{"id":108,"date":"2022-04-12T12:07:57","date_gmt":"2022-04-12T12:07:57","guid":{"rendered":"http:\/\/8thclass.deltapublications.in\/?page_id=108"},"modified":"2025-10-14T07:00:55","modified_gmt":"2025-10-14T07:00:55","slug":"f-10-multiplication-and-division-with-exponents","status":"publish","type":"page","link":"https:\/\/8thclass.deltapublications.in\/index.php\/f-10-multiplication-and-division-with-exponents\/","title":{"rendered":"F.10 Multiplication and division with exponents"},"content":{"rendered":"\n<h2 class=\"wp-block-heading has-text-align-center has-text-color\" style=\"color:#00056d;text-transform:uppercase\"><strong>Multiplication and division with exponents<\/strong><\/h2>\n\n\n\n<p class=\"has-text-color has-link-color has-large-font-size wp-elements-09eb8d27d0d70f35bd20b1cf47293f11\" style=\"color:#74008b\"><strong>Key Notes :<\/strong><\/p>\n\n\n\n<h2 class=\"wp-block-heading has-primary-color has-text-color has-link-color has-large-font-size wp-elements-653e962ea770d99aa0bb7d6d7d6a5177\">\ud83c\udf1f <strong>Multiplication and Division with Exponents<\/strong> \ud83c\udf1f<\/h2>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-text-color has-link-color has-fixed-layout\" style=\"color:#000060\"><tbody><tr><td>\u2733\ufe0f <strong>What are Exponents?<\/strong><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p>An <strong>exponent<\/strong> tells how many times a number (called the <strong>base<\/strong>) is multiplied by itself.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\ud83d\udc49 <strong>Example<\/strong>: 2<sup>4<\/sup>=2\u00d72\u00d72\u00d72=16<\/li>\n<\/ul>\n\n\n\n<p>Here, <strong>2<\/strong> is the base and <strong>4<\/strong> is the exponent.<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-text-color has-link-color has-fixed-layout\" style=\"color:#000060\"><tbody><tr><td>\u2734\ufe0f<strong> Multiplying Powers with the Same Base<\/strong><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p>When <strong>bases are the same<\/strong>, <strong>add<\/strong> the exponents. <\/p>\n\n\n\n<p class=\"has-text-align-center\">a<sup>m<\/sup>\u00d7a<sup>n<\/sup>=a<sup>m+n<\/sup><\/p>\n\n\n\n<p>\ud83d\udc49 <strong>Example<\/strong>: 3<sup>2<\/sup>\u00d73<sup>4<\/sup>=3<sup>2+4<\/sup>=3<sup>6<\/sup>=729<\/p>\n\n\n\n<p>\ud83e\udde0 <strong>Remember:<\/strong> Base must be the same!<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-text-color has-link-color has-fixed-layout\" style=\"color:#000060\"><tbody><tr><td>\u2733\ufe0f <strong>Dividing Powers with the Same Base<\/strong><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p>When <strong>bases are the same<\/strong>, <strong>subtract<\/strong> the exponents. <\/p>\n\n\n\n<p class=\"has-text-align-center\">a<sup>m<\/sup>\/a<sup>n<\/sup>=a<sup>m\u2212n<\/sup><\/p>\n\n\n\n<p>\ud83d\udc49 <strong>Example<\/strong>: 5<sup>7<\/sup>\u00f75<sup>3<\/sup>=5<sup>7\u22123<\/sup>=5<sup>4<\/sup>=625<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-text-color has-link-color has-fixed-layout\" style=\"color:#000060\"><tbody><tr><td>\u2734\ufe0f <strong>Power of a Power Rule<\/strong><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p>When you raise a power to another power, <strong>multiply<\/strong> the exponents. <\/p>\n\n\n\n<p class=\"has-text-align-center\">(a<sup>m<\/sup>)<sup>n<\/sup>=a<sup>m\u00d7n<\/sup><\/p>\n\n\n\n<p>\ud83d\udc49 <strong>Example<\/strong>: (2<sup>3<\/sup>)<sup>4<\/sup>=2<sup>3\u00d74<\/sup>=2<sup>12<\/sup><\/p>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-text-color has-link-color has-fixed-layout\" style=\"color:#000060\"><tbody><tr><td>\u2733\ufe0f <strong>Power of a Product Rule<\/strong><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p>When multiplying two bases with an exponent outside, <strong>apply the exponent to each base<\/strong>. <\/p>\n\n\n\n<p class=\"has-text-align-center\">(ab)<sup>m<\/sup>=a<sup>m<\/sup>\u00d7b<sup>m<\/sup><\/p>\n\n\n\n<p>\ud83d\udc49 <strong>Example<\/strong>: (2\u00d73)<sup>2<\/sup>=2<sup>2<\/sup>\u00d73<sup>2<\/sup>=4\u00d79=36<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-text-color has-link-color has-fixed-layout\" style=\"color:#000060\"><tbody><tr><td>\u2734\ufe0f <strong>Power of a Quotient Rule<\/strong><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p>When dividing two bases with an exponent outside, <strong>apply the exponent to both numerator and denominator.<\/strong> <\/p>\n\n\n\n<p class=\"has-text-align-center\">(a\/b)<sup>m<\/sup>=a<sup>m<\/sup>\/b<sup>m<\/sup><\/p>\n\n\n\n<p>\ud83d\udc49 <strong>Example<\/strong>: (4\/2)<sup>3<\/sup>=4<sup>3<\/sup>\/2<sup>3<\/sup>=64\/8=8<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-text-color has-link-color has-fixed-layout\" style=\"color:#000060\"><tbody><tr><td>\u2733\ufe0f <strong>Zero and Negative Exponents<\/strong><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Zero Exponent Rule:<\/strong> a0=1(a\u22600)<\/li>\n\n\n\n<li><strong>Negative Exponent Rule:<\/strong> a<sup>\u2212n<\/sup>=1\/a<sup>n<\/sup><\/li>\n<\/ul>\n\n\n\n<p>\ud83d\udc49 <strong>Example<\/strong>: 2<sup>\u22123<\/sup>=1\/2<sup>3<\/sup>=1\/8<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-text-color has-link-color has-fixed-layout\" style=\"color:#000060\"><tbody><tr><td>\ud83d\udca1 <strong>Quick Tips to Remember<\/strong><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p>\u2705 Same base \u2192 Add or subtract exponents (depending on \u00d7 or \u00f7).<\/p>\n\n\n\n<p>\u2705 Different base \u2192 Apply exponent separately.<\/p>\n\n\n\n<p>\u2705 Power to power \u2192 Multiply exponents.<\/p>\n\n\n\n<p>\u2705 Always simplify at the end!<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-text-color has-link-color has-fixed-layout\" style=\"color:#000060\"><tbody><tr><td>\ud83e\uddee <strong>Practice Examples<\/strong><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p>1\ufe0f\u20e3 6<sup>3<\/sup>\u00d76<sup>2<\/sup>=6<sup>5<\/sup>=7776<\/p>\n\n\n\n<p>2\ufe0f\u20e3 8<sup>5<\/sup>\u00f78<sup>3<\/sup>=8<sup>2<\/sup>=64<\/p>\n\n\n\n<p>3\ufe0f\u20e3 (3<sup>2<\/sup>)<sup>3<\/sup>=3<sup>6<\/sup>=729<\/p>\n\n\n\n<p>4\ufe0f\u20e3 (2\u00d75)<sup>2<\/sup>=2<sup>2<\/sup>\u00d75<sup>2<\/sup>=100<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-text-color has-link-color has-fixed-layout\" style=\"color:#000060\"><tbody><tr><td>\ud83c\udfaf<strong> Summary Chart<\/strong><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><thead><tr><th>Operation<\/th><th>Rule<\/th><th>Example<\/th><\/tr><\/thead><tbody><tr><td>Multiply powers<\/td><td>a<sup>m<\/sup>\u00d7a<sup>n<\/sup>=a<sup>m+n<\/sup><\/td><td>2<sup>3<\/sup>\u00d72<sup>4<\/sup>=2<sup>7<\/sup><\/td><\/tr><tr><td>Divide powers<\/td><td>a<sup>m<\/sup>\u00f7a<sup>n<\/sup>=a<sup>m\u2212n<\/sup><\/td><td>5<sup>6<\/sup>\u00f75<sup>2<\/sup>=5<sup>4<\/sup><\/td><\/tr><tr><td>Power of a power<\/td><td>(a<sup>m<\/sup>)<sup>n<\/sup>=a<sup>m\u00d7n<\/sup><\/td><td>(3<sup>2<\/sup>)<sup>3=<\/sup>3<sup>6<\/sup><\/td><\/tr><tr><td>Power of a product<\/td><td>(ab)<sup>m<\/sup>=a<sup>m<\/sup>\u00d7b<sup>m<\/sup><\/td><td>(2\u00d73)<sup>2<\/sup>=2<sup>2<\/sup>\u00d73<sup>2<\/sup><\/td><\/tr><tr><td>Power of a quotient<\/td><td>(a\/b)<sup>m<\/sup>=a<sup>m<\/sup>\/b<sup>m<\/sup><\/td><td>(4\/2)<sup>3<\/sup>=64\/8<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\"has-text-align-center has-text-color has-link-color has-large-font-size wp-elements-3fefd20f8ce119a518e5f53333ab5486\" style=\"color:#105000\"><strong>Learn with an example<\/strong><\/p>\n\n\n\n<div class=\"wp-block-group has-primary-color has-text-color has-background has-link-color has-normal-font-size wp-elements-0f07d77a7f2329d18adcc6a11cc08ac5\" style=\"background-color:#f7f2b4\"><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-text-color has-background has-link-color wp-elements-7499a130558bed72e962b4849e7e3fe1\" style=\"color:#b00012\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<p><strong>Simplify.&nbsp;Express your answer as a single term, without a&nbsp;denominator.<\/strong><\/p>\n\n\n\n<p class=\"has-text-align-center has-foreground-color has-text-color has-link-color wp-elements-db4569dfa6b8b17080dafa0a8e296ac9\">t<sup>2<\/sup>.t<sup>4<\/sup> \/ t<sup>6<\/sup><\/p>\n<\/div><\/div>\n\n\n\n<p>First,&nbsp;simplify the&nbsp;numerator.<\/p>\n\n\n\n<p>t<sup>2<\/sup>.t<sup>4<\/sup> \/ t<sup>6<\/sup><\/p>\n\n\n\n<p class=\"has-normal-font-size\">t<sup>2+4<\/sup> \/ t<sup>6<\/sup>    &#8211;&gt; Multiply,&nbsp;remembering to add the&nbsp;exponents<\/p>\n\n\n\n<p>t<sup>6<\/sup> \/ t<sup>6<\/sup><\/p>\n\n\n\n<p>The&nbsp;numerator is simplified. Now, divide the numerator by the&nbsp;denominator.<\/p>\n\n\n\n<p>t<sup>6-6<\/sup>  &#8212;&gt; <em>Divide,&nbsp;remembering to subtract the&nbsp;exponents<\/em><\/p>\n\n\n\n<p>t<sup>0<\/sup>= 1   &#8211;&gt; Any&nbsp;number to the zeroth power is equal to&nbsp;one.<\/p>\n\n\n\n<p class=\"has-text-align-center has-normal-font-size\">t<sup>2<\/sup>.t<sup>4<\/sup> \/ t<sup>6<\/sup>  = 1<\/p>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group has-primary-color has-text-color has-background has-link-color has-normal-font-size wp-elements-b8e5b55c8c76030749356cc3b2ce7c25\" style=\"background-color:#c0b6fa\"><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-text-color has-background has-link-color wp-elements-818a46c489063c0c5dc188a5de571ad7\" style=\"color:#b00012\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<p><strong>Simplify.&nbsp;Express your answer as a single term, without a&nbsp;denominator.<\/strong><\/p>\n\n\n\n<p class=\"has-text-align-center has-foreground-color has-text-color has-link-color wp-elements-adb1d2d5edbfd299fb41c4b65b662610\">a<sup>3<\/sup>.a<sup>4<\/sup>.a<sup>5<\/sup><\/p>\n<\/div><\/div>\n\n\n\n<p>Simplify.<\/p>\n\n\n\n<p>a<sup>3<\/sup>.a<sup>4<\/sup>.a<sup>5<\/sup><\/p>\n\n\n\n<p>a<sup>3+4+5<\/sup>  &#8212;&gt; <em>Multiply,&nbsp;remembering to add the&nbsp;exponents<\/em><\/p>\n\n\n\n<p class=\"has-normal-font-size\">a<sup>12<\/sup><\/p>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group has-primary-color has-text-color has-background has-link-color has-normal-font-size wp-elements-625cb7345f6bdda5b3fc64b59667a4c9\" style=\"background-color:#c7fbe5\"><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-text-color has-background has-link-color wp-elements-07c1b838126017c4ab6c1f6bfd283742\" style=\"color:#b00012\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<p><strong>Simplify.&nbsp;Express your answer as a single term, without a&nbsp;denominator.<\/strong><\/p>\n\n\n\n<p class=\"has-text-align-center has-foreground-color has-text-color has-link-color wp-elements-384a996265f9282435b41bf37f6c9d5a\">v<sup>4<\/sup> \/ v<sup>7<\/sup>.v<sup>5<\/sup><\/p>\n<\/div><\/div>\n\n\n\n<p>First,&nbsp;simplify the&nbsp;denominator.<\/p>\n\n\n\n<p>v<sup>4<\/sup> \/ v<sup>7<\/sup>.v<sup>5<\/sup><\/p>\n\n\n\n<p>v<sup>4<\/sup> \/ v<sup>7+5<\/sup>  &#8212;&gt;<em>Multiply,&nbsp;remembering to add the&nbsp;exponents<\/em><\/p>\n\n\n\n<p>v<sup>4<\/sup>\/v<sup>12<\/sup><\/p>\n\n\n\n<p>The&nbsp;denominator is simplified. Now, divide the numerator by the&nbsp;denominator.<\/p>\n\n\n\n<p class=\"has-normal-font-size\">v<sup>4-12<\/sup>  &#8212;&gt; <em>Divide,&nbsp;remembering to subtract the&nbsp;exponents<\/em><\/p>\n\n\n\n<p class=\"has-normal-font-size\">v<sup>-8<\/sup><\/p>\n<\/div><\/div>\n\n\n\n<p class=\"has-text-color has-large-font-size\" style=\"color:#d90000\"><strong>Let&#8217;s practice!\ud83d\udd8a\ufe0f<\/strong><\/p>\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\/36489\/625\/956\"><img loading=\"lazy\" decoding=\"async\" width=\"500\" height=\"500\" src=\"https:\/\/8thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-302.png\" alt=\"\" class=\"wp-image-9604\" srcset=\"https:\/\/8thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-302.png 500w, https:\/\/8thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-302-300x300.png 300w, https:\/\/8thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-302-150x150.png 150w\" 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\/33893\/150\/356\"><img loading=\"lazy\" decoding=\"async\" width=\"500\" height=\"500\" src=\"https:\/\/8thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-309.png\" alt=\"\" class=\"wp-image-9605\" srcset=\"https:\/\/8thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-309.png 500w, https:\/\/8thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-309-300x300.png 300w, https:\/\/8thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-309-150x150.png 150w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/a><\/figure>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Multiplication and division with exponents Key Notes : \ud83c\udf1f Multiplication and Division with Exponents \ud83c\udf1f \u2733\ufe0f What are Exponents? An exponent tells how many times a number (called the base) is multiplied by itself. Here, 2 is the base and 4 is the exponent. \u2734\ufe0f Multiplying Powers with the Same Base When bases are the<a class=\"more-link\" href=\"https:\/\/8thclass.deltapublications.in\/index.php\/f-10-multiplication-and-division-with-exponents\/\">Continue reading <span class=\"screen-reader-text\">&#8220;F.10 Multiplication and division with 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,"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"class_list":["post-108","page","type-page","status-publish","hentry","entry"],"aioseo_notices":[],"_hostinger_reach_plugin_has_subscription_block":false,"_hostinger_reach_plugin_is_elementor":false,"_links":{"self":[{"href":"https:\/\/8thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/108","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/8thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/8thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/8thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/8thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/comments?post=108"}],"version-history":[{"count":24,"href":"https:\/\/8thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/108\/revisions"}],"predecessor-version":[{"id":21848,"href":"https:\/\/8thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/108\/revisions\/21848"}],"wp:attachment":[{"href":"https:\/\/8thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/media?parent=108"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}