Download - 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
Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.
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.
Lambrick Park Secondary
Section 1.1 - Number Systems ♦ 7
Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.
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:
a) Naturalnumbers b) Wholenumbers
c) Integers d) Rationalnumbers
e) Irrationalnumbers f) Realnumbers
4. Considerthelistofnumbers: , , , . , . , , . ,64 64 0 008 0 04 0 494
7218
903 3- - .Listall:
a) Naturalnumbers b) Wholenumbers
c) Integers d) Rationalnumbers
e) Irrationalnumbers f) Realnumbers
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8 ♦ Chapter 1 - Real Numbers
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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
Lambrick Park Secondary
Section 1.2 - Greatest Common Factor and Least Common Multiple ♦ 9
Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.
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
Lambrick Park Secondary
10 ♦ Chapter 1 - Real Numbers
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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
Lambrick Park Secondary
Section 1.2 - Greatest Common Factor and Least Common Multiple ♦ 11
Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.
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
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12 ♦ Chapter 1 - Real Numbers
Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.
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
Lambrick Park Secondary
Section 1.2 - Greatest Common Factor and Least Common Multiple ♦ 13
Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.
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
3 18 54# = isnotamultipleof15
4 18 72# = isnotamultipleof15
5 18 90# = isamultipleof15,6 15 90# =
ThereforetheLCMof15and18is90.
Example 5
Lambrick Park Secondary
14 ♦ Chapter 1 - Real Numbers
Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.
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
Lambrick Park Secondary
Section 1.2 - Greatest Common Factor and Least Common Multiple ♦ 15
Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.
Finding the Least Common Multiple (Method 3) (Forusewith3ormorenumbers)
1. Dividebyaprimefactorthatcandividetwoormoreofyournumbers.Bringdownanynumberthatdoes notfactor. 2. Repeat,untilnoremainingnumbershaveanyprimefactorsincommon.
FindtheLCMof15,18and24.
►Solution: , ,3 15 18 24 allthreenumbersaredivisibleby3 2 , ,5 6 8 6and8aredivisibleby2,bringdownthe5 5,3,4 noremainingcommonprimefactors
ThereforetheLCMof15,18and24is3 2 5 3 4 360# # # # =
FindtheLCMof27,30and36.
►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
Lambrick Park Secondary
16 ♦ Chapter 1 - Real Numbers
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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
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Section 1.2 - Greatest Common Factor and Least Common Multiple ♦ 17
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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
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18 ♦ Chapter 1 - Real Numbers
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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
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Section 1.2 - Greatest Common Factor and Least Common Multiple ♦ 19
Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.
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
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20 ♦ Chapter 1 - Real Numbers
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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
Lambrick Park Secondary
Section 1.2 - Greatest Common Factor and Least Common Multiple ♦ 21
Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.
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?
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22 ♦ Chapter 1 - Real Numbers
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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
Lambrick Park Secondary
Section 1.3 - Squares and Square Roots ♦ 23
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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 #= =
Method2:ContinuousDivision
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
Lambrick Park Secondary
24 ♦ Chapter 1 - Real Numbers
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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
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Section 1.3 - Squares and Square Roots ♦ 25
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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 # #= =
Method2:ContinuousDivision
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
Lambrick Park Secondary
26 ♦ Chapter 1 - Real Numbers
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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
Lambrick Park Secondary
Section 1.3 - Squares and Square Roots ♦ 27
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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
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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.
Lambrick Park Secondary
Section 1.4 - Rational and Irrational Numbers ♦ 29
Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.
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
Lambrick Park Secondary
30 ♦ Chapter 1 - Real Numbers
Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.
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
Lambrick Park Secondary
Section 1.4 - Rational and Irrational Numbers ♦ 31
Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.
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
Lambrick Park Secondary
32 ♦ Chapter 1 - Real Numbers
Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.
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
Lambrick Park Secondary
Section 1.4 - Rational and Irrational Numbers ♦ 33
Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.
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
Lambrick Park Secondary
34 ♦ Chapter 1 - Real Numbers
Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.
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
Lambrick Park Secondary
Section 1.5 - Exponential Notation ♦ 35
Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.
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
Lambrick Park Secondary
36 ♦ Chapter 1 - Real Numbers
Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.
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
Lambrick Park Secondary
Section 1.5 - Exponential Notation ♦ 37
Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.
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
Lambrick Park Secondary
38 ♦ Chapter 1 - Real Numbers
Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.
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
Lambrick Park Secondary
Section 1.5 - Exponential Notation ♦ 39
Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.
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# #= = = = =+
Lambrick Park Secondary
40 ♦ Chapter 1 - Real Numbers
Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.
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
Lambrick Park Secondary
Section 1.5 - Exponential Notation ♦ 41
Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.
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
=
Lambrick Park Secondary
42 ♦ Chapter 1 - Real Numbers
Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.
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
-
-
Lambrick Park Secondary
Section 1.5 - Exponential Notation ♦ 43
Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.
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
4. Simplify.Expresswithoutbracketsornegativeexponents.
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
Lambrick Park Secondary
44 ♦ Chapter 1 - Real Numbers
Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.
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
Lambrick Park Secondary
Section 1.5 - Exponential Notation ♦ 45
Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.
6. Simplify.Expresswithoutbracketsornegativeexponents.
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
Lambrick Park Secondary
46 ♦ Chapter 1 - Real Numbers
Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.
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
-
Lambrick Park Secondary
Section 1.5 - Exponential Notation ♦ 47
Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.
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
Lambrick Park Secondary
48 ♦ Chapter 1 - Real Numbers
Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.
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 $` j
j)25
5 125x
x x
3
2$-
Lambrick Park Secondary
Section 1.5 - Exponential Notation ♦ 49
Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.
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= =
Lambrick Park Secondary
50 ♦ Chapter 1 - Real Numbers
Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.
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
Lambrick Park Secondary
Section 1.6 - Irrational Numbers ♦ 51
Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.
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
Lambrick Park Secondary
52 ♦ Chapter 1 - Real Numbers
Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.
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
Lambrick Park Secondary
Section 1.6 - Irrational Numbers ♦ 53
Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.
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-
Lambrick Park Secondary
54 ♦ Chapter 1 - Real Numbers
Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.
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
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Section 1.6 - Irrational Numbers ♦ 55
Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.
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--
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56 ♦ Chapter 1 - Real Numbers
Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.
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#
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Section 1.6 - Irrational Numbers ♦ 57
Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.
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.
Lambrick Park Secondary
58 ♦ Chapter 1 - Real Numbers
Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.
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
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Section 1.7 - Chapter Review ♦ 59
Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.
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
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60 ♦ Chapter 1 - Real Numbers
Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.
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
8. Simplify.Expresswithoutbracketsornegativeexponents.
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
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Section 1.7 - Chapter Review ♦ 61
Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.
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#
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62 ♦ Chapter 1 - Real Numbers
Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.
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# #
Lambrick Park Secondary