unit 4 exponents and exponential...

Unit4ExponentsandExponentialFunctions

TestDate:________________

Name:___________________________________________________________________________________Bytheendofthisunit,youwillbeableto…

• Multiplyanddividemonomialsusingpropertiesofexponents• Simplifyexpressionscontainingexponents• Differentiatetheoutcomebetweenanegativesigninthebaseorinthepowerofanexpression

withexponents• Understandtherelationshipbetweenrationalexponentsandnthroots• UsethePowerPropertyofEqualitytosolveexponentialequations• Distinguishbetweenalinearandexponentialfunctionintheequation,table,andgraph• Describethedomainandrangeforanexponentialfunction• Graphanexponentialgrowth/decayfunction

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TableofContentsMultiplicationPropertiesofExponents.......................................................................................................................................3

DivisionPropertiesofExponents...................................................................................................................................................6

SquareRootsasExponents............................................................................................................................................................10

nthRoots..................................................................................................................................................................................................11

RationalExponents............................................................................................................................................................................12

SolvingExponentialEquations.....................................................................................................................................................13

ExponentialFunctions......................................................................................................................................................................15

IdentifyingExponentialBehavior...........................................................................................................................................16

ExponentialGrowthvs.Decay......................................................................................................................................................17

ExponentialFunctionsPractice....................................................................................................................................................18

Summarize:GraphsofExponentialFunctions.......................................................................................................................20

ExponentialGrowthandDecay....................................................................................................................................................21

ExponentialGrowth......................................................................................................................................................................21

CompoundInterest.......................................................................................................................................................................22

ExponentialDecay.........................................................................................................................................................................22

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4.1MultiplicationPropertiesofExponentsAmonomialisanexpressionwith____________________________________________________connectedonlybymultiplicationanddivision.

• No________________________________________

• No________________________inthedenominatorAconstantisamonomialwhichisa_______________________________________.Examples:Monomial NotaMonomial

Trythis!Expandandevaluatethefollowing:

1. 2" ∙ 2$

2. 4& ∙ 4"

3. 𝑥( ∙ 𝑥Whatdoyounotice?ProductofPowersProperty:Examples:

1. 5" ∙ 5& 2.𝑎(𝑎,)(𝑎&) 3.𝑥𝑦 ∙ 𝑥𝑦

4. (6𝑛&)(2𝑛1) 5. 6𝑐𝑑( 5𝑐(𝑑" 6.(−4𝑥𝑦"𝑧&)(−6𝑥(𝑦"𝑧)

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Trythis!Expandandevaluatethefollowing:

1. 3" $

2. 2( "

3. 𝑥$ &Whatdoyounotice?PowerofaPowerProperty:Examples:

1. 2& " 2. 3& $ (

3. 𝑥( , 4. 𝑥" & "Trythis!Expandandevaluatethefollowing:

1. 𝑥𝑦 &

2. 2𝑧 $Whatdoyounotice?PowerofaProductProperty:Examples:

1. 𝑥𝑦$ , 2. −3𝑝(𝑡, $

3. 4𝑎$𝑏:𝑐 " 4. −4𝑥"𝑦(𝑧; &CHALLENGE:

1. Simplify 5𝑥𝑦& −3𝑥"𝑦" & " 2. Simplify −3𝑥( $ 𝑥"𝑦& ( ,

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WarmUp:𝑏& ∗ 𝑏>> =_____________ 2( & =_____________ −2𝑥𝑦" ( =_______________

Reminder:

WhenMULTIPLYINGpowerswiththesamebase,ADDtheexponents.

Whenraisingapowertoapower,MULTIPLYtheexponents.

Whenthere’salotgoingon,followtheorderofoperations:

• P:Takecareofanythinginsideparentheses.Startwiththeinnermostsetofparentheses.• E:Takecareofexponents.Raiseeverythinginsideparenthesestothepower!• M:Multiplyeverythingtogether.

o Combineliketermso Addexponents

Examples

1. 2𝑎& $ 𝑎& &

2. 𝑐& " −3𝑐( "

3. 5x"𝑦 " 2𝑥𝑦&𝑧 &(4𝑥𝑦𝑧)

4. 2𝑥"𝑦 & ( 3𝑦 "

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DivisionPropertiesofExponents1. QuotientofPowersProperty

ExpandandSimplify:"A

"B= CB

CD=

Inwords:Todividetwopowerswiththesamebase,____________________theexponents.Insymbols:Foranynonzeronumbera,andanyintegersmandp,Examples:

1. EFF

EG 2.

HDIJ

HIK

3. LJ

LK

4. MANFOPMJNDP

2. PowerofaQuotientProperty

ExpandandSimplify:&$

&= E

Q

"=

Inwords:Tofindthepowerofaquotient,findthepowerofthenumeratorandthedenominator.Insymbols:Foranyrealnumbersaandbnotequaltozero,andanyintegerm,Examples:

1. &(

$ 2.

&PD

1

"

3. &RB

$

&

4. "SK

&RD

"

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3. ZeroExponentPropertyExpandandSimplify: UsetheQuotientofPowerProperty:&J

&J= &J

&J=

Inwords:Azeroexponentisanynonzeronumberraisedtothezeropower.Itisalwaysequalto1.Insymbols:Foranynonzeronumbera,Examples:

1. TE

U 2.

RJS0

RD

3. "RDSRKS

U

4. "TAEO

TK

4. NegativeExponentProperty

ExpandandSimplify: UsetheQuotientofPowersProperty:EK

EJ= EK

EJ=

Inwords:Fora(anotzero)andn(anynumber),𝑎WXand𝑎Xarereciprocals.

Insymbols:Foranynonzeronumberaandanyintegern,Examples:

1. 2W$ 2. >YZB

3. >&ZK

4. 𝑥W>U

5. XZDPB

LZK

6.

[ZD\RK

\SZJ

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Directions: Simplify each of the following.

1. 𝟖𝟖

𝟖𝟒 2. 𝒂

𝟒𝒃𝟔

𝒂𝒃𝟑 3. 𝟓𝒄𝟐𝒅𝟑

W𝟒𝒄𝟐𝒅

4. 𝟒𝒇𝟑𝒈𝟑𝒉𝟔

𝟑 5. W$RK

"$RJ 6. ,\J

1PjLD

"

7. 𝒙𝟑(𝒚W𝟓)(𝒙W𝟖) 8. &1

W" 9. ""LDmK

>>LKmZD

10. 𝟔𝒇Z𝟐𝒈𝟑𝒉𝟓

𝟓𝟒𝒇Z𝟐𝒈Z𝟓𝒉𝟑 11. W>"C

ZFnJRZB

"CZDnRJ 12. ("o

ZKT)ZD

(oKTB

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Directions: Simplify each of the following.

1. 𝒎𝟓𝒏𝒑𝒎𝟒𝒑

2. 𝟓𝒄𝟐𝒅𝟑

W𝟒𝒄𝟐𝒅 3. 𝟖𝒚𝟕𝒛𝟔

𝟒𝒚𝟔𝒛𝟓

4. 𝟒𝒇𝟑𝒈𝟑𝒉𝟔

𝟑 5. W$RK

"$RJ 6. ,\J

1PjLD

"

7. 𝒙𝟑(𝒚W𝟓)(𝒙W𝟖) 8. &1

W" 9. ""LDmK

>>LKmZD

10. 𝟔𝒇Z𝟐𝒈𝟑𝒉𝟓

𝟓𝟒𝒇Z𝟐𝒈Z𝟓𝒉𝟑 11. W>"C

ZFnJRZB

"CZDnRJ 12. N

ZKXZJ

NBXD ZF

13. 𝒋Z𝟏𝒌𝟑

Z𝟒

𝒋𝟑𝒌𝟑 14. ("o

ZKT)ZD

(oKTB 15. "RDSKx

&RBSxZK

W"

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SquareRootsasExponentsDoNow:Useyourcalculatortoevaluatethefollowing.16= (16)

FK =

(100)

FK = 100 =

Whatdoyounotice?____________________________________________________________________________________________________________Whyisthishappening?

Checkitout: 𝑏FK"=________________________________

Examples:Writeeachexpressioninradicalform,orwriteeachradicalinexponentialform.Example1: 25

FK

Example2: 18

Example3:5𝑥FK

Example4: 8𝑝

Example5: 49FK

Example6: 22

Example7: 7𝑤FK

Example8:2 𝑥

Definition:

CalculatorTutorial#1Useparenthesestoevaluateexpressionsinvolvingrationalexponentsonagraphingcalculator.Forexample,tofind125

FD,press

125[^][(]1[÷]3[)][ENTER].

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nthRootsUseyourcalculatortoevaluatethefollowing.6&= 216D = 2, = 64j =Whatdoyounotice?______________________________________________

______________________________________________Weknowthatif8" = 64,then 64 = 8.Similarly,if2$ = 16,then 16B = 2.Definition:Foranyrealnumbersaandbandanypositiveintegern,if𝑎X = 𝑏,thenaisannthrootofb.Examples:Evaluate.Example1: 27D

Example2: 32J

Example3: 64D

Example4: 10,000B

Likesquareroots,nthrootscanberepresentedbyrationalexponents.Definition(Part2):Examples:Usethenthrootdefinitiontoconvertformsandevaluate.Example1: 125

FD

Example2: 1296FB

Example3: 27FD

Example4: 256FB

CalculatorTutorial#2Touseexponents,pressthecaretsymbol(^)toraiseanumbertoapower.CalculatorTutorial#3Tofindnthroots,enteryournumbern,thenpress[MATH]andchoose√� .(5)

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RationalExponentsSimplifytheseexpressionsusingMultiplicationProperties:

36FK&=________________

32$

FJ =________________

Simplifytheseexpressionsusingthenthrootdefinition:

36FK&=________________

32$

FJ =________________

Definition: Examples:Convertformsandevaluatethefollowingexpressions.Example1: 8

KD

Example2: 64KD

Example3: 36DK

Example4: 27KD

Example5: 256JB

Example6: 81JK

Example7: 7𝑤DK

Example8:2 𝑥J &

ChallengeProblems:

1. −8KD

2. 81 WJB

3. 𝑥"𝑦$ WFK

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SolvingExponentialEquationsWarmUp:Answerthefollowingquestions.

1. 2towhatpoweris32? 2. 6towhatpoweris216?

3. 5towhatpoweris625?

Findasolutiontothefollowingequations.

4. 2R = 32 5. 6R = 216 6. 5R = 625ThePowerPropertyofEqualityAslongasbisarealnumbergreaterthanzeroandnotequalto1,then𝑏R = 𝑏Sifandonlyif𝑥 = 𝑦.Examples:

1. If5R = 5&,then𝑥 = 3. 3.If𝑛 = >",then4X = 4

FK.

2. 7R = 343 4.3&R�> = 81

Thispropertyhelpsuswhensolvingmorecomplicatedexponentialequations(likeexample4).AnotherExample:25RW> = 5

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Examples:Solveeachequationforx.1. 2&R = 512

2. 3"R = 9R�>

3. 36RW> = 6

4. 2$R = 32R�>

5. 16R = >"

6. >&,

R�>= 216

7. >"1

R= 81

8. 25R = >>"(

1. Thesunprotectionfactor(SPF)ofasunscreenindicateshowwellitprotectsyoufromthesun’s

harmfulrays.SunscreenwithanSPFof𝑓absorbsabout𝑝percentoftheUV-Brays,where𝑝 = 50𝑓U.".FindtheSPFthatabsorbs100%ofUV-Brays.

2. Thepopulationpofaculturethatbeginswith40bacteriaanddoublesevery8hoursismodeledby𝑝 = 40 2

�G,wheretistimeinhours.Findtif𝑝 = 20,480.

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ExponentialFunctionsThezombiesarehere…Eachnight,everyzombiewillinfectanewperson…Howmanynightsdoyouthinkitwilltaketoinfectthewholeroom?_______Writeafunctionthatrepresentsthisscenario:AnexponentialfunctionhastheformThefollowingrestrictionsapply:1.__________________2.___________________3._____________________Note:Thebaseisa___________________.Theexponentisa_________________________.Directions:Useyourtableabovetographthefunction.

1. Whatisthey-interceptofthefunction?Whatdoesitrepresentinthisscenario?

2. Whatisthedomainofthefunction?

3. Whatistherangeofthefunction?Summarize:Howdoyoufindthey-intercept?Howdoyoufindthedomainandrange?

Night #ofzombies

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IdentifyingExponentialBehaviorUpuntilnow,wehavebeenworkingwithlinearfunctions.Thegraphofalinearfunctionis_______________,andalinearfunctionhasa_________________________________________.Thereare2methodswecanusetodeterminewhetherafunctionislinearvs.exponential:1. Graphing

Example:Graphthedatainthetable.Determinewhethertherelationshipislinearorexponential.

x y

-2 9

-1 3

0 1

1 13

2. Lookingforaconstantratio

Example:Exponentialfunctionshaveconstantratiosinsteadofaconstantrateofchange.Thismeansthatifthex-valuesareatregularintervalsandthey-valuesdifferbyacommonfactor,thedataisprobablyexponential.Inthisexample,theconstantratiois_______.Summarize:Howcanyoudeterminewhetherafunctionislinearorexponential?

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ExponentialGrowthvs.DecayAfterthezombieoutbreak,ourclassisnowfullofzombies.Theschooladministrationfiguresoutwhat’sgoingonandsendsPrincipalWaynetoclearourclassofthezombieepidemic.PrincipalWaynecancureonehalfoftheremainingzombieseachdaywithavaccinecreatedinMr.Benters’BiologyLab.Whenwillourentireclassbecured?_________________________Writeafunctionthatrepresentsthisscenario:Useyourtabletographthefunctionbelow.

1. Whatisthey-intercept?Whatdoesthatrepresentinthisscenario?

2. Whatisthedomain?

3. Whatistherange?

Aslightlymorerealisticbiologyexample:Acertainbacteriapopulationdoublesinsizeevery20minutes.Beginningwith10cellsinaculture,thepopulationcanberepresentedbythefunction𝐵 = 10 2 C ,where𝐵isthenumberofbacteriacellsand𝑡isthetimein20minuteincrements.Howmanybacteriacellswilltherebeafter2hours?

Day #ofzombies

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ExponentialFunctionsPracticeCreateatableandgraphthefunction.Youwillneedtochoosewhichvaluestouseinyourtable.Identifythey-intercept,domain,andrangeofeachfunction.Alsoidentifywhetherthefunctionrepresentsexponentialgrowthordecay.USEPENCIL!1. 𝑦 = 2R

Growthordecay?(circleone)y-intercept:__________Domain:__________Range:__________2. 𝑦 = 2R − 1

Growthordecay?(circleone)y-intercept:__________Domain:__________Range:__________3. 𝑦 = 2R + 3

Growthordecay?(circleone)y-intercept:__________Domain:__________Range:__________ClassDiscussion:

x y

x y

x y

19

4.𝑦 = >"

RW>

Growthordecay?(circleone)y-intercept:__________Domain:__________Range:__________

4. 𝑦 = >"

R�"

Growthordecay?(circleone)y-intercept:__________Domain:__________Range:__________

5. 𝑦 = >"

RW"+ 6

Growthordecay?(circleone)y-intercept:__________Domain:__________Range:__________ClassDiscussion:

x y

x y

x y

20

Summarize:GraphsofExponentialFunctionsExponentialGrowthFunctions ExponentialDecayFunctionsEquation:Domain:Range:Intercepts:Endbehavior:Sketchofgraph:

Equation:Domain:Range:Intercepts:Endbehavior:Sketchofgraph:

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ExponentialGrowthandDecayExponentialGrowthThenumberofonlineblogshasrapidlyincreasedinthelast15years.Infact,thenumberofblogsincreasedatamonthlyrateofabout13.7%over21months,startingwith1.1millionblogsinNovember2003.Theaveragenumberofblogspermonthfrom2003-2005canbemodeledbytheequation𝑦 = 1.1 1 + 0.137 Cor𝑦 = 1.1 1.137 CwhereyrepresentsthetotalnumberofblogsinmillionsandtisthenumberofmonthssinceNovember2003.Labelthediagrambelowwithwhateachvariableorconstantrepresents.

𝑦 = 1.1 1 + 0.137 CIngeneral,theequationforexponentialgrowthisasfollows:

𝑦 = 𝑎 1 + 𝑟 CExample1:Theprizeforaradiostationcontestbeginswitha$100giftcard.Onceaday,anameisannounced.Thepersonhas15minutestocallortheprizeincreasesby2.5%forthenextday.

a. Writeanequationtorepresenttheamountofthegiftcardindollarsaftertdayswithnowinners.

b. Howmuchwillthegiftcardbeworthifnoonewinsafter10days?Example2:Acollege’stuitionhasrisen5%eachyearsince2000.Ifthetuitionin2000was$10,850,writeanequationfortheamountofthetuitiontyearsafter2000.Predictthecostoftuitionforthiscollegein2020.

CalculatorTutorial#4Whensolvingexponentialequations,youwilloftenencounter“unfriendly”decimals.Ifyouroundthesebeforeyourfinalanswer,youmaygetaslightlyincorrectanswer.Onyourcalculator,usethe[2nd][(-)]keystoget[Ans],yourEXACTpreviousanswer.

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CompoundInterestCompoundinterestisaspecialkindofexponentialgrowth.Itisinterestearnedorpaidbothontheinitialinvestmentandpreviouslyearnedinterest.Ingeneral,theequationforcompoundinterestisasfollows:

𝐴 = 𝑃 1 +𝑟𝑛

XC

Example3:Maria’sparentsinvested$14,000at6%peryearcompoundedmonthly.Howmuchmoneywilltherebeintheaccountafter10years?Example4:Determinetheamountofaninvestmentif$300isinvestedataninterestrateof3.5%compoundedeveryothermonthfor22years.ExponentialDecayIngeneral,theequationforexponentialdecayisasfollows:

𝑦 = 𝑎 1 − 𝑟 CExample5:Afullyinflatedchild’sraftforapoolislosing6.6%ofitsaireveryday.Theraftoriginallycontained4500cubicinchesofair.

a. Writeanequationtorepresentthelossofair.

b. Estimatetheamountofairintheraftafter7days.Example6:ThepopulationofCampbellCounty,Kentuckyhasbeendecreasingatanaveragerateofabout0.3%peryear.In2000,itspopulationas88,647.Writeanequationtorepresentthepopulationsince2000.Ifthetrendcontinues,predictthepopulationin2018.