{"id":186,"date":"2022-04-12T09:48:56","date_gmt":"2022-04-12T09:48:56","guid":{"rendered":"http:\/\/7thclass.deltapublications.in\/?page_id=186"},"modified":"2025-02-10T09:24:10","modified_gmt":"2025-02-10T09:24:10","slug":"i-5-exponents-with-decimal-and-fractional-bases","status":"publish","type":"page","link":"https:\/\/7thclass.deltapublications.in\/index.php\/i-5-exponents-with-decimal-and-fractional-bases\/","title":{"rendered":"I.5 Exponents with decimal and fractional bases"},"content":{"rendered":"\n<h2 class=\"wp-block-heading has-text-align-center has-text-color\" style=\"color:#00056d;text-transform:uppercase\"><strong> Exponents with decimal and fractional bases<\/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-a33fa1f009d74ee3346d2db7f66fbce8\" style=\"color:#000060\"><strong>Understanding Exponents<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list has-large-font-size\">\n<li>An exponent represents repeated multiplication of a base.<\/li>\n\n\n\n<li>Example: 2<sup>3<\/sup> = 2 \u00d7 2 \u00d7 2 = 8.<\/li>\n<\/ul>\n\n\n\n<p class=\"has-text-color has-link-color has-large-font-size wp-elements-25ca0518f2bb72633209d07a3b410f95\" style=\"color:#000060\"><strong>Decimal Bases with Exponents<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list has-large-font-size\">\n<li>Decimal numbers can be used as bases in exponent expressions.<\/li>\n\n\n\n<li>Example: (0.2)<sup>3<\/sup> = 0.2 \u00d7 0.2 \u00d7 0.2 = 0.008.<\/li>\n\n\n\n<li>The value gets smaller when raising a decimal (between 0 and 1) to a positive exponent.<\/li>\n<\/ul>\n\n\n\n<p class=\"has-text-color has-link-color has-large-font-size wp-elements-bd5ad4a1800ea3fe9df35dcf9a3a6f0e\" style=\"color:#000060\"><strong>Fractional Bases with Exponents<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list has-large-font-size\">\n<li>Fractions can be used as bases and follow the same exponent rules.<\/li>\n\n\n\n<li>Example: (1\/2)<sup>3<\/sup> = 1\/2 \u00d7 1\/2 \u00d7 1\/2 = 18\u200b.<\/li>\n\n\n\n<li>The numerator and denominator are both raised to the exponent.<\/li>\n\n\n\n<li>Example: (3\/5)<sup>2<\/sup> = 3<sup>2<\/sup>\/5<sup>2<\/sup> = 9\/25\u200b.<\/li>\n<\/ul>\n\n\n\n<p class=\"has-text-color has-link-color has-large-font-size wp-elements-d734b05324af0cb8d2855c90ca9b5767\" style=\"color:#000060\"><strong>Negative Exponents with Decimal and Fractional Bases<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list has-large-font-size\">\n<li>A negative exponent means taking the reciprocal of the base.<\/li>\n\n\n\n<li>Example: (0.2)<sup>\u22122<\/sup> = 1 \/ (0.2)<sup>2<\/sup> = 1 \/ 0.04 = 25<\/li>\n\n\n\n<li>Example: (2\/3)<sup>-2<\/sup> = (32)<sup>2<\/sup> = 9\/4.<\/li>\n<\/ul>\n\n\n\n<p class=\"has-text-color has-link-color has-large-font-size wp-elements-b81f235154053740bddef0ebfe89c06f\" style=\"color:#000060\"><strong>Exponent Rules Apply<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list has-large-font-size\">\n<li><strong>Product Rule<\/strong>: a<sup>m<\/sup> \u00d7 a<sup>n<\/sup> = a<sup>m+n<\/sup><\/li>\n\n\n\n<li><strong>Quotient Rule<\/strong>: a<sup>m<\/sup> \/ a<sup>n<\/sup>=a<sup>m\u2212n<\/sup><\/li>\n\n\n\n<li><strong>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>: Any nonzero base raised to the power of 0 is 1 (e.g., (0.5)<sup>0<\/sup> = 1).<\/li>\n<\/ul>\n\n\n\n<p class=\"has-text-color has-link-color has-large-font-size wp-elements-2911c724c8990704d25d755832e74ba3\" style=\"color:#000060\"><strong>Real-World Applications<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list has-large-font-size\">\n<li>Used in scientific notation for very small numbers (e.g., 3.2 \u00d7 10<sup>\u22124<\/sup>).<\/li>\n\n\n\n<li>Growth and decay problems involve fractional exponent bases (e.g., interest rates and bacteria growth).<\/li>\n<\/ul>\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 has-large-font-size\" style=\"background-color:#fcdcf2\"><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\">\u27a1\ufe0f <strong>Evaluate. Write your answer as a fraction or whole number.<\/strong><\/p>\n\n\n\n<p>( 1\/4 )<sup>4<\/sup> = _____<\/p>\n<\/div><\/div>\n\n\n\n<p>( 1\/4 )<sup>4<\/sup> = 1\/4 x 1\/4 x 1\/4 x 1\/4<\/p>\n\n\n\n<p>= 1 \/ 4 x 4 x 4 x 4<\/p>\n\n\n\n<p>= 1 \/ 256<\/p>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group has-background has-large-font-size\" style=\"background-color:#dbe6f6\"><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\">\u27a1\ufe0f<strong> Evaluate. Write your answer as a fraction or whole number.<\/strong><\/p>\n\n\n\n<p>( 3\/4 )<sup>3<\/sup> =______<\/p>\n<\/div><\/div>\n\n\n\n<p>( 3\/4 )3 = 3\/4 x 3\/4 x 3\/4<\/p>\n\n\n\n<p>= 3 x 3 x 3 \/ 4 x 4 x 4<\/p>\n\n\n\n<p>= 27 \/ 64<\/p>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group has-background has-large-font-size\" style=\"background-color:#e5fbd1\"><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\">\u27a1\ufe0f <strong>Evaluate. Write your answer as a decimal or whole number.<\/strong><\/p>\n\n\n\n<p>(0.01)<sup>3<\/sup>&nbsp;=____<\/p>\n<\/div><\/div>\n\n\n\n<p>The base is 0.01 and the exponent is 3. Use 0.01 as a factor 3 times.<\/p>\n\n\n\n<p>(0.01)<sup>3<\/sup> = 0.01 \u00b7 0.01 \u00b7 0.01<\/p>\n\n\n\n<p>= 0.000001<\/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\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\/85887\/268\/795\"><img loading=\"lazy\" decoding=\"async\" width=\"500\" height=\"500\" src=\"https:\/\/7thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-2-278.png\" alt=\"\" class=\"wp-image-7447\" srcset=\"https:\/\/7thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-2-278.png 500w, https:\/\/7thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-2-278-300x300.png 300w, https:\/\/7thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-2-278-150x150.png 150w, https:\/\/7thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-2-278-400x400.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\/86163\/151\/944\"><img loading=\"lazy\" decoding=\"async\" width=\"500\" height=\"500\" src=\"https:\/\/7thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-1-286.png\" alt=\"\" class=\"wp-image-7448\" srcset=\"https:\/\/7thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-1-286.png 500w, https:\/\/7thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-1-286-300x300.png 300w, https:\/\/7thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-1-286-150x150.png 150w, https:\/\/7thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-1-286-400x400.png 400w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/a><\/figure>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Exponents with decimal and fractional bases Key notes: Understanding Exponents Decimal Bases with Exponents Fractional Bases with Exponents Negative Exponents with Decimal and Fractional Bases Exponent Rules Apply Real-World Applications Learn with an example \u27a1\ufe0f Evaluate. Write your answer as a fraction or whole number. ( 1\/4 )4 = _____ ( 1\/4 )4 = 1\/4<a class=\"more-link\" href=\"https:\/\/7thclass.deltapublications.in\/index.php\/i-5-exponents-with-decimal-and-fractional-bases\/\">Continue reading <span class=\"screen-reader-text\">&#8220;I.5 Exponents with decimal and fractional bases&#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-186","page","type-page","status-publish","hentry","entry"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/7thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/186","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=186"}],"version-history":[{"count":17,"href":"https:\/\/7thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/186\/revisions"}],"predecessor-version":[{"id":15636,"href":"https:\/\/7thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/186\/revisions\/15636"}],"wp:attachment":[{"href":"https:\/\/7thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/media?parent=186"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}