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number 332 - 333 Glossary
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b. Conductthe experimentand recordthe data. Thenuse your calculatoror curvefittingsoftwareto find the modelthat seemsto fit the data pattern well.
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c. Comparethe modelssuggestedby logicalanalysisof the experimentand
by fittingof a functionto actualdata. Decidewhichis the bettermodelof the
experimentand explainyour choice.
b. How would you use the definitionof exponentor otherreasoningto
convinceanotherstudent your answerto Part a is correct?
c. Whatwould you expectto seeas the mostcommonerrorsin evaluating
quotientsof powerslike _812? Explainhow you would help someonewho
84
made thoseerrorscorrecttheir understandingof how exponentswork.
INVESTIGATION 4
Properties
of Exponents
5 Useyour answersto Problem3 Partsg and h and Problem4 Part a to explain
why it is reasonableto defineb0= 1 for any baseb(b≠ 0).
II
In studyingthe reboundheightof a bouncingball,you calculatedpowersof the
fraction_2. Youcancalculatea powerlike ()_2 4by repeatedmultiplication
3
3
()(
_2 _2)(_2)()_2 . Butthereis a shortcutrule for suchcalculationswith exponents.
3 3 3 3
Asyou workon the problemsin this investigation,makenotesof answersto this
question:
Whatexponentpropertiesprovideshortcutrulesfor
calculatingpowersoffractions,quotientsof
powers,andnegativeexponents?
NegativeExponents Supposethat you werehired as a sciencelab assistantto
monitoran ongoingexperimentstudyingthe growthof an insectpopulation.If the
populationwhen you tookoverwas 48and it was expectedto doubleeveryday,
you couldestimatethe populationfor any timein the futureor the past with the
functionp = 48(2x).
Futureestimatesare easy:Oneday fromnow,the populationshouldbe about
48(21) = 96;two days fromnow it shouldbe about48(22) = 48(2)(2)= 192,and
so on.
Estimatesof the insectnumbersin the populationbeforeyou tookoverrequire
division:Oneday earlier,the populationshouldhavebeenabout
1 = 24;two days ago,it shouldhavebeenabout:
48(2−1) = 48÷ 2 = 48(_)
2
48(2−2) = (48÷ 2)÷ 2
Powers of a Fraction Asyou workon the nextcalculations,lookfor a pattern
suggestingwaysto writepowersof fractionsin usefulequivalentforms.
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4 Examinethe resultsof your workon Problem3.
a. Whatpattern seemsto relatetask and resultin everycase?
1 Findvaluesof x and y that willmaketheseequationstrue statements:
cx
a. (_3) 3= _3x
b. (_c) 2= _5y
5y
5
5
( )
cx (n ≠ 0)
4 5= _4x(n ≠ 0)
c. (_)
d. _c2 3= _ny
ny
n
n
= 48÷ 22
= 48(_1 )
22
= 12
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Thiskind of reasoningaboutexponentialgrowthsuggestsa generalrule that for
any nonzeronumberband any integern, b−n= _1n.
b
2 Examinethe resultsof your workon Problem1.
a. Whatpattern seemsto relatetask and resultin everycase?
b. How would you use the definitionof exponentor otherreasoningto
convinceanotherstudent that your answerto Part a is correct?
c. Whatwould you expectto seeas the mostcommonerrorsin evaluating
powersof a fractionlike ()_3 4? Explainhow you would help someonewho
5
made thoseerrorscorrecttheir understandingof how exponentswork.
Quotients of Powers Sincemanyusefulalgebraicfunctionsrequiredivisionof
quantities,it is helpfulto be ableto simplifyexpressionsinvolvingquotientsof
b (b≠ 0).
powerslike _x
by
3 Findvaluesfor x,y, and z that willmaketheseequationstrue statements.
a. _210= 2z
b. _36= 3z
c. _109= 10z
d. _2x= 27
23
32
103
25
e. _7x= 72
f. _b5= bz
g. _35= 3z
h. _bx= bz
7y
b3
35
bx
6 Therule for operatingwith negativeintegerexponentsalsofollowslogically
fromthe propertyaboutquotients of powersand the definitionb0= 1.Justify
eachstep in the reasoningbelow.
_1 = _b0
(1)
bn bn
= b0- n (2)
= b-n
(3)
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7 Usethe relationshipbetweenfractionsand negativeintegerexponentsto
writeeachof the followingexpressionsin a differentbut equivalentform.
In Partsa–f,writean equivalentfractionthat doesnot use exponentsat all.
2 -1
a. 5-3
b. 6-1
c. 2-4
d. (_)
5
1 -3
2 -2
e. (_)
f. (_)
g. x-3
h. _1
2
5
a4
m
aylelA
ka
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332 UNIT5 ExponentialFunctions
AUDIO ICONS:
Lesson2 ExponentialDecay 333
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332 - 333
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