simplify expressions involving indices use the rules of indices to simplify expressions involving indices use negative and fractional indices.


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1 Indices or Powers mctyindicespowers A knowledge of powers, or indices as they are often called, is essential for an understanding ofmostalgebraicprocesses. Inthissectionoftextyouwilllearnaboutpowersandrulesfor manipulating them through a number of worked examples. In order to master the techniques explained here it is vital that you undertake plenty of practice exercises so that they become second nature. Afterreadingthistext,and/orviewingthevideotutorialonthistopic,youshouldbeableto: simplify expressions involving indices use the rules of indices to simplify expressions involving indices use negative and fractional indices. Contents. Introduction 2 2. Thefirstrule: a m a n = a m+n. Thesecondrule: (a m ) n = a mn 4. Thethirdrule: a m a n = a m n 4 5. Thefourthrule: a 0 = 4 6. Thefifthrule: a = a and a m = a m 5 7. Thesixthrule: a 2 = aand a q = q a 6 8. Afinalresult: a p q = (a p ) q = q ap, a p q = (a q ) p = ( q a) p 9. Further examples c mathcentre 2009
2 . Introduction Inthesectionwewillbelookingatindicesorpowers.Eithernamecanbeused,andbothnames mean the same thing. Basically, they are a shorthand way of writing multiplications of the same number. So, suppose we have Wewritethisas 4tothepower : So = 4 Thenumberiscalledthepowerorindex.Notethatthepluralofindexisindices. Anindex,orpower,isusedtoshowthataquantityisrepeatedlymultipliedbyitself. Thiscanbedonewithlettersaswellasnumbers.So,wemighthave: a a a a a Sincetherearefive a smultipliedtogetherwewritethisas atothepower5. So a 5 a a a a a = a 5. Whatifwehad 2x 2 raisedtothepower4?thismeansfourfactorsof 2x 2 multipliedtogether, that is, This can be written 2x 2 2x 2 2x 2 2x x 2 x 2 x 2 x 2 whichwewillseeshortlycanbewrittenas 6x 8. Useofapowerorindexissimplyaformofnotation,thatis,awayofwritingsomethingdown. Whenmathematicianshaveawayofwritingthingsdowntheyliketousetheirnotationinother ways.forexample,whatmightwemeanby a 2 or a 2 or a 0? Toproceedfurtherweneedrulestooperatewithsowecanfindoutwhatthesenotations actually mean. 2 c mathcentre 2009
3 Exercises. Evaluate each of the following. a) 5 b) 7 c) 2 9 d) 5 e) 4 4 f) 8 2. The first rule Supposewehave a andwewanttomultiplyitby a 2.Thatis a a 2 = a a a a a Altogethertherearefive a smultipliedtogether. Clearly,thisisthesameas a 5. Thissuggests our first rule. Thefirstruletellsusthatifwearemultiplyingexpressionssuchasthesethenweaddtheindices together.so,ifwehave a m a n weaddtheindicestoget a m a n = a m+n a m a n = a m+n. The second rule Supposewehad a 4 andwewanttoraiseitalltothepower.thatis This means (a 4 ) a 4 a 4 a 4 Nowourfirstruletellsusthatweshouldaddtheindicestogether.Sothatis a 2 Butnotealsothat2is 4. Thissuggeststhatifwehave a m allraisedtothepower nthe resultisobtainedbymultiplyingthetwopowerstoget a m n,orsimply a mn. c mathcentre 2009
4 (a m ) n = a mn 4. The third rule Considerdividing a 7 by a. a 7 a = a7 a = a a a a a a a a a a Wecannowbegindividingoutthecommonfactorsof a. Threeofthe a satthetopandthe three a satthebottomcanbedividedout,sowearenowleftwith a 4 thatis a 4 Thesameanswerisobtainedbysubtractingtheindices,thatis, 7 = 4. Thissuggestsour thirdrule,that a m a n = a m n. a m a n = a m n 5. What can we do with these rules? The fourth rule Let shavealookat a dividedby a. Weknowtheanswertothis. Wearedividingaquantity byitself,sotheanswerhasgottobe. a a = Let sdothisusingourrules;rulewillhelpusdothis. Ruletellsusthattodividethetwo quantities we subtract the indices: a a = a = a 0 Weappeartohaveobtainedadifferentanswer. Wehavedonethesamecalculationintwo differentways.wehavedoneitcorrectlyintwodifferentways.sotheanswersweget,evenif theylookdifferent,mustbethesame.so,whatwehaveis a 0 =. 4 c mathcentre 2009
5 a 0 = Thismeansthatanynumberraisedtothepowerzerois.So ( ) = (, 000, 000) 0 = = ( 6) 0 = 2 However,notethatzeroitselfisanexceptiontothisrule. 0 0 cannotbeevaluated.anynumber, apartfromzero,whenraisedtothepowerzeroisequalto. 6. The fifth rule Let shavealooknowatdoingadivisionagain. Consider a dividedby a 7. a a 7 = a a = a a a 7 a a a a a a a Again,wecannowbegindividingoutthecommonfactorsof a.thea satthetopandthree ofthe a satthebottomcanbedividedout,sowearenowleftwith a a 7 = a a a a = a 4 Nowlet suseourthirdruleanddothesamecalculationbysubtractingtheindices. a a 7 = a 7 = a 4 Wehavedonethesamecalculationintwodifferentways. Wehavedoneitcorrectlyintwo differentways.sotheanswersweget,eveniftheylookdifferent,mustbethesame.so a 4 = a 4 Soanegativesignintheindexcanbethoughtofasmeaning over. a = a andmoregenerally a m = a m 5 c mathcentre 2009
6 Now let s develop this further in the following examples. Inthenexttwoexampleswestartwithanexpressionwhichhasanegativeindex,andrewriteit sothatithasapositiveindex,usingtherule a m = a m. s 2 2 = 2 2 = 4 5 = 5 = 5 Wecanreversetheprocessinordertorewritequantitiessothattheyhaveanegativeindex. s a = a = a 7 = Oneyoushouldtrytorememberis a = a asyouwillprobablyuseitthemost. Butnowwhataboutanexamplelike 7 2. Usingtheabove,weseethatthismeans /72. Herewearedividingbyafraction,andtodividebyafractionweneedtoinvertand multiply so: 7 = 2 /7 = 2 7 = 72 2 = 72 This illustrates another way of writing the previous keypoint: = am a m Exercises 2. Evaluate each of the following leaving your answer as a proper fraction. a) 2 9 b) 5 c) 4 4 d) 5 e) 7 f) 8 7. The sixth rule Sofarwehavedealtwithintegerpowersbothpositiveandnegative.Whatwouldwedoifwehad afractionforapower,like a 2.Toseehowtodealwithfractionalpowersconsiderthefollowing: 6 c mathcentre 2009
7 Suppose we have two identical numbers multiplying together to give another number, as in, for example 7 7 = 49 Thenweknowthat7isasquarerootof49.Thatis,if 7 2 = 49 then 7 = 49 Now suppose we found that a p a p = a Thatis,whenwemultiplied a p byitselfwegottheresult a.thismeansthat a p mustbeasquare rootof a. However, lookatthisanotherway: notingthat a = a, andalsothat, fromthefirstrule, a p a p = a 2p weseethatif a p a p = athen a 2p = a from which 2p = andso p = 2 Thisshowsthat a /2 mustbethesquarerootof a.thatis a 2 = a thepower/2denotesasquareroot: a 2 = a Similarly and More generally, a = a a 4 = 4 a thisisthecuberootof a thisisthefourthrootof a a q = q a 7 c mathcentre 2009
8 Work through the following examples: Whatdowemeanby 6 /4? Forthisweneedtoknowwhatnumberwhenmultipliedtogetherfourtimesgives6.Theanswer is2.so 6 /4 = 2. Whatdowemeanby 8 /2? Forthisweneedtoknowwhatnumberwhenmultipliedbyitself gives8.theansweris9.so 8 2 = 8 = 9. Whatabout 24 5?Whatnumberwhenmultipliedtogetherfivetimesgivesus24?Ifweare familiarwithtimestableswemightspotthat 24 = 8,andalsothat 8 = 9 9.So 24 /5 = ( 8) /5 = ( 9 9) /5 = ( ) /5 So multiplied by itself five times equals 24. Hence 24 /5 = Noticeindoingthishowimportantitistobeabletorecognisewhatfactorsnumbersaremade upof.forexample,itisimportanttobeabletorecognisethat: 6 = 2 4, 6 = 4 2, 8 = 9 2, 8 = 4 andsoon. You will find calculations much easier if you can recognise in numbers their composition as powers ofsimplenumberssuchas2,,4and5.onceyouhavegotthesefirmlyfixedinyourmind,this sort of calculation becomes straightforward. Exercises. Evaluate each of the following. a) 25 / b) 24 /5 c) 256 /4 d) 52 /9 e) 4 / f) 52 / 8. A final result Whathappensifwetake a 4? Wecanwritethisasfollows: a 4 = (a 4 ) usingthe2ndrule (a m ) n = a mn Whatdowemeanby 6 4? 6 4 = (6 4 ) = (2) = c mathcentre 2009
9 Wecanalsothinkofthiscalculationperformedinaslightlydifferentway.Notethatinsteadof writing (a m ) n = a mn wecouldwrite (a n ) m = a mn because mnisthesameas nm. Whatdowemeanby 8 2?Onewayofcalculatingthisistowrite 8 2 = (8 ) 2 = (2) 2 = 4 Alternatively, 8 2 = (8 2 ) = (64) = 4 Additional note Doing this calculation the first way is usually easier as it requires recognising powers of smaller numbers. Forexample,itisstraightforwardtoevaluate 27 5/ as 27 5/ = (27 / ) 5 = 5 = 24 because,atleastwithpractice,youwillknowthatthecuberootof27is. Whereas,evaluationinthe following way 27 5/ = (27 5 ) / = / would require knowledge of the cube root of Writing these results down algebraically we have the following important point: a p q = (a p ) q = q a p a p q = (a q ) p = ( q a) p Both results are exactly the same. 9 c mathcentre 2009
10 Exercises 4. Evaluate each of the following. a) 4 2/ b) 52 2/ c) 256 /4 d) 25 4/ e) 52 7/9 f) 24 6/5 5. Evaluate each of the following. a) 52 7/9 b) 24 6/5 c) 256 /4 d) 25 4/ e) 4 2/ f) 52 2/ 9. Further examples The remainder of this unit provides examples illustrating the use of the rules of indices. Write 2x 4usingapositiveindex. Write 4x 2 a usingpositiveindices. 2x 4 = 2 x 4 = 2 x 4 Write 4x 2 a = 4 x 2 a = 4a x 2 4a 2usingapositiveindex. Simplify a 2a 2. 4a = 2 4 a = 2 4 a2 = a2 4 a 2a 2 = 2a a 2 = 2a 5 6 addingtheindices = 2 a 5 6 = 2 a 5 6 Simplify 2a 2. a c mathcentre 2009
11 2a 2 a 2 = 2a 2 a 2 = 2a 2 ( /2) subtractingtheindices Simplify a 2 2 a. = 2a 2 = 2 a 2 a2 2 a = a 2 a 2 = a 6 byaddingtheindices Simplify 6 4. Simplify = (6 4 ) = 2 = 8 Simplify = = (4 2) 5 = 2 5 = 2 Simplify = (25 ) 2 = 5 2 = 25 Simplify 8 2 = 8 2 = (8 ) 2 = 2 2 = = 252 = c mathcentre 2009
12 Simplify (24) 5. ( ) 8 4 Simplify. 6 (24) 5 = (24 5 ) = = 27 Exercises ( ) = = = ( 8 6 ) 4 ( 6 8 ( (6 ) 4 8 ( ) 2 = = 8 27 ) ) 4 6. Evaluate each of the following. ( a) 4 ) 2 ( b) 5 ) ( c) ) ( d) 8 ) ( e) 5 ) ( f) ) 7. Evaluate each of the following. ( a) 4 ) 2 ( b) 5 ) ( c) ) ( d) 8 ) ( e) 5 ) ( f) ) 8. Evaluate each of the following. a) d) ( 2 6/5 24) b) ( 26 4) / e) ( 6 /4 8) c) ( 25 52) 2/ f) ( 625 ) /4 256 ( 25 ) 2/ Eachofthefollowingexpressionscanbewrittenas a n forsomevalueof n. Ineachcase determine the value of n. a) a a a a b) c) a a a a 5 e) a a 5 a f) 6 d) g) (a 4 ) 2 h) a 2 a 5 (a ) i) a 2 a a 2 j) a /2 a 2 k) a a 2 l) (a 2 ) 2 c mathcentre 2009
13 0. Simplifyeachofthefollowingexpressionsgivingyouranswerintheform Cx n,where C and n are numbers. a) x 2 2x 4 b) 5x 4x 5 c) (2x ) 4 8x d) 6 e) 4x 5 f) 2x 8 2x x 2 x 2 g) (5x ) h) (9x 4 ) /2 i) 2x 6 4x 2 j) 2x 4 k) (2x) 4 l) 6x x 5 x 5 (2x) Answers a) 24 b) 4 c) 52. d) 25 e) 256 f) a) b) c) d) e) f) a) 5 b) c) 4 d) 2 e) 7 f) 8 a) 49 b) 64 c) 64 d) 625 e) 28 f) 729 a) b) c) d) e) f) a) 6 8 b) 25 4 c) d) e) f) a) b) c) d) e) f) a) b) c) d) e) f) a) 4 b) c) 0 5 d) e) 8 f) 4 5 g) 8 h) 2 i) 2 5 j) k) 5 l) 6 2 a) 6x 6 b) 20x 6 c) 6x 2 d) 4x e) 2x f) 4x 6 g) 5 x h) x 2 i) 2 x8 j) 2x k) 6x l) 2x 4 c mathcentre 2009
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