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  • MCR3UUnit4:ExponentialRelationsLesson1 Date:___________Learninggoal:Icansimplifyandevaluateexpressionsinvolvingnegativeandrationalexpressionsusingexponentlaws.

    ExponentLaws

    POWERS

    Powersareaconvenientwayofwritingrepeatedmultiplication.

    Apowerconsistsoftwoparts:

    1. Abasetellsuswhichvaluetorepeatedlymultiply,and

    2. Anexponenttellsushowmanytimestoperformthemultiplication.

    !!

    Thepowercanbewritteninexponentialformas34,inexpandedformas(3)(3)(3)(3),orevaluatedformas8.

    Recallthedifferencebetweennegativebasesandcoefficients.

    3 ! = 3 3 = 9,but3! = 3 3 = 9.

    EXPONENTIALLAWS

    Example1:Simplifythefollowing.

    a)(5!!!!!!)(2!!!!!!) b)(3!!!!)! c) (!"!!!!!!!!!!!!!!)!

    ExponentLaw

    !!!! = !!!! !!

    !! = !!!!, ! 0

    !" ! = !!!! !!!= !

    !

    !! , ! 0

    !! ! = !mn !! = 1

    !!

    !!= !! ,!, ! 0

    !!

    !!= !!

    !,!, ! 0

  • d)(!!!!! )!! e)(!!!

    !!!!!)(!!!!!!!!)!!"!!!!! f)

    (!!!!!!)!!(!!!!!)!(!!!!!!!)!!

    g) 3!(3!! 3!!!!) h)!!!!!!!!!! RATIONALEXPONENTS

    Example2:Useacalculatortocompletethefollowing.

    a) 9!! =and 9 =

    b) 64!! =and 64! =

    c) 16!! =and 16! =

    RationalExponents

    1. !!! = !! (if! > 1and! 0)!! iscalledaradicalandmeansthenthrootofb.

    2. !!! = !!! !!or!

    !! = !!!

    (ifmandnarebothpositiveintegersand! 0)

  • Example3:Writeeachexpressioninradicalform.Donotevaluate.

    a) !!!

    b) !!! c) 125!

    !!

    Example4:Writeeachexpressionasapowerusingrationalexponents.Donotevaluate.a) 12!

    b) ( 13! )! c) !!! ( !!! )

    Example5:Evaluatethefollowingwithacalculator.Givefinalanswersinexactform.

    a) 64!!

    b) ( !"!"#)!! c) (!"!")

    !!!

    d) (32)!!!

    e) 125!!! f) 16!.!"

  • g) (9!!9

    !!)!"

    h) 64! i) ( 5!! )( 5! )

    HW:Pg.79#1-4withoutcalculator,14,16,17,pg.85#1-6,13,14(#5cans !!! )

    SuccessCriteriatoEvaluatingPowerswithRationalExponents!!!

    1.

    2.

    3.

    4.

  • MCR3UUnit4:ExponentialRelationsLesson2 Date:___________Learninggoal:Icansimplifyradicalsandrationalizedenominators.

    RationalizingDenominatorsRECALL:Wecanmultiply/divideradicalstogether,creatingasingleradicalconsistingoftheproduct/quotientofthebases.

    2 5 = 25= 10

    Ingeneral ! ! = !!

    102

    = 2 52 = 5

    Ingeneral !! =!!

    Example1:Simplifythefollowingradicals.

    a)2 50 b)9 24 14 c) !"!"

    d)!!" !"#! ! e)!"! !"

    ! f)!!"!! !"#

    !

    ADDING&SUBTRACTINGRADICALSJustlikeweaddorsubtractliketermswhensimplifyingexpressionswithrationalnumbers,wecandothesamewithirrationalnumbers.Radicalswiththesamebasecanbecollected.Example2:Simplify.

    a)10 2 7 3 + 2 2 + 4 3 b) 24 + 54 + 150 c) 3 48 4 8 + 4 27 2 72

  • RATIONALIZINGDENOMINATORS

    Thereareconventionsinmath,likeinEnglish.InEnglishwestarteverynewsentencewithacapitalletter.Inmath,wedontleavenegativesindenominators.Anotherconventionis,weleavenoirrationalpartinadenominator.Inotherwordswerationalizethedenominator.

    Torationalizethedenominatorwhenyouhaveamonomialdenominator,wemultiplythenumeratoranddenominatorbytheradicalappearinginthedenominator.

    Example3:

    a)! !! b)! !! ! c)

    !! 72!

    Torationalizethedenominatorwhenyouhaveabinomialdenominator,wemultiplythenumeratoranddenominatorbytheradicalconjugateofthedenominator.Theradicalconjugatehasthesamebinomialpartsasthedenominator,butthesecondsignisreversed.

    Forexample,theradicalconjugateof2+ 3 5is2 3 5.Whenabinomialismultipliedbyitsconjugate,themiddletermsofexpandingcanceleachotherout.

    Example4:Simplify.

    a) !! !! ! b)! !! !! !!! !

    HW:Simplifying&RationalizingWorksheet

  • Simplifying&RationalizingWorksheet

    1. Simplify.a)4 75 2 12 + 8 48 2 8 b)3 63 + 88 2 112 c)!!! !"!"

    2. Rationalizethedenominatorforeachofthefollowing:

    a) !! b)!! c)

    !! !!! ! d)

    ! !!! !!! !

    e)!!! !!! ! f)!

    !!!!! g)!10

    !! !!!

    3. Simplify.

    a)! !! b)! !! !

    ! c)!

    10!! d)! !

    ! !!!

    e) !! !!! ! f)!

    ! 27!! 12 g)!!!

    ! 27!! !

    Answers

    1a) 48 3 4 2 b) 13 7+ 2 22 c)!!! !! 2a)! !! b)

    !! c)5 2 6 d)

    16!! 10! e)

    !10! !! !!! 15! f)

    !!!!!!!! g)2 ! + 5+ !

    3a) 21 b)3 3 c) !! !! d)12!! !

    ! e)!! 21! f)

    !13 g)

    !! !!! !!! !!

  • MCR3UUnit4:ExponentialRelationsLesson3 Date:___________Learninggoal:Icansolveequationswithavariableintheexponent.

    SolvingExponentialEquations

    RECALL: 2! 3! cantbesimplifiedthroughlaws.Powerlawsonlyapplyifbasesarethesame.However,sometimesbasescanbechanged.

    3! 9! = 3! 3! !

    = 3! 3!= 3!

    Example1:Writethefollowingpowerswithabaseof2.

    a)!! b)4!! c) !16

    !

    SOLVINGEXPONENTIALEQUAITONSExponentialequationshaveunknownsintheexponent.Example2:Solve.

    a)9!!! = 3!!! b)25!!!! = ( !!"#)! c) 2!!!! = 8

    SuccessCriteriaforSolvingExponentialEquations

    1. Writeexpressionsaspowersofthesamebase.

    2. Dropthebase,settheexponentsequaltooneanother

    3. Solve

  • d) 2 !!! ! = 64 e) 4 7!!!! = 1372 f) 9! + 3! = 12

    HW:Pg.94#1,4,6-9,13,22-24

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