{"id":9205,"date":"2017-05-02T16:40:23","date_gmt":"2017-05-02T16:40:23","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/prealgebra\/?post_type=chapter&#038;p=9205"},"modified":"2017-08-09T19:59:33","modified_gmt":"2017-08-09T19:59:33","slug":"identifying-expressions-and-equations","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/csn-prealgebra\/chapter\/identifying-expressions-and-equations\/","title":{"raw":"Identifying Expressions and Equations","rendered":"Identifying Expressions and Equations"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\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>Identify the difference between an expression and an equation<\/li>\r\n \t<li>Use exponential notation to express repeated multiplication<\/li>\r\n \t<li>Write an exponential expression in expanded form<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2 data-type=\"title\">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 data-effect=\"italics\">expressions<\/em> and <em data-effect=\"italics\">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 \" data-label=\"\">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th data-align=\"left\">Expression<\/th>\r\n<th data-align=\"left\">Words<\/th>\r\n<th data-align=\"left\">Phrase<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]3+5[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]3\\text{ plus }5[\/latex]<\/td>\r\n<td data-align=\"left\">the sum of three and five<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]n - 1[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]n[\/latex] minus one<\/td>\r\n<td data-align=\"left\">the difference of [latex]n[\/latex] and one<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]6\\cdot 7[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]6\\text{ times }7[\/latex]<\/td>\r\n<td data-align=\"left\">the product of six and seven<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]\\frac{x}{y}[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]x[\/latex] divided by [latex]y[\/latex]<\/td>\r\n<td data-align=\"left\">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 \" data-label=\"\">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th data-align=\"left\">Equation<\/th>\r\n<th data-align=\"left\">Sentence<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]3+5=8[\/latex]<\/td>\r\n<td data-align=\"left\">The sum of three and five is equal to eight.<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]n - 1=14[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]n[\/latex] minus one equals fourteen.<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]6\\cdot 7=42[\/latex]<\/td>\r\n<td data-align=\"left\">The product of six and seven is equal to forty-two.<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]x=53[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]x[\/latex] is equal to fifty-three.<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]y+9=2y - 3[\/latex]<\/td>\r\n<td data-align=\"left\">[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=\".\" data-label=\"\">\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<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144735&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"280\"><\/iframe>\r\n\r\n<\/div>\r\n&nbsp;\r\n<h2 data-type=\"title\">Simplify Expressions with 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 \" data-media-type=\"image\/png\" \/>\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 \" data-media-type=\"image\/png\" \/>\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 data-align=\"left\">Exponential Notation<\/th>\r\n<th data-align=\"center\">In Words<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]{7}^{2}[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]7[\/latex] to the second power, or [latex]7[\/latex] squared<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]{5}^{3}[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]5[\/latex] to the third power, or [latex]5[\/latex] cubed<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]{9}^{4}[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]9[\/latex] to the fourth power<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]{12}^{5}[\/latex]<\/td>\r\n<td data-align=\"left\">[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=\".\" data-label=\"\">\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<iframe id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144737&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"280\"><\/iframe>\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<iframe id=\"mom3\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144744&amp;theme=oea&amp;iframe_resize_id=mom3\" width=\"100%\" height=\"280\"><\/iframe>\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=\".\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td data-align=\"center\">[latex]{3}^{4}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Expand the expression.<\/td>\r\n<td data-align=\"center\">[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 data-align=\"center\">[latex]9\\cdot 3\\cdot 3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td data-align=\"center\">[latex]27\\cdot 3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td data-align=\"center\">[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<iframe id=\"mom4\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144745&amp;theme=oea&amp;iframe_resize_id=mom4\" width=\"100%\" height=\"280\"><\/iframe>\r\n\r\n<\/div>\r\n&nbsp;","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\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>Identify the difference between an expression and an equation<\/li>\n<li>Use exponential notation to express repeated multiplication<\/li>\n<li>Write an exponential expression in expanded form<\/li>\n<\/ul>\n<\/div>\n<h2 data-type=\"title\">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 data-effect=\"italics\">expressions<\/em> and <em data-effect=\"italics\">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\" data-label=\"\">\n<thead>\n<tr valign=\"top\">\n<th data-align=\"left\">Expression<\/th>\n<th data-align=\"left\">Words<\/th>\n<th data-align=\"left\">Phrase<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]3+5[\/latex]<\/td>\n<td data-align=\"left\">[latex]3\\text{ plus }5[\/latex]<\/td>\n<td data-align=\"left\">the sum of three and five<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]n - 1[\/latex]<\/td>\n<td data-align=\"left\">[latex]n[\/latex] minus one<\/td>\n<td data-align=\"left\">the difference of [latex]n[\/latex] and one<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]6\\cdot 7[\/latex]<\/td>\n<td data-align=\"left\">[latex]6\\text{ times }7[\/latex]<\/td>\n<td data-align=\"left\">the product of six and seven<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]\\frac{x}{y}[\/latex]<\/td>\n<td data-align=\"left\">[latex]x[\/latex] divided by [latex]y[\/latex]<\/td>\n<td data-align=\"left\">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\" data-label=\"\">\n<thead>\n<tr valign=\"top\">\n<th data-align=\"left\">Equation<\/th>\n<th data-align=\"left\">Sentence<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]3+5=8[\/latex]<\/td>\n<td data-align=\"left\">The sum of three and five is equal to eight.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]n - 1=14[\/latex]<\/td>\n<td data-align=\"left\">[latex]n[\/latex] minus one equals fourteen.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]6\\cdot 7=42[\/latex]<\/td>\n<td data-align=\"left\">The product of six and seven is equal to forty-two.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]x=53[\/latex]<\/td>\n<td data-align=\"left\">[latex]x[\/latex] is equal to fifty-three.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]y+9=2y - 3[\/latex]<\/td>\n<td data-align=\"left\">[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=\".\" data-label=\"\">\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=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144735&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"280\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<h2 data-type=\"title\">Simplify Expressions with 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\" data-media-type=\"image\/png\" \/><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\" data-media-type=\"image\/png\" \/><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 data-align=\"left\">Exponential Notation<\/th>\n<th data-align=\"center\">In Words<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]{7}^{2}[\/latex]<\/td>\n<td data-align=\"left\">[latex]7[\/latex] to the second power, or [latex]7[\/latex] squared<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]{5}^{3}[\/latex]<\/td>\n<td data-align=\"left\">[latex]5[\/latex] to the third power, or [latex]5[\/latex] cubed<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]{9}^{4}[\/latex]<\/td>\n<td data-align=\"left\">[latex]9[\/latex] to the fourth power<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]{12}^{5}[\/latex]<\/td>\n<td data-align=\"left\">[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=\".\" data-label=\"\">\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=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144737&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"280\"><\/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-1\" 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=\"mom3\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144744&amp;theme=oea&amp;iframe_resize_id=mom3\" width=\"100%\" height=\"280\"><\/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=\".\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-align=\"center\">[latex]{3}^{4}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Expand the expression.<\/td>\n<td data-align=\"center\">[latex]3\\cdot 3\\cdot 3\\cdot 3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply left to right.<\/td>\n<td data-align=\"center\">[latex]9\\cdot 3\\cdot 3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-align=\"center\">[latex]27\\cdot 3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td data-align=\"center\">[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=\"mom4\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144745&amp;theme=oea&amp;iframe_resize_id=mom4\" width=\"100%\" height=\"280\"><\/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-9205\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Example: Write Repeated Multiplication in Exponential Form. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/HkPGTmAmg_s\">https:\/\/youtu.be\/HkPGTmAmg_s<\/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>Question ID: 144735, 144737, 144744, 144745. <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><\/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":4,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Example: Write Repeated Multiplication in Exponential Form\",\"author\":\"James Sousa 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