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MCR3U–Unit4:ExponentialRelations–Lesson1 Date:___________Learninggoal:Icansimplifyandevaluateexpressionsinvolvingnegativeandrationalexpressionsusingexponentlaws.
ExponentLaws
POWERS
Powersareaconvenientwayofwritingrepeatedmultiplication.
Apowerconsistsoftwoparts:
1. Abasetellsuswhichvaluetorepeatedlymultiply,and
2. Anexponenttellsushowmanytimestoperformthemultiplication.
!!
Thepowercanbewritteninexponentialformas34,inexpandedformas(3)(3)(3)(3),orevaluatedformas8.
Recallthedifferencebetweennegativebasesandcoefficients.
−3 ! = −3 −3 = 9,but−3! = − 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!!!
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4.
MCR3U–Unit4:ExponentialRelations–Lesson2 Date:___________Learninggoal:Icansimplifyradicalsandrationalizedenominators.
RationalizingDenominatorsRECALL:Wecanmultiply/divideradicalstogether,creatingasingleradicalconsistingoftheproduct/quotientofthebases.
2× 5 = 2×5= 10
Ingeneral… !× ! = !×!
102
= 2× 52
= 5Ingeneral… !
! =!!
Example1:Simplifythefollowingradicals.
a)2 50 b)9 2×4 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,wedon’tleavenegativesindenominators.Anotherconventionis,weleavenoirrationalpartinadenominator.Inotherwordswerationalizethedenominator.
Torationalizethedenominatorwhenyouhaveamonomialdenominator,wemultiplythenumeratoranddenominatorbytheradicalappearinginthedenominator.
Example3:
a)! !! b)! !
! ! c)!! 72!
Torationalizethedenominatorwhenyouhaveabinomialdenominator,wemultiplythenumeratoranddenominatorbytheradicalconjugateofthedenominator.Theradicalconjugatehasthesamebinomialpartsasthedenominator,butthesecondsignisreversed.
Forexample,theradicalconjugateof2+ 3 5is2− 3 5.Whenabinomialismultipliedbyitsconjugate,themiddletermsofexpanding“cancel”eachotherout.
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)!! !!! !!! !
!
MCR3U–Unit4:ExponentialRelations–Lesson3 Date:___________Learninggoal:Icansolveequationswithavariableintheexponent.
SolvingExponentialEquations
RECALL: 2! 3! can’tbesimplifiedthroughlaws.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