{"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":"2024-04-29T18:44:19","modified_gmt":"2024-04-29T18:44:19","slug":"identifying-expressions-and-equations","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/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 symbols and words<\/li>\r\n \t<li>Identify and write mathematical equations using symbols and words<\/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\nYou have simplified many expressions so far using the four main mathematical operations. 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:\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;\">However, there are other mathematical notations used to simplify the numbers we are working with. 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 \u201cbase\u201d and the number three is labeled as \u201cexponent\u201d.\" width=\"220\" height=\"26\" 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 \u201cbase\u201d and the letter n is labeled as \u201cexponent\u201d. Below this is the letter a with the letter n, in superscript, to the right of the a set equal to n factors of a.\" width=\"224\" height=\"104\" 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>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Identify and write mathematical expressions using symbols and words<\/li>\n<li>Identify and write mathematical equations using symbols and words<\/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>You have simplified many expressions so far using the four main mathematical operations. 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;\">However, there are other mathematical notations used to simplify the numbers we are working with. 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 loading=\"lazy\" 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 \u201cbase\u201d and the number three is labeled as \u201cexponent\u201d.\" width=\"220\" height=\"26\" 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 loading=\"lazy\" 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 \u201cbase\u201d and the letter n is labeled as \u201cexponent\u201d. Below this is the letter a with the letter n, in superscript, to the right of the a set equal to n factors of a.\" width=\"224\" height=\"104\" 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\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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