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


 Jeffry Warner
 2 years ago
 Views:
Transcription
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 0 8 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) = 8 8 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 2 0 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 = 625 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
MEP Y9 Practice Book A
1 Base Arithmetic 1.1 Binary Numbers We normally work with numbers in base 10. In this section we consider numbers in base 2, often called binary numbers. In base 10 we use the digits 0, 1, 2, 3, 4, 5,
More informationIf A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C?
Problem 3 If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C? Suggested Questions to ask students about Problem 3 The key to this question
More informationhow to use dual base log log slide rules
how to use dual base log log slide rules by Professor Maurice L. Hartung The University of Chicago Pickett The World s Most Accurate Slide Rules Pickett, Inc. Pickett Square Santa Barbara, California 93102
More informationHow Old Are They? This problem gives you the chance to: form expressions form and solve an equation to solve an age problem. Will is w years old.
How Old Are They? This problem gives you the chance to: form expressions form and solve an equation to solve an age problem Will is w years old. Ben is 3 years older. 1. Write an expression, in terms of
More informationYou know from calculus that functions play a fundamental role in mathematics.
CHPTER 12 Functions You know from calculus that functions play a fundamental role in mathematics. You likely view a function as a kind of formula that describes a relationship between two (or more) quantities.
More informationSelecting a Subset of Cases in SPSS: The Select Cases Command
Selecting a Subset of Cases in SPSS: The Select Cases Command When analyzing a data file in SPSS, all cases with valid values for the relevant variable(s) are used. If I opened the 1991 U.S. General Social
More informationCritical points of once continuously differentiable functions are important because they are the only points that can be local maxima or minima.
Lecture 0: Convexity and Optimization We say that if f is a once continuously differentiable function on an interval I, and x is a point in the interior of I that x is a critical point of f if f (x) =
More informationWHAT IS CBT? What are the Principles of CBT?
WHAT IS CBT? CBT, or CognitiveBehavioural Therapy, is a psychological treatment that was developed through scientific research. That is, all of the components of CBT have been tested by researchers to
More informationRevised Version of Chapter 23. We learned long ago how to solve linear congruences. ax c (mod m)
Chapter 23 Squares Modulo p Revised Version of Chapter 23 We learned long ago how to solve linear congruences ax c (mod m) (see Chapter 8). It s now time to take the plunge and move on to quadratic equations.
More informationApproximating functions by Taylor Polynomials.
Chapter 4 Approximating functions by Taylor Polynomials. 4.1 Linear Approximations We have already seen how to approximate a function using its tangent line. This was the key idea in Euler s method. If
More informationWHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT?
WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT? introduction Many students seem to have trouble with the notion of a mathematical proof. People that come to a course like Math 216, who certainly
More informationSwitching Algebra and Logic Gates
Chapter 2 Switching Algebra and Logic Gates The word algebra in the title of this chapter should alert you that more mathematics is coming. No doubt, some of you are itching to get on with digital design
More informationExcel s Business Tools: WhatIf Analysis
Excel s Business Tools: Introduction is an important aspect of planning and managing any business. Understanding the implications of changes in the factors that influence your business is crucial when
More informationProgressions for the Common Core State Standards in Mathematics (draft)
Progressions for the Common Core State Standards in Mathematics (draft) cthe Common Core Standards Writing Team 2 April 202 K 5, Number and Operations in Base Ten Overview Students work in the baseten
More informationCritical analysis. Be more critical! More analysis needed! That s what my tutors say about my essays. I m not really sure what they mean.
Critical analysis Be more critical! More analysis needed! That s what my tutors say about my essays. I m not really sure what they mean. I thought I had written a really good assignment this time. I did
More information1. Graphing Linear Inequalities
Notation. CHAPTER 4 Linear Programming 1. Graphing Linear Inequalities x apple y means x is less than or equal to y. x y means x is greater than or equal to y. x < y means x is less than y. x > y means
More informationINTRODUCTION FRAME WORK AND PURPOSE OF THE STUDY
STUDENTS PERCEPTIONS ABOUT THE SYMBOLS, LETTERS AND SIGNS IN ALGEBRA AND HOW DO THESE AFFECT THEIR LEARNING OF ALGEBRA: A CASE STUDY IN A GOVERNMENT GIRLS SECONDARY SCHOOL KARACHI Abstract Algebra uses
More informationInteraction effects between continuous variables (Optional)
Interaction effects between continuous variables (Optional) Richard Williams, University of Notre Dame, http://www.nd.edu/~rwilliam/ Last revised February 0, 05 This is a very brief overview of this somewhat
More informationIntroduction to Differential Calculus. Christopher Thomas
Mathematics Learning Centre Introduction to Differential Calculus Christopher Thomas c 1997 University of Sydney Acknowledgements Some parts of this booklet appeared in a similar form in the booklet Review
More informationProbability Review Solutions
Probability Review Solutions. A family has three children. Using b to stand for and g to stand for, and using ordered triples such as bbg, find the following. a. draw a tree diagram to determine the sample
More informationGuide for Texas Instruments TI83, TI83 Plus, or TI84 Plus Graphing Calculator
Guide for Texas Instruments TI83, TI83 Plus, or TI84 Plus Graphing Calculator This Guide is designed to offer stepbystep instruction for using your TI83, TI83 Plus, or TI84 Plus graphing calculator
More informationHow many numbers there are?
How many numbers there are? RADEK HONZIK Radek Honzik: Charles University, Department of Logic, Celetná 20, Praha 1, 116 42, Czech Republic radek.honzik@ff.cuni.cz Contents 1 What are numbers 2 1.1 Natural
More informationMath 2001 Homework #10 Solutions
Math 00 Homework #0 Solutions. Section.: ab. For each map below, determine the number of southerly paths from point to point. Solution: We just have to use the same process as we did in building Pascal
More informationJust want the standards alone? You can find the standards alone at http://corestandards.org/thestandards
4 th Grade Mathematics Unpacked Content For the new Common Core State Standards that will be effective in all North Carolina schools in the 201213 school year. This document is designed to help North
More informationTEACHING ADULTS TO MAKE SENSE OF NUMBER TO SOLVE PROBLEMS USING THE LEARNING PROGRESSIONS
TEACHING ADULTS TO MAKE SENSE OF NUMBER TO SOLVE PROBLEMS USING THE LEARNING PROGRESSIONS Mā te mōhio ka ora: mā te ora ka mōhio Through learning there is life: through life there is learning! The Tertiary
More informationA Modern Course on Curves and Surfaces. Richard S. Palais
A Modern Course on Curves and Surfaces Richard S. Palais Contents Lecture 1. Introduction 1 Lecture 2. What is Geometry 4 Lecture 3. Geometry of InnerProduct Spaces 7 Lecture 4. Linear Maps and the Euclidean
More informationMaximum Range Explained range Figure 1 Figure 1: Trajectory Plot for AngledLaunched Projectiles Table 1
Maximum Range Explained A projectile is an airborne object that is under the sole influence of gravity. As it rises and falls, air resistance has a negligible effect. The distance traveled horizontally
More informationNCTM Content Standard/National Science Education Standard
Title: BASEic Space Travel Brief Overview: This unit introduces the concepts of bases and exponents (or powers) in order to gain a deeper understanding of place value. Students will assume the role of
More informationIntroduction to. Hypothesis Testing CHAPTER LEARNING OBJECTIVES. 1 Identify the four steps of hypothesis testing.
Introduction to Hypothesis Testing CHAPTER 8 LEARNING OBJECTIVES After reading this chapter, you should be able to: 1 Identify the four steps of hypothesis testing. 2 Define null hypothesis, alternative
More information