*I E1* I E1. Mathematics Grade 12. Numbers and Number Relationships. I Edition 1

Transcription

1 *I000-E* I000-E Mathematics Grade Numbers ad Number Relatioships I000 Editio

2 MATHEMATICS GRADE Numbers ad Number Relatioships CONTENTS PAGE How to work through this study uit Learig Outcomes ad Assessmet Stadards Itroductio to this study uit Lesso : Idices (revisio) 7 Lesso : Solvig epoetial equatios (revisio) 9 Lesso : Surds (revisio) 6 Lesso : Logarithms: Basic priciples ad applicatios 60 Lesso : Arithmetic sequeces ad series 7 Lesso 6: Geometric sequeces ad series 0 Lesso 7: Sigma otatio Lesso 8: Compoud icrease ad decrease 6 Lesso 9: Simple ad compoud iterest 89 Coclusio 0 Refereces 0 Please read the Addedum at the ed of this study uit before you start Lesso. Numbers ad Number Relatioships / ICG

3 Writte by: Alle Taylor PRINTING HISTORY st Editio Jauary 00 INTERNATIONAL COLLEGES GROUP (ICG) PTY. LTD Strad Street, Cape Tow, 800, South Africa This documet cotais proprietary iformatio that is protected by copyright. All rights are reserved. No part of this documet may be photocopied, reproduced, electroically stored or trasmitted, or traslated without the writte permissio of ICG. Numbers ad Number Relatioships / ICG / (ii)

4 HOW TO WORK THROUGH THIS STUDY UNIT Welcome to Mathematics Grade : Numbers ad Number Relatioships. We have desiged the lessos i this uit i such a way that each builds o the kowledge gaied from the previous lessos. Therefore, it's essetial that you esure that you have a good uderstadig of each lesso. It's also importat that you work through the lessos i the order i which we preset them i the uit. The icos used i this study uit Read the descriptios of icos below. Look out for these icos as you work through the study uit. They will show you at a glace where you eed to work through activities, defiitios, self-assessmet questios, ad so o. Ico Descriptio Learig Outcomes: This sigals the Learig Outcomes of the uit. You must be able to show competece i the Outcomes after you have worked through the study uit. Competece meas that you must be able to demostrate that you ca meet a Outcome with skill ad kowledge. Defiitio: This sigals a importat defiitio that you should uderstad ad remember. Mathematical formula: This is a formula that you will use i calculatios. It is importat that you kow what values to substitute ito the formula. Importat statemet: This sigals a importat poit that you must grasp before you cotiue with the rest of the lesso. It could also sigal a iterestig sippet of iformatio. Activity: This sigals a activity that you should complete as part of your learig. We have iterspersed activities throughout your lessos. Self-assessmet questios: This sigals the questios that will help you to aalyse your uderstadig of the theory that was covered i the lesso. We coclude each lesso with a set of self-assessmet questios. Aswers to self-assessmet questios: This sigals the suggested aswers to the self-assessmet questios. Please do ot look at the aswers before you have tried to aswer the questios yourself. Competece checklist: This sigals a checklist to help you discover whether you ca meet all the Assessmet Stadards i the lesso. Numbers ad Number Relatioships / ICG / Page

5 The best way to study To esure that you get the full beefit of this study uit, we recommed that you do the followig: Carefully read the et sectio. It provides you with the Learig Outcomes ad Assessmet Stadards for each of the lessos i the study uit. Carefully ad diligetly work through each lesso, keepig i mid the Outcomes that you have to achieve. Esure that you complete all the activities i the lessos. Esure that you aswer all the self-assessmet questios i the lessos. Compare your aswers to the aswers that we provide i this study uit. If you ecouter ay words that you do ot uderstad, make a list of them, ad the either look them up i a dictioary or ask your tutor for their meaigs. Numbers ad Number Relatioships / ICG / Page

6 LEARNING OUTCOMES AND ASSESSMENT STANDARDS Numbers ad Number Relatioships has bee developed accordig to the Natioal Curriculum Statemet subject guidelies for Mathematics Grade. We also icluded the Natioal Curriculum Statemet subject guidelies for Mathematics Grade as revisio as this will also be assessed i the fial eamiatio. Read through the lists below. Your aim is to complete the Learig Outcome ad Assessmet Stadards for this uit successfully. I this uit, you will complete oe Learig Outcome ad si Assessmet Stadards for Grade (Revisio) ad five Assessmet Stadards for Grade. You eed to have this iformatio because you must keep checkig your learig goals. Learig Outcomes ad Assessmet Stadards After you have worked through the cotet covered i this study uit, you should be able to do all the tasks we have listed below: Learig Outcome : Numbers ad Number Relatioships Whe solvig problems, the learer is able to recogise, describe, represet ad work cofidetly with umbers ad their relatioships to estimate, calculate ad check solutios. Assessmet Stadards (Grade Revisio).. Uderstad that ot all umbers are real. (This requires the recogitio, but ot the study of o-real umbers).. (a) Simplify epressios usig the laws of epoets for ratioal epoets. (b) Add, subtract, multiply ad divide simple surds. (c) Demostrate a uderstadig of error margis... Ivestigate umber patters (icludig, but ot limited to those where there is a costat secod differece betwee cosecutive terms i a umber patter, ad the geeral term is therefore quadratic) ad hece: (a) (b) make cojectures ad geeralisatios provide eplaatios ad justificatios, ad attempt to prove cojectures... Use simple ad compoud decay formulae ( A P ( i) ad A P ( i) ) to solve problems (icludig, straight lie depreciatio ad depreciatio o a reducig balace)... Demostrate a uderstadig of differet periods of compoudig growth ad decay (icludig, effective compoudig growth ad decay ad icludig effective ad omial iterest rates)...6 Solve o-routie, usee problems. Numbers ad Number Relatioships / ICG / Page

7 Assessmet Stadards (Grade ).. Demostrate a uderstadig of the defiitio of a logarithm ad ay laws eeded to solve real-life problems (for eample, growth ad decay)... (a) Idetify ad solve problems ivolvig umber patters (icludig, but ot limited to) arithmetic ad geometric sequeces ad series. (b) Correctly iterpret sigma otatio. (c) Prove ad correctly select the formula for, ad calculate the sum of series, icludig: i i i i ( + ) i a + ( i ) d a r i [ a + ( ) d] a( r ) ; r r i a r i a ; < r > r (d) Correctly iterpret recursive formulae (for eample, T + T + T )... (a) Calculate the value of i the formula A P (± i). (b) Apply kowledge of geometric series to solvig auity, bod repaymet ad sikig fud problems, with or without the use of the formulae: F ( ( i) + ) ( (+ i) ) ad P i i.. Critically aalyse ivestmet ad loa optios, ad make iformed decisios as to the best optio(s), icludig, pyramid ad micro-leders schemes...6 Solve o-routie, usee problems. Numbers ad Number Relatioships / ICG / Page

8 INTRODUCTION TO THIS STUDY UNIT I this study uit, Mathematics Grade : Numbers ad Number Relatioships, we'll cover Learig Outcome, as outlied i the Natioal Curriculum Statemet for Mathematics. Learig Outcome reads as follows: Whe solvig problems, the learer is able to recogise, describe, represet ad work cofidetly with umbers ad their relatioships to estimate, calculate ad check solutios. We will break the Learig Outcome Statemet i this uit ito differet lessos. Some of the lessos iclude revisio of Grade work, ad they will be highlighted as such. I this study uit, you'll lear about umbers, ad the relatioships betwee umbers. The first lesso is a revisio of Grade topics, ad deals with idices or epoets. We will revise the laws of idices. We will also revise surds. The we'll itroduce you to the cocept of a logarithm. Here, you'll lear how to apply the laws of idices to logarithms. We will discuss the relatioship betwee idices ad logarithms, ad eplai whe it is ecessary to use logarithms. The secod task is for you to apply what you have leart about umbers ad umber relatioships to problems. You'll lear how to estimate ad calculate aswers to problems, both with ad without a calculator. Specifically, we'll look at problems related to busiess trasactios like bods, hire purchase ad auities. This will help you to develop the ability to eplore real-life mathematical problems. A basic cocer for ayoe resposible for maagig moey is to determie the future value of curret ivestmets. The oly reaso that you ivest moey is so that it will be worth more i the future. The time value of moey refers to the guaratee that R today is worth more tha R at some time i the future. I this uit, you will lear how ad why moey is worth more i the future by discussig the effect of iterest. We will eplore simple ad compoud iterest, which is revisio of Grade work, i order to make sese of: effective ad omial iterest rates; ad appreciatio ad depreciatio. Iterest refers to a charge made for the use of someoe else's moey. Whe we borrow moey from other people or istitutios, the they charge us iterest. However, whe we ivest moey with a fiacial istitutio, the they pay us iterest. Baks usually charge iterest as a percetage of the amout origially borrowed or ivested. You should read your guide before you start workig through this study uit. Your guide tells you whe ad how to work through this study uit. Your guide refers to this study uit i the study schedule, ad at the start of the relevat study sessio. This study uit is your prescribed learig material. Numbers ad Number Relatioships / ICG / Page

9 To esure the best learig eperiece, it is very importat that you do all the activities ad aswer all the self-assessmet questios that appear i this study uit. To help you remember ad to esure that you have read all iitial eplaatios ad defiitios it is a good idea to highlight the key words or formulas especially i log eplaatios. Numbers ad Number Relatioships / ICG / Page 6

10 LESSON : INDICES (REVISION) Learig Outcomes for Lesso After you have worked through Lesso, you should be able to do the followig: LO LO AS.. Assessmet Stadard Uderstad that ot all umbers are real. (This requires the recogitio, but ot the study of o-real umbers) AS.. (a) Simplify epressios usig the laws of epoets for ratioal epoets. AS..6 Solve o-routie, usee problems. AS..6 Solve o-routie, usee problems. We would like to help you to remai i complete cotrol of your studies. So we give you a opportuity to check your competece at the ed of each lesso. We've provided you with a checkbo to tick at the ed of the lesso. Whe you tick the Assessmet Stadards, you'll see what you've achieved i each lesso! Itroductio Lesso is all about idices. After completig this lesso, you will: kow the four basic defiitios of idices; kow the four laws of idices; be able to apply the defiitios ad laws without usig a calculator; ad be able to simplify epressios cotaiig idices. If you multiply a umber, say, twelve times, we write it as follows:. As you ca see, it takes up a lot of space to write the epressio. Mathematicias have foud a easier ad shorter way to write this epressio, by usig idices. Idices are a eample of how mathematicias have foud a clever way of epressig comple epressios i a relatively simple way. This ide otatio eables you to simplify mathematical epressios, ad to solve complicated equatios. The mathematical epressios that we'll work with i this lesso will cosist of either oe term or will be a polyomial. A polyomial is a mathematical epressio that has more tha oe term. Numbers ad Number Relatioships / ICG / Page 7

11 You'll lear how to write epressios cotaiig oly oe term as factors of prime umbers. This helps to simplify the epressio by usig the rules of idices. You will also lear how simplify epressios cotaiig a polyomial, by: workig with each term idividually i the polyomial (if there are o variables i the epressio); or factorisig the polyomial, if there are variables i the epressio. Let's begi by eamiig the basic priciples of idices i more detail. Basic priciples of idices We also refer to a ide as a epoet. The plural of ide is idices. If is a positive iteger, ad is ay real umber, the we defie ( to the power ) as follows:... ( factors). The followig are eamples of epressios cotaiig a base ad a ide: [base is ; ide is ] 8 [base is ; ide is ] 7 a a a a a a a a [base is a; ide is 7] Idices have certai laws that ca be applied to simplify epressios. Let's work through these laws of idices to make sure that you are familiar with the laws, ad that you kow how to apply them. The laws of idices There are four laws of idices. The four laws relate to the followig situatios: the multiplicatio of powers that have the same base; the divisio of powers that have the same base; idices separated by brackets; ad the power of a product. Let's look at each of these laws i more detail. The multiplicatio of powers that have the same base a m a a m + As you ca see, whe multiplyig two powers that have the same base, add the idices together. Numbers ad Number Relatioships / ICG / Page 8

12 Let's look at a few eamples of how to apply this law. EXAMPLE Simplify the followig epressios without usig a calculator: Aswers The divisio of powers that have the same base a a m a m Whe dividig two powers that have the same base, subtract the ide of the deomiator from the ide of the umerator. Let's look at a few eamples of how to apply this law. EXAMPLE Simplify the followig epressios without usig a calculator:... 7 a + a Numbers ad Number Relatioships / ICG / Page 9

13 Aswers a + a a + (a ) a + a + 7 Idices separated by brackets ( a m ) a m Wheever there are idices separated by brackets, we must multiply these idices together. Let's look at a few eamples of how to apply this law: EXAMPLE Simplify the followig epressios without usig a calculator:. ) ( +. ( ). ( a + ) Aswers. ) ( Alteratively: ( ) ( ) ( ) ( ) + + Numbers ad Number Relatioships / ICG / Page 0

14 +. ( ) ( ) ) ( a a ( )( + ) 6 a The power of a product m ( ab ) a m b m The power (m) of a product (ab) is equal to the product of the powers of the factors. ( a m b m ) Let's look at a few eamples of how to apply this law: EXAMPLE Simplify the followig epressios without usig a calculator:. (6). ( y z ) Aswers. (6) ( ) ( ). ( y z ) y y 6 z z Additioal defiitios I additio to the basic defiitio of a ide ad a base give above, three special circumstaces eed further eplaatio ad defiitio. These additioal defiitios cover situatios i which a epressio has a ide that is: zero; a egative umber; or a ratioal umber. Numbers ad Number Relatioships / ICG / Page

15 I our earlier defiitio of a ide, we specified that must always be a positive iteger. If this is the case, the what do the followig epressios mea? 0 a ; m a ; ad m a. A epressio that has a ide of 0 If a is ay o-zero real umber, the 0 a is always equal to. Eamples of epressios that have 0 as a ide iclude: ( a + b) () 0 We ca prove that 0 a m a m 0 a is equal to, as follows: a a m m The epressio 0 0 is udefied. I other words, a caot be equal to 0. A epressio that has a ide that cotais a egative umber If is a egative umber, the we ca rewrite the epressio so that becomes positive. If a is ay o-zero real umber, the: Ad: m a m a m a m a You ca always use this rule to write a epressio with a positive ide. Let's look at a few eamples to help make this clear. Numbers ad Number Relatioships / ICG / Page

16 EXAMPLE Simplify the followig epressios without usig a calculator:..... a m m 0 a Aswers. 8. a a. m 0 0 m m 7 m m 7.. a a A epressio that has a ide that is a ratioal umber The third special case that we idicated earlier is oe i which the ide of a m epressio is a ratioal umber, amely. Note that both m ad are real umbers, ad a ad must both be greater tha zero (positive values). m a a m (m, Ζ, > 0, a > 0) Numbers ad Number Relatioships / ICG / Page

17 The th root of a umber b, writte as b, is a umber c so that b b c b. To write m a i epoetial form, a is the base; the ide is a fractio where the umerator is the ide of a (which is ) ad the umerator is the root m (which is m). a. The square root ( a a ) is the secod root of a umber. Therefore: The followig eamples show how to apply this formula. EXAMPLE Simplify the followig epressios without usig a calculator:... a b Aswers. ( ). ( ) Numbers ad Number Relatioships / ICG / Page

18 . a b a b a b, which ca also be writte as.a. b Now, let's do a few eamples where we apply all the laws together. EXAMPLE Simplify the followig epressios without usig a calculator:. ( ). (k).. 0 ( 07). (a) 6. a 7. ( 8) 8. () 9. ( y 8 ) ( a b ) ( b a ). 8 Aswer. ( ) ( ) (). (k) [() (k) ] [8 k ] 8k. ( 07) 0 Numbers ad Number Relatioships / ICG / Page

19 .. (a) (a ) (a ) (a ) or ( ) ( a ) a 6. a a a 7. ( 8) 8 alteratively ( ) (usig prime factors) 6 or 6 or 6 or 8. () () () or ( ) (usig prime factors) ( y 8 ) 8 y 6 0 y Numbers ad Number Relatioships / ICG / Page 6

20 ( ) ( a b ) ( b a ) [( ) (a ) (b 6 ) ] [( ) (a ) (b ) ] (a 8 b ) ( 8a 9 b 9 ) ( 8) (a 8+9 ) (b +9 ) 7 00a b. 8 8 ( ) 7 Now work carefully through the followig activity to practise what you have leart about idices. Activity Simplify the followig epressios (if possible) without usig a calculator. Write dow all aswers with positive idices a 6b 8. ( + y ) Numbers ad Number Relatioships / ICG / Page 7

21 Aswer ( ) ( ) (prime factors) (apply law ) a 6b 8 a 8 b 8 (Prime factors) ( ( ) 8 ) ( a ( b ) 8 ) (apply law ) a b (apply law ) b a (write with positive idices) b a. ( + y ) You caot simplify the epressio ( y ) +. Note that the laws of idices do ot make provisio for additio ad subtractio. Numbers ad Number Relatioships / ICG / Page 8

22 Simplifyig epressios cosistig of oe term I the first part of this lesso, you leart about the defiitios ad laws of idices. I the last two sectios of the lesso, we'll look at usig the defiitios ad laws to simplify epressios that are more comple. I this sectio, we focus specifically o epressios cosistig of oe term oly. Keep the followig guidelies i mid whe simplifyig epressios: Covert all fractios or decimals to proper fractios or improper fractios. Always give aswers with positive epoets. Wheever possible, reduce umbers i the epressio to their prime factors, for eample, 7 or () 6. Look for like terms whe addig or subtractig epoets. Look for commo factors whe addig or subtractig terms. Let's work through a eample that uses the laws ad defiitios of idices to simplify epressios cotaiig oe term. EXAMPLE Simplify each of the followig epressios: 9. ( ) ( ) 6 Aswers 9. ( ) 6 9 ( ) 6 ( ) (covert to improper fractios) (write as positive idices) 0 7 Numbers ad Number Relatioships / ICG / Page 9

23 . 6 ) ( (use prime factors) (simplify the brackets) ( ) 8 (covert from surd to epoetial form) (remember that ). ( ) (covert to improper fractios) (use prime factors) 6 (simplify the brackets) 6 (simplify the brackets) 6 79 I the fial sectio of this lesso, we'll demostrate how to simplify epoetial epressios. Simplificatio of epoetial epressios Ofte, the most importat step i the case of epoetial epressios is to break up all the bases ito powers of prime factors. After that, you ca apply the laws of idices by workig systematically, oe step at a time. Let's work through a few eamples. Numbers ad Number Relatioships / ICG / Page 0

24 Numbers ad Number Relatioships / ICG / Page EXAMPLE Simplify the followig epressios (without usig a calculator) by applyig the laws of idices. Write all your aswers with positive idices Aswers ( ) ( ) ( ) (apply law ) ) ( ) ( ) ( ) ( (apply laws ad ) (write as a positive ide) (apply law i reverse order) ( ) ) ( (factorise) 6 (cacel equal factors) or (Hit: The deomiator ad umerator cosist of factors oly) (Hit: The deomiator ad umerator cosist of terms) (Hit: The deomiator ad umerator cosist of factors oly)

25 (. ). (factorise).... (apply laws ad ) +. (arrage terms with the same bases ad apply laws ad ). (simplify) (covert to positive idices) Now work carefully through the followig activity to practise what you have leart about simplifyig epressios without usig a calculator. Activity Simplify the followig epressios without usig a calculator. Give all aswers with positive idices ( ) 7 a y 9. a 7 a + +. a + a a + a. + y + y Aswers 6 9. ( 7 y ) 6 9 ( y ) (prime factors) ( ) 6 ( ) 9 ( ) y (Apply law ) 6 y (Apply law ) 9y 6 (Write with positive idices) Numbers ad Number Relatioships / ICG / Page

26 Numbers ad Number Relatioships / ICG / Page a a a a a a (Write i epoetial form) a a a (Prime factors; apply law ) ( ) 6 ) a ( a a (Apply laws ad ) ( ) (Apply law ) (Write with a positive ide). + + a a a a + a a a a (law i reverse order). ) ( + a a. ) ( + (cacel a ). ) 6 ( + 8. ) 6 ( (take out commo factor of a i umerator)

27 + y. + y + y + y Write as positive idices Simplify the deomiator + y y + y Remember that y + + y + y y + y y Summary I Lesso, you leart about idices, which was revisio of Grade topics. The first part of the lesso itroduced you to the basic cocepts of idices, icludig: the four defiitios of idices; ad the four laws of idices. The secod part of the lesso focused o applyig the laws ad defiitios to simplifyig mathematical epressios cotaiig idices. Here, you leart how to simplify epressios cotaiig: oly factors; oly oe term; ad epressios cotaiig polyomials (more tha oe term). Remember to use the followig guidelies whe decidig o how to simplify a epressio: Oly work with proper fractios or improper fractios. I other words, always covert decimal values or mied umbers to fractios before tryig to simplify the epressio. Always give aswers with positive epoets. If the epressio cotais oe or more terms, the you may first eed to factorise the epressio. Wheever possible, reduce umbers i the epressio to their prime factors. This may ofte help you to simplify a epressio by addig or subtractig epoets for values that have the same base. You may also fid that both the umerator ad deomiator have factors i commo, which you ca the cacel out. Look for like terms whe addig or subtractig epoets. Look for commo factors whe addig ad subtractig terms. Numbers ad Number Relatioships / ICG / Page

28 Self-assessmet Questios Test your kowledge of this lesso by completig the self-assessmet questios below. Whe you aswer the questios, do't look at the suggested aswers that we give straight away. Look at them oly after you've writte your aswers dow, ad the compare your aswers with our aswers. (Complete the assessmet first. If you are ot satisfied, redo the assessmet.) Simplify the followig epressios without usig a calculator. 8.. ( 0,006). ( y ). ( ) 8. 6a b 9ab a + a ( ) 7. a a p + p + p p Suggested aswers to Self-assessmet Questios 8. ( ( ) ) Numbers ad Number Relatioships / ICG / Page

29 Numbers ad Number Relatioships / ICG / Page 6. ( ), ( ) y ( ) y ( ) ( ) ( ).. y y 0 0 y 0 y. ( ) If we look at the top lie: The deomiators simplify: +

30 Numbers ad Number Relatioships / ICG / Page ab b a ab b a b a b a Simplify the factors i a : a a a a a ab b a ( ) ( ) ( ) ) ( ( ) a a a a a a a a a Simplify: ( ) ( ) a a a a a a a a (Note that terms are separated by a egative sig, so we caot add the ide. We take out a commo factor.)

31 8. + ( ) 9 7 ( ) p + p + p p p p ( ) ( + ) Check your competece Now that you have worked through this lesso, please check that you ca perform the tasks below: I uderstad that ot all umbers are real. I ca simplify epressios usig the laws of epoets for ratioal epoets. I ca solve o-routie, usee problems. The et lesso If you are sure that you uderstad the cotets covered i this lesso, ad have achieved your Assessmet Stadards, start Lesso. Numbers ad Number Relatioships / ICG / Page 8

32 LESSON : SOLVING EXPONENTIAL EQUATIONS (REVISION) Learig Outcomes for Lesso After you have worked through Lesso, you should be able to do the followig: LO LO AS.. Assessmet Stadard Uderstad that ot all umbers are real. (This requires the recogitio, but ot the study of o-real umbers) AS.. (a) Simplify epressios usig the laws of epoets for ratioal epoets. AS..6 Solve o-routie, usee problems. AS..6 Solve o-routie, usee problems. We would like to help you to remai i complete cotrol of your studies. So we give you a opportuity to check your competece at the ed of this lesso. Itroductio I Lesso, we'll discuss how to solve epoetial equatios. I particular, we'll look at two types of the followig form: equatios with the ukow i the base, for eample 6; ad equatios with the ukow i the ide, for eample 8. Equatios with the ukow i the base I this sectio, we'll deal with the first type of epoetial equatios, amely equatios with the ukow i the base, for eample 6. We will the work through some eamples. Fially, we will ed the lesso with a activity. Let's begi! To solve a equatio, we say, for eample: or: a We ote that both idices of the ukow is. Therefore, the aim is to get the ide of the ukow equal. Numbers ad Number Relatioships / ICG / Page 9

33 To achieve this, we multiply the ide by the opposite of the ide of the ukow. The opposite of the ide is also kow as the reciprocal of the ide of the ukow. So, if we multiply the ide with its opposite, we will get. Accordig to the rules of mathematics, if we multiply the left-had side (LHS) of the equatio with the opposite of its ide, the we also eed to multiply the right-had side (RHS) of the equatio with that same ide. For eample, the opposite of the value is. Similarly, the opposite of is (also called the reciprocal or iverse.) The opposite of is. To check if you determied the correct opposite of a ide, the product of the ide ad its opposite are always equal to. For eample: Remember, all idices that are writte as improper fractios must be coverted to proper fractios. Now let's work through a few eamples to see how this helps us to solve epoetial equatios that have a ukow i the base. Solve for without usig a calculator:. 6 EXAMPLE. 7 8 Aswers. Here, the ukow is, ad the ide of the ukow is. Therefore, the opposite of the ukow is. Numbers ad Number Relatioships / ICG / Page 0

34 Therefore, we have: 6 ( 6) (raise both sides to the power of ) ( 6) (apply law o the LHS) 6 ( ) (prime factor) 8 (apply law o the RHS) 6. Here, the ukow is, ad the ide of the ukow is Therefore, the reciprocal of the ukow is Therefore, we have: (isolate the ukow) 8 7 (use the reciprocal of the ide) ( ) ( ) (write as a positive ide) (use prime factors) 9 Now work carefully through the followig activity to practise what you have leart about solvig equatios with a ukow i the base. Numbers ad Number Relatioships / ICG / Page

35 Activity Solve for without usig a calculator: Aswer. Here, the ukow is, ad the ide of the ukow is. Therefore, the reciprocal of the ukow is, or. Therefore, we have: (isolate the ukow) ( ) or ( ) 6 6. Here, the ukow is, ad the ide of the ukow is Therefore, the reciprocal of the ukow is Therefore, we have: (isolate the ukow) 8 Numbers ad Number Relatioships / ICG / Page

36 ( ) () (8) 6 Equatios with the ukow i the epoet I the previous sectio, you leart how to solve equatios that had the ukow i the base. I this sectio, we'll solve what is kow as epoetial equatios. The key to solvig epoetial equatios lies i the followig priciple: If m a a, the m, provided that a 0 ad a ±. I other words, if the bases are equal, the the epoets of the base should also be equal. Therefore, to solve epoetial equatios, you'll eed to follow these steps: Step : Rewrite the bases o the LHS ad the RHS of the equatio so that they are the same. Step : If the base o LHS of the equatio is equal to the base o the RHS of the equatio, the bases ca be igored such that the idices are left over. The we solve the idices usig methods of equatios. Step : Always check the calculated solutio i the origial equatio to esure the solutio is valid. Numbers ad Number Relatioships / ICG / Page

37 The most effective way to make the bases o both sides of the equatio equal is to epress the values usig prime factors. Let's work through a few eamples to see how to solve epoetial equatios with the ukow i the epoet. Solve for without usig a calculator:. 6 EXAMPLE.. ( - ) 8 0 0,. ( ) + ( ). ( ) Aswers. 6. ( - ) 8 ( ) Therefore: ( ) Therefore: or 0 ( + )( ) 0 To check if the solutios are valid, we substitute the values of i the origial equatio to check if the LHS RHS. If the LHS RHS, the the solutios are valid. Numbers ad Number Relatioships / ICG / Page

38 Therefore, we have: For : LHS: ( ) Ad for : 8 RHS LHS: () 8 RHS Therefore, both solutios are valid.. 0, 0 The first step is to epress 0, as a fractio ad the to try to write the fractio as a base of a whole umber. Now 0, 0. As a base of a whole umber 0 becomes 0. Therefore, the origial equatio ca be rewritte as: 0 0 Therefore: Agai, let s check the validity of the solutio: For : LHS: () RHS Therefore, the solutio is valid. Numbers ad Number Relatioships / ICG / Page

39 . ( ) + - ( ) ( + - ) (take out a commo factor of ) + (simplify the brackets) (simplify the brackets) (multiply both sides by ) 6 Let s check the validity of the solutio i the origial equatio: LHS: ( - ) + - ( - ) RHS Therefore, the solutio - is valid.. ( ) + 0 ( ) ( ) + 0 ( ) ( ) + 0 ( ) ( ) + 0 If we substitute a a + 0 ( a )( a ) 0 a, the the equatio becomes: a or a Numbers ad Number Relatioships / ICG / Page 6

40 Remember, we substituted solutios a or a. a. So ow we substitute a back i the Also ote that we eed to solve for ad ot a. So, if a, our first solutio becomes: Ad for a, we have: 0 0 Fially, we oly eed to check the validity of the solutios by substitutig our solutio for i the origial equatio ad check if the LHS RHS. So, for, we have: LHS: ( ) + Ad, for 0, we have: LHS: ( ) + ( ) RHS 0 ( 0 ) + () RHS Therefore, both solutios for are valid Let 7 a, the our equatio becomes: a 8 + a 7 0 Numbers ad Number Relatioships / ICG / Page 7

41 If we multiply the equatio by a, we get a 8a ( a 7)( a ) 0 a 7 or a So ow we substitute 7 a back i the solutios a 7 or a. So if a 7, the our first solutio becomes: Ad for a, we have: We oly eed to check the validity of the solutios by substitutig our solutio for i the origial equatio ad check if the LHS RHS. So, for, we have: LHS: Ad for 0, we have: RHS Therefore, both solutios for are valid. Now work carefully through the followig activity to practise what you have leart about solvig epoetial equatios. Numbers ad Number Relatioships / ICG / Page 8

42 Activity Solve for without usig a calculator: ( ).. 0, Aswers. 8 8 ( ) 7 (make the bases the same) (or ) 6 (if the bases are the same, the idices are equal). ( ) Therefore: 0 ( )( + ) 0 or You eed to check these possible solutios i the origial equatio. Therefore, for, you have: LHS: ( ) ( ) ( ) Therefore, is valid. 7 RHS Numbers ad Number Relatioships / ICG / Page 9

43 Ad, for, you have: LHS: ( ) ( ) ( ) Therefore, is valid, ad both solutios for are valid.. 0,7 0, (divide both sides by ) 8 0, Therefore,.. 8 ( 8 ) (remove commo factor) (multiply both sides by ) 8 Therefore, ( ) 8 ( ) (multiply both sides by ) ( ) ( ) 8 0 ( ) ( ) 8 0 ( 8)( + ) 0 Put k k k k 8 0 ( + )( k 8) k 0 k 8 0 or k 8 0 k - or k 8 Numbers ad Number Relatioships / ICG / Page 0

44 Therefore, there are two possible solutios to the equatio, both of which are epoetial equatios. The first solutio is: 8 Therefore: The secod solutio is: The secod solutio is ivalid, because > 0 for all values of. Remember that the ide must always be a positive umber. Therefore, the fial solutio to the origial equatio is oly. Summary I Lesso, we eamied how to solve equatios that cotai idices. You leart how to solve two types of equatios, amely: equatios that cotai a ukow i the base; ad equatios that cotai a ukow i the ide. You also leart that equatios that have a ukow i the ide (or epoet) are also kow as epoetial equatios. Sometime more tha oe law ca get you to your aswer. Do the oe that is easier for you. We solve equatios that cotai a ukow i the base by raisig both sides of the equatio to the reciprocal of the ide of the ukow. We solve equatios that cotai the ukow i the ide by applyig the priciple m that if a a, the m, provided that a 0 ad a ±. Self-assessmet Questios Test your kowledge of this lesso by completig the self-assessmet questios below. Whe you aswer the questios, do't look at the suggested aswers that we give straight away. Look at them oly after you've writte your aswers dow, ad the compare your aswers with our aswers. Solve for without usig a calculator:.. 0 Numbers ad Number Relatioships / ICG / Page

45 ( + ) Suggested aswers to Self-assessmet Questios (7) (7) ( ) 9 Numbers ad Number Relatioships / ICG / Page

46 Put k k k 6k + 0 ( )( k ) k 0 k 0 or k 0 k k or k Therefore, there are two possible solutios to the equatio. The first possible solutio is: 0 The secod possible solutio is: 0 (). 8 8 Therefore: (if the bases are the same, the idices are equal) Numbers ad Number Relatioships / ICG / Page

47 . ( + ) + + ( ) + ( ) (multiply each term by ) ( ) + ( ) 0 ( ) + ( ) (multiply each term by ) If you let a 0 a + a a + a 0 0 (a )(a + 0) a or a 0 Therefore, there are two possible solutios. The first possible solutio is: Therefore,. The secod possible solutio is: 0 Sice must be positive, there is o value of that will satisfy the secod solutio. Therefore, the oly valid solutio to the equatio is. Numbers ad Number Relatioships / ICG / Page

48 Check your competece Now that you have worked through this lesso, please check that you ca perform the tasks below: I uderstad that ot all umbers are real. I ca simplify epressios usig the laws of epoets for ratioal epoets. I ca solve o-routie, usee problems. The et lesso If you are sure that you uderstad the cotets covered i this lesso, ad have achieved your Assessmet Stadards, start Lesso. Numbers ad Number Relatioships / ICG / Page

49 LESSON : SURDS (REVISION) Learig Outcomes for Lesso After you have worked through Lesso, you should be able to do the followig: LO LO AS.. Assessmet Stadard Uderstad that ot all umbers are real. (This requires the recogitio, but ot the study of o-real umbers) AS.. (a) Simplify epressios usig the laws of epoets for ratioal epoets. (b) (c) Add, subtract, multiply ad divide simple surds. Demostrate a uderstadig of error margis. AS..6 Solve o-routie, usee problems. AS..6 Solve o-routie, usee problems. As before, we would like to help you to remai i complete cotrol of your studies. So, oce agai, we provide a checkbo at the ed of this lesso. Itroductio Lesso is all about surds. By the ed of this lesso, you'll kow eactly what a surd is, ad how to work with surds i mathematical calculatios. There are two mai sectios to the lesso, amely: the basic priciples of surds; ad calculatios with surds. Let's begi! The basic priciples of surds Before we ca defie a surd, we eed to kow the differece betwee a ratioal umber ad a irratioal umber. A ratioal umber is ay real umber that we ca epress as a ratio of two itegers. Eamples of ratioal umbers iclude ad. Numbers ad Number Relatioships / ICG / Page 6

50 A irratioal umber is ay real umber that is ot a ratioal umber. A eample of a irratioal umber is. Most real umbers are irratioal umbers. Now, let's defie a surd. A surd is a irratioal umber i the form a surds as radicals. m. We sometimes also refer to Therefore, 8 is ot a surd, because 8, which is a ratioal umber. However, 8 is a surd. Fially, recall from Lesso that a m a. Now let's have a look at the four surd laws. Surd law a b ab, (a, b m + Ζ, N) For eample: Ad: Surd law 7 9 For eample: a a, (a, b + Ζ, N) b b Ad: Numbers ad Number Relatioships / ICG / Page 7

51 Surd law ( ) a m a m, (a + Ζ, m, N) For eample: Ad: ( 9) ( 9 ) ( ) Surd law m a m a, (a + Ζ, m, N) For eample: ( 66) 8 ( 8 ) 8 Ad: ( 79) 6 ( 6 ) 6 I the followig sectio, you will lear how to apply the surd laws whe solvig problems that iclude surds. Numbers ad Number Relatioships / ICG / Page 8

52 Calculatios with surds I this sectio, we'll eplai how to work with surds. Specifically, we'll discuss the three operatios that you'll ecouter most ofte whe workig with surds. These operatios eable you to: simplify surds usig prime factors; make use of the order of a surd; ad ratioalise a deomiator cotaiig surds. Simplify surds usig prime factors Surds ca be simplified usig prime factors. To do so, reduce each value i the epressio to prime factors, ad the look for ways to group these factors together. These groupigs ofte eable you to simplify the overall mathematical epressio. Let's work through a few eamples to see how to simplify epressios cotaiig surds. Simplify without usig a calculator: EXAMPLE Aswers. 6 (prime factors) ( ) () (group as cubes) (prime factors) 7 7 Numbers ad Number Relatioships / ICG / Page 9

53 7 (simplified surds) 7 (cacel equal factors) (simplify) (or, takig out a commo factor of we get ( 6 0) + (0) 0) The order of a surd Remember that a surd is a irratioal umber i the form a m. I this form, is the order of the surd. If two surds have the same order, the they must have the same value for. For eample,, 9 ad 7 are surds of the same order. Specifically, these surds are of order. Oe importat use of the order of a surd is that it eables you to arrage them i either ascedig order or descedig order of value Let s look at the et eample shows. EXAMPLE Arrage the followig three surds i descedig order: ; ; Aswers To arrage the surds i descedig order, we eed to: first rewrite them i epoetial form; the rewrite each epoet where the deomiators are the same; ad the covert each epoetial form back to surd form. Numbers ad Number Relatioships / ICG / Page 0

54 Let s epress each oe i epoetial form: ( ) ; ( ) ; ad ( ). Net, we eed to chage the fractios so that they all have the same deomiator. The lowest commo multiple (LCM) of the deomiators of the idices, ad is. Therefore, we eed to rewrite all the idices with deomiator, ad the rewrite the epoetial epressio back to surd form: ( ) 6 ( ) 6 6 ( ) ( ) ( ) ( ) 6 Now, sice the order of each of the surds is the same, we ca directly compare the values represeted by m i the geeral form of a surd. Therefore: 6 > > 6 Ad, if we epress the surds i their origial form, we have: > >. Numbers ad Number Relatioships / ICG / Page

55 Ratioalise a deomiator cotaiig surds Sice a surd is a irratioal umber, ay umber epressios that iclude a surd are also irratioal. Therefore, epressios such as ad have irratioal deomiators. To ratioalise a deomiator of a surd, it meas the deomiator cotais o surds. There are essetially two situatios that you may fid whe tryig to ratioalise a deomiator, amely: the deomiator cotais oe term oly; or the deomiator cotais more tha oe term. If the deomiator oly cotais a sigle term, the it ca be ratioalised by multiplyig the epressio by: deomiator deomiator However, if there is more tha oe term i the deomiator, the the epressio eeds to be multiplied by: sum or differece of two terms of the deomiator sum or differece of two terms of the deomiator What this mea is if the deomiator cotais a sum of two terms, the the epressio eeds to be multiplied by: differece of two terms of the deomiator differece of two terms of the deomiator Coversely, if the deomiator cotais a differece of two terms,the the epressio eeds to be multiplied by: sum of two terms of the deomiator. sum of two terms of the deomiator For eample, is a eample of the differece of two terms. The the epressio must be multiplied by is the deomiator. + + to ratioalise the deomiator if Similarly, + multiplied by is a eample of the sum of two terms. The the epressio must be to ratioalise the deomiator if + is the deomiator. The umerator remais irratioal. Oly the deomiator must be ratioal. Let s look at followig eamples. Numbers ad Number Relatioships / ICG / Page

56 EXAMPLE Ratioalise the deomiator i each of the followig epressios: Aswers. 7 7 ( ) ( ) Now work carefully through the followig activity to practise what you have leart about workig with surds. Activity. Simplify without usig a calculator: (a) (b) Arrage the followig surds i ascedig order (without usig a calculator): ; 8; 0. Ratioalise the deomiator of the followig epressios: (a) (b) 8 7 Numbers ad Number Relatioships / ICG / Page

57 Aswers. (a) (oe of the factors has to be a perfect square) 6 (cacel out like terms) 6 (b) Rewrite all surd form to epoetial form: ; ; ( ) ; ; ( ) ; () ; () () 8 ; () ; () 8 8 ; ; 8 Now you ca compare the values to each other, ad order them as follows: < 8 8 < Ad epressed i the origial form of each surd, you have: < < Numbers ad Number Relatioships / ICG / Page

58 . (a) There is oly oe term i the deomiator ( ) 8 (take out a commo factor) (b) There are two terms i the deomiator Summary I Lesso, you leart that a surd is a irratioal umber of the form a m. There are four surd laws that we derived directly from the ide laws. These laws are: Surd law : a b ab, (a, b + Ζ, N) a a Surd law :, (a, b + Ζ, N) b b Surd law : ( ) m a m a, (a Surd law : m a m a, (a + Ζ, m, N) + Ζ, m, N) Fially, you leart about some of the ways that you are able to work with surds. Specifically, you leart how to: simplify surds usig prime factors; make use of the order of a surd; ad ratioalise a deomiator cotaiig surds. Numbers ad Number Relatioships / ICG / Page

59 Self-assessmet Questios Test your kowledge of this lesso by completig the self-assessmet questios below. Whe you aswer the questios, do't look at the suggested aswers that we give straight away. Look at them oly after you've writte your aswers dow, ad the compare your aswers with our aswers. Remember, o calculators must be used.. Simplify the epressio ( 0 ).. Simplify ad ratioalise the deomiator of the epressio Simplify the epressio Simplify the epressio Solve for i the followig equatio:. 6. Simplify the epressio Simplify the epressio Solve for i the followig equatio: + 0. Suggested aswers to Self-assessmet Questios. ( 0 ) ( 6 ) ( ) ( ). 8 6 Numbers ad Number Relatioships / ICG / Page 6

60 (simplify) (multiplyig by Therefore: (7 ) 7 7(6) 7 6 Numbers ad Number Relatioships / ICG / Page 7

61 ( ) ( ) (isolate the surd) ( ) (square both sides) ( )( ) 0 Therefore, there are two possible solutios to the equatio: or Test each of the two possible aswers for validity. For, you have: LHS: -( ) + ( ) RHS Therefore, is ot a valid solutio. For, you have: LHS: () + () RHS Therefore, is a valid solutio. Numbers ad Number Relatioships / ICG / Page 8

62 Check your competece Now that you have worked through this lesso, please check that you ca perform the tasks below: I uderstad that ot all umbers are real. I ca simplify epressios usig the laws of epoets for ratioal epoets. I ca add, subtract, multiply ad divide simple surds. I ca demostrate a uderstadig of error margis. I ca solve o-routie, usee problems. The et lesso If you are sure that you uderstad the cotets covered i this lesso, ad have achieved your Assessmet Stadards, start Lesso. Numbers ad Number Relatioships / ICG / Page 9

63 LESSON : LOGARITHMS: BASIC PRINCIPLES AND APPLICATIONS Learig Outcomes for Lesso After you have worked through Lesso, you should be able to do the followig: LO LO AS.. Assessmet Stadard Uderstad that ot all umbers are real. (This requires the recogitio, but ot the study of o-real umbers) AS.. (a) Simplify epressios usig the laws of epoets for ratioal epoets. (b) Add, subtract, multiply ad divide simple surds. (c) Demostrate a uderstadig of error margis. AS..6 Solve o-routie, usee problems. AS.. Demostrate a uderstadig of the defiitio of a logarithm ad ay laws eeded to solve real-life problems (for eample, growth ad decay). AS..6 Solve o-routie, usee problems. Remember to use the checkbo oce you've completed this lesso. You'll fid it right at the ed of the lesso. Itroductio I Lesso, you'll lear about logarithms. We'll begi by defiig a logarithm, ad you'll see how logarithms are closely related to umbers epressed i epoetial form. As with surds, there are laws that gover logarithms. These laws follow directly from the laws of idices, just as was the case with the surd laws. I the last two sectios of the lesso, we'll discuss how to apply the logarithm laws. Specifically, you'll lear how to: simplify epressios without usig a calculator; ad solve epoetial equatios by usig a calculator if the bases caot be made the same. So, the mai sectios of this lesso are as follows: The defiitio of a logarithm The logarithmic laws Simplifyig epressios cotaiig logarithms Solvig epoetial equatios usig logarithms Numbers ad Number Relatioships / ICG / Page 60

64 The defiitio of a logarithm The word 'logarithm' is a little cumbersome, ad so you'll ofte fid it abbreviated to the word 'log'. We'll use the same covetio i this uit as well, ad will use the two words iterchageably. The formal defiitio of a logarithm is as follows: b log b if ad oly if a, for a > 0, a ad > 0. a Let's look more closely at this defiitio. Logarithms use the same termiology as epoetial epressios. The defiitio of a logarithm says that the logarithm of, if the base is a, is the ide or epoet to which a must be raised to produce. For eample, the epressio log 6 meas that log of 6 is the ide to which must be raised to produce 6. I other words, eeds to be raised to the power of i order to produce the value 6. Therefore, the epoetial form of the same epressio is 6. Table shows a few more eamples of umbers epressed i both the logarithmic form ad the epoetial form. TABLE LOGARITHMIC AND EXPONENTIAL FORMS Logarithmic form Epoetial form log log 8 8 log ( ) A egative value for either a or i the defiitio of a log is ot defied. For eample, cosider the epressio log ( 8). There is o power of (or ay positive umber) that will be egative. Therefore, log ( 8) does ot eist. I geeral, the epressio log a is oly defied for values a ad greater tha 0, ad for a ot equal to. Before the geeral availability of calculators, logs were a very importat aid to doig complicated calculatios. Origially, mathematicias always took the base of a log to be 0. These days, we still refer to logs with base 0 as commo logs. If a epressio does ot show the base of a log, the assume that the base is 0. For eample, log 7 is the same as log0 7. Although logarithms do ot play the same role i calculatios as they used to, they are still of fudametal importace i mathematical calculatios. No traiig i basic mathematics will be comprehesive if it does ot iclude the theory of logarithms. I the et sectio, we'll look at the laws that gover logarithms. Numbers ad Number Relatioships / ICG / Page 6

65 The logarithmic laws As metioed i the itroductio to this lesso, the log laws follow directly from the laws of idices. There are four log laws. Log law log ( cd) a a For eample: log c + loga d log 0 log + log However, ote that: Ad: loga ( c + d) loga c + loga log c log d log ( cd) a Log law a c log a loga c loga d d For eample: log 7 0 log 7 00 log 7 However, ote that: Ad: ( c d) log a a a d log c loga d log log a a c d c log a d Log law log a (c t ) t loga For eample: log log log However, ote that: c t (log a c ) t loga c Numbers ad Number Relatioships / ICG / Page 6

66 Log law logb c log log for eample, a a c b log log log 0 0 Now let's see how to apply the logarithmic laws to help simplify mathematical epressios. Simplifyig epressios cotaiig logarithms The log laws eable us to simplify umber epressios that cotai logarithms, or to rewrite them i aother form. The followig eamples illustrate how to apply the log laws whe simplifyig epressios. EXAMPLE Simplify the followig epressios by usig the log laws, ad without usig a calculator:. log + log log. log log log. log () -.. log 9 + log log 6 log log log log 9 6. log 7 + log 6 6 log 7 9 Aswers. log + log log log + log log (log law ) log 9 + log log 8 log 9 8 log 8 (log laws ad ). log log log log (log + log ) log log ( ) (log law ) log 8 (log law ) Numbers ad Number Relatioships / ICG / Page 6

67 . log () log ( ) log () log (Log law ) (Note: loga a, sice a a). log 9 + log log 6 log (9 ) log 6 (log law ) log 6 log 6 log 6 log 6 log 6 log 6 (log law ) (cacel equal factors). There are two ways to solve this problem. Method : log log log log 9 log log log log log log log log log log (log log ) Method : log log log log 9 log log log log log log ( ) ( ) 9 ( ) ( ) ( ) ( ) Numbers ad Number Relatioships / ICG / Page 6

68 6. There are two ways to solve this problem. Method : log 7 + log 6 6 log 7 9 log 7 log + log 6 log 6 log 9 log 7 log + log log 7 log 7 log log + log 7 log 7 + Method : Therefore: log 7 log log log 6 6 log 7 9 log 7 7 log 7 7 log 7 + log6 6 log Now work carefully through the followig activity. Activity 6. Without a calculator, fid the value of the followig epressios: (a) log 0, (b) log (c) log (d) log 0. Rewrite the followig epressios i epoetial form: (a) log 6 6 (b) log 8 Numbers ad Number Relatioships / ICG / Page 6

69 . Rewrite the followig epressios i log form: (a) 9 (b) (c). Simplify without a calculator: (a) (b) log log 6 log log 6 log log 7 log (log 00) 9 (c) log 8 + log 8. If log 0, ad log 0,8, calculate the value of log 0,7. Aswers. (a) log 0, log log 8 () log (b) log log () log (c) log 0 (sice 0 0 ) (d). (a) log 0 is udefied 6 6 (b) () 8 Numbers ad Number Relatioships / ICG / Page 66

70 . (a) log 9 (b) log 000 (c) log. (a) log log log log 6 log 6 log log log (log log ) (log log ) (b) log log (log() ) log log ( log ) (remember, log 00 ) log log (c) log 8 log + log log 8 log log + log log log 0,7 log 00 log log log log log 0,8 (0,0) (substitute give values) 0, I the last sectio of this lesso, we'll eplai how to solve epoetial equatios usig logs. Numbers ad Number Relatioships / ICG / Page 67