[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-207812-en":3,"doc-seo-207812-105":30,"detail-sidebar-cat-0-en-105":90},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},207812,549758146520,"Patrick","https://ap-avatar.wpscdn.com/avatar/80002397d8c0411e94?_k=1775819394049821470",4,"Exam","Indices or Powers - Worked Examples and Rules for Manipulating Indices","A focused learning section on powers (indices) and how they are used to rewrite repeated multiplication in both numbers and letters. It builds understanding from foundational notation, then applies a sequence of rules for manipulating indices, including multiplication and exponent-of-a-power relationships, division with index subtraction, the identity for the zero index, and negative/fractional indices. Worked examples and practice exercises support simplification of algebraic expressions.","Indices or Powers  \nmc-TY-indicespowers-2009-1  \nA knowledge of powers, or indices as they are often called, is essential for an understanding of most algebraic processes. In this section of text you will learn about powers and rules for manipulating them through a number of worked examples.  \nIn order to master the techniques explained here it is vital that you undertake plenty of practice exercises so that they become second nature.  \nAfter reading this text, and/or viewing the video tutorial on this topic, you should be able to:  \n• simplify expressions involving indices  \n• use the rules of indices to simplify expressions involving indices  \n• use negative and fractional indices.  \nContents  \n1. Introduction 2  \n2. The first rule: am × an = am+n 3  \n3. The second rule: (am )n = amn 3  \n4. The third rule: am ÷ an = am − n 4  \n5. The fourth rule: a0 = 1 4  \n6. The fifth rule: a −1 = 1a and a − m = a~~ ~~ 5  \n7. The sixth rule: a 12 = √a and a 1q = √q a 6  \n8. A final result: a pq = (ap ) 1q = √q ap  ,  \na pq = (a1q)p = ( √q a )p 8  \n9. Further examples 10  \n[www.mathcentre.ac.uk 1](www.mathcentre.ac.uk 1) 􀀍c mathcentre 2009   \n1. Introduction  \nIn the section we will be looking at indices or powers. Either name can be used, and both names mean the same thing.  \nBasically, they are a shorthand way of writing multiplications of the same number. So, suppose we have  \n4 × 4 × 4  \nWe write this as ‘4 to the power 3’:  \n43  \nSo  \n4 × 4 × 4 = 43  \nThe number 3 is called the power or index. Note that the plural of index is indices.  \n Key Point  \nAn index, or power, is used to show that a quantity is repeatedly multiplied by itself.  \nThis can be done with letters as well as numbers. So, we might have:  \na × a × a × a × a  \nSince there are five a’s multiplied together we write this as ‘a to the power 5’.  \na5  \nSo  \na × a × a × a × a = a5 .  \nWhat if we had 2x2 raised to the power 4 ? This means four factors of 2x2 multiplied together,  \nthat is,  \n2x2 × 2x2 × 2x2 × 2x2  \nThis can be written  \n2 × 2 × 2 × 2 × x2 × x2 × x2 × x2  \nwhich we will see shortly can be written as 16x8 .  \nUse of a power or index is simply a form of notation, that is, a way of writing something down. When mathematicians have a way of writing things down they like to use their notation in other ways. For example, what might we mean by  \na−2 or a 12 or a0 ?  \nTo proceed further we need rules to operate with so we can find out what these notations actually mean.  \n[www.mathcentre.ac.uk 2](www.mathcentre.ac.uk 2) 􀀍c mathcentre 2009   \nExercises  \n1. Evaluate each of the following.  \na) 35 b) 73 c) 29  \nd) 53 e) 44 f) 83  \n2. The first rule  \nSuppose we have a3 and we want to multiply it by a2 . That is  \na3 × a2 = a × a × a × a × a  \nAltogether there are five a’s multiplied together. Clearly, this is the same as a5 . This suggests our first rule.  \nThe first rule tells us that if we are multiplying expressions such as these then we add the indices together. So, if we have  \nam × an  \nwe add the indices to get  \nam × an = am+n  \n Key Point  \nam × an = am+n  \n3. The second rule  \nSuppose we had a4 and we want to raise it all to the power 3 . That is  \n(a4 )3  \nThis means  \na4 × a4 × a4  \nNow our first rule tells us that we should add the indices together. So that is  \na12  \nBut note also that 12 is 4 × 3. This suggests that if we have am all raised to the power n the result is obtained by multiplying the two powers to get am ×n , or simply amn .  \n[www.mathcentre.ac.uk 3](www.mathcentre.ac.uk 3) 􀀍c mathcentre 2009   \n Key Point  \n(am )n = amn  \n4. The third rule  \nConsider dividing a7 by a3 .  \na7   a × a × a × a × a × a × a  \na7 ÷ a3 = =  \na3 a × a × a  \nWe can now begin dividing out the common factors of a. Three of the a’s at the top and the three a’s at the bottom can be divided out, so we are now left with  \na41~~ ~~ that is a4  \nThe same answer is obtained by subtracting the indices, that is, 7 − 3 = 4 . This suggests our third rule, that am ÷ an = am−n .  \n Key Point","cbCais8sn5VQhjfz","https://ap.wps.com/l/cbCais8sn5VQhjfz","pdf",273386,1,13,"English","en",105,"# Introduction\n## Powers (Indices) as Repeated Multiplication\n# Exercises\n## Evaluate Powers\n# Rules of Indices\n## Multiplying Powers: am × an = am+n\n## Power of a Power: (am)n = amn\n## Dividing Powers: am ÷ an = am−n\n## Zero Index: a0 = 1\n## Negative Indices: a−1 = 1/a","[{\"question\":\"What do indices (powers) mean in algebraic notation?\",\"answer\":\"An index shows how many times a quantity is multiplied by itself. For example, a5 means a × a × a × a × a.\"},{\"question\":\"How do you multiply expressions involving indices?\",\"answer\":\"When multiplying powers with the same base, add the indices: am × an = am+n.\"},{\"question\":\"What is the rule for dividing powers with the same base?\",\"answer\":\"When dividing powers with the same base, subtract the indices: am ÷ an = am−n.\"}]","Indices or Powers - Worked Examples and Rules for Manipulating Indices | PDF",1788601221,33,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":85,"head_meta":87,"extra_data":89,"updated_unix":28},"indices-or-powers-worked-examples-and-rules-for-manipulating-indices","",{"@graph":36,"@context":84},[37,53,67],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/exam/",3,{"item":52,"name":13,"@type":43,"position":11},"https://docshare.wps.com/document/indices-or-powers-worked-examples-and-rules-for-manipulating-indices/207812/",{"url":52,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":61,"encodingFormat":60,"isAccessibleForFree":62,"interactionStatistic":63},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-09-05",true,{"@type":64,"interactionType":65,"userInteractionCount":4},"InteractionCounter",{"@type":66},"ViewAction",{"@type":68,"mainEntity":69},"FAQPage",[70,76,80],{"name":71,"@type":72,"acceptedAnswer":73},"What do indices (powers) mean in algebraic notation?","Question",{"text":74,"@type":75},"An index shows how many times a quantity is multiplied by itself. For example, a5 means a × a × a × a × a.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"How do you multiply expressions involving indices?",{"text":79,"@type":75},"When multiplying powers with the same base, add the indices: am × an = am+n.",{"name":81,"@type":72,"acceptedAnswer":82},"What is the rule for dividing powers with the same base?",{"text":83,"@type":75},"When dividing powers with the same base, subtract the indices: am ÷ an = am−n.","https://schema.org",{"og:url":52,"og:type":86,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":88,"canonical":52},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":91},[92,96,100,103,108,113,118,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":46,"category_name":93,"show_sort_weight":94,"slug":95},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":97,"show_sort_weight":98,"slug":99},"Literature",80,"literature",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":101,"slug":102},70,"exam",{"id":104,"doc_module":4,"doc_module_name":46,"category_name":105,"show_sort_weight":106,"slug":107},5,"Comic",60,"comic",{"id":109,"doc_module":4,"doc_module_name":46,"category_name":110,"show_sort_weight":111,"slug":112},6,"Technology",50,"technology",{"id":114,"doc_module":4,"doc_module_name":46,"category_name":115,"show_sort_weight":116,"slug":117},7,"Healthcare",40,"healthcare",{"id":119,"doc_module":4,"doc_module_name":46,"category_name":120,"show_sort_weight":121,"slug":122},8,"Research & Report",30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":104,"slug":138},19,"General","general"]