td>\n<td>[latex]18\\div 6+4(\\color{red}{3})[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Exponents? No.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Multiplication or division? Yes.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Divide first because we multiply and divide left to right.<\/td>\n<td>[latex]\\color{red}{3}+4(3)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Any other multiplication or division? Yes.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]3+\\color{red}{12}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Any other multiplication or division? No.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Any addition or subtraction? Yes.<\/td>\n<td>[latex]15[\/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=\"ohm144758\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144758&theme=oea&iframe_resize_id=ohm144758&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the video below we show another example of how to use the order of operations to simplify a mathematical expression.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Ex: Evaluate an Expression Using the Order of Operations\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/qFUvF5-w9o0?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>&nbsp;<\/p>\n<p>When there are multiple grouping symbols, we simplify the innermost parentheses first and work outward.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>[latex]\\text{Simplify: }5+{2}^{3}+3\\left[6 - 3\\left(4 - 2\\right)\\right][\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q221697\">Show Solution<\/span><\/p>\n<div id=\"q221697\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1164754129434\" class=\"unnumbered unstyled\" summary=\"The image shows the expression five plus two cubed plus three, open bracket, six minus three, open parentheses, four minus two close parentheses, close bracket. Are there any parentheses (or other grouping symbols)? Yes, there are parentheses within brackets. Start with the innermost grouping, the parentheses. Inside the parentheses subtract two from four to get two. The expression becomes five plus two cubed plus three, open bracket, six minus three, in parentheses, two, close bracket. Within the brackets is six minus three, in parentheses, two. Multiply three by two since multiplication comes before subtraction. The expression becomes five plus two cubed plus three, in brackets, six minus six. Finish simplifying inside the brackets by performing the subtraction. Six minus six leaves zero. The expression is now five plus two cubed plus three, in brackets, zero. Are there exponents? Yes, two cubed is eight and the expression becomes five plus eight plus three, in brackets, zero. Is there any multiplication or division? Yes, multiply three by zero to get zero. The expression is now five plus eight plus zero. The remaining operations are both addition. Perform the addition from left to right. five plus eight is thirteen. Thirteen plus zero is thirteen.\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]5+{2}^{3}+ 3[6-3(4-2)][\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Are there any parentheses (or other grouping symbol)? Yes.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Focus on the parentheses that are inside the brackets.<\/td>\n<td>[latex]5+{2}^{3}+ 3[6-3\\color{red}{(4-2)}][\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtract.<\/td>\n<td>[latex]5+{2}^{3}+3[6-\\color{red}{3(2)}][\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Continue inside the brackets and multiply.<\/td>\n<td>[latex]5+{2}^{3}+3[6-\\color{red}{6}][\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Continue inside the brackets and subtract.<\/td>\n<td>[latex]5+{2}^{3}+3[\\color{red}{0}][\/latex]<\/td>\n<\/tr>\n<tr>\n<td>The expression inside the brackets requires no further simplification.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Are there any exponents? Yes.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Simplify exponents.<\/td>\n<td>[latex]5+\\color{red}{{2}^{3}}+3[0][\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Is there any multiplication or division? Yes.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]5+8+\\color{red}{3[0]}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Is there any addition or subtraction? Yes.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Add.<\/td>\n<td>[latex]\\color{red}{5+8}+0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add.<\/td>\n<td>[latex]\\color{red}{13+0}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]13[\/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=\"ohm144759\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144759&theme=oea&iframe_resize_id=ohm144759&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the video below we show another example of how to use the order of operations to simplify an expression that contains exponents and grouping symbols.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-4\" title=\"Example 3:  Evaluate An Expression Using The Order of Operation\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/8b-rf2AW3Ac?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify: [latex]{2}^{3}+{3}^{4}\\div 3-{5}^{2}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q199030\">Show Solution<\/span><\/p>\n<div id=\"q199030\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1164754324162\" class=\"unnumbered unstyled\" summary=\"The image shows the expression two cubed plus three to the fourth divided by three minus five squared. Are there any parentheses? No. Are there any exponents? Yes, several. Simplify each exponent. Two cubed is eight, three to the fourth is eighty-one, and five squared is twenty-five. The expression becomes eight plus eighty-one divided by three minus twenty-five. Is there any multiplication or division? Yes, just division. Divide eighty-one by three to get twenty-seven. The expression is now eight plus twenty-seven minus five. There is both addition and subtraction left. Perform these from left to right. Eight plus twenty-seven is thirty-five. Now the expression is thirty-five minus twenty five which leaves ten.\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]{2}^{3}+{3}^{4}\\div 3-{5}^{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>If an expression has several exponents, they may be simplified in the same step.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Simplify exponents.<\/td>\n<td>[latex]\\color{red}{{2}^{3}}+\\color{red}{{3}^{4}}\\div 3-\\color{red}{{5}^{2}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Divide.<\/td>\n<td>[latex]8+\\color{red}{81\\div 3}-25[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add.<\/td>\n<td>[latex]\\color{red}{8+27}-25[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtract.<\/td>\n<td>[latex]\\color{red}{35-25}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]10[\/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=\"ohm144762\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144762&theme=oea&iframe_resize_id=ohm144762&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-4622\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Write Inequalities as Words. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/q2ciQBwkjbk\">https:\/\/youtu.be\/q2ciQBwkjbk<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Question ID: 144651, 144652, 144653, 144654, 144655, 144729, 144735, 144737, 144744, 144745, 144748, 144751, 144756, 144758, 144759, 144762. <strong>Authored by<\/strong>: Alyson Day. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Ex: Evaluate an Expression Using the Order of Operations. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/qFUvF5-w9o0\">https:\/\/youtu.be\/qFUvF5-w9o0<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Example 3: Evaluate An Expression Using The Order of Operation. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/8b-rf2AW3Ac\">https:\/\/youtu.be\/8b-rf2AW3Ac<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Prealgebra. <strong>Provided by<\/strong>: OpenStax. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":17533,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Prealgebra\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757\"},{\"type\":\"original\",\"description\":\"Write Inequalities as Words\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/q2ciQBwkjbk\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Question ID: 144651, 144652, 144653, 144654, 144655, 144729, 144735, 144737, 144744, 144745, 144748, 144751, 144756, 144758, 144759, 144762\",\"author\":\"Alyson Day\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"},{\"type\":\"cc\",\"description\":\"Ex: Evaluate an Expression Using the Order of Operations\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/qFUvF5-w9o0\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Example 3: Evaluate An Expression Using The Order of Operation\",\"author\":\"James Sousa 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