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!!!

1.

2.

3.

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