The Improving Mthemtics Eduction in Schools (TIMES) Project FRACTIONS AND THE INDEX LAWS IN ALGEBRA A guide for techers - Yers 8 9 June 2011 NUMBER AND ALGEBRA Module 32 8YEARS 9
Frctions nd the Index Lws in Alger (Numer nd lger : Module 32) For techers of Primry nd Secondry Mthemtics 510 Cover design, Lyout design nd Typesetting y Clire Ho The Improving Mthemtics Eduction in Schools (TIMES) Project 2009 2011 ws funded y the Austrlin Government Deprtment of Eduction, Employment nd Workplce Reltions. The views expressed here re those of the uthor nd do not necessrily represent the views of the Austrlin Government Deprtment of Eduction, Employment nd Workplce Reltions. The University of Melourne on ehlf of the Interntionl Centre of Excellence for Eduction in Mthemtics (ICE EM), the eduction division of the Austrlin Mthemticl Sciences Institute (AMSI), 2010 (except where otherwise indicted). This work is licensed under the Cretive Commons Attriution- NonCommercil-NoDerivs 3.0 Unported License. 2011. http://cretivecommons.org/licenses/y-nc-nd/3.0/
The Improving Mthemtics Eduction in Schools (TIMES) Project FRACTIONS AND THE INDEX LAWS IN ALGEBRA A guide for techers - Yers 8 9 June 2011 NUMBER AND ALGEBRA Module 32 Peter Brown Michel Evns Dvid Hunt Jnine McIntosh Bill Pender Jcqui Rmgge YEARS 8 9
{4} A guide for techers FRACTIONS AND THE INDEX LAWS IN ALGEBRA ASSUMED KNOWLEDGE The rithmetic of integers nd of frctions. Bsic methods of mentl rithmetic. The HCF nd LCM in rithmetic. Algeric expressions with whole numers nd negtives. The index lws in rithmetic with nonzero whole numers indices. The three index lws involving products in lger. Liner equtions MOTIVATION The vrious uses of lger require systemtic skills in mnipulting lgeric expressions. This module is the third of four modules tht provide systemtic introduction to sic lgeric skills. In the module, Algeric Expressions, we introduced lger using only whole numers nd the occsionl frction for the pronumerls. Then in the module Negtives nd the Index Lws in Alger we extended the methods of tht module to expressions involving negtives. The present module extends the methods of lger so tht ll negtive frctions nd decimls cn lso e sustituted into lgeric expressions, nd cn pper s the solutions of lgeric equtions. Although erlier modules occsionlly used negtive frctions, this module provides the first systemtic ccount of them, nd egins y presenting the four opertions of rithmetic, nd powers, in the context of negtive frctions nd decimls. The resulting numer system is clled the rtionl numers. This system is sufficient for ll the norml clcultions of every life, ecuse dding, sutrcting, multiplying nd dividing rtionl numers (prt from division y zero), nd tking whole numer powers of rtionl numers, lwys produces nother rtionl numers. In prticulr, scientific clcultions often involve mnipultion of equtions with decimls.
The Improving Mthemtics Eduction in Schools (TIMES) Project {5} The module lso egins the discussion of lgeric frctions, which routinely cuse prolems for students. In this module, the discussion is quickly restricted y excluding numertors nd denomintors with more thn one term. These more complicted lgeric frctions re then covered in more detil in the fourth module, Specil Expnsions nd Algeric Frctions. Once frctions hve een introduced into lger, the two index lws deling with quotients cn e stted in more systemtic form, nd then integrted with lger. This completes the discussion of the index lws in rithmetic nd lger s fr s nonzero whole numer indices re concerned. CONTENT THE RATIONAL NUMBERS Our rithmetic egn with the whole numers 0, 1, 2, 3, 4, We then constructed extensions of the whole numers in two different directions. First, in the module, Frctions we dded the positive frctions, such s 2 3 nd 9 2 = 4 1 2, which cn e written s the rtio of whole numer nd non zero whole numer, nd we showed how to dd, sutrct, multiply nd divide frctions. This gve system of numers consisting of zero nd ll positive frctions. Within this system, every non zero numer hs reciprocl, nd division without reminder is lwys possile (expect y zero). Only zero hs n opposite, however, nd sutrction is only possile when. Secondly, in the module, Integers we strted gin with the whole numers nd dded the opposites of the whole numers. This gve system of numers clled the integers, consisting of ll whole numers nd their opposites,, 3, 2, 1, 0, 1, 2, 3, nd we showed how to dd, sutrct, multiply nd divide integers. Within this system of integers, every numer hs n opposite, nd sutrction is lwys possile. Only 1 nd 1 hve reciprocls, however, nd division without reminder is only possile in situtions such s 21 7 when the dividend is n integer multiple of the divisor.
{6} A guide for techers Arithmetic needs to e done in system without such restrictions we wnt oth sutrction nd division to e possile ll the time (except for division y zero). To chieve this, we strt with the integers nd positive frctions tht we hve lredy. Then we dd the opposites of the positive frctions, so tht our numer system now contin ll numers such s 2 3 nd 9 2 = 4 1 2 All these numers together re clled the rtionl numers the word rtionl is the djective from rtio. We hve ctully een using negtive frctions in these notes for some time. These remrks re intended to formlise the sitution. The usul lws of signs pply to multiplying nd dividing in this lrger system, so tht frction with negtive integers in the numertor, or in the denomintor, or in oth, cn e written s either positive or negtive frction, 2 3 nd 2 3 = 2 3 nd 2 3 = 2 3 Becuse of these identities, it is usul to define rtionl numer s the rtio of n integer nd non zero integer: A rtionl numer is numer tht cn e written s frction, where is n integer nd is non zero integer. Thus positive nd negtive mixed numerls, terminting decimls, nd recurring decimls re rtionl numers ecuse they cn e written s frctions of integers. For exmple, 9 1 5 = 46 5 23 917 12, 23.917 = 1000, 1.09 = 11 In lter modules on topics such s surds, circles, trigonometry nd logrithms, we will need to introduce further numers, clled irrtionl numers, tht cnnot e written s frctions. Some exmples of such numers re 2, π, sin 22, log 2 5. The module, The Rel Numers will discuss the resulting more generl system of rithmetic. Show tht ech numer elow is rtionl y writing it s frction, where nd re integers with 0. 5 3 7 c 5 1 3 d 0.002 e 4.7 f 1.3 5 = 5 1 3 7 = 3 7 c 51 3 = 16 3 d 0.002 = 2 1000 e 4.7 = 47 10 f 1.3 = 4 3
The Improving Mthemtics Eduction in Schools (TIMES) Project {7} Constructing the rtionl numers There re two wys to construct the rtionl numers. The first wy is to construct the positive frctions first nd then the negtives, which is roughly wht hppened historiclly. This involves three step procedure: 1 First construct the whole numers 0, 1, 2, 3, 4, 2 Then construct the non negtive frctions, such s 3 4 nd 9 4, s the rtio of two whole numers, where the second is non zero. This process involves identifying frction such s 2 6, where the two whole numers hve common fctor, with its simplest form 1 3, nd identifying frction such s 5 1, where the denomintor is 1, with the whole numer 5. 3 Lstly construct the rtionl numers y constructing the opposites of every positive frction nd whole numer. The other method is to construct the integers first nd then the frctions. The three steps re now: 1 First construct the whole numers 0, 1, 2, 3, 4,.. 2 Then construct the integers y constructing the opposites, 4, 3, 2, 1 of ll the nonzero whole numers. 3 Lstly, construct the rtionl numers y constructing ll the frctions such s 3 4 nd 9 4 the rtio of two integers, where the second is non zero. This process gin involves identifying equivlent frctions, nd identifying frctions with denomintor 1 s integers. These two procedures re eqully cceptle mthemticlly, nd their finl results re identicl in structure. Students lern out rtionl numers, however, y working with exmples of them in rithmetic nd lger, nd there is little to e gined y such explntions. ADDING AND SUBTRACTING NEGATIVE FRACTIONS AND DECIMALS When dding nd sutrcting frctions nd decimls tht my e positive or negtive, the rules re comintion of the rules for integers nd the rules for dding nd sutrcting positive frctions nd decimls: To sutrct negtive numer, dd its opposite. Perform ddition nd sutrction from left to right. When deling with mixed numerls, it is usully esier to seprte the integer prt nd the frction.
{8} A guide for techers The following exmples demonstrte such clcultions first in rithmeticl exmples, then in sustitutions nd equtions. Simplify: 5 8 + 3 4 1 4 32.6 + ( 23.6) ( 26.4) 5 8 + 3 4 1 4 = 5 8 6 8 + 2 8 32.6 + ( 23.6) ( 26.4) = 32.6 23.6 + 26.4 = 1 8. = 35.4. Evlute x y + z when x = 20 2 3, y = 25 1 2 nd z = 353 4. x = 3.6, y = 5.6 nd z = 2.07. x y + z = 20 2 3 25 1 2 + 353 4 = 20 + 8 12 25 6 12 35 9 12 = 40 7 12 = 40 7 12 Bewre tht in nottion 40 7 7 12, the negtive sign pplies to the whole numer 4012, so tht 40 7 7 12 mens 40 + 12 = 40 7 12. x y + z = ( 3.6) 5.6 + ( 2.07) = 3.6 5.6 2.07 = 4.07 EXERCISE 1 Simplify 1 + 1 2 1 3 + 1 4 1 5 + 1 6.
The Improving Mthemtics Eduction in Schools (TIMES) Project {9} MULTIPLYING NEGATIVE FRACTIONS AND DECIMALS As with ddition nd sutrction, we need to comine the rules lredy discussed: First determine the sign of the product, then del with the numers. The product of two positives or two negtives is positive. The product of positive nd negtive is negtive. In the sence of rckets, the order of opertions is: 1 Powers, 2 Multipliction nd division from left to right, 3 Addition nd sutrction from left to right. When multiplying frctions, first convert them to improper frctions. When multiplying decimls, convert them to frctions whose denomintors re powers of 10. (Some people prefer to use the rule, Before discrding triling zeroes, the numer of deciml plces in the nswer equls the totl numer decimls plces in the question. ) The following exmples demonstrte these procedures first in rithmeticl exmples, then with sustitutions nd equtions. Simplify: 2 1 2 3 1 2 3 3 5 3 1.2 ( 0.2) 3 The product is negtive ecuse there re five minus signs in the product. 2 1 2 3 1 2 3 3 3 5 = 5 2 10 3 10 3 3 5 3 5 3 5 = 6 The product is positive ecuse there re four minus signs in the product. 1.2 ( 0.2) 3 = 12 10 2 3 103 = 96 104 = 0.0096
{10} A guide for techers Simplify 3x 2 yz when: x = 1 1 2, y = 2 1 2 nd z = 4 1 5 x = 1.1, y = 0.4 nd z = 12.1 3x 2 yz = 3 3 2 3 2 + 5 2 21 5 (The 3 is not squred indices come first) = 27 4 + 21 2 = 6 3 4 + 10 1 2 = 6 3 4 + 10 + 2 4 (Cre with mixed numerls: 63 4 mens 6 3 4 ). = 4 1 4 3x 2 yz = = 3 3 4 3 112 100 4 10 121 10 = 363 100 484 100 = 1.21 Solve ech eqution: x 3 + 43 4 = 53 4 x 1.2 + 2.01 = 2 x 3 + 43 4 = 53 4 x 3 = 10 1 2 x= 31 1 2 4 3 4 ( 3) In the lst line, it ws esier to clculte 3 10 1 2 = 30 + 1 1 2 thn to use improper frctions x 1.2 + 2.01 = 2 x 1.2 = 0.01 2.01 x= 1 100 12 10 1.2 = 0.012 EXERCISE 2 How mny fctors must e tken in the product ( 0.8) 0.9 ( 1.0) for the product to exceed 1?
The Improving Mthemtics Eduction in Schools (TIMES) Project {11} DIVIDING NEGATIVE FRACTIONS AND DECIMALS We hve lredy reviewed the order of opertions. The other necessry rules re: First determine the sign of the quotient, then del with the numers. The quotient of two positives or two negtives is positive. The quotient of positive nd negtive is negtive. When dividing y frction, multiply y its reciprocl. An improper frction must first e converted to n improper frction. To divide y deciml, write the clcultion s frction, then multiply top nd ottom y power of 10 lrge enough to mke the divisor whole numer. (Some people prefer to chnge oth decimls to frctions nd then use frction methods.) Simplify: 1 1 2 2 1 2 0.9 ( 0.08) ( 0.06) 2 1 1 2 2 1 2 = 3 2 2 5 = 3 5. 0.9 ( 0.08) ( 0.06) 2 = 0.072 0.0036 = 720 36 (Multiply top nd ottom y 10000.) = 20. Simplify 3 c when: = 8 9, = 1 1 10 nd c = 2. = 0.09, = 0.027 nd c = 0.1. 3 c = 3 1 8 9 10 11 + 2 3 c = 3 0.09 0.027 ( 0.1) = 80 90 33 + 2 = 3 27 + 0.1 = 14 33 = 10 + 0.1 = 9.9
{12} A guide for techers The reciprocl We hve seen tht the reciprocl of positive frction such s 2 5 is the frction 5 2 formed y interchnging the numertor nd denomintor, ecuse 2 5 5 2 = 1 Similrly, the reciprocl of negtive frction such s 2 5 is 5 2, ecuse 2 5 5 2 = 1 To form the reciprocl, interchnge numertor nd denomintor, nd do not chnge the sign. Reciprocls of positive nd negtive frctions re importnt in vrious lter pplictions, prticulrly perpendiculr grdients. Find, s mixed numerl when pproprite, the reciprocls of: 4 13 20 107 c 5 1 6 d 7 7 10 The reciprocl of 4 13 is 13 4 = 3 1 6. The reciprocl of 20 107 107 is 20 = 5 7 20. c The reciprocl of 5 1 6 = 31 6 is 6 31. d The reciprocl of 7 7 10 = 77 10 10 is 77. Compound frctions 2 3 + 3 4 There is very strightforwrd wy to simplify compound frction such s 2 1 2 + 1 6 To simplify compound frction, multiply top nd ottom y the lowest common denomintor of ll the frctions. In the exmple ove, the lowest common denomintor of ll the frctions is 12, so multiplying top nd ottom y 12 2 3 + 3 4 2 1 2 + 1 6 = 8 + 9 30 + 2 = 17 32 It is lso possile to rewrite the frction s division 2 3 + 3 4 2 1 2 + 1 6, ut such n pproch requires more steps.
The Improving Mthemtics Eduction in Schools (TIMES) Project {13} Simplify 1 1 3 + 1 1 2 1 1 3 1 1 2 Evlute x + 2 x when x = 3 1 3 nd = 3 5 Multiply top nd ottom y 6. 1 1 3 + 1 1 2 1 1 3 1 1 2 = 8 + 9 8 9 = 1 17 Multiply top nd ottom y 15. 1 3 3 + 6 5 x + 2 x = 3 1 3 3 5 = 50 + 18 50 9 = 32 59 = 39 59 Solving equtions with frctions An eqution with severl frctions cn e solved y the stndrd method, Move every term in x to one side, nd ll the constnts to the other. The resulting frction clcultions, however, cn e quite elorte. There is fr quicker pproch tht gets rid of ll the frctions in one single step, If n eqution involves frctions, multiply y the lowest common denomintor. Similrly, if n eqution involves decimls, multiply through y suitle power of 10. The following exmple uses first this quicker pproch, then the stndrd method, to solve two equtions. Compre the pproches nd decide for yourself. Use the quicker pproch ove, nd then the stndrd pproch, to solve these equtions. Solve: 5 6 x + 1 2 = 3 1 3 0.024x + 0.1 = 0.04 5 6 x + 1 2 = 3 1 3 Multiply y 6 to remove ll frctions. 5x + 3 = 20 6 5x = 23 3 x = 4 3 5 ( 5) or
{14} A guide for techers 5 6 x + 1 2 = 3 1 3 5 6 x = 3 2 6 3 6 1 2 5 6 x = 35 6 x = 6 5 23 6 6 5 = 4 3 5 0.024x + 0.1 = 0.04 Multiply y 1000 to remove ll decimls. 24x + 100 = 40 1000 24x = 60 100 x = 2.5 ( 24) or 0.024x + 0.1 = 0.04 0.024x = 0.06 0.1 = 0.06 0.024 ( 0.024) = 60 24 = 2.5 EXERCISE 3 Use n efficient method to find the men of the numers 1, 0.9, 0.8,, 2. The system of rtionl numers We hve now introduced four numer systems in turn. First, we introduced the whole numers 0, 1, 2, 3, 4, The set of whole numers is closed under ddition nd multipliction, which mens tht when we dd or multiply two whole numers, we otin nother whole numer. The whole numers re not closed under either sutrction or division, however, ecuse, for exmple, 7 10 is not whole numer nd 7 10 is not whole numer. Next, we dded the positive frctions to the whole numers, so tht our numer system now contined ll non negtive rtionl numers, including numers such s 4, 2 3, nd 55 7. This system is closed under division (except y zero), ut is still not closed under sutrction.
The Improving Mthemtics Eduction in Schools (TIMES) Project {15} Then we introduced the integers, 3, 2, 1, 0, 1, 2, 3, consisting of ll whole numers nd their opposites. This system is closed under sutrction, ut not under division. Now tht we re working in the rtionl numers the integers together with ll positive nd negtive frctions we finlly hve system tht is closed under ll four opertions of ddition, sutrction, multipliction nd division (except y zero). They re lso closed under the opertion of tking whole numer powers. This set of rtionl numers is therefore very stisfctory system for doing rithmetic, nd is quite sufficient for ll everydy needs. When we move to the even lrger system of rel numers, every rel numer cn e pproximted s close s we like y rtionl numers (indeed y terminting decimls), so tht the rtionl numers will remin extremely importnt. CANCELLING, MULTIPLYING AND DIVIDING ALGEBRAIC FRACTIONS An lgeric frction is frction involving pronumerls, x 3, 5 x, + c, 3r s t. The pronumerls in the frction represent numers, so we del with lgeric frctions using exctly the sme procedures tht we use in rithmetic. Sustitution into lgeric frctions We hve seen in rithmetic tht the denomintor in frction cnnot e zero. This mens tht: In the expression 5 x, we cnnot sustitute x = 0. In the expression + c, we cnnot sustitute c = 0. In the expression 3r s t, we cnnot sustitute the sme vlue for s nd t. Aprt from this qulifiction, sustitution into lgeric frctions works exctly the sme s sustitution into ny lgeric expression. For exmple, if r = 5, s = 7 nd t = 4, then 3r s t = 15 7 4 = 5 Sustitution of negtive numers requires the usul cre with signs. If r = 2, s = 7 nd t = 5, 3r s t = 6 7 5 = 6 12 = 1 2
{16} A guide for techers Wht vlue of cnnot e sustituted into + 3 3? Wht vlue of mkes + 3 3 zero? c Evlute + 3 3 when = 2. When = 3, the denomintor is zero, nd the expression cnnot e evluted. When = 3, the numertor is zero ut the denomintor is not zero, so the expression is zero. c When = 2, + 3 3 = 2 + 3 2 3 = 1 5 = 1 5 The rest of this module will exclude denomintors, such s s t nd 3, tht hve two or more terms. Such expressions will e considered in the lter module, Specil Expnsions nd Algeric Frctions. Cncelling lgeric frctions Becuse pronumerls re just numers, we cn cncel oth pronumerls nd numers in n lgeric frction. For exmple, 3xy 6x = y 2 nd 2 =. We cnnot cncel zero, so strictly we should exclude zero vlues of the pronumerl when we cncel. This would men writing 3xy 6x = y 2 provided x 0 nd 2 = provided 0 nd 0. Such qulifictions ecome importnt much lter in clculus. At this stge, however, it is not pproprite to ring the mtter up, unless students insist. Simplify ech lgeric frction y cncelling ny common fctors: 5 10 9 12c c 42 6 d 12xy2 z 12x2yz 5 10 = 1 2 9 12c = 3 4c c 42 6 = 2 3 d 12xy2 z 12x2yz = y x
The Improving Mthemtics Eduction in Schools (TIMES) Project {17} Dividing lgeric expressions When we divide, the lws of signs re the sme s the lws for multiplying: The quotient of two negtives or two positives is positive. The quotient of positive nd negtive is negtive. The following exmples illustrte ll the possiilities: 6x2 2x = 3x 6x2 2x = 3x 6x2 2x = 3x 6x2 2x = 3x As with multipliction, there re three steps in hndling more complicted expressions: First del with the signs. Then del with the numerls. Then del with ech pronumerl in turn. 3xy = y 6x2 2x nd (21c 2 ) ( 7 2 c) = 3c The lst nswer cn lso e written s 3c nswer 3c, with the negtive sign out the front of the frction. or even s 3c, ut the est wy to write the Frction nottion is usully used for quotients, ut the division sign cn lso e used, s in the second exmple ove. Simplify ech quotient: ( 12) ( 3c) 8x2 4x c 3c 2c3 d 92 62 c ( 12) ( 3c) = 4 c 3c = 2 2c3 2c2 8x2 4x = 2x d 92 62 = 32 22 Multiplying lgeric frctions We hve seen how to multiply frctions in rithmetic, 15 5 10 21 = 5 7 10 7 = 50 49 First del with the signs. Then cncel ny common fctors from numertor nd denomintor. Then multiply the numertors, nd multiply the denomintors.
{18} A guide for techers Exctly the sme method pplies to multiplying lgeric frctions. For exmple, 3 x x = 3 1 1 = 3 5x 6y 6xy 25 = x 1 x 5 = x 2 5 Simplify: 2x 3 6 x 2 c c 2x 3 6 x = 2 1 2 1 2 c c = 2 c c = 4 = 3 c2 Dividing y n lgeric frction We know the lw of signs for division, nd we know how to divide y frction. To divide y n lgeric frction, we pply these sme lws in succession 1 First determine the sign of the quotient. 2 To divide y frction, multiply y its reciprocl For exmple, 4 x 6 x = 4 x x 6 = 2 1 1 3 32 c 32 c = 32 c c 32 = 1 1 = 2 3 = Simplify ech quotient: 12 x 3 x 6x 2 y 4x y2 c 3 c ( 6c) d 22 c 2 c2 12 x 3 x = 12 x x 3 6x2 y c 3 c = 4 1 1 1 4x = 6x2 y2 y y 2 4y = 6x2 y y 2 4y = 4 = 3xy 2 3 ( 6c) = c 1 6c d 22 c 2 c2 = 22 c c2 2 = c 1 1 2c = 1 c 1 = 1 2c2 = c
The Improving Mthemtics Eduction in Schools (TIMES) Project {19} The reciprocl of n lgeric frction Reciprocls of lgeric frctions re formed in exctly the sme wy s ws done in rithmetic. The reciprocl of 2x y is y 2x 2x, ecuse y y 2x = 1. The reciprocl of 3 is 1 3, ecuse ( 3 ) 1 3 = 1. The reciprocl of 7xy 3t is 3t 7xy, ecuse 7xy 3t 3t 7xy = 1. To form the reciprocl, interchnge numertor nd denomintor, nd do not chnge the sign. Write down the reciprocl of ech lgeric frction: 2 2 c 6x2 7yz d 6x2 7yz 2 2 c 7yz d 7yz 6x 2 6x 2 THE INDEX LAWS FOR FRACTIONS The previous lger module Negtives nd the Index Lws in Alger delt with the index lws for product of powers of the sme se, for power of power, nd for power of product: To multiply powers of the sme se, dd the indices: m n = m+n. To rise power to power, dd the indices: ( m ) n = mn. The power of product os the product of the powers: () n = n n. In these identities, nd re ny numers, nd m nd n re ny nonzero whole numers. We now turn to the remining two index lws, which involve quotients. The index lw for the power of quotient To see wht hppens when we tke power of frction, we simply write out the power. 2 3 2 5 = 5 2 5 2 5 5 = = 23 5 3 = 5 5
{20} A guide for techers The generl sitution cn thus e expressed s: The power of quotient is the quotient of the powers: n = n n where nd 0 re numers, nd n is nonzero whole numer. The following exmples illustrte the use of the lw in rithmetic nd lger. Simplify: 1 2 5 3 4 c 1 2 d 1 3 10 4 2 3 3 1 5 1 2 = 5 2 5 = 1 32 = c 4 1 2 2 9 2 = 2 d 3 1 3 3 4 3 10 = 4 10 4 3 = 10 3 81 10000 = 81 4 = 100 27 = 20 1 4 = 37 1 27 When pplied to n lgeric frction, the lws previously introduced for the power of product nd for the power of power, re often required s well. For exmple, 3 2x 3 8x 3y = 3 27y 3 nd 1 2 1 = 5 3 25 6 Expnd the rckets: 2 2x 3y 5 3 c 2 2 3 d 3 4 2 4 2 2x 4x 3y = 2 5 3 125 9y2 = 3 3 c 2 2 = 4 3 6 d 3 4 2 4 81 = 16 16 The index lw for the quotient of powers of the sme se The remining index lw from rithmetic dels with dividing powers of the sme se. The module Multiples, Fctors nd Powers developed the lw in the form, To divide powers of the sme se, sutrct the indices. For exmple, 7 5 7 3 = (7 7 7 7 7) (7 7 7) = 7 2. Frctions, however, hd not een introduced t tht stge, so the divisor could not e
The Improving Mthemtics Eduction in Schools (TIMES) Project {21} higher power thn the dividend. Now tht we hve frctions, we cn express the lw in frctionl nottion nd do wy with the restriction: 7 5 7 = 7 7 7 7 7 3 7 7 7 7 4 7 = 7 7 7 7 4 7 7 7 7 7 3 7 7 7 = 7 5 7 7 7 7 7 = 7 2 = 1 = 1 7 2 The se cn lso e pronumerl, nd the lw cn now e written s: To divide powers of the sme se, sutrct the indices. For exmple, 5 3 = 2 nd 4 = 1 nd 3 4 = 1 5 2. Lter, when zero nd negtive indices hve een introduced in the module The Index Lws, we will e le to stte this lw in its usul, more concise form: m n = m n. Simplify ech expression: 3x 6 x4 8 6 6 c 34 44 d 3 2 6 6 c 6 8 48 38 3x 6 = 3x2 x 4 8 6 = 2 6 c 34 44 = 1 8 2 48 38 4 d 3 2 6 6 c 6 1 = 4 3 4 c 6 ADDING AND SUBTRACTING ALGEBRAIC FRACTIONS When set of lgeric frctions hve common denomintor, it is strightforwrd to dd them. As lwys, we collect like terms, 3 x + x 7 x + 4 x = 5 x 4 x (which is perhps written s 5 x 4 ) Simplify ech expression: 5 x x 4 x 7 2x 11 2x 20 2x 5 x x 4 x = 1 x x, or 1 x 7 2x 11 2x 20 2x = 24 2x = 12 x
{22} A guide for techers Using common denomintor When the frctions do not hve common denomintor, we need to find the lowest common denomintor, just s we did for rithmetic frctions. We shll restrict the discussion to numericl denomintors t this stge, nd leve denomintors with pronumerls to the module Specil Expnsions nd Algeric Frctions. x 4 x 6 = 3x 12 2x 12 = x 12. Simplify ech expression: x 2 x 3 + x 4 + x 12 3x _ 5x 8 6 x 2 x 3 + x 4 + x 12 = 6x _ 4x 12 12 + 3x 12 + x 12 3x _ 5x 8 6 = 9x _ 20x 24 24 = 6x 12 = 11x 24 = x 2 LINKS FORWARD As mentioned in the Motivtion section, this module is the third of four modules deling with sic mnipultion of lgeric expressions. The four modules re: Algeric Expressions Negtives nd the Index Lws in Alger Frctions nd the Index Lws in Alger (the present module) Specil Expnsions nd Algeric Frctions Algeric frctions hve een introduced in this module, ut the lger of lgeric frctions cn cuse considerle difficulty if it is introduced in full strength efore more fundmentl skills hve een ssimilted. Algeric frctions will therefore e considered in more detils in the fourth module Specil Expnsions nd Algeric Frctions. The index lws will e gretly expnded in scope once negtive nd frctionl indices re introduced in the module Indices. One piece of clumsiness in the present module concerned the quotient of the difference of powers, where we needed to write down three different exmples x5 = x 2 x 3 nd x4 x 4 = 1 nd x3 x 5 = 1 x 2. Once the zero index nd negtive indices re rought into lger, ll this cn e replced
The Improving Mthemtics Eduction in Schools (TIMES) Project {23} with the single lw xm xn = x m n where m nd n re integers. Squre roots nd cue roots cn lso e represented using frctionl indices for exmple, 5 is written s 5 1 2 nd 3 5 is written s 5 1 3 nd yet the five index lws still hold even when the index is ny rtionl numer. Negtive nd frctionl indices re introduced in the module, Indices nd Logrithms. Lter in clculus, n index cn e ny rel numer. ANSWERS TO EXERCISES EXERCISE 1 The lowest common denomintor is 60. Grouping the numers in pirs efore dding them ll reduces the size of the numertor. Sum = 1 + 1 2 + 1 3 + 1 4 + 1 5 + 1 6 = 1 2 1 12 1 30 = 30 60 5 60 2 60 = 37 60 EXERCISE 2 We cn ignore ll negtive products. Since ( 0.8) 0.9 ( 1.0) 1.1 is less thn 1, the first product greter thn 1 is ( 0.8) 0.9 ( 1.0) 1.1 ( 1.2) 1.3 ( 1.4). EXERCISE 3 One pproch is to notice tht when the 31 numers re dded, the first 21 numers cncel out. The remining 10 numers cn then e dded y grouping them in pirs: 1 + 0.9 + 0.8 + + ( 2) = 1.1 1.2 1.3 2. = ( 1.1 2) +( 1.2 1.9) + ( 1.3 1.8) + ( 1.4 1.7) + ( 1.5 1.6) = 3.1 5 = 15.5. Hence men = 15.5 31 = 31 62 = 0.5. A etter pproch is relise tht 0.5 is the middle numer, nd tht the other 30 numers fll into pirs eqully spced one to the left nd one to the right of 0.5. Hence the men is 0.5.
The im of the Interntionl Centre of Excellence for Eduction in Mthemtics (ICE-EM) is to strengthen eduction in the mthemticl sciences t ll levelsfrom school to dvnced reserch nd contemporry pplictions in industry nd commerce. ICE-EM is the eduction division of the Austrlin Mthemticl Sciences Institute, consortium of 27 university mthemtics deprtments, CSIRO Mthemticl nd Informtion Sciences, the Austrlin Bureu of Sttistics, the Austrlin Mthemticl Society nd the Austrlin Mthemtics Trust. The ICE-EM modules re prt of The Improving Mthemtics Eduction in Schools (TIMES) Project. The modules re orgnised under the strnd titles of the Austrlin Curriculum: Numer nd Alger Mesurement nd Geometry Sttistics nd Proility The modules re written for techers. Ech module contins discussion of component of the mthemtics curriculum up to the end of Yer 10. www.msi.org.u