Lambrick Park Secondary 1.1 Number Systems

Section 1.1 - Number Systems ♦ 5

Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.

Inthiscourse,wewillworkwithmanynumbersystems.Wewillsolveproblemsinvolvingprofitandloss,time expressedasfractions,lengthsoftrianglesexpressedassquareroots,andmore.

Inthissection,wewilllookatnumbersthatarepartofalargercollectionofnumberscalledthereal numbers.

Natural Numbers: 1,2,3..." ,

Whole Numbers: 0,1,2,3..." ,

Integers: ... 3, 2, 1,0,1,2,3...- - -" ,

Rational Numbers: Allthenumbersthatcanbewrittenasafractionwiththedenominatornotequaltozero.

Examples: 3, 0, 5,32, , . , .710 2 35 2 35- - -

Irrational Numbers: Allthenumbersthatcannotbewrittenasafraction,aterminatingdecimal,ora repeatingdecimal.

Examples: , , . ...2 1 62789r

Real Numbers: Alltherationalnumbersandirrationalnumberscombined.

Towhichnumbersystemsdothefollowingnumbersbelong?

, , , , ,2 0 4 132 2 r-$ .

►Solution: Naturalnumbers: 4 Wholenumbers: 0,4 Integers: −2,0,4 Rationalnumbers: −2,0,4,1

32

Irrationalnumbers: 2 ,r Realnumbers: −2,0,4,1

32 , 2 ,r

NaturalNumbers

WholeNumbers

Integers

RationalNumbers

RealNumbers

IrrationalNumbers

Example 1

Number Systems1.1

Lambrick Park Secondary

6 ♦ Chapter 1 - Real Numbers


1.1 Exercise Set

1. Statewhethereachstatementistrueorfalse.

a) Everynaturalnumberispositive. b) Everywholenumberispositive.

c) Everyintegerisawholenumber. d) Everywholenumberisaninteger.

e) Everyrealnumberisarationalnumber. f) Everyintegerisarationalnumber.

g) Everydecimalnumberisarationalnumber. h) Zeroisarationalnumber.

i) Allrealnumbersareintegers. j) Allintegersarerealnumbers.

k) Allwholenumbersarerealnumbers. l) Thereciprocalofanon-zerointegerisalsoaninteger.

m) Thereciprocalofanon-zerorationalnumberis n) Anirrationalnumbertimesadifferentirrational alsoarationalnumber. numbercouldberational.

o) Everyrealnumberiseitherrationalorirrational. p) Ifanumberisrealthenitisirrational.

q) Thereareaninfinitenumberofrealnumbers r) Arationalnumbertimesanirrationalnumberis between0and1 alwaysirrational.

s) Betweenanytwoirrationalnumbersisanother t) Anirrationalnumbertimesanirrationalnumber irrationalnumber. isalwaysanirrationalnumber.


Section 1.1 - Number Systems ♦ 7


2. Considerthelistofnumbers: 12, 2.7, 0, , , , 4.21, 501332 r- - .Listall:

a) Naturalnumbers b) Wholenumbers

c) Integers d) Rationalnumbers

e) Irrationalnumbers f) Realnumbers

3. Considerthelistofnumbers: 4, 0. , 0, 0.121121112 , 2.3535 , 2 , 12,3 1031f f- .Listall:




4. Considerthelistofnumbers: , , , . , . , , . ,64 64 0 008 0 04 0 494

7218

903 3- - .Listall:







5. Statethenumbersystemseachofthefollowingbelongto.(Natural,Whole,Integer,Rational,Irrational,Real)

a) 16 b) r

c) 0 d) .2 34

e) 4.010010001f f) −3.181818f

g) 1227 h) .0 0004

6. Listeachofthefollowing:

a) Naturalnumberslessthan4 b) Naturalnumbersgreaterthan5

c) Wholenumberslessthan2 d) Integersgreaterthan−3

e) Positiveintegersgreaterthan−4 f) Wholenumberslessthan0

g) Non-negativeintegerslessthan4 h) Non-positiveintegersgreaterthan−4

i) Negativeintegersgreaterthan−3 j) Positiveandnegativeintegersbetween−2and2

k) Naturalnumberslessthan1 l) Wholenumberslessthan1

m) Negativeintegersgreaterthan−3.205 n) Positivewholenumberslessthan2.758

o) Naturalnumbersbetween−3and2 p) Wholenumbersbetween−3and2


Section 1.2 - Greatest Common Factor and Least Common Multiple ♦ 9


AkeyconceptinfindingtheGCFandLCMisfactoring.Factoringisthedecompositionofanumberinto aproductofothernumbers,orfactors,whichwhenmultipliedtogethergivetheoriginalvalue.

Forexample: 4isafactorof12because4 3 12# = 4isnotafactorof18becausethereisnowholenumberksuchthat k4 18# =

Next,considertwoveryspecialtypesofwholenumberscalledprimenumbersandcompositenumbers.

Prime Numbers and Composite Numbers

Aprimenumberisawholenumberthathasexactlytwofactors:1anditself.

Acompositenumberisawholenumbergreaterthan1thathasadivisorotherthanonoritself,or inotherwords,isnotprime.Everycompositenumberhasmorethantwofactors.

List of Prime Numbers Less Than 100

2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97

Note: Thesetofprimenumbersisinfinite.AsofAugust23,2008thelargestprimenumberthathasbeen discoveredis2 143112609

- .Thisnumberhas12978189digits!(Printingthisnumberwouldtake about2200pages.)

Zero and One

Thewholenumbers0and1areneitherprimenorcomposite.

Zeroisnotaprimenumberbecauseithasaninfinitenumberofdivisors.Oneisnotaprimenumber becauseitdoesnothavetwodifferentpositivewholenumberdivisors.

Whichofthefollowingarecompositenumbers:3,10,18,23,29

►Solution: 10 2 5#= and18 2 3 3# #= ,therefore10and18arecomposite.Theothersareprime.

Example 1

Greatest Common Factor and Least Common Multiple1.2




Divisibility Properties

Divisibilityby2: Awholenumberisdivisibleby2ifitslastdigitis0,2,4,6or8. Divisibilityby3: Awholenumberisdivisibleby3ifthesumofitsdigitsaredivisibleby3. Divisibilityby4: Awholenumberisdivisibleby4ifthelasttwodigitsaredivisibleby4. Divisibilityby5: Awholenumberisdivisibleby5ifthelastdigitis0or5. Divisibilityby6: Awholenumberisdivisibleby6ifitisanevennumberthatisdivisibleby3. Divisibilityby9: Awholenumberisdivisibleby9ifthesumofitsdigitsaredivisibleby9. Divisibilityby10: Awholenumberisdivisibleby10ifitendsina0.

Determinewhichofthe4digitnumbersaredivisiblebythegiven1or2digitnumber.

a)4735,2638;2b)2783,5271;3c)4384,6394;4d)3753,8240;5

e)7936,2898;6f)3564,5239;9g)9625,3780;10

►Solution: a) 2638endsinanevennumber,therefore2638isdivisibleby2.

4735endsinanoddnumber,therefore4735isnotdivisibleby2.

b) Thedigitsof5271addto15 5 2 7 1 15+ + + =^ h,whichisdivisibleby3,therefore 5721isdivisibleby3.

Thedigitsof2783addto20 2 7 8 3 20+ + + =^ h,whichisnotdivisibleby3,therefore 2783isnotdivisibleby3

c) Thelasttwodigitsof4384are84,whichisdivisibleby4,therefore4384isdivisibleby4.

Thelasttwodigitsof6394are94,whichisnotdivisibleby4,therefore6394isnot divisibleby4.

d) Thelastdigitof8240is0,therefore8240isdivisibleby5.

Thelastdigitof3753is3,therefore3753isnotdivisibleby5.

e) Thedigitsof2898addto27 2 8 9 8 27+ + + =^ h,whichisdivisibleby3,and2898isan evennumber,therefore2898isdivisibleby6.

Thedigitsof7936addto25 7 9 3 6 25+ + + =^ h,whichisnotdivisibleby3,therefore 7936isnotdivisibleby6.

f) Thedigitsof3564addto18 3 5 6 4 18+ + + =^ h,therefore3564isdivisibleby9.

Thedigitsof5239addto19 5 2 3 9 19+ + + =^ h,therefore5239isnotdivisibleby9.

g) Thelastdigitof3780is0,thereforeitisdivisibleby10.

Thelastdigitof9625is5,thereforeitisnotdivisibleby10.

Example 2




Finding Prime Factors of Composite Numbers

Threemethodsforfindingtheprimefactorsofthenumber72willbeshown.

Method1:FactorTree

or or

Therefore72factorsinto2 2 2 3 3# # # # .

Method2:DividingByPrimesUntilTheQuotientIsAPrimeNumber

Startwiththeprimenumber2andrepeatedlydivideintoeachresultingdividenduntilitisnolongerpossible.

2 7236g ,2 36

18g ,2 189g

Onceitisnolongerpossibletodividewith2trythenextprimenumber3.

3 93g

Thedivisorsareprimenumbersandthequotientisprime,therefore72factorsinto2 2 2 3 3# # # # .

Method3:ContinuousDivision

2 72 ←2dividesinto72.Since36isnotprime,continue. 2 36 ←2dividesinto36.Since18isnotprime,continue. 2 18 ←2dividesinto18.Since9isnotprime,continue. 3 9 ←3dividesinto9.Sincethequotientisprime,stop. 3

Therefore72factorsinto2 2 2 3 3# # # # .

72

2

4

2 3

9

32

36

72

4

2

3

2

18

2 9

3

72

8

34

2

3

2

2

9




Theprimefactorsoftwoormorecompositenumberscanbeusedtofindtheirgreatestcommonfactor.

Forexample: 60 2 2 3 5# # #= 72 2 2 3 3# # #=

Sincetwo2’sanda3arecommontoboth,thegreatestcommonfactorof60and72is2 2 3 12# # = .

Greatest Common Factor (GCF)

Thelargestnumberthatdivideseachofthegivennumbersexactly.

Finding the Greatest Common Factor

1. Findtheprimefactorsofeachnumber. 2. Listeachcommonfactortheleastnumberoftimesitappearsinanyonenumber. 3. Findtheproductofthefactors.

FindtheGCFof36,48,60.

►Solution: 36 2 2 3 3# # #= 2 32 2#=

48 2 2 2 2 3# # # #= 2 34#=

60 2 2 3 5# # #= 2 3 52# #=

Thecommonfactorsare2and3.

Theleastnumberof2’sis2² Theleastnumberof3’sis3.

ThereforetheGCFof36,48,60is2 3 122# = .

FindtheGCFof32,40,60.

►Solution: 3 2 22 2 2 2# # # #= 25=

4 2 2 20 5# # #= 2 53#=

60 2 2 3 5# # #= 2 3 52# #=

Theonlycommonfactoris2.Theleastnumberof2’sis2².

ThereforetheGCFof32,40,60is2²=4.

Example 3

Example 4




Theprimefactorsoftwoormorecompositenumberscanbeusedtofindtheirleastcommonmultiple.

Forexample: 60 2 2 3 5# # #= 72 2 2 3 3# # #=

Theonlyuniqueprimefactoris5.Thehighestpowerof2is2².Thehighestpowerof3is3².

Theleastcommonmultipleof60and72is5 2 3 1802 2# # = .

Multiple

Theproductofanumberbywholenumber

Forexample,themultiplesof7are:0,7,14,21,28...

Least Common Multiple (LCM)

Thesmallestcommonnon-zeromultipleoftwoormorewholenumbers,orthesmallest numberthatisdivisiblebyallthenumbers.

Finding the Least Common Multiple (Method 1)

1. Determineifthelargestnumberisamultipleofthesmallernumber.Ifso,thelargernumberistheLCM. 2. Ifnot,writemultiplesofthelargernumberuntilyoufindonethatisamultipleofthesmallernumbers.

FindtheLCMof15and18.

►Solution: 15isnotamultipleof18

2 18 36# = isnotamultipleof15



5 18 90# = isamultipleof15,6 15 90# =

ThereforetheLCMof15and18is90.

Example 5




Finding the Least Common Multiple (Method 2)

1. Writeeachnumberasaproductofitsprimefactors. 2. Selecttheprimesthatoccurthegreatestnumberoftimesinanyonefactor. 3. TheLCMistheproductoftheseprimes.

FindtheLCMof60and72.

►Solution: 60 2 2 3 5# # #= 2 3 52# #= 72 2 2 2 3 3# # # #= 2 33 2#=

Theuniqueprimefactorsare2,3and5

Thehighestpowerof2is23. Thehighestpowerof3is32. Thehighestpowerof5is5.

ThereforetheLCMof60and72is2 3 5 3603 2# # = .

FindtheLCMof35,60and75.

►Solution: 35 5 7#= 60 2 2 3 5# # #= 2 3 52# #= 7 35 5 5# #= 3 52#=

Theuniqueprimefactorsare2,3,5and7

Thehighestpowerof2is22. Thehighestpowerof3is3. Thehighestpowerof5is52. Thehighestpowerof7is7.

ThereforetheLCMof35,60and75is2 3 5 72 2# # # =2100.

Example 6

Example 7




Finding the Least Common Multiple (Method 3) (Forusewith3ormorenumbers)

1. Dividebyaprimefactorthatcandividetwoormoreofyournumbers.Bringdownanynumberthatdoes notfactor. 2. Repeat,untilnoremainingnumbershaveanyprimefactorsincommon.


►Solution: , ,3 15 18 24 allthreenumbersaredivisibleby3 2 , ,5 6 8 6and8aredivisibleby2,bringdownthe5 5,3,4 noremainingcommonprimefactors

ThereforetheLCMof15,18and24is3 2 5 3 4 360# # # # =


►Solution: , ,2 27 30 36 30and36aredivisibleby2,bringdownthe27 , ,3 27 15 18 allthreenumbersaredivisibleby3 , ,3 9 5 6 6and9aredivisibleby3,bringdownthe5 3,5,2 noremainingcommonprimefactors

ThereforetheLCMof27,30and36is2 3 3 3 5 2 540# # # # # =

Simplify728140 usingprimes.

►Solution: 140 2 2 5 7# # #=

728 2 2 2 7 13# # # #=

Therefore728140

2 2 2 7 13

2 2 5 7

265

# # # #

# # #= =

Example 8

Example 9

Example 10




1.2 Exercise Set

1. Withouttheuseofacalculator,determinewhichofthefollowingnumbers:63,126,280,396,575,2610,7800 aredivisibleby:

a) 2 b) 3

c) 4 d) 5

e) 6 f) 9

g) 10

2. Thenumber530hasamissinglastdigit.Determinethesmallestdigitsothatthe4digitnumberis divisibleby:

a) 2 b) 3

c) 4 d) 5

e) 6 f) 9

g) 10




3. Thenumber3689hasamissinglastdigit.Determinethesmallestdigitsothatthe5digitnumberis divisibleby

a) 2 b) 3

c) 4 d) 5

e) 6 f) 9

g) 10

4. Bytestingnumbers,determinewhichofthefollowingstatementsaretrue.

a) Anumberisdivisibleby8ifthelastthreedigitsaredivisibleby8. T/F

b) Anumberisdivisibleby11if,startingfromthefirstdigit,youfindthesumof T/F everyalternatedigit,andthensubtractthesumoftheremainingdigitstoget anewvaluewhichisdivisibleby11.

c) Anumberisdivisibleby12ifitisdivisiblebyboth3and4. T/F

d) Anumberisdivisibleby15ifitisdivisiblebyboth3and5. T/F

e) Anumberisdivisibleby18ifitisdivisiblebyboth2and9. T/F




5. Listthefirsteightmultiples.

a) 3 b) 6

c) 8 d) 12

6. Decidewhetherthenumberisprimeorcomposite.Ifthenumberiscomposite,factoritintoprimes.

a) 19 b) 51

c) 87 d) 101

e) 117 f) 199

g) 611 h) 997

i) 629 j) 551

7. Simplifyusingprimefactorization.

a)455385 b)

11881155

c)23101848 d)

57754950

e)35752600 f)

26182210




8. Completelyfactoreachofthenumbers.

a) 36 b) 78

c) 84 d) 169

e) 178 f) 425

g) 1000 h) 6250

i) 2431 j) 5681

9. Findthegreatestcommonfactor.

a) 12,28 b) 54,66

c) 48,136 d) 65,169

e) 81,108 f) 30,45,60

g) 12,27,42 h) 28,42,84

i) 52,130,182 j) 66,165,231




10. Findtheleastcommonmultiple.

a) 5,10 b) 12,16

c) 21,48 d) 24,56

e) 30,55 f) 6,12,15

g) 12,16,24 h) 20,36,48

i) 7,11,13 j) 28,35,42

k) 22,33,66 l) 4,36,225

m) 8,27,125 n) 14,84,98

o) 2,8,12,18 p) 9,15,25,45

11. Fillinthemissingnumbers.

Product 6 48 18 72 108 32 75

Factor 2 7 7 11

Factor 3 9 6 9

Sum 5 16 11 22 31 12 13 12 20




12. Mercury,VenusandEarthrevolvearoundtheSun 13. Earth,Jupiter,Saturn,andUranusrevolvearound every3,7and12monthsrespectively.Ifthethree theSunevery1,12,30,and84yearsrespectively. planetsarecurrentlylinedup,howmanymonths Ifthefourplanetscurrentlylineup,howmany willpassbeforethishappensagain? yearswillpassbeforetheywouldlineupagain?

14. Tom,DickandHarrygethalfdaysoffwithpay. 15. Thereare6playersonavolleyballteam,9players Tomgetsahalfdayoffwithpayeveryeightdays, onabaseballteam,and11playersonasoccerteam. Dickevery10daysandHarryevery12days.If Whatisthesmallestnumberofstudentsinaschool allthreeareofftogetheronApril1,whatisthe thatcanbesplitevenlyintoanyofthethreeteams? nextdatewhenallthreewillbeofftogetheragain?

16. ThereisauniquewayoffindingtheLCMoftwo 17. FindtheLCMof72,108bytakingtheproductof numbers.Taketheproductofbothnumbers,and bothnumbersanddividingthatproductbytheGCF. dividethatproductbytheGCF.FindtheLCM of18,24bythismethod.

18. TheFieldsMedal,thehighestscientificawardfor 19. Whatisthesmallestwholenumberdivisibleby Mathematics,isawardedevery4years.Itwas everywholenumberfrom1to10? awardedin2010.TheBirkhoffPrizeforApplied Mathematicsisawardedevery3years.Itwas awardedin2009.Whatarethefirsttwotimes bothawardsweregiveninthesameyearinthe 21stcentury?




Tosquareanumbermeanstoraisethenumbertothesecondpower.

Forexample: 4 4 4 162 #= =

9 9 9 812 #= =

Somenumberscanbewrittenastheproductoftwoidenticalfactors.

Forexample: 25 5 5#=

64 8 8#=

Theseidenticalfactorsarecalledthesquare rootofthenumber.Thesymbol (calledaradicalsign)isused toindicatesquareroots. 25 5= and 64 8= .

Allnumberswithsquarerootsthatarerationalarecalledperfect squares.

A List of Perfect Square Whole Numbers

0,1,4,9,16,25,36,49,64,81,100,121,144,169,196,225,256,289,324,361,400,441,484,529,576,625

Determinewhichareperfectsquares.

a)4b)94 c)7 d)

154

►Solution: a) Yes,because7 7 49# = .

b) Yes,because323294

# = .

c) No,because7cannotbewrittenastheproductoftwoidenticalrationalnumbers.

d) No,because154 cannotbewrittenastheratiooftwoidenticalrationalnumbers.

Example 1

Squares and Square Roots1.3


Section 1.3 - Squares and Square Roots ♦ 23


Finding Square Roots Without a Calculator or Table

Method1:FactorTree

Example:Determiningthesquarerootof196.

or

Therefore 196 2 2 7 7 2 2 7 7 2 7 14# # # # # # #= = = =

Note: Forwholenumbers x x x x2 #= =


Example:Determiningthesquarerootof441

441isnotdivisibleby2,butisdivisibleby3so:

3 441 ←3dividesinto441.Since147isnotprime,continue. 3 147 ←3dividesinto147.Since49isnotprime,continue. 7 49 ←7dividesinto49.Since7isprime,stop. 7

Therefore 441 3 3 7 7 3 3 7 7 3 7 21# # # # # # #= = = =

196

2

7

98

2 49

7

7

49

7

196

4

2 2




Cubes and Cube Roots

Tocubeanumbermeanstoraisethenumbertothethirdpower,ortomultiplythenumberbyitselfthreetimes.

Forexample: 4 4 4 4 643 # #= =

7 7 7 7 3433 # #= =

Somenumberscanbewrittenastheproductofthreeidenticalfactors.

Forexample: 27 3 3 3# #=

125 5 5 5# #=

Theseidenticalfactorsarecalledthecube rootofthenumber.Thesymbol3 (calledaradicalsign)isused toindicatecuberoots. 27 33

= ,and 125 53=

A List of Perfect Cube Whole Numbers

0,1,8,27,64,125,216,343,512,729,1000

Determinewhichareperfectcubes.

a) 8b)6427 c)25 d)

98

►Solution: a) Yes,because2 2 2 8# # = .

b) Yes,because4343436427

# # = .

c) No,because25cannotbewrittenastheproductofthreeidenticalintegers.

d) No,becauseitcannotbewrittenastheproductofthreeidenticalrationalnumbers.

Note: Intheexpression ak ,wecallktheindex,andassumek 2$ .Iftheindexisnotwritten,theexpression isassumed tobeasquareroot,i.e.k=2.

eg. 32 25 = because2 2 2 2 2 32# # # # =

Example 2




Finding Cube Roots Without a Calculator or Table

Method1:FactorTree

Example:Findthecuberootof216.

or

Therefore 216 2 2 2 3 3 3 2 2 2 3 3 3 2 3 63 3 3 3# # # # # # # # # # #= = = =

Note: Forwholenumbers, x x x x x33 3 # #= =


Example:Findthecuberootof729.

729isnotdivisibleby2,butisdivisibleby3so:

3 729 ←3dividesinto729.Since243isnotprime,continue. 3 243 ←3dividesinto243.Since81isnotprime,continue. 3 81 ←3dividesinto81.Since27isacuberootnumber,stop. 27

Therefore 729 3 3 3 27 3 3 3 27 3 3 93 3 3 3# # # # # # #= = = =

216

2

2

2

108

54

27

3

3

9

3

216

2

2

27

3

3

8

4

2

9

3




1.3 Exercise Set

1. Findthesquarerootoftheperfectsquareswithoutacalculator.

a) 100 b) 441

c) 225 d) 361

e) 529 f) 2890000

g) 25000000 h) 72 900-

i) 9 610000- j) .0 000004

2. Findthecuberootoftheperfectcubeswithoutacalculator.

a) 273 b) 10003

c) 3433 d) 17283

e) 33753 f) 80003

g) 1250000003 h) 2160003-

i) 640000000003- j) .0 0000083




3. Findtheperfectsquareroot,ifitexists,withoutacalculator.

a) 25 b) 29

c) 80 d) 81

e) 169 f) 99

g) 1600 h) 900

i)40081 j)

188

4. Findtheperfectcuberoot,ifitexists,withoutacalculator.

a) 8 b) 9

c) 64 d) 81

e) 100 f) 216

g) 1000 h) 144

i) 625 j) 729




5. Theareaofabaseballdiamondis8100ft². 6. PoliceusetheformulaS x2 5= todeterminethe speed,Sinmilesperhourofaskidoflengthxft. a) Howfarisitfromhomeplatetofirstbase? Determinethespeedofacarwhoseskidlengthis

a) 45ft.

b) Howfarisitfrom1stbaseto3rdbase? b) 180ft.

7. Theareaofarectanglewithlengthtwiceaslong 8. Acubehasavolumeof216cm³.Determinethe asthewidthis1250m².Determinethelength lengthofeachsideofthecube. andwidthoftherectangle.

9. Arectangularsolidhasalengththreetimesthe 10. Thevolumeofasphereisgivenbytheformula widthandaheighttwiceitswidth.Ifthevolume V r34

3r= ,withrtheradiusofthesphere. oftherectangularsolidis384in³,determinethe Determinetheradiusofaspherewithvolume dimensionsoftherectangularsolid. 972r mm³.


Section 1.4 - Rational and Irrational Numbers ♦ 29


Ifarealnumberisnotrational,itmustbeirrational.Allnumberswithsquarerootsthatarerationalmustbe perfectsquares.Thisisalsotrueforcuberoots.

Forexample: 25 Thisisarationalnumbersince25isaperfectsquare; 25 5=

253 Thisisanirrationalnumbersince25isnotaperfectcube.

Remember,arationalnumberisanumberthatcanberepresentedasafraction.Anirrationalnumberisa non-repeating,ornon-terminatingdecimalvalue.Therefore,whenwritingavalueforanirrationalnumber,it isjustanapproximation.

Irrationalnumbersdonothavetobethen throotofaninteger.Herearesomeotherexamplesofirrational numbers:

r ,e,1.010010001..., 2

Approximating Irrational Numbers

Withtheuseofacalculator,anapproximationofthefollowingvalueswereobtained.

.7 2 650 .70 8 370 .700 26 50 .7000 83 70

Noticethat 7 and 700 havethesamenumerals,butdifferentdecimalpointanswers.Thisisbecause 700 7 100 7 10# #= = .Thesameisalsotruefor 70 and 7000 70 100#= .

Withouttheuseofacalculator,anapproximationofann throotcanbefoundbydeterminingwherethevalue liesonanumberline.

Betweenwhattwoconsecutiveintegersis 7 ?

►Solution: Letaandbbeconsecutiveintegerssothata b71 1 .

Thereforea b72 21 1 .

Since2 42= and3 92

= :2 7 3 2 7 32 2"1 1 1 1

So 7 liesbetween2and3.

Example 1

Rational and Irrational Numbers1.4




Betweenwhattwoconsecutiveintegersis 1003 ?

►Solution: Letaandbbeconsecutiveintegerssothata b10031 1 .

Thereforea b1003 31 1 .

Since4 643= and5 1253

= :4 100 5 4 100 53 3 3"1 1 1 1

So 1003 liesbetween4and5.

Further Approximation of Irrational Numbers

Thepreviousexamplesshowroughapproximationsofirrationalnumbers.Addingsomeadditionalstepsto theprocesswillallowourestimatestobeclosertotheactualvalue.

Approximate 11 toonedecimalplace.

►Solution: Letaandbbeconsecutiveintegerssothata b111 1 .Thereforea b112 21 1 .

Since3 92= and4 162

= :3 11 4 3 11 42 2"1 1 1 1

Then9 11 161 1 showsus11is2unitsfrom9,and5unitsfrom16.

Byratios: .2 52 0 30+

units

Therefore .11 3 30 (Check: . .3 3 10 892 - )

Approximate 1403 toonedecimalplace.

►Solution: Letaandbbeconsecutiveintegerssothata b14031 1 .Thereforea b1403 31 1 .

Since5 1253= and6 2163

= :5 140 6 5 140 63 3 3"1 1 1 1

Then125 140 2161 1 showsus140is15unitsfrom125and76unitsfrom216.

Byratios: .15 7615 0 20+

units

Therefore .140 5 23 0 (Check: . .5 2 140 63 - )

Example 2

Example 3

Example 4




1.4 Exercise Set

1. Withoutacalculator,decideifthenumbersarerationalorirrational.

a) 81 b) 810

c) 40 d) 400

e) .6 4 f) .0 64

g) .0 004 h) .0 0004

i) 22500 j) .22 5

k) 322 l) .0 322

2. Withoutacalculator,decideifthenumbersarerationalorirrational.

a) 13 b) 103

c) 1003 d) 10003

e) 83 f) 803

g) 8003 h) 80003

i) .0 83 j) .0 083

k) .0 0083 l) .0 00083




3. Usingacalculator,approximatetheirrationalnumbersto2decimalplaces.

a) .2 3 b) 23

c) 230 d) .0 23

e) 2300 f) .0 023

g) .7 93 h) 793

i) 7903 j) 79003

k) .0 793 l) .0 00793

4. Given .21 4 580 and .210 14 490 ,determinethevalueofthesquareroot.

a) .2 1 b) .0 21

c) 2100 d) .0 021

e) 21000 f) .0 0021

g) 210 000 h) .0 00021

i) 2100 000 j) .0 000021




5. Given .21 2 763 0 , .210 5 943 0 and .2100 12 813 0 ,determinethevalueofthecuberoot.

a) 210003 b) .2 13

c) 2100003 d) .0 213

e) .0 0213 f) .0 00213

g) 2100 0003 h) .0 000213

i) 21000 0003 j) .0 0000213

6. Estimatethepositionofeachirrationalnumberonthenumberlineprovided.

a) 11

b) 32-

c) 29

d) .1 9-

e) 1003

f) 873-

g) 1533

0 4 5 6321

0 4 5 6321

0 4 5 6321

0 4 5 6321

0 4 5 6321

0 4 5 6321

0 4 5 6321




7. Withoutusingtherootfunctionsofacalculator,approximatetheirrationalnumbertoonedecimalpoint.

a) 30 b) 58-

c) 88 d) 76-

e) 130 f) 983

g) 743- h) 43

i) 113- j) 193

8. Ordertheirrationalnumbersonthenumberlineprovided.

a) , , , ,18 27 30 22r - -

b) , , , ,e73 130 30 1103 3 3 3-

c) , , , , ,10 10 17 17 28 283 3 3- -

0 4 5 6321

0 4 5 6321

0 4 5 6321


Section 1.5 - Exponential Notation ♦ 35


Anexponenttellshowmanytimesthebaseisusedasafactor.Inthestatementa a a a a a5# # # # = ,the exponentis5andthebaseisa.Thisisread“atothefifthpower”.

Forexample: 24 means 2 2 2 2# # #

a3 2^ h means a a3 3#

y 3-^ h means y y y# #- - -^ ^ ^h h h

10p3 means p p p10 # # #

One and Zero as Exponents

Considerthefollowingexample:

3 3 3 3 34# # # = 3 3 3 33# # = 3 3 32# =

Ontheleftsideoftheequationeachstepisbeingdividedby3.Ontherightsideoftheequation,theexponent decreasesby1oneachstep.

Tocontinuethepattern:

3 31= 1 30=

Exponents of 0 and 1

a a1= ,foranynumbera.

a 10= ,foranynon-zeronumbera.

Note:00isnotdefined.

Examples: 5 10=

17 171=

32 10

=` j

5 10- =-

5 10- =^ h

Exponential Notation1.5




The Product Rule

Considerthefollowingexample:a a a a aa a a2 3 5# # # # #= =^ ^h h

Theexponentintheexpressiona5isthesumoftheexponentsintheexpressiona a2 3# . Therefore:a a a a2 3 2 3 5# = =+ .

Product Rule

Foranynumbersaandbwithexponentsmandn:

,a a a a 0m n m n# != +

Examples: 3 3 3 34 5 4 5 9#- - = - = -+^ ^ ^ ^h h h h

32

32

32

325 1 5 6

# = =+` ` ` `j j j j

22 8 2 2 2 2 23x x x x x x x xx3 2 3 3 6 3 6 92# # #= = = =+^ h

The Quotient Rule

Considerthefollowingexample:aa

a a a

a a a a a a aa a a a a3

74

# #

# # # # # ## # #= = =

Theexponentintheexpressiona4isthedifferenceoftheexponentsintheexpressionaa3

7

. Therefore:

aa a a3

77 3 4

= =- .

Quotient Rule

Foranynumberawithexponentsmandn:

,aa a a 0n

mm n != -

Examples:4

44 43

8

8 3 5

-

-= - = --

^^ ^ ^hh h h

5353

53

53

6

6 1 5

= =-

`

`` `

j

jj j




The Power Rule

Anexpressionsuchas 52 3^ h means5 5 52 2 2# # .Bytheproductrule5 5 5 5 52 2 2 2 2 2 6# # = =+ + .Butwecan alsogetto56bymultiplyingtheexponents: 5 5 52 3 2 3 6

= =#^ h .

Power Rule

Foranynumbersaandbwithexponentsmandn:

a am n m n= #^ h

Examples: 3 3 35 4 5 4 20= =#^ h

2 2 23 3 3 3 9- = - = -#^ ^ ^h h h6 @

52

52

523 4 3 4 12

= =#

` ` `j j j8 B

Raising a Product to a Power

Anexpressionsuchas x2 3^ h means x x x2 2 2# # . Thisexpressioncanbewritten: x x x x x2 2 2 2 83 3 3# # # # # #= =^ ^h h .

A Product to a Power

Foranynumbersaandbwithexponentn:

ab a bn n n#=^ h

Examples: x x x3 3 273 3 3 3#= =^ h

y y y2 2 162 4 4 2 4 8#= =#^ h

a b a b a b3 3 93 4 2 2 3 2 4 2 6 8# #- = - =# #^ ^h h




Raising a Fraction to a Power

Anexpressionsuchas32 4` j means

32323232

# # # .Multiplyingthefractionsgives:3 3 3 32 2 2 2

32

8116

4

4

# # ## # # = = .

A Fraction to a Power

Foranynumbersaandb, b 0! ,withexponentn:

ba

ban

n

n

=` j

Examples:x x x2 2 83

3

3

3= =` j

a a a1 1 14

2

4 2

2

8= =#c m

Negative Exponents

Considerthefollowingexample:55

5 5 5 5 5 5

5 55 5 5 5

151

6

2

4# # # # #

## # #

= = = .

Solvingthepreviousexamplewiththequotientrulegives:55 5 56

22 6 4

= =- -

Therefore51 54

4= -

Negative Exponents

Foranynumbera, 0a ! ,withexponentn:

aa1nn=-

Examples: 221

813

3= =-

xx x

221

1614

4 4= =-^ ^h h




Changing from Negative to Positive Exponents

Considertheexpression213- .Bythenegativeexponentrule,thisisequivalentto2 23 3

=- -^ h .

Changing from Negative to Positive Exponents

Foranynon-zeronumbersaandb,withexponentsmandn:

ba

ab

n

m

m

n

=-

-

andba

abm m

=-` `j j

Examples:

32

23

8273 3

= =-` `j j

43

34

964

3

2

2

3

= =-

-

Rational Exponents: a 1n

Considerthesquarerootexample: 2 2 2# = .

Nowconsidertheexponentruleexample:2 2 2 2 221

21

2121

1# = = =+ .

Since 2 2# and2 221

21

# equal2, 2 shouldequal221

.

Rational Exponents: a 1n

Foranynon-negativerealnumberaandanypositiveintegern.

a ann

1

=

Examples: x x21

=

224 41

=

2 2 2 2 2 2 2 8421

41

2141

43

34 4# #= = = = =+




Rational Exponents: a nm

Thisisamoregeneralversionofrationalexponents.

Bythepowerrule,8 2 168 834

31 4 3 4 4

= = = =^ ^h h

However,834

canalsobewrittenas 168 8 4096431

43 3= = =^ h

Rational Exponents: a nm

Foranynon-negativerealnumberaandanypositiveintegern.

a anm mn=

Examples:4 2 2 2 823

223

223

3= = = =#^ ch m

32 32 2 2 165 454

554

4= = = =^ ^h h

162

121

81

43

43

43= = =-

^ h

2

4

2

4

2

2

2

2 2 2 2 324

3

41

31

41

2

41

32

3241

128

123

51231

12= = = = = = =- -^ h

9 27 9 27 3 3 3 3 3 3 3 2 2 2 2 323 431

41

231

341

32

43

3243

128

129

1217

1712 12 512 12$ $ $ $$ = = = = = = = = =+ +^ ^h h




Summary of Exponent Rules

Foranyintegersmandn:

Exponentof1 a a1= 3 31

=

Exponentof0 ,a a1 00 != 5 10- =^ h

ProductRule ,a a a a 0m n m n# != + 2 2 2 23 4 3 4 7# = =+

QuotientRule ,aa a a 0n

mm n != -

33 3 33

55 3 2

= =-

PowerRules

a am n m n= #^ h

ab a bn n n#=^ h

ba

ban

n

n

=` j

2 2 23 4 3 4 12= =#^ h

x x2 23 3 3#=^ h

32

324

4

4

=` j

NegativeExponents

aa1nn=-

ba

abn n

=-` `j j

ba

ab

n

m

m

n

=-

-

2213

3=-

43

342 2

=-` `j j

32

23

4

3

3

4

=-

-

RationalExponentsa an

n1

=

a amnnm

=

553 31

=

553443

=




1.5 Exercise Set

1. Multiply.Leaveanswerinexponentialform.

a) 2 23 4# b) 3 35 7#

c) 4 43 2#- d) 5 50 3#

e) a a a2 3# # f) y y y3 2# #-

g) 8 8 80 1 2# # h)32

323 4

#` `j j

i) 3 3 34 2 1# #- - -^ ^ ^h h h j)21

21

215 3

# #- - --` ` `j j j

2. Divide.Leaveanswerinexponentialform

a)553

6

b)444

8

c)222

8

d)333

9

e)tt2

6

f)xx7

7

g)6

63

4

-

--^

^hh h)

9

96

3

-

--

-

^^hh

i)x

x

2

24

3

-

--^

^hh j)

zz6

2

-

-




3. Simplify.Expresswithoutbracketsornegativeexponents.

a) 24 2^ h b) 53 2-^ h

c) 3 4 2- -^ h d) x3 2 0- -^ h

e) x2 3^ h f) x3 4 2-^ h

g) a2 4 3-^ h h) x y3 4 2 4-^ h

i) a b4 3 2 2- - -^ h j) x y2 3 2 3- - -^ h


a)33 3

5

4 7# b)2 224 3

5

#

c)44 4

1

3#-

-

d)5 55 53 1

4 2

##

-

-

e)7 77 7

2

0 3

##

-

-

f)1111 11

1

2 3#-

g)xx32

3 2

-

^ h h)xx33

2 3-^ h

i) a b c2 2 4 5 3- -^ h j)ba32

4

2 3-

c m




5. Evaluate.

a) 32 b) 3 2-

c)31 2` j d)

31 2-` j

e) 32- f) 3 2-^ h

g)31 2

- -` j h)31 2

-` j

i)31 2

--` j j)

31 2

- --` j

k) 23 l) 2 3-

m)21 3` j n)

21 3-` j

o) 23- p) 2 3-^ h

q)21 3

- -` j r)21 3

-` j

s)21 3

--` j t)

21 3

- --` j





a)a b

a b ab2 43 4

2 3 2 1 3#-

- -

^^ ^

hh h b)

x yx y x y

1 2

5 2 2 2 2 3#- -

- -^ ^h h

c)m n

m n m n

2

5 23 2 1

1 2 2 2 3 3#-

- - -

^^ ^

hh h d)

a b

a b a b

3

3 32 2 3

2 3 2 1 4 1#- -

- - - -

^^ ^

hh h

e)x y

x y x y

4

3 52 3 2

1 2 1 2 4 2#

- -

- - - -

^^ ^

hh h f)

a b

a b a b

3

3 43 4 2

1 1 2 2 3 4 2#

- -

- - - - - -

^^ ^

hh h

g)x y

x y

x y

x y4 82

2 2 3 3

3 1

1 3 2

-

- -

-

- - -

c cm m h)a bab

a ba b

89

23

2 2

1 2

2 1

2 2 3

-

- -

-

-

c cm m

i)x y

x y x y

12

2 42 2

1 2 2 3 2- - -

^^ ^

hh h j)

x y

x y x y

15

5 62 4

3 4 2 2 5 2

-

- - - -^ ^h h; E




7. Evaluate.

a) 1643

b) 16 43-

c) 832

d) 8 32-

e) 2734

f) 27 34-

g) 1645

- h) 16 4

5

- -

i) 3254

- j) 32 54

- -

k) 21632

l) 216 32-

m) 12534

- n) 125 34

- -

o) 64 67

p) 64 67

-

q) 49 23

- r) 49 23

- -

s) 128 75

t) 128 75

-

u) 243 56

- v) 243 56

- -

w) 81 45

x) 81 45

-




8. Simplify.Leaveanswerwithpositiveexponents.

a) 2 241

45

# b) 3 332

37

#

c) 4 441

43

# - d) 5 532

31

#- -

e)6

6

45

43

f)7

7

51

52

-

g)8

8 8

73

72

74

#-

-

h)9 9

9

52

54

53

# -

i) a a43

45

# j) b b65

31

# -

k)c

c

65

32

l)d

d

21

31

-

m)49 23

` j n)49 2

3-` j

o)1681 4

3

` j p)1681 4

3-` j

q) a b3 4132^ h r) x y4 2

134^ h

s) a b c32

65

2176

^ h t) x y z34

4325

512

-^ h




9. Simplifyeachradical.Assumevariablesarepositive.

a) 44 b) 863

c) 1638 d) 2723

e) 9312 f) 428

g) a24 h) b39

i) c46 j) d28

10. Simplify.

a) 2 23# b) 3 34#

c) 2 23 4# d)4

44

3

e)9

273

f)8

164

3

g)4

8x

x x

21 $` j

h)93 27

x

x x$

i)27

81x

x x

31 $` j

j)25

5 125x

x x

3

2$-




11. Identifytheerrors,thencorrecttheresult.

a) 2 2 25 3 8+ = b) 3 3 94 2 6# =

c) 4 4 43 5 15# = d)22 24

123

=

e)22 14

128

= f) x x2 3 5=^ h

g) x x3 3=-- h)

216 82

3

=c m

i) 3 814- = j) x x2 82 4 4

=^ h

k) 2 10- =-^ h l) 0 10=

m) 2 2 11+ =-^ h n)6416

64

16

41= =




Tofactorasquareroot,theproductrulecanbeused.

The Product Rule for Square Roots

Foranyrealnumbers A and B : A B A B# #=

Theproductruleisusedwhenthereisaperfectsquareasafactor.

Consider 72 .Tosimplifythisexpression,therearemanywaystofactor72.

Method1: 72 = 4 18#

= 4 18#

= 2 18#

= 2 9 2# #

= 2 9 2# #

= 2 3 2# #

= 6 2

Analternativewaytoapproachthisproblemistolookforthelargestperfectsquarefactorof72,whichis36.

Method2: 72 = 36 2#

= 36 2#

= 6 2

Simplifyingismucheasierifthelargestperfectsquarefactorisidentifiedinthefirststep.

Simplify 48 .

►Solution: 48 = 16 3#

= 16 3#

= 4 3

Example 1

Irrational Numbers1.6


Section 1.6 - Irrational Numbers ♦ 51


Squarerootexpressionssuchas 14 or 2 cannotbesimplified,sinceneitherhaveperfectsquarefactors. Sometimes,aftermultiplication,itispossibletosimplifytheproduct.

Forexample: 2 14# = 28

= 4 7#

= 4 7#

=2 7

Tofactoracuberoot,theproductruleisusedagain.

The Product Rule for Cube Roots

Foranyrealnumbers A3 and B3 : A B A B3 3 3# #=

Theproductruleisusedwhenthereisaperfectcubeasafactor.

Simplify 403 .

►Solution: 403 = 8 53 #

= 8 53 3#

= 2 53

Simplifyingproductsofcuberootsfollowsthemethodusedforsimplifyingproductsofsquareroots.

Forexample: 9 63 3# = 543

= 27 23 #

= 27 23 3#

= 3 23

Entire Root

Anexpressionsuchas2 7 iscalledamixed root,andtheexpression 28 iscalledanentire root.Both expressionshavethesamevalue.Anymixedrootcanbechangedtoanentireroot.

Forexample: 9 3 = 9 9 3# # 2 53 = 2 2 2 53 # # #

= 243 = 403

Example 2




1.6 Exercise Set

1. Findeachproduct.

a) 3 5# b) 7 2#

c) 13 13# d) 5 6#

e) 2 11# f) 2 3 5# #

g) 4 53 3# h) 2 2 23 3 3# #

i) 2 3 53 3 3# # j) 6 7 53 3 3# #

2. Whichoneoftheentirerootscannotbesimplifiedtoamixedroot?

a) , , ,44 46 48 50 b) , , ,18 20 21 24

c) , , ,40 81 100 1253 3 3 3 d) , , ,16 36 54 1283 3 3 3

e) , , ,32 32 100 1003 3 f) , , ,64 64 75 753 3

g) , , ,27 27 50 503 3 h) , , ,8 10 12 20

i) , , ,36 5424 723 3 3 3 j) , , ,12525 25 1253 3




3. Simplifytoamixedroot.

a) 20 b) 72

c) 45 d) 24

e) 75 f) 125

g) 140 h) 128

i) 80- j) 160-

4. Simplify.

a) 2 9 b) 4 25

c) 6 40 d) 3 8

e) 4 27 f) 6 50

g)25 32- h) 2

31 72-

i) .0 8 125- j) .1 25 128-




5. Simplifyeachroot.

a) 403 b) 483

c) 543 d) 1353

e) 1283 f) 1923

g) 2 273 h) 3 163-

i)21 643 j)

53 2503-

6. Multiply,andsimplifyifpossible.

a) 3 6# b) 7 14#

c) 3 27# d) 2 8#

e) 3 24# f) 10 20#

g) 5 6 2 18# h) 4 10 21#-

i) 2 10 3 50# j) 3 12 2 18#- -^ ^h h




7. Multiply,andsimplifyifpossible.

a) 4 63 3# b) 9 243 3#

c) 5 53 3# d) 4 543 3#

e) 2 12 303 3# f) 3 25 4 753 3#-

g) 2 10 3 503 3# h) 3 12 2 183 3- -^ ^h h

i) 3 4 2 323 3- -^ ^h h j) 5 49 2 563 3-^ ^h h

8. Withoutacalculator,compareeachexpressionusing>,<or=. (Hint:Converteachexpressiontoanentirerootfirst)

a) 4 1514 b) 162 9 2

c) 3 11 7 2 d) 12 11 13#

e) 54 23 f) 2 7 563 3

g) 2 15 1253 3 h) 5 3 4 43 3

i) 25 67633 3-- j) 7 5523 3--




9. Expressasanentireroot.

a) 4 3 b) 2 5

c) 7 6 d) 12 2

e) 5 11 f) 4 14

g) 6 3 h) 2 3 3 2#

i) 2 5 3# j) 4 5 3 3#

10. Expressasanentireroot.

a) 3 23 b) 4 33

c) 5 43 d) 3 53

e) 6 63 f) 4 73

g) 7 83 h) 3 2 4 33 3#

i) 2 4 5 53 3# j) 3 6 73 3#




11. Asquarehasanareaof150mm².Whatarethe 12. Acubehasavolumeof192cm³.Whatarethe lengthsofthesidesofthesquare? lengthsofeachedgeofthecube?

13. Thedimensionsofarectangleare9 30 cmby 14. Thedimensionsofarectangleare5 6 cmby 4 105 cm.Calculatetheareaoftherectangle. 34 cm.Calculatetheareaoftherectangle.

15. Thebaseofatriangleis6 14 ftanditsheightis 16. Thebaseofatriangleis5 33 manditsheightis 3 21 ft.Calculatetheareaofthetriangle. 6 55 m.Calculatetheareaofthetriangle.

17. Thedimensionsofarectangularprismare: 18. Thedimensionsofarectangularprismare: length2 10 cm,width3 14 cm,andheight length3 110 m,width2 22 mand 35 cm.Determinethevolumeoftherectangular height4 15 m.Determinethevolumeof prism. therectangularprism.

19. Thedimensionsofarectangularbasepyramidare: 20. Thedimensionsofarectangularbasepyramidare: length2 42 cm,width3 30 cm,andheight length3 26 m,width2 39 m,andheight 4 70 cm.Determinethevolumeofthepyramid. 4 42 m.Determinethevolumeofthepyramid.




Section 1.1

1. Considerthelistofnumbers: , . , , . , . , , ,2 0 4 0 0 343343334 4 22235 7 2f f- .Listall:

a) Naturalnumbers

b) Wholenumbers

c) Integers

d) Rationalnumbers

e) Irrationalnumbers

f) Realnumbers

Section 1.2

2. Simplifythecompositenumberstoaproductofprimenumbers.

a) 4950 b) 1848

c) 2618 d) 264264

3. Findthegreatestcommonfactor.

a) 126,588 b) 1755,2475

c) 7007,13013 d) 544,600,2250

Chapter Review1.7


Section 1.7 - Chapter Review ♦ 59


4. Findtheleastcommonmultiple.

a) 56,196 b) 90,300

c) 15,20,30 d) 30,45,84

Section 1.3

5. Determinetherootswithoutacalculator.

a) 64 b) 643

c) 729 d) 7293

e) 1296 f) 27443

g) 1764 h) 58323

Section 1.4

6. Withoutacalculator,determineifthenumberisarationalorirrationalnumber.

a) .0 4 b) .0 04

c) 90 d) 900

e) .0 273 f) .0 0273

g) 8003 h) 80003




7. Using . , . , . , . , .18 4 24 180 13 42 18 2 62 180 5 64 1800 12 163 3 30 0 0 0 0 ,determinethevalueofthe radical.

a) .1 8 b) .0 18

c) .0 018 d) 1800

e) 18 000 f) .1 83

g) .0 183 h) .0 0183

i) 18 0003 j) 180 0003

Section 1.5


a)22 2

5

4 3# b)3

3 35

2

2

3 4 2#^

^ ^h

h h

c)x

x22

3 2

4-

-

^^hh d)

yx23

3

2 2-

c m

e) a b c3 2 3 4 2- - -^ h f)ba23

3

2 4- -

c m

g)x y

x y x y

2

2 22 2

2 3

3

2 1 4 1#-

-

-

- -

^^ ^

hh h h)

x y

x y

x y

x y

4

3

2

32 2

1 2

2

2 2 3

-

- -

-

-

c cm m


Section 1.7 - Chapter Review ♦ 61


9. Simplify.Evaluateifpossible.

a) 32 54

b) 32 54

-

c) 625 43

- d) 625 43

- -

e) 27 64 34

#^ h f) 27 64 34

# -^ h

g)8116 4

3

` j h)8116 4

3-

` j

i) x63 j) ,x x 0412 $

k)2

23

l)2

43

m)3

34

3

n)2

43

4

o)2

2 24

3# p)27

3 94

3#




Section 1.6

10. Simplifyeachradical.

a) 108 b) 1083

c) 288 d) 2883

e) 3 54 f) 3 543

g) 2 14 28# h) 2 14 283 3#

i) 5 12 54#- j) 5 12 543 3#-

11. Expressasanentireradical.

a) 3 2 b) 3 23

c) 2 5- d) 2 53-

e) 5 2 2 3# f) 5 2 2 33 3#

g) 3 3 2 5 4 2# # h) 4 3 3 4 2 23 3 3# #


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Lambrick Park Secondary 1.1 Number Systems