{"id":156524,"date":"2022-03-26T01:51:07","date_gmt":"2022-03-25T20:21:07","guid":{"rendered":"https:\/\/infinitylearn.com\/surge\/laws-of-exponents\/"},"modified":"2022-12-22T21:42:12","modified_gmt":"2022-12-22T16:12:12","slug":"laws-of-exponents","status":"publish","type":"post","link":"https:\/\/infinitylearn.com\/surge\/maths\/laws-of-exponents\/","title":{"rendered":"Laws of Exponents"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_37 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" style=\"display: none;\"><label for=\"item\" aria-label=\"Table of Content\"><span style=\"display: flex;align-items: center;width: 35px;height: 30px;justify-content: center;\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/label><input type=\"checkbox\" id=\"item\"><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1' style='display:block'><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/laws-of-exponents\/#What_are_Exponents\" title=\"What are Exponents?\">What are Exponents?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/laws-of-exponents\/#Rules_of_Exponents_With_Examples\" title=\"Rules of Exponents With Examples\">Rules of Exponents With Examples<\/a><ul class='ez-toc-list-level-3'><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/laws-of-exponents\/#Product_With_the_Same_Bases\" title=\"Product With the Same Bases\">Product With the Same Bases<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/laws-of-exponents\/#Quotient_with_Same_Bases\" title=\"Quotient with Same Bases\">Quotient with Same Bases<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/laws-of-exponents\/#Power_Raised_to_a_Power\" title=\"Power Raised to a Power\">Power Raised to a Power<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/laws-of-exponents\/#Product_to_a_Power\" title=\"Product to a Power\">Product to a Power<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/laws-of-exponents\/#Quotient_to_a_Power\" title=\"Quotient to a Power\">Quotient to a Power<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/laws-of-exponents\/#Zero_Power\" title=\"Zero Power\">Zero Power<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/laws-of-exponents\/#Negative_Exponent_Rule\" title=\"Negative Exponent Rule\">Negative Exponent Rule<\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<h2><span class=\"ez-toc-section\" id=\"What_are_Exponents\"><\/span>What are Exponents?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><a href=\"https:\/\/infinitylearn.com\/surge\/maths\/laws-of-exponents\/\">Exponents<\/a> are used to show repeated multiplication of a number by itself. For example, 7 \u00d7 7 \u00d7 7 can be represented as <span class=\"latex-for-amp\">7<sup>3<\/sup><\/span>. Here, the exponent is \u20183\u2019 which stands for the number of times the number 7 is multiplied. 7 is the base here which is the actual number that is getting multiplied. So basically exponent or power denotes the number of times a number can be multiplied. If the power is 2, that means the base number is multiplied two times with itself. Some of the examples are:<\/p>\n<ul>\n<li>3<span class=\"latex-for-amp\"><sup>4<\/sup><\/span> = 3\u00d73\u00d73\u00d73<\/li>\n<li>10<span class=\"latex-for-amp\"><sup>5<\/sup><\/span> = 10\u00d710\u00d710\u00d710\u00d710<\/li>\n<li>16<span class=\"latex-for-amp\"><sup>3<\/sup><\/span> = 16 \u00d7 16 \u00d7 16<\/li>\n<\/ul>\n<p>Suppose, a number \u2018a\u2019 is multiplied by itself n-times, then it is represented as a<span class=\"latex-for-amp\"><sup>n<\/sup><\/span> where a is the base and n is the exponent.<\/p>\n<p><img loading=\"lazy\" class=\" aligncenter size-full wp-image-11511\" title=\"Capture\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2018\/11\/maths\/2016\/02\/08061632\/Capture.png\" alt=\"laws of exponents\" width=\"223\" height=\"41\" \/><\/p>\n<p><a href=\"https:\/\/en.wikipedia.org\/wiki\/Exponentiation\">Exponents<\/a> follow certain rules that help in simplifying expressions which are also called its laws. Let us discuss the laws of exponents in detail.<br \/>\n<a id=\"Laws of Exponents\" name=\"Laws of Exponents\"><\/a><\/p>\n<h2><span class=\"ez-toc-section\" id=\"Rules_of_Exponents_With_Examples\"><\/span>Rules of Exponents With Examples<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>As discussed earlier, there are different  laws or rules defined for exponents. The important laws of exponents are given below:<\/p>\n<p>&nbsp;<\/p>\n<ul>\n<li aria-level=\"1\">a<sup>m<\/sup>\u00d7a<sup>n<\/sup> = a<sup>m+n<\/sup><\/li>\n<li aria-level=\"1\">a<sup>m<\/sup>\/a<sup>n<\/sup> = a<sup>m-n<\/sup><\/li>\n<li aria-level=\"1\">(a<sup>m<\/sup>)<sup>n<\/sup> = a<sup>mn<\/sup><\/li>\n<li aria-level=\"1\">a<sup>n<\/sup>\/b<sup>n<\/sup> = (a\/b)<sup>n<\/sup><\/li>\n<li aria-level=\"1\">a<sup>0<\/sup> = 1<\/li>\n<li aria-level=\"1\">a<sup>-m<\/sup> = 1\/a<sup>m<\/sup><\/li>\n<li aria-level=\"1\">\n<div class=\"mathjax-scroll\"><\/div>\n<\/li>\n<\/ul>\n<p><strong>Also, read: <\/strong>Now let us discuss all the laws one by one with examples here.<\/p>\n<div class=\"table-responsive\"><\/div>\n<p><a id=\"Powers with Same Base\" name=\"Powers with Same Base\"><\/a><\/p>\n<h3><span class=\"ez-toc-section\" id=\"Product_With_the_Same_Bases\"><\/span>Product With the Same Bases<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>As per this law, for any non-zero term a,<\/p>\n<ul>\n<li><strong>a<span class=\"latex-for-amp\"><sup>m<\/sup><\/span><span class=\"latex-for-amp\">\u00d7<\/span>a<span class=\"latex-for-amp\"><sup>n <\/sup><\/span>= a<span class=\"latex-for-amp\"><sup>m+n<\/sup><\/span><\/strong><\/li>\n<\/ul>\n<p>where <span class=\"latex-for-amp\">m<\/span> and <span class=\"latex-for-amp\">n<\/span> are real no.<\/p>\n<p><strong>Example 1: What is the simplification of <span class=\"latex-for-amp\">5<sup>5<\/sup> \u00d7 5<sup>1<\/sup> <\/span>?<\/strong><\/p>\n<p>Solution: <span class=\"latex-for-amp\">5<sup>5<\/sup> \u00d7 5<sup>1<\/sup> <\/span>= <span class=\"latex-for-amp\">5<sup>5+1<\/sup><\/span> = 5<span class=\"latex-for-amp\"><sup>6<\/sup><\/span><\/p>\n<p><strong>Example 2: What is the simplification of (\u22126)<span class=\"latex-for-amp\"><sup>-4<\/sup><\/span> \u00d7 (\u22126)<span class=\"latex-for-amp\"><sup>-7<\/sup><\/span>?<\/strong><\/p>\n<p>Solution: (\u22126)<span class=\"latex-for-amp\"><sup>-4<\/sup><\/span> \u00d7 (\u22126)<span class=\"latex-for-amp\"><sup>-7 <\/sup><\/span>= (-6)<span class=\"latex-for-amp\"><sup>-4-7 <\/sup><\/span>= (-6)<span class=\"latex-for-amp\"><sup>-11<\/sup><\/span><\/p>\n<p><strong>Note:<\/strong> We can state that the law is applicable for negative terms also. Therefore the term m and n can be any integer.<br \/>\n<a id=\"Quotient with Same Base\" name=\"Quotient with Same Base\"><\/a><\/p>\n<h3><span class=\"ez-toc-section\" id=\"Quotient_with_Same_Bases\"><\/span>Quotient with Same Bases<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>As per this rule,<\/p>\n<ul>\n<li><strong>a<span class=\"latex-for-amp\"><sup>m<\/sup><\/span>\/a<span class=\"latex-for-amp\"><sup>n<\/sup><\/span> = a<span class=\"latex-for-amp\"><sup>m-n<\/sup><\/span><\/strong><\/li>\n<\/ul>\n<p>where a is a non-zero term and m and n are integers.<\/p>\n<p><strong>Example 3: Find the value when <span class=\"latex-for-amp\">10<sup>-5<\/sup><\/span> is divided by 10<span class=\"latex-for-amp\"><sup>-3<\/sup><\/span>.<\/strong><\/p>\n<p>Solution: As per the question;<\/p>\n<p><span class=\"latex-for-amp\">10<sup>-5<\/sup><\/span>\/10<span class=\"latex-for-amp\"><sup>-3 <\/sup><\/span><\/p>\n<p>= 10<span class=\"latex-for-amp\"><sup>-5-(-)3<\/sup><\/span><\/p>\n<p>= 10<span class=\"latex-for-amp\"><sup>-5+3<\/sup><\/span><\/p>\n<p>= 10<span class=\"latex-for-amp\"><sup>-2<\/sup><\/span><\/p>\n<p>= 1\/100<br \/>\n<a id=\"Power of a Power\" name=\"Power of a Power\"><\/a><\/p>\n<h3><span class=\"ez-toc-section\" id=\"Power_Raised_to_a_Power\"><\/span>Power Raised to a Power<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>According to this law, if \u2018a\u2019 is the base, then the power raised to the power of base \u2018a\u2019 gives the product of the powers raised to the base \u2018a\u2019, such as;<\/p>\n<ul>\n<li><strong>(a<span class=\"latex-for-amp\"><sup>m<\/sup><\/span>)<span class=\"latex-for-amp\"><sup>n<\/sup><\/span> = a<span class=\"latex-for-amp\"><sup>mn<\/sup><\/span><\/strong><\/li>\n<\/ul>\n<p>where a is a non-zero term and <span class=\"latex-for-amp\">m<\/span> and n are integers.<\/p>\n<p><strong>Example 4: Express <span class=\"latex-for-amp\">8<sup>3<\/sup><\/span> as a power with base 2.<\/strong><\/p>\n<p>Solution: We have, <span class=\"latex-for-amp\">2\u00d72\u00d72<\/span> = <span class=\"latex-for-amp\">8<\/span> = 2<span class=\"latex-for-amp\"><sup>3<\/sup><\/span><\/p>\n<p>Therefore, 8<span class=\"latex-for-amp\"><sup>3<\/sup><\/span>= <span class=\"latex-for-amp\">(2<sup>3<\/sup>)<sup>3<\/sup><\/span> = 2<span class=\"latex-for-amp\"><sup>9<\/sup><\/span><br \/>\n<a id=\"Product to a Power\" name=\"Product to a Power\"><\/a><\/p>\n<h3><span class=\"ez-toc-section\" id=\"Product_to_a_Power\"><\/span>Product to a Power<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>As per this rule, for two or more different bases, if the power is same, then;<\/p>\n<ul>\n<li>a<span class=\"latex-for-amp\"><sup>n <\/sup><\/span>b<span class=\"latex-for-amp\"><sup>n <\/sup><\/span>= (ab)<span class=\"latex-for-amp\"><sup>n<\/sup><\/span><\/li>\n<\/ul>\n<p>where a is a non-zero term and n is the integer.<\/p>\n<p><strong>Example 5: Simplify and write the exponential form of: 1\/8 x 5<span class=\"latex-for-amp\"><sup>-3<\/sup><\/span><\/strong><\/p>\n<p>Solution: We can write, 1\/8 = 2<span class=\"latex-for-amp\"><sup>-3<\/sup><\/span><\/p>\n<p>Therefore, <span class=\"latex-for-amp\">2<sup>-3<\/sup> x 5<sup>-3<\/sup><\/span> = <span class=\"latex-for-amp\">(2 \u00d7 5)<sup>-3<\/sup><\/span> = <span class=\"latex-for-amp\">10<sup>-3<\/sup><\/span><br \/>\n<a id=\"Quotient to a Power\" name=\"Quotient to a Power\"><\/a><\/p>\n<h3><span class=\"ez-toc-section\" id=\"Quotient_to_a_Power\"><\/span>Quotient to a Power<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>As per this law, the fraction of two different bases with the same power is represented as;<\/p>\n<ul>\n<li> a<span class=\"latex-for-amp\"><sup>n<\/sup><\/span>\/b<span class=\"latex-for-amp\"><sup>n <\/sup><\/span>= (a\/b)<span class=\"latex-for-amp\"><sup>n<\/sup><\/span><\/li>\n<\/ul>\n<p>where a and <span class=\"latex-for-amp\">b<\/span> are non-zero terms and n is an integer.<\/p>\n<p><strong>Example 6:<\/strong> Simplify the expression and find the value:15<span class=\"latex-for-amp\"><sup>3<\/sup><\/span>\/5<span class=\"latex-for-amp\"><sup>3<\/sup><\/span><\/p>\n<p>Solution: We can write the given expression as;<\/p>\n<p>(15\/5)<span class=\"latex-for-amp\"><sup>3<\/sup><\/span>= 3<span class=\"latex-for-amp\"><sup>3<\/sup><\/span> = 27<br \/>\n<a id=\"Zero Power Rule\" name=\"Zero Power Rule\"><\/a><\/p>\n<h3><span class=\"ez-toc-section\" id=\"Zero_Power\"><\/span>Zero Power<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>According to this rule, when the power of any integer is zero, then its value is equal to 1, such as;<\/p>\n<p>a<span class=\"latex-for-amp\"><sup>0 <\/sup><\/span>= 1<\/p>\n<p>where \u2018a\u2019 is any non-zero term.<\/p>\n<p><strong>Example 7: What is the value of <span class=\"latex-for-amp\">5<sup>0<\/sup> + 2<sup>2<\/sup> + 4<sup>0<\/sup> + 7<sup>1<\/sup> \u2013 3<sup>1<\/sup><\/span> ?<\/strong><\/p>\n<p>Solution: <span class=\"latex-for-amp\">5<sup>0<\/sup> + 2<sup>2<\/sup> + 4<sup>0<\/sup> + 7<sup>1<\/sup> \u2013 3<sup>1<\/sup><\/span> = 1+4+1+7-3= 10<br \/>\n<a id=\"Negative Exponent Rule\" name=\"Negative Exponent Rule\"><\/a><\/p>\n<h3><span class=\"ez-toc-section\" id=\"Negative_Exponent_Rule\"><\/span>Negative Exponent Rule<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>According to this rule, if the exponent is negative, we can change the exponent into positive by writing the same value in the denominator and the numerator holds the value 1.<\/p>\n<p>The negative exponent rule is given as:<\/p>\n<p><strong>a<sup>-m<\/sup> = 1\/a<sup>m<\/sup><\/strong><\/p>\n<p><img loading=\"lazy\" class=\"aligncenter wp-image-156523 size-full\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/laws-of-exponents.jpg\" alt=\"Laws of Exponents\" width=\"606\" height=\"428\" srcset=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/laws-of-exponents.jpg?v=1648239666 606w, https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/laws-of-exponents-300x212.jpg?v=1648239666 300w\" sizes=\"(max-width: 606px) 100vw, 606px\" \/><\/p>\n","protected":false},"excerpt":{"rendered":"<p>What are Exponents? Exponents are used to show repeated multiplication of a number by itself. For example, 7 \u00d7 7 [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_yoast_wpseo_focuskw":"Laws of Exponents","_yoast_wpseo_title":"","_yoast_wpseo_metadesc":"laws of Exponents are used to show repeated multiplication of a number by its exponent or power denotes the number","custom_permalink":"maths\/laws-of-exponents\/"},"categories":[13],"tags":[],"table_tags":[],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v17.9 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Laws of Exponents - Infinity Learn by Sri Chaitanya<\/title>\n<meta name=\"description\" content=\"laws of Exponents are used to show repeated multiplication of a number by its exponent or power denotes the number\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/laws-of-exponents\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Laws of Exponents - 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