A_Level_数学1

©BoardworksLtd20051of38©BoardworksLtd20051of38AS-LevelMaths:Core1forEdexcelC1.1Algebraandfunctions1ThisiconindicatestheslidecontainsactivitiescreatedinFlash.Theseactivitiesarenoteditable.Formoredetailedinstructions,seetheGettingStartedpresentation.©BoardworksLtd20052of38Contents©BoardworksLtd20052of38UsingandmanipulatingsurdsRationalizingthedenominatorTheindexlawsZeroandnegativeindicesFractionalindicesSolvingequationsinvolvingindicesExamination-stylequestionsUsingandmanipulatingsurds©BoardworksLtd20053of38TypesofnumberWecanclassifynumbersintothefollowingsets:Thesetofnaturalnumbers,:Positivewholenumbers{0,1,2,3,4…}Thesetofintegers,:Positiveandnegativewholenumbers{0,±1,±2,±3…}Thesetofrationalnumbers,:Numbersthatcanbeexpressedintheform,wherenandmareintegers.Allfractionsandallterminatingandrecurringdecimalsarerationalnumbers;forexample,¾,–0.63,0.2.nmThesetofrealnumbers,:Allnumbersincludingirrationalnumbers;thatis,numbersthatcannotbeexpressedintheform,wherenandmareintegers.Forexample,and.nm2©BoardworksLtd20054of38169=1001691.69=100IntroducingsurdsNumberswritteninthisformarecalledsurds.Whenthesquarerootofanumber,forexample√2,√3or√5,isirrational,itisoftenpreferabletowriteitwiththerootsign.2,3or5,√1.69isnotasurdbecauseitisnotirrational.1.69=1.313=10Canyouexplainwhy√1.69isnotasurd?1.69Thisusesthefactthat=aabb©BoardworksLtd20055of38ManipulatingsurdsWhenworkingwithsurdsitisimportanttorememberthefollowingtworules:Youshouldalsorememberthat,bydefinition,√ameansthepositivesquarerootofa.aaabb=andabab=?Also:?aaa©BoardworksLtd20056of38SimplifyingsurdsStartbyfindingthelargestsquarenumberthatdividesinto50.Thisis25.Wecanusethistowrite:WecandothisusingthefactthatForexample:abab=?=5250=25?=25?Weareoftenrequiredtosimplifysurdsbywritingthemintheform.abSimplifybywritingitintheform50.ab©BoardworksLtd20057of38=9?SimplifyingsurdsSimplifythefollowingsurdsbywritingthemintheforma√b.=3545=9?1)452)9833)40=7298=49?=49?3=2533×40=8533=8?.ab©BoardworksLtd20058of38Simplifyingsurds©BoardworksLtd20059of38AddingandsubtractingsurdsSurdscanbeaddedorsubtractedifthenumberunderthesquarerootsignisthesame.Forexample:Simplify45+80.Startbywritingandintheirsimplestforms.4580=3545=9?=9?=4580=16?=16?45+80=35+45=75©BoardworksLtd200510of38ExpandingbracketscontainingsurdsSimplifythefollowing:1)(42)(1+32)2)(72)(7+2)+12226=4=1122+27272=7=5Problem2)demonstratesthefactthat(a–b)(a+b)=a2–b2.Ingeneral:()(+)ababab©BoardworksLtd200511of38Contents©BoardworksLtd200511of38UsingandmanipulatingsurdsRationalizingthedenominatorTheindexlawsZeroandnegativeindicesFractionalindicesSolvingequationsinvolvingindicesExamination-stylequestionsRationalizingthedenominator©BoardworksLtd200512of38RationalizingthedenominatorWhenafractioncontainsasurdasthedenominatorweusuallyrewriteitsothatthedenominatorisarationalnumber.Thisiscalledrationalizingthedenominator.Forexample:Simplifythefraction.52Inthisexamplewerationalizethedenominatorbymultiplyingthenumeratorandthedenominatorby2.5=2252×2×2©BoardworksLtd200513of38RationalizingthedenominatorSimplifythefollowingfractionsbyrationalizingtheirdenominators.3)3471)232)5234=7×7×7×5×5×3×32=3323=525102837©BoardworksLtd200514of38RationalizingthedenominatorWhenthedenominatorinvolvessumsofdifferencesbetweensurdswecanusethefactthat(a–b)(a+b)=a2–b2torationalizethedenominator.Forexample:Simplify1.52115+2=?52525+25+2=54=5+2©BoardworksLtd200515of38Working:23131RationalizingthedenominatorMoredifficultexamplesmayincludesurdsinboththenumeratorandthedenominator.Forexample:Simplify231.3+1(32311)(31(231)=3+1(3+1))733=31=6233+1=733733=2©BoardworksLtd200516of38Contents©BoardworksLtd200516of38UsingandmanipulatingsurdsRationalizingthedenominatorTheindexlawsZeroandnegativeindicesFractionalindicesSolvingequationsinvolvingindicesExamination-stylequestionsTheindexlaws©BoardworksLtd200517of38IndexnotationSimplify:a×a×a×a×a=a5atothepowerof5a5hasbeenwrittenusingindexnotation.anThenumberaiscalledthebase.Thenumberniscalledtheindex,powerorexponent.Ingeneral:an=a×a×a×…×anofthese©BoardworksLtd200518of38IndexnotationEvaluatethefollowing:0.62=0.6×0.6=0.3634=3×3×3×3=81(–5)3=–5×–5×–5=–12527=2×2×2×2×2×2×2=128(–1)5=–1×–1×–1×–1×–1=–1(–4)4=–4×–4×–4×–4=256Whenweraiseanegativenumbertoanoddpowertheanswerisnegative.Whenweraiseanegativenumbertoanevenpowertheanswerispositive.©BoardworksLtd200519of38ThemultiplicationruleForexample:a4×a2=(a×a×a×a)×(a×a)=a×a×a×a×a×a=a6Whenwemultiplytwotermswiththesamebasetheindicesareadded.=a(4+2)Ingeneral:am×an=a(m+n)©BoardworksLtd200520of38ThedivisionruleForexample:a5÷a2=a×a×a×a×aa×a=a34p6÷2p4=2p2=a(5–2)=2p(6–4)Whenwedividetwotermswiththesamebasetheindicesaresubtracted.Ingeneral:am÷an=a(m–n)24×p×p×p×p×p×p2×p×p×p×p=©BoardworksLtd200521of38Forexample:(y3)2=(pq2)4=Thepowerruley3×y3=(y×y×y)×(y×y×y)=y6pq2×pq2×pq2×pq2=p4×q(2+2+2+2)=p4×q8=p4q8Whenatermisraisedtoapowerandtheresultraisedtoanotherpower,thepowersaremultiplied.Ingeneral:=y3×2=p1×4q2×4(am)n=amn©BoardworksLtd200522of38Usingindexlaws©BoardworksLtd200523of38Contents©BoardworksLtd20052