Icebergs and Exponents Lesson 19-1 Basic Exponent Properties
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1 Icebergs nd Eponents Lesson 19-1 Bsic Eponent Properties Lerning Trgets: Develop bsic eponent properties. Simplif epressions involving eponents. SUGGESTED LEARNING STRATEGIES: Crete Representtions, Predict nd Confirm, Look for Pttern, Think-Pir-Shre, Discussion Groups, Shring nd Responding An iceberg is lrge piece of freshwter ice tht hs broken off from glcier or ice shelf nd is floting in open sewter. Icebergs re clssified b size. The smllest sized iceberg is clled growler. A growler ws found floting in the ocen just off the shore of Greenlnd. Its volume bove wter ws pproimtel 7 cubic meters. 1. Reson quntittivel. Two icebergs flot ner this growler. One iceberg s volume is times greter thn the growler. The second iceberg s volume is 8 times greter thn the growler. Which iceberg hs the lrger volume? Eplin.. Wht is the mening of nd 8? Wh do ou think eponents re used when writing numbers? CONNECT TO Becuse ice is not s dense s sewter, bout one-tenth of the volume of n iceberg is visible bove wter. It is difficult to tell wht n iceberg looks like underwter simpl b looking t the visible prt. Growlers got their nme becuse the sound the mke when the re melting sounds like growling niml. MATH TERMS ACTIVITY 19 GEOLOGY GROUP DISCUSSION TIPS Work with our peers to set rules for: discussions nd decision-mking cler gols nd dedlines individul roles s needed 01 College Bord. All rights reserved.. Suppose the originl growler s volume under the wter is 9 times the volume bove. How much of its ice is below the surfce?. Write our solution to Item using powers. Complete the eqution below. Write the missing terms s power of. volume bove wter = volume below the surfce = The epression is power. The bse is nd the eponent is. The term power m lso refer to the eponent. 5. Look t the eqution ou completed for Item. Wht reltionship do ou notice between the eponents on the left side of the eqution nd the eponent on the right? Activit 19 Eponent Rules 87
2 ACTIVITY 19 Bsic Eponent Properties 6. Use the tble below to help verif the pttern ou noticed in Item 5. First write ech product in the tble in epnded form. Then epress the product s single power of the given bse. The first one hs been done for ou. Originl Product Epnded Form Single Power Epress regulrit in repeted resoning. Bsed on the pttern ou observed in the tble in Item 6, write the missing eponent in the bo below to complete the Product of Powers Propert for eponents. m n = 5 8. Use the Product of Powers Propert to write s single power. CONNECT TO SCIENCE The formul for densit is D= M V where D is densit, M is mss, nd V is volume. 9. The densit of n iceberg is determined b dividing its mss b its volume. Suppose growler hd mss of 59,09 kg nd volume of 81 cubic meters. Compute the densit of the iceberg. 10. Write our solution to Item 9 using powers of 9. Mss Volume = Densit 01 College Bord. All rights reserved. 11. Wht pttern do ou notice in the eqution ou completed for Item 10? 88 SpringBord Mthemtics Algebr 1, Unit Eponents, Rdicls, nd Polnomils
3 Bsic Eponent Properties ACTIVITY Use the tble to help verif the ptterns ou noticed in Item 11. First write ech quotient in the tble below in epnded form. Then epress the quotient s single power of the given bse. The first one hs been done for ou. Originl Quotient Epnded Form Single Power 5 = Bsed on the pttern ou observed in Item 1, write the missing eponent in the bo below to complete the Quotient of Powers Propert for eponents. m =, where 0 n 1. Use the Quotient of Powers Propert to write s single power College Bord. All rights reserved. The product nd quotient properties of eponents cn be used to simplif epressions. Emple A Simplif: 5 5 Step 1: Group powers with the sme bse. 5 5 = 5 5 Step : Product of Powers Propert = Step : Simplif the eponent. = 10 9 Solution: 5 5 = 10 9 Activit 19 Eponent Rules 89
4 ACTIVITY 19 Bsic Eponent Properties Emple B Simplif: 5 5 Step 1: Group powers with the sme bse. = 5 Step : Quotient of Powers Propert = 5 1 Step : Simplif the eponents. = Solution: 5 = Tr These A B Simplif ech epression.. ( )( 5 ) b. 5 b c b c 6 c. 18 Check Your Understnding 15. Simplif z 5 z. 16. Simplif 1 7fg 17. A growler hs mss of kg nd volume of 7 cubic meters. Compute the densit of the iceberg b completing the following. Write our nswer using powers of. 5 = f g 7. MATH TIP Use grphic orgnizer to record the properties of eponents ou lern in this ctivit. 18. Which epression hs the greter vlue? Eplin our resoning b. 19. The mss of n object is 15 grms. Its volume is 9 cm. Wht is the object s densit? 0. The densit of n object is 10 grms/cm. Its volume is cm. Wht is the object s mss? 1 7 ( ) 1. Simplif the epression ( ). ( ). Mke sense of problems. Tnik sks Tob to multipl the epression Tob ss he doesn t know how to do it, becuse he believes the Product of Powers Propert works with onl two eponentil terms, nd this problem hs three terms. Eplin how Tob could use the Product of Powers Propert with three eponentil terms. 01 College Bord. All rights reserved. 90 SpringBord Mthemtics Algebr 1, Unit Eponents, Rdicls, nd Polnomils
5 Negtive nd Zero Powers ACTIVITY 19 Lerning Trgets: Understnd wht is ment b negtive nd zero powers. Simplif epressions involving eponents. SUGGESTED LEARNING STRATEGIES: Look for Pttern, Discussion Groups, Shring nd Responding, Think-Pir-Shre, Close Reding, Note Tking 1. Attend to precision. Write ech quotient in epnded form nd simplif it. Then ppl the Quotient of Powers Propert. The first one hs been done for ou. Originl Quotient Epnded Form Single Power 5 8 = = = College Bord. All rights reserved.. Bsed on the pttern ou observed in Item 1, write the missing eponent in the bo below to complete the Negtive Power Propert for eponents. 1= n, where 0. Write ech quotient in epnded form nd simplif it. Then ppl the Quotient of Powers Propert. The first one hs been done for ou. Originl Quotient Epnded Form Single Power = = 1 = CONNECT TO AP In clculus, n epression contining negtive eponent is often preferble to one written s quotient. For emple, 1 is written. Activit 19 Eponent Rules 91
6 ACTIVITY 19 Negtive nd Zero Powers. Bsed on the pttern ou observed in Item, fill in the bo below to complete the Zero Power Propert of eponents. 0 =, where 0 5. Use the properties of eponents to evlute the following epressions.. b c. 5 0 d. (.75) 0 When evluting nd simplifing epressions, ou cn ppl the properties of eponents nd then write the nswer without negtive or zero powers. Emple A 0 Simplif 5 z nd write without negtive powers. 0 Step 1: Commuttive Propert 5 z Step : Step : Appl the eponent rules. Simplif the eponents. = 5 1 z 0 = z 0 = 15 1 Step : Write without negtive eponents. = 15 0 Solution: 5 z 15 = Tr These A Simplif nd write without negtive powers.. b 5b b c. ( 5 ) 0 01 College Bord. All rights reserved. 9 SpringBord Mthemtics Algebr 1, Unit Eponents, Rdicls, nd Polnomils
7 Negtive nd Zero Powers ACTIVITY 19 Check Your Understnding Simplif ech epression. Write our nswer without negtive eponents. 6. (z) 7. 1(z) ( b) 0 1. For wht vlue of v is v = 1, if 0? 1. For wht vlue of w is b w = 1 9, if b 0? b 1. For wht vlue of is 1 = 9? 15. For wht vlue of z is z = 1? 16. Determine the vlues of n nd m tht would mke the eqution 7 n 7 m = 1 true sttement. Assume tht n m. 17. For wht vlue of is =? 18. Reson bstrctl. Wht is the vlue of ? Wht is the vlue of n multipliction problem in which ll of the fctors re rised to power of 0? Eplin. 01 College Bord. All rights reserved. Activit 19 Eponent Rules 9
8 ACTIVITY 19 Additionl Properties of Eponents Lerning Trgets: Develop the Power of Power, Power of Product, nd the Power of Quotient Properties. Simplif epressions involving eponents. SUGGESTED LEARNING STRATEGIES: Note Tking, Look for Pttern, Crete Representtions, Think-Pir-Shre, Shring nd Responding, Close Reding 1. Write ech epression in epnded form. Then write the epression using single eponent with the given bse. The first one hs been done for ou. Originl Epression Epnded Form Single Power ( ) = 8 (5 5 ) ( ). Bsed on the pttern ou observed in Item 1, write the missing eponent in the bo below to complete the Power of Power Propert for eponents. m n ( ) =. Use the Power of Power Propert to write s single power.. Write ech epression in epnded form nd group like terms. Then write the epression s product of powers. The first one hs been done for ou. Originl Epression Epnded Form () = Product of Powers 01 College Bord. All rights reserved. ( ) ( ) 9 SpringBord Mthemtics Algebr 1, Unit Eponents, Rdicls, nd Polnomils
9 Additionl Properties of Eponents ACTIVITY Bsed on the pttern ou observed in Item, write the missing eponents in the boes below to complete the Power of Product Propert for eponents. ( b) m = b 6. Use the Power of Product Propert to write c 1 d powers. 8 1 s product of 7. Mke use of structure. Use the ptterns ou hve seen. Predict nd write the missing eponents in the boes below to complete the Power of Quotient Propert for eponents. m ( ) =, where b 0 b b 1 8. Use the Power of Quotient Propert to write s quotient of 6 powers. 01 College Bord. All rights reserved. You cn ppl these power properties nd the eponent rules ou hve lred lerned to simplif epressions. Emple A Simplif ( 5 ) ( ) nd write without negtive powers. Step 1: Step : Step : Step : Step 5: Power of Power Propert ( 5 ) ( ) = 5 Simplif the eponents nd the numericl terms. Commuttive Propert = = Product of Powers Propert = 8 Simplif the eponents = MATH TIP Crete n orgnized summr of the properties used to simplif nd evlute epressions with eponents Solution: ( ) ( ) = 8 9 Activit 19 Eponent Rules 95
10 ACTIVITY 19 Additionl Properties of Eponents Emple B Simplif z Step 1:. Power of Quotient Propert z = z 6 Step : Simplif the eponents. = z Step : Negtive Power Propert = z Solution: z = z 6 6 Tr These A B Simplif nd write without negtive powers.. ( ) ( ) b. b(5b c) c. d e. ( ) ( z)(6z ) 1 Check Your Understnding Simplif ech epression. Write our nswer without negtive eponents. 9. ( 1 ) ( b c) (b c 5 )(z) 0 1. (fg ) ( fg h) (gh ) 1 1. b b ( ) 1. 7nm Simplif ( ) b. ( ) 16.. () b. () 17.. ( 5 ) b. ( 5 ) Model with mthemtics. The formul for the re of squre is A = s, where s is the side length. A squre grden hs side length of. Wht is the re of the grden? 01 College Bord. All rights reserved. 96 SpringBord Mthemtics Algebr 1, Unit Eponents, Rdicls, nd Polnomils
11 Icebergs nd Eponents ACTIVITY College Bord. All rights reserved. Write our nswers on notebook pper. Show our work. Lesson 19-1 For Items 1 5, evlute the epression. Write our nswer without negtive powers b b. (6 b)( b ) The volume of n iceberg tht is below the wter line is 5 cubic meters. The volume tht is bove the wter line is cubic meters. How mn times greter is the volume below the wter line thn bove it? A..5 B. C. 7 D A megbte is equl to 0 btes, nd gigbte is equl to 0 btes. How mn times lrger is gigbte thn megbte? 8. A jckpot is worth 10 5 dollrs. The contestnt who wins the jckpot hs the opportunit to put it ll on the line with the single spin of prize wheel. If the contestnt spins the number 7 on the wheel, she will win 10 times more mone. How mn dollrs will the contestnt win if she risks her prize mone nd spins 7? The number of erthqukes of given mgnitude tht re likel to occur in n given er is represented b the formul 10 (8 M), where M is the mgnitude. Use this formul for Items 9 nd How mn erthqukes of mgnitude 8 re likel to occur net er? 10. If n erthquke of mgnitude 10 occurred lst er, how mn ers will it be before nother one of tht mgnitude is likel to occur? Lesson Which of the following epressions is not equl to 1? A. B C. b b D. 1. Which of the following epressions is equl to? A. B. C. D. Determine whether ech sttement is lws, sometimes, or never true. 1. For 0, the vlue of 1 is positive. 1. If n is n integer, then n n equls If 6 p > 0, then p > equls If m is n integer, then the vlue of m is negtive. Activit 19 Eponent Rules 97
12 ACTIVITY 19 Icebergs nd Eponents 18. For wht vlue of is w = 1, if w 0? 19. For wht vlue of b is p b 1 = 5 1, if p 0? p For ech of the following, give the vlue of the epression or stte tht the epression is undefined when = 0 1. when = 0. 1p when = 0 nd p > 0. 0 n 0 n when n is n integer Lesson 19-. The re of squre is given b the formul A = s, where s is the length of the side. Wht is the re of the squre shown? b The volume of cube is given b the formul V = s, where s is the length of the side. Use this formul for Items Wht is the volume of the cube shown? Simplif ech epression. Write our nswer without negtive eponents ( ) 9. c d c 0. ( z 1 ) (z )( ) ( m n ) m. ( ). Which of the following is true sttement bout the epression 1 ( ), given tht 0? A. The epression is lws equl to 1. B. The vlue of the epression is positive. C. If is negtive, then the vlue of the epression is lso negtive. D. The epression cnnot be simplified n further. MATHEMATICAL PRACTICES Construct Vible Arguments nd Critique the Resoning of Others. Aln ss tht (b) (b) is the sme s [(b) ]. Is Aln correct? Justif our response. 6. Wht is the volume of the cube shown? c d 01 College Bord. All rights reserved. 7. The volume of cube is 7 cubic inches. Wht epression represents the length of one side of the cube? Justif our resoning. 98 SpringBord Mthemtics Algebr 1, Unit Eponents, Rdicls, nd Polnomils
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