Exponents. Learning Objectives. Pre-Activity

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1 Sectio. Pre-Activity Preparatio Epoets A Chai Letter Chai letters are geerated every day. If you sed a chai letter to three frieds ad they each sed it o to three frieds, who each sed it o to three frieds, how may chai letters are set? st roud: d roud: rd roud: How may chai letters are there after te rouds? What if you origially sed the letter to seve frieds? How may rouds will it take to reach 7,69 people? Compoud iterest: Suppose you have ivested $000 i a iterest bod. If you kow the iterest rate per coversio period is 0.0 ad that the accout has bee compouded 8 times, you ca figure out how much is i the accout by calculatig $000(.0) 8. The accout has $7.66 after 8 coversios. Geometry: Fid the volume of a cube if each side measures cm: Scietific Notatio: I oe year, light travels approimately.9 0 miles. The mass of a hydroge atom is.7 0. The speed of light is.86 0 miles per secod. Oe gram is tos. ( cm) cm Each of these eamples demostrates the use ad utility of epoets. Learig Objectives Demostrate how the rules for epoetiatio work Epad a epoetial epressio Evaluate epoetial epressios Simplify epoetial epressios 7

2 8 Chapter Evaluatig Epressios Termiology Previously Used Commutative Property costat Distributive Property epressio factor simplify variable New Terms to Lear base epaded form epoet power Buildig Mathematical Laguage A base is a umber that is beig used as a factor. A epoet is the umber of times the base factor appears i a product of base factors. 6 : : : a a : a : a : a : a : a factors 6 factors of a Power is aother word for epoet. Powers of ad 0 A base to the power oe is the same as the base aloe; the base is to be used as a factor time: ( 7) 7 Coversely, o epoet meas a epoet of (ot zero): ( 7) y y Ay o-zero umber raised to the zero power is : 0 (-99) 0 z 0 base epoet Other otatios: ^ ad ()^ This symbol ^ is called a caret ad idicates that the umber followig it is a epoet. This otatio is used whe showig a superscript (smaller ad raised up) umber as a epoet is ot possible. It is also ofte used o calculators to deote the epoet will follow. Two special cases: 0 0 is udefied ad 0 0 is udefied. Epoetial Notatio A epressio writte i epoetial otatio uses epoets to cout or show how may times a base is used as a factor. Epoet otatio is prevalet i algebra to such a etet that there are special rules that gover its use ad simplificatio procedures. Evaluatig epressios with epoets, iterpretig formulas with epoets, ad havig a uderstadig of the magitude of a umber squared or cubed or preseted i scietific otatio are ecessary for progress through your required math ad sciece courses as well as good preparatio for uderstadig comple cosumer iformatio such as amortizatio. NOTE: The idea that epoets are couts of factors will be eteded beyod whole umbers to iclude egative umbers, fractios, ad variables.

3 Sectio. Epoets 9 Properties ad Priciples of Epoets The epoet rules preseted below are used to simplify epressios with epoets. The first two rules i the chart are to be used as defiitios. The Product, Power, ad Quotiet Rules apply to LIKE BASES ONLY. By the umeric eamples i Table, you ca see how the rules were derived. Table Epoet Rule Symbolic Represetatio Numeric Eample Key Observatios Zero Epoet Rule A zero epoet shows o factors of the base. Do ot cofuse this with a base raised to a epoet of such as 6 6. Negative Epoet Rule -m m - ad - Ay umber raised to a egative epoet is defied to be the iverse of the umber to that epoet. The egative sig of the epoet idicates the idea of iverse locatio, ot opposite value. m m+ Product Rule ( ) Power Rule m m: 7 : 7 (7 : 7)(7 : 7 : 7) ( ) ( )( )( )( ) $ 8 Whe multiplyig, make sure the bases are the same umber (or variable) before addig epoets. Thik: multiplicatio is liked to additio. This rule is ofte called the power to power rule. Multiply the epoets; keep the base uchaged. Quotiet Rule m m : : : - Validate: Whe the bases are the same, divisio is performed by subtractig epoets. Divisio is repeated subtractios: make the metal coectio by likig divisio with subtractio. Product to Power Rule ( ) ab a : b ( : ) ( : )( : ) : Validate: : 9 : Product to power rule is used whe differet bases are raised to the same epoet. It is ot the same as the distributive property o additio is preset but it does distribute the epoet over multiplicatio. Use the commutative property to rearrage factors. cotiued o et page

4 0 Chapter Evaluatig Epressios Epoet Rule Symbolic Represetatio Numeric Eample Key Observatios Quotiet to Power Rule a a b b 9 Validate: 9 This rule allows fractios to be raised to powers. Notice it is ot like the quotiet rule because the bases i the umerator ad deomiator are differet umbers. Combied Rules a by m m m a m b y m 6 s s t t 8 Combied rules show a logical etesio of the quotiet to power rule with the product to power rule. Situatios where epoet rules are combied occur frequetly i algebra. Epadig the rule table to iclude algebraic eamples uderscores the importace of beig able to apply the rules to umeric or algebraic epressios. I Table you ca agai see how the rules were derived. Table Epoet Rule Symbolic Represetatio Numeric Eample Algebraic Eample Zero Epoet Rule p 0 Negative Epoet Rule -m m 7 ad y ad y y y Product Rule m m+ : ()( : : : ) : : Power Rule ( ) m m : ( 7 ) ( 7 )( 7 )( 7 ) 7 7 : ( ) ( )( )( )( )( ) $

5 Sectio. Epoets Epoet Rule Symbolic Represetatio Numeric Eample Algebraic Eample Quotiet Rule m m : 0 : 0 : 0 : 0 0 : p p 9 p p p p p p p p ppppp -9 - p p p Product to Power Rule Quotiet to Power Rule ( ) ab a : b ( : ) a a b b : Validate: 6 96 ad : ( : y) : y y y y y y Combied Rules m m m a a m m by b y ( : ) ( : ) 0 : : : y y 9y A epressio is simplified whe there is o more tha oe of each differet base raised to a sigle positive epoet. Not Simplified y ( )(9 ) ( ) z Simplified 9 y 8 7 z 8

6 Chapter Evaluatig Epressios Scietific Notatio Oe importat applicatio of epoets is scietific otatio. Scietists work with very large umbers, like the distace from oe star to aother, ad with very small umbers, like the weight of a sigle atom. Workig with umbers with may zeroes ca be very cumbersome. We ca choose to write these umbers i a shorteed way that is easier to read ad use for calculatios. Numbers writte i scietific otatio look like: b 0, where: b is a umber betwee ad 0 (mathematically, b < 0 ) (the epoet) is a iteger ad shows the umber of decimal places the decimal must be moved to show the umber i stadard otatio. (Note: whe a umber i stadard otatio is less tha oe, its epoet is egative i scietific otatio a applicatio of the egative epoet rule.) Stadard Form THINK Scietific Notatio is b ad it s betwee ad 0. The iitial umber is greater tha so the epoet is positive. The decimal moves places: so is 6.7 is b ad it s betwee ad 0. The iitial umber is less tha so the epoet is egative. The decimal moves 6 places: so is Write the followig umbers i Scietific Notatio. a) b) c) Write the followig i stadard form. a). 0 b). 0 8 c) Look it up! Use ay resource to look up the values of the followig:. Legth of a Agstrom i meters. Plack s costat. The mass of a electro. Oe electro volt i ergs

7 Sectio. Epoets Models Model Note: To write a epoetial epressio i epaded form meas to rewrite it without usig epoets. Problem Epaded Form Evaluated Validated : : : : : : : : : The bases are ot the same so evaluate each base with its correspodig epoet separately. : : : 8 8 : : : 7 : : 8 ad 9 8 : 9 7 : : : : : : : : 9 Go backwards: : 9 ( ) ( ) ( )( )( ) : : : : : : : : The bases are the same umber; use the power rule ad the quotiet rule: ( ) 6 : 6 ( ) ( )( )( ) : : : : : : : : Model Problem Epaded Form Simplified Validated ( ) () ()()() ( ) 6 6 ( ) ( ) ( ) ( ) : : : : : 6

8 Chapter Evaluatig Epressios Problem Epaded Form Simplified Validated : : : : : : : : : + ( ) : : : : : : : : :??? Why ca we do this? :??? Why ca we do this? I may of the previous cases, there is more tha oe way of thikig about the simplificatio process. You ca use the rules for epoets or you ca apply rules you have already leared (or both). You ca use the fact that the epoet applies to the epressio withi paretheses or multiply withi paretheses first: the result is the same. Work the problem oe way ad the use a alterative approach to validate your aswer. Addressig Commo Errors Issue Icorrect Process Resolutio Correct Process Validatio Not determiig the correct base whe simplifyig egative epoet epressios Write implied operatios with their appropriate symbol to emphasize each base. For, the implied operatio is multiplicatio. : : Noe required. Not rememberig to iclude the costat factor whe raisig a epressio to a power (6 ) 6 6 Apply each epoet rule step-by-step, showig the appropriate factors for costats as well as variables. (6 ) (6 ) (6 ) 6 : 6 : : 6 6 (6 ) (6 ) (6 : : )

9 Sectio. Epoets Issue Icorrect Process Resolutio Correct Process Validatio Assumig that a umber raised to a egative epoet is always a egative umber or Use the defiitio of egative epoets: b m b m The result is ot always egative. For eample: or a positive umber. Combiig bases as well as epoets whe multiplyig with ulike bases 7 9 The epoet rules apply to LIKE BASES ONLY. The epressio 7 is already i simplified form because the bases are distict ( ad 7) Evaluate it with a calculator. 7 0 : 0 8, whereas 9 79,80,06,8 Misapplyig the power to power rule ( ) Write out the factors whe usure of the epoets rules. ( ) ( )( ) ( : : )( : : ) 6 Evaluate with a calculator: ( ),6 (),6 Preparatio Ivetory Before proceedig, make sure that you ca: Epad epoetial epressios Evaluate umerical epoet epressios Evaluate ay umber raised to the zero power Uderstad how to apply a egative epoet to a umber or variable Use the epoet rules to simplify epressios i epoetial form

10 Sectio. Activity Epoets Performace Criteria Writig epoetial epressios i epaded form appropriate base factors correct umber of factors of each base validatio of the aswer with alterative iterpretatio Evaluatig umeric epoetial epressios accuracy demostrated use of epoet rules validatio of the aswer Simplifyig algebraic epressios cotaiig epoets demostrated use of epoet rules aswer preseted i its simplest form validatio of the aswer Critical Thikig Questios. How do you determie the base(s) i a epoetial epressio?. How do you use the epoet i epadig a epoetial epressio?. How do you validate that a epressio is simplified? 6

11 Sectio. Epoets 7. How does the ame give to the Product to Power rule relate to the meaig of the rule?. By aalyzig the commo errors, what is the relatioship betwee paretheses ad the Epoet Rules that use them? 6. How does the Quotiet Rule for epoets justify the rule that ay o-zero umber raised to the zero power is equal to? 7. Are there sigificat differeces betwee the umeric eamples ad algebraic eamples i Table? Eplai your aswer. Tips for Success Show your work oe step at a time as you are developig the solutio to a problem Start with the simplest possible eamples ad the move o to more comple oes Whe workig a comple problem, model it usig simpler or more cocrete compoets. For eample, if ad y are used, see what happes if you echage for ad for y.

12 8 Chapter Evaluatig Epressios Demostrate Your Uderstadig. Use metal math to evaluate the followig: a) b) 0 c) 7 d) (9 7) 0 aswer: aswer: aswer: aswer:. Use the defiitio of egative epoets to rewrite the followig with positive epoets: a) b) c) d) 6 aswer: aswer: aswer: aswer:. Evaluate the followig epoetial epressios. Problem Epaded Form Worked Solutio Validatio a) : b) 6 c) ( ) ( ) d) e)

13 Sectio. Epoets 9. Simplify the followig algebraic epressios: Problem Epaded Form Worked Solutio Validatio a) ( )( )() b) ( ) c) a 6b 0 d) e) () f) ( b ): ( b ). Eted the cocept: What values for make the statemet true? a) 8 b) c) : d) 6

14 0 Chapter Evaluatig Epressios Idetify ad Correct the Errors Idetify ad correct the errors i the followig problems. Worked Solutio List the Errors Correct Process Validatio ) 7(0 ) 7( 0)( 0)( 0) 7000 ) (6 ) ) 0 ) 6 ) 7 6