{"id":5123,"date":"2023-06-26T18:39:45","date_gmt":"2023-06-26T18:39:45","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=5123"},"modified":"2024-10-18T20:53:29","modified_gmt":"2024-10-18T20:53:29","slug":"cryptography-background-youll-need-1","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/cryptography-background-youll-need-1\/","title":{"raw":"Cryptography: Background You\u2019ll Need 1","rendered":"Cryptography: Background You\u2019ll Need 1"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Substitute a number in for a variable and simplify<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Evaluate Algebraic Expressions<\/h2>\r\n<p>Any variable in an algebraic expression may take on or be assigned different values. When that happens, the value of the algebraic expression changes. To evaluate an algebraic expression means to determine the value of the expression given the value of each variable in the expression.<\/p>\r\n<section class=\"textbox questionHelp\">\r\n<p><strong>How to: Evaluate Algebraic Expressions<\/strong><\/p>\r\n<ol>\r\n\t<li><strong>Identify Variables and Values<\/strong>: Look at the problem to determine which variables are present and what values they have been assigned.<\/li>\r\n\t<li><strong>Replace Variables with Values<\/strong>: Substitute each variable in the expression with the given value. Be sure to maintain any coefficients or operators attached to the variables.<\/li>\r\n\t<li><strong>Follow the Order of Operations<\/strong>: After substitution, simplify the expression by following the order of operations, also known by the acronym PEMDAS:\r\n\r\n<ul>\r\n\t<li><strong>P<\/strong>arentheses: Start by simplifying expressions within parentheses or other grouping symbols.<\/li>\r\n\t<li><strong>E<\/strong>xponents: Next, calculate the powers or exponents.<\/li>\r\n\t<li><strong>M<\/strong>ultiplication and <strong>D<\/strong>ivision: Then perform all multiplication and division from left to right.<\/li>\r\n\t<li><strong>A<\/strong>ddition and <strong>S<\/strong>ubtraction: Finally, complete any addition and subtraction from left to right.<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li><strong>Combine Like Terms<\/strong>: If the expression has like terms (terms with the same variable and exponent), combine them by adding or subtracting the coefficients.<\/li>\r\n<\/ol>\r\n<\/section>\r\n<p>In the next example we show how to substitute various types of numbers into a mathematical expression.<\/p>\r\n<section class=\"textbox example\">Evaluate [latex]x+7[\/latex] when:\r\n\r\n<ol>\r\n\t<li>[latex]x=3[\/latex]<\/li>\r\n\t<li>[latex]x=12[\/latex]<\/li>\r\n<\/ol>\r\n<p>[reveal-answer q=\"180574\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"180574\"]<\/p>\r\n<p>1. To evaluate, substitute [latex]3[\/latex] for [latex]x[\/latex] in the expression, and then simplify.<\/p>\r\n<table id=\"eip-id1166566546426\" class=\"unnumbered unstyled\" summary=\"The image shows the given expression x plus 7. Substitute 3 for x. The expression becomes 3 plus x which is 10.\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td>&nbsp;<\/td>\r\n<td>[latex]x+7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute.<\/td>\r\n<td>[latex]\\color{red}{3}\\color{black}+7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add.<\/td>\r\n<td>[latex]10[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>When [latex]x=3[\/latex], the expression [latex]x+7[\/latex] has a value of [latex]10[\/latex].<\/p>\r\n<p>2. To evaluate, substitute [latex]12[\/latex] for [latex]x[\/latex] in the expression, and then simplify.<\/p>\r\n<table id=\"eip-id1166566410105\" class=\"unnumbered unstyled\" summary=\"The image shows the given expression x plus 7, substitute 12 for x. The expression becomes 12 plus x which is 19.\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td>&nbsp;<\/td>\r\n<td>[latex]x+7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute.<\/td>\r\n<td>[latex]\\color{red}{12}\\color{black}+7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add.<\/td>\r\n<td>[latex]19[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>When [latex]x=12[\/latex], the expression [latex]x+7[\/latex] has a value of [latex]19[\/latex].<\/p>\r\n<p>Notice that we got different results for parts 1 and 2 even though we started with the same expression. This is because the values used for [latex]x[\/latex] were different. When we evaluate an expression, the value varies depending on the value used for the variable.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">Evaluate the expression [latex]2x + 7[\/latex] for each value for [latex]x[\/latex]<em>.<\/em>\r\n<ol>\r\n\t<li>[latex]x=0[\/latex]<\/li>\r\n\t<li>[latex]x=1[\/latex]<\/li>\r\n<\/ol>\r\n<p>[reveal-answer q=\"346830\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"346830\"]<\/p>\r\n<ol>\r\n\t<li>Substitute [latex]0[\/latex] for [latex]x[\/latex].<br \/>\r\n[latex]\\begin{array}{ll}2x+7 \\hfill&amp; = 2\\left(\\color{red}{0}\\color{black}\\right)+7 \\\\ \\hfill&amp; =0+7 \\\\ \\hfill&amp; =7\\end{array}[\/latex]\r\n\r\n<p>&nbsp;<\/p>\r\n<\/li>\r\n\t<li>Substitute [latex]1[\/latex] for [latex]x[\/latex].<br \/>\r\n[latex]\\begin{array}{ll}2x+7 \\hfill&amp; = 2\\left(\\color{red}{1}\\color{black}\\right)+7 \\\\ \\hfill&amp; =2+7 \\\\ \\hfill&amp; =9\\end{array}[\/latex]<\/li>\r\n<\/ol>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]8800[\/ohm2_question]<\/section>\r\n<section class=\"textbox example\">Evaluate [latex]{2}^{x}[\/latex] when [latex]x=5[\/latex].[reveal-answer q=\"824631\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"824631\"]In this expression, the variable is an exponent.\r\n\r\n<table id=\"eip-id1168469574741\" class=\"unnumbered unstyled\" summary=\"The image shows the given expression 2 to the power of x. Substitute 5 for x. The expression becomes 2 to the fifth power. By the definition of exponents, 2 to the fifth power is 2 times 2 times 2 times 2 times 2, or 5 factors of 2. Multiply from left to right to get 32.\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td>&nbsp;<\/td>\r\n<td>[latex]2^x[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute [latex]\\color{red}{5}[\/latex] for [latex]x[\/latex].<\/td>\r\n<td>[latex]{2}^{\\color{red}{5}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the definition of exponent.<\/td>\r\n<td>[latex]2\\cdot2\\cdot2\\cdot2\\cdot2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]32[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>When [latex]x=5[\/latex], the expression [latex]{2}^{x}[\/latex] has a value of [latex]32[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]8801[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Substitute a number in for a variable and simplify<\/li>\n<\/ul>\n<\/section>\n<h2>Evaluate Algebraic Expressions<\/h2>\n<p>Any variable in an algebraic expression may take on or be assigned different values. When that happens, the value of the algebraic expression changes. To evaluate an algebraic expression means to determine the value of the expression given the value of each variable in the expression.<\/p>\n<section class=\"textbox questionHelp\">\n<p><strong>How to: Evaluate Algebraic Expressions<\/strong><\/p>\n<ol>\n<li><strong>Identify Variables and Values<\/strong>: Look at the problem to determine which variables are present and what values they have been assigned.<\/li>\n<li><strong>Replace Variables with Values<\/strong>: Substitute each variable in the expression with the given value. Be sure to maintain any coefficients or operators attached to the variables.<\/li>\n<li><strong>Follow the Order of Operations<\/strong>: After substitution, simplify the expression by following the order of operations, also known by the acronym PEMDAS:\n<ul>\n<li><strong>P<\/strong>arentheses: Start by simplifying expressions within parentheses or other grouping symbols.<\/li>\n<li><strong>E<\/strong>xponents: Next, calculate the powers or exponents.<\/li>\n<li><strong>M<\/strong>ultiplication and <strong>D<\/strong>ivision: Then perform all multiplication and division from left to right.<\/li>\n<li><strong>A<\/strong>ddition and <strong>S<\/strong>ubtraction: Finally, complete any addition and subtraction from left to right.<\/li>\n<\/ul>\n<\/li>\n<li><strong>Combine Like Terms<\/strong>: If the expression has like terms (terms with the same variable and exponent), combine them by adding or subtracting the coefficients.<\/li>\n<\/ol>\n<\/section>\n<p>In the next example we show how to substitute various types of numbers into a mathematical expression.<\/p>\n<section class=\"textbox example\">Evaluate [latex]x+7[\/latex] when:<\/p>\n<ol>\n<li>[latex]x=3[\/latex]<\/li>\n<li>[latex]x=12[\/latex]<\/li>\n<\/ol>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q180574\">Show Answer<\/button><\/p>\n<div id=\"q180574\" class=\"hidden-answer\" style=\"display: none\">\n<p>1. To evaluate, substitute [latex]3[\/latex] for [latex]x[\/latex] in the expression, and then simplify.<\/p>\n<table id=\"eip-id1166566546426\" class=\"unnumbered unstyled\" summary=\"The image shows the given expression x plus 7. Substitute 3 for x. The expression becomes 3 plus x which is 10.\" data-label=\"\">\n<tbody>\n<tr>\n<td>&nbsp;<\/td>\n<td>[latex]x+7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute.<\/td>\n<td>[latex]\\color{red}{3}\\color{black}+7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add.<\/td>\n<td>[latex]10[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>When [latex]x=3[\/latex], the expression [latex]x+7[\/latex] has a value of [latex]10[\/latex].<\/p>\n<p>2. To evaluate, substitute [latex]12[\/latex] for [latex]x[\/latex] in the expression, and then simplify.<\/p>\n<table id=\"eip-id1166566410105\" class=\"unnumbered unstyled\" summary=\"The image shows the given expression x plus 7, substitute 12 for x. The expression becomes 12 plus x which is 19.\" data-label=\"\">\n<tbody>\n<tr>\n<td>&nbsp;<\/td>\n<td>[latex]x+7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute.<\/td>\n<td>[latex]\\color{red}{12}\\color{black}+7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add.<\/td>\n<td>[latex]19[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>When [latex]x=12[\/latex], the expression [latex]x+7[\/latex] has a value of [latex]19[\/latex].<\/p>\n<p>Notice that we got different results for parts 1 and 2 even though we started with the same expression. This is because the values used for [latex]x[\/latex] were different. When we evaluate an expression, the value varies depending on the value used for the variable.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">Evaluate the expression [latex]2x + 7[\/latex] for each value for [latex]x[\/latex]<em>.<\/em><\/p>\n<ol>\n<li>[latex]x=0[\/latex]<\/li>\n<li>[latex]x=1[\/latex]<\/li>\n<\/ol>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q346830\">Show Answer<\/button><\/p>\n<div id=\"q346830\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>Substitute [latex]0[\/latex] for [latex]x[\/latex].<br \/>\n[latex]\\begin{array}{ll}2x+7 \\hfill& = 2\\left(\\color{red}{0}\\color{black}\\right)+7 \\\\ \\hfill& =0+7 \\\\ \\hfill& =7\\end{array}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/li>\n<li>Substitute [latex]1[\/latex] for [latex]x[\/latex].<br \/>\n[latex]\\begin{array}{ll}2x+7 \\hfill& = 2\\left(\\color{red}{1}\\color{black}\\right)+7 \\\\ \\hfill& =2+7 \\\\ \\hfill& =9\\end{array}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm8800\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=8800&theme=lumen&iframe_resize_id=ohm8800&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox example\">Evaluate [latex]{2}^{x}[\/latex] when [latex]x=5[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q824631\">Show Answer<\/button><\/p>\n<div id=\"q824631\" class=\"hidden-answer\" style=\"display: none\">In this expression, the variable is an exponent.<\/p>\n<table id=\"eip-id1168469574741\" class=\"unnumbered unstyled\" summary=\"The image shows the given expression 2 to the power of x. Substitute 5 for x. The expression becomes 2 to the fifth power. By the definition of exponents, 2 to the fifth power is 2 times 2 times 2 times 2 times 2, or 5 factors of 2. Multiply from left to right to get 32.\" data-label=\"\">\n<tbody>\n<tr>\n<td>&nbsp;<\/td>\n<td>[latex]2^x[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute [latex]\\color{red}{5}[\/latex] for [latex]x[\/latex].<\/td>\n<td>[latex]{2}^{\\color{red}{5}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the definition of exponent.<\/td>\n<td>[latex]2\\cdot2\\cdot2\\cdot2\\cdot2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]32[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>When [latex]x=5[\/latex], the expression [latex]{2}^{x}[\/latex] has a value of [latex]32[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm8801\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=8801&theme=lumen&iframe_resize_id=ohm8801&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":16,"menu_order":2,"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\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":2245,"module-header":"background_you_need","content_attributions":[{"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"}],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/5123"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/users\/16"}],"version-history":[{"count":23,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/5123\/revisions"}],"predecessor-version":[{"id":15342,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/5123\/revisions\/15342"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/parts\/2245"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/5123\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/media?parent=5123"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=5123"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/contributor?post=5123"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/license?post=5123"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}