{"id":182,"date":"2022-04-12T09:48:16","date_gmt":"2022-04-12T09:48:16","guid":{"rendered":"http:\/\/7thclass.deltapublications.in\/?page_id=182"},"modified":"2025-02-10T04:34:59","modified_gmt":"2025-02-10T04:34:59","slug":"i-3-solve-equations-with-variable-exponents","status":"publish","type":"page","link":"https:\/\/7thclass.deltapublications.in\/index.php\/i-3-solve-equations-with-variable-exponents\/","title":{"rendered":"I.3 Solve equations with variable exponents"},"content":{"rendered":"\n<h2 class=\"wp-block-heading has-text-align-center has-text-color\" style=\"color:#00056d;text-transform:uppercase\"><strong> Solve equations with variable exponents<\/strong><\/h2>\n\n\n\n<p class=\"has-text-color has-huge-font-size\" style=\"color:#74008b;text-transform:capitalize\">Key notes:<\/p>\n\n\n\n<p class=\"has-text-color has-link-color has-large-font-size wp-elements-581fb3587ad9978c516dd5dce0861aa3\" style=\"color:#000060\"><strong>Understanding Exponents<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list has-large-font-size\">\n<li>Exponents represent repeated multiplication (e.g., 2<sup>3<\/sup> = 2 \u00d7 2 \u00d7 2 = 8 ).<\/li>\n\n\n\n<li>The base is the number being multiplied, and the exponent tells how many times to multiply it.<\/li>\n<\/ul>\n\n\n\n<p class=\"has-text-color has-link-color has-large-font-size wp-elements-81997c4f145a4f0e911fb1d5831658c0\" style=\"color:#000060\"><strong>Equations with Variable Exponents<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list has-large-font-size\">\n<li>These are equations where the exponent contains a variable, such as x<sup>n<\/sup> = a or 2<sup>x<\/sup> = 16.<\/li>\n\n\n\n<li>The goal is to find the value of the variable that makes the equation true.<\/li>\n<\/ul>\n\n\n\n<p class=\"has-text-color has-link-color has-large-font-size wp-elements-cd3b3bd5d9a5296263447b3c44796928\" style=\"color:#000060\"><strong>Solving Equations with the Same Base<\/strong><\/p>\n\n\n\n<p class=\"has-large-font-size\">If the equation has the same base on both sides, set the exponents equal to each other.<\/p>\n\n\n\n<ul class=\"wp-block-list has-large-font-size\">\n<li>Example: 3<sup>x<\/sup> = 3<sup>4<\/sup> \u27f9 x = 4.<\/li>\n<\/ul>\n\n\n\n<p class=\"has-text-color has-link-color has-large-font-size wp-elements-237c765ec7be1a97c6ba6f03363f7221\" style=\"color:#000060\"><strong>Using Logarithms (Basic Introduction)<\/strong><\/p>\n\n\n\n<p class=\"has-large-font-size\">If the equation is not easily rewritten with the same base, logarithms can be used to solve for the exponent.<\/p>\n\n\n\n<ul class=\"wp-block-list has-large-font-size\">\n<li>Example: 2<sup>x<\/sup> =10 \u27f9 x = log\u2061<sub>2<\/sub> 10 (Grade 7 may use trial and error for estimating x).<\/li>\n<\/ul>\n\n\n\n<p class=\"has-text-color has-link-color has-large-font-size wp-elements-d11155fdf048f9982157041c7ff35ff2\" style=\"color:#000060\"><strong>Exponent Rules for Solving Equations<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list has-large-font-size\">\n<li><strong>Multiplication Rule:<\/strong> a<sup>m<\/sup> \u00d7 a<sup>n<\/sup> = a<sup>m+n<\/sup><\/li>\n\n\n\n<li><strong>Division Rule:<\/strong> a<sup>m<\/sup> \/ a<sup>n<\/sup> = a<sup>m\u2212n<\/sup>.<\/li>\n\n\n\n<li><strong>Power of a Power Rule:<\/strong> (a<sup>m<\/sup>)<sup>n<\/sup> = a<sup>m\u00d7n<\/sup>.<\/li>\n\n\n\n<li><strong>Zero Exponent Rule:<\/strong> a<sup>0<\/sup> = 1 (where a \u2260 0).<\/li>\n<\/ul>\n\n\n\n<p class=\"has-text-color has-link-color has-large-font-size wp-elements-612deb7c6e1308fdc620cafad7f78a14\" style=\"color:#000060\"><strong>Checking Solutions<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li class=\"has-large-font-size\">Substitute the found value of the variable back into the original equation to verify the solution.<\/li>\n<\/ul>\n\n\n\n<p class=\"has-text-color has-link-color has-large-font-size wp-elements-db8268305644bf0019c080e0f8e41651\" style=\"color:#000060\"><strong>Applications in Real Life<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list has-large-font-size\">\n<li>Growth and decay problems (e.g., population growth, radioactive decay).<\/li>\n\n\n\n<li>Financial calculations involving compound interest.<\/li>\n<\/ul>\n\n\n\n<div class=\"wp-block-group has-large-font-size\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<div class=\"wp-block-group has-background\" style=\"background-color:#f9d3d3\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<p>An&nbsp;<strong>exponent<\/strong>&nbsp;tells you how many times its&nbsp;<strong>base<\/strong>&nbsp;is used as a&nbsp;factor.<\/p>\n\n\n\n<p>Exponents&nbsp;are used to write repeated&nbsp;multiplication.<\/p>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group has-background\" style=\"background-color:#f9d8cf\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<p>An&nbsp;<strong>exponent<\/strong>&nbsp;tells you how many times its&nbsp;<strong>base<\/strong>&nbsp;is used as a&nbsp;factor.<\/p>\n\n\n\n<p>When&nbsp;10&nbsp;is raised to a whole number exponent, its value has a leftmost digit of&nbsp;1&nbsp;and all other digits are&nbsp;0.&nbsp;The exponent tells you how many&nbsp;0s&nbsp;come after the&nbsp;1.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"700\" height=\"300\" src=\"https:\/\/7thclass.deltapublications.in\/wp-content\/uploads\/2023\/01\/Untitled__700___300px_-removebg-preview.png\" alt=\"\" class=\"wp-image-4287\" style=\"width:448px;height:191px\" srcset=\"https:\/\/7thclass.deltapublications.in\/wp-content\/uploads\/2023\/01\/Untitled__700___300px_-removebg-preview.png 700w, https:\/\/7thclass.deltapublications.in\/wp-content\/uploads\/2023\/01\/Untitled__700___300px_-removebg-preview-300x129.png 300w\" sizes=\"auto, (max-width: 700px) 100vw, 700px\" \/><\/figure><\/div><\/div><\/div>\n\n\n\n<p class=\"has-text-align-center has-text-color has-large-font-size\" style=\"color:#105000\"><strong>Learn with an example<\/strong><\/p>\n\n\n\n<div class=\"wp-block-group has-background\" style=\"background-color:#faf6c8\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<div class=\"wp-block-group has-background-background-color has-background\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<p class=\"has-text-color\" style=\"color:#b00012\">\ud83c\udf00 <strong>Solve&nbsp;for&nbsp;u.<\/strong><\/p>\n\n\n\n<p>7<sup>u<\/sup> = 49<\/p>\n\n\n\n<p>u = _____<\/p>\n<\/div><\/div>\n\n\n\n<p>Look&nbsp;at the&nbsp;equation.<\/p>\n\n\n\n<p>7<sup>u<\/sup>=49<\/p>\n\n\n\n<p>The&nbsp;base is&nbsp;7.&nbsp;The exponent,&nbsp;u,&nbsp;is the number of times&nbsp;7&nbsp;is used in the repeated&nbsp;multiplication.<\/p>\n\n\n\n<p>Compute&nbsp;powers of&nbsp;7&nbsp;until you get&nbsp;49.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"600\" height=\"300\" src=\"https:\/\/7thclass.deltapublications.in\/wp-content\/uploads\/2023\/01\/Untitled_design-removebg-preview.png\" alt=\"\" class=\"wp-image-4284\" style=\"width:476px;height:238px\" srcset=\"https:\/\/7thclass.deltapublications.in\/wp-content\/uploads\/2023\/01\/Untitled_design-removebg-preview.png 600w, https:\/\/7thclass.deltapublications.in\/wp-content\/uploads\/2023\/01\/Untitled_design-removebg-preview-300x150.png 300w\" sizes=\"auto, (max-width: 600px) 100vw, 600px\" \/><\/figure><\/div>\n\n\n<p>So, u = 2.<\/p>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group has-background\" style=\"background-color:#d8defa\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<div class=\"wp-block-group has-background-background-color has-background\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<p class=\"has-text-color\" style=\"color:#b00012\">\ud83c\udf00 <strong>Solve&nbsp;for&nbsp;w.<\/strong><\/p>\n\n\n\n<p>10<sup>w<\/sup> = 1,000<\/p>\n\n\n\n<p>w=____<\/p>\n<\/div><\/div>\n\n\n\n<p>To&nbsp;solve for&nbsp;w,&nbsp;look at the right side of the equation and count the number of&nbsp;0s&nbsp;after&nbsp;1.<\/p>\n\n\n\n<p>10<sup>w<\/sup> = 1,000<\/p>\n\n\n\n<p>There&nbsp;are&nbsp;3&nbsp;of&nbsp;them.<\/p>\n\n\n\n<p>So,&nbsp;w=3. <\/p>\n\n\n\n<p>10<sup>3<\/sup>=1,000<\/p>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group has-background\" style=\"background-color:#f8cbf9\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<div class=\"wp-block-group\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<div class=\"wp-block-group has-background-background-color has-background\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<p class=\"has-text-color\" style=\"color:#b00012\">\ud83c\udf00 <strong>Solve&nbsp;for&nbsp;r.<\/strong><\/p>\n\n\n\n<div class=\"wp-block-group\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<p>6<sup>r<\/sup> = 36<\/p>\n\n\n\n<p>r=_____<\/p>\n<\/div><\/div>\n<\/div><\/div>\n\n\n\n<p>Look&nbsp;at the&nbsp;equation.<\/p>\n\n\n\n<p>6<sup>r<\/sup> = 36<\/p>\n\n\n\n<p>The&nbsp;base is&nbsp;6.&nbsp;The exponent,&nbsp;r,&nbsp;is the number of times&nbsp;6&nbsp;is used in the repeated&nbsp;multiplication.<\/p>\n\n\n\n<p>Compute&nbsp;powers of&nbsp;6&nbsp;until you get&nbsp;36.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"700\" height=\"300\" src=\"https:\/\/7thclass.deltapublications.in\/wp-content\/uploads\/2023\/01\/Untitled__700___300px___1_-removebg-preview.png\" alt=\"\" class=\"wp-image-4288\" style=\"width:531px;height:227px\" srcset=\"https:\/\/7thclass.deltapublications.in\/wp-content\/uploads\/2023\/01\/Untitled__700___300px___1_-removebg-preview.png 700w, https:\/\/7thclass.deltapublications.in\/wp-content\/uploads\/2023\/01\/Untitled__700___300px___1_-removebg-preview-300x129.png 300w\" sizes=\"auto, (max-width: 700px) 100vw, 700px\" \/><\/figure><\/div><\/div><\/div>\n\n\n\n<p>So, r=2.<\/p>\n<\/div><\/div>\n\n\n\n<p class=\"has-text-color has-large-font-size\" style=\"color:#d90000\">let&#8217;s practice! \ud83d\udd8a\ufe0f<\/p>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<figure class=\"wp-block-image size-full\"><a href=\"https:\/\/wordwall.net\/play\/84636\/542\/509\"><img loading=\"lazy\" decoding=\"async\" width=\"500\" height=\"500\" src=\"https:\/\/7thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-2-276.png\" alt=\"\" class=\"wp-image-7441\" srcset=\"https:\/\/7thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-2-276.png 500w, https:\/\/7thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-2-276-300x300.png 300w, https:\/\/7thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-2-276-150x150.png 150w, https:\/\/7thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-2-276-400x400.png 400w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/a><\/figure>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<figure class=\"wp-block-image size-full\"><a href=\"https:\/\/wordwall.net\/play\/86157\/225\/596\"><img loading=\"lazy\" decoding=\"async\" width=\"500\" height=\"500\" src=\"https:\/\/7thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-1-284.png\" alt=\"\" class=\"wp-image-7442\" srcset=\"https:\/\/7thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-1-284.png 500w, https:\/\/7thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-1-284-300x300.png 300w, https:\/\/7thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-1-284-150x150.png 150w, https:\/\/7thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-1-284-400x400.png 400w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/a><\/figure>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Solve equations with variable exponents Key notes: Understanding Exponents Equations with Variable Exponents Solving Equations with the Same Base If the equation has the same base on both sides, set the exponents equal to each other. Using Logarithms (Basic Introduction) If the equation is not easily rewritten with the same base, logarithms can be used<a class=\"more-link\" href=\"https:\/\/7thclass.deltapublications.in\/index.php\/i-3-solve-equations-with-variable-exponents\/\">Continue reading <span class=\"screen-reader-text\">&#8220;I.3 Solve equations with variable exponents&#8221;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"om_disable_all_campaigns":false,"footnotes":""},"class_list":["post-182","page","type-page","status-publish","hentry","entry"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/7thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/182","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/7thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/7thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/7thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/7thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/comments?post=182"}],"version-history":[{"count":14,"href":"https:\/\/7thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/182\/revisions"}],"predecessor-version":[{"id":15622,"href":"https:\/\/7thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/182\/revisions\/15622"}],"wp:attachment":[{"href":"https:\/\/7thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/media?parent=182"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}