54 (5 ) Chpter 5 Polynoils nd Eponents 5.6 POSITIVE INTEGRAL EXPONENTS In this section The product rule for positive integrl eponents ws presented in Section 5., nd the quotient rule ws presented in Section 5.5. In this section we review those rules nd then further investigte the properties of eponents. The Product nd Quotient Rules Rising n Eponentil Epression to Power Power of Product Power of Quotient Sury of Rules The Product nd Quotient Rules The rules tht we hve lredy discussed re surized elow. The following rules hold for nonnegtive integers nd n nd 0. n n Product rule n n if n Quotient rule n n if n 0 Zero eponent CAUTION The product nd quotient rules pply only if the ses of the epressions re identicl. For eple, 4 6, ut the product rule cnnot e pplied to 5 4. Note lso tht the ses re not ultiplied: 4 9 6. Note tht in the quotient rule the eponents re lwys sutrcted, s in 7 4 nd y 8 y5 y. If the lrger eponent is in the denointor, then the result is plced in the denointor. E X A M P L E Using the product nd quotient rules Use the rules of eponents to siplify ech epression. Assue tht ll vriles represent nonzero rel nuers. ) ) () 0 (5 )(4) 9 8 ( ) c) 5 d) ( 5 6 ) ) Becuse the ses re oth, we cn use the product rule: 5 Product rule Siplify. ) () 0 (5 )(4) 5 4 Definition of zero eponent 0 Product rule 8 4 c) 5 Quotient rule
5.6 Positive Integrl Eponents (5 ) 55 study tip Keep trck of your tie for one entire week. Account for how you spend every hlf hour. Add up your totls for sleep, study, work, nd recretion. You should e sleeping 50 60 hours per week nd studying hours for every hour you spend in the clssroo. d) First use the product rule to siplify the nuertor nd denointor: 9 ( ) 5 0 8 ( 6 ) 6 Product rule 8 6 Quotient rule Rising n Eponentil Epression to Power When we rise n eponentil epression to power, we cn use the product rule to find the result, s shown in the following eple: (w 4 ) w 4 w 4 w 4 Three fctors of w 4 ecuse of the eponent w Product rule By the product rule we dd the three 4 s to get, ut is lso the product of 4 nd. This eple illustrtes the power rule for eponents. Power Rule If nd n re nonnegtive integers nd 0, then ( ) n n. In the net eple we use the new rule long with the other rules. E X A M P L E Using the power rule Use the rules of eponents to siplify ech epression. Assue tht ll vriles represent nonzero rel nuers. ) ( ) 5 ) ( 5) 4 7 9 c) 5 4 ( ) 5 ) ( ) 5 5 Power rule 7 Product rule ) ( 5) 4 7 9 4 9 7 Power rule nd product rule 4 Product rule 5 Evlute 5. c) 5 4 0 ( ) 5 5 5 Quotient rule Power of Product Consider n eple of rising onoil to power. We will use known rules to rewrite the epression. () Definition of eponent Couttive nd ssocitive properties Definition of eponents
56 (5 4) Chpter 5 Polynoils nd Eponents Note tht the power is pplied to ech fctor of the product. This eple illustrtes the power of product rule. Power of Product Rule If nd re rel nuers nd n is positive integer, then () n n n. E X A M P L E Using the power of product rule Siplify. Assue tht the vriles re nonzero. ) (y ) 5 ) () c) ( y z 7 ) ) (y ) 5 5 (y ) 5 Power of product rule 5 y 5 Power rule ) () () Power of product rule 7 ()()() 7 c) ( y z 7 ) ( ) (y ) (z 7 ) 8 9 y 6 z helpful hint Note tht these rules of eponents re not solutely necessry. We could siplify every epression here y using only the definition of eponent. However, these rules ke it lot sipler. Power of Quotient Rising quotient to power is siilr to rising product to power: 5 5 5 5 Definition of eponent 5 5 5 5 Definition of ultipliction of frctions Definition of eponents The power is pplied to oth the nuertor nd denointor. This eple illustrtes the power of quotient rule. Power of Quotient Rule If nd re rel nuers, 0, nd n is positive integer, then n n n. E X A M P L E 4 Using the power of quotient rule Siplify. Assue tht the vriles re nonzero. ) 4 y ) 5 ) 5 (5 ) Power of quotient rule 5 4 6 (5 ) 5 ( ) 5 6 c) 5 7 4
5.6 Positive Integrl Eponents (5 5) 57 ) 4 y 9 y Power of quotient nd power of product rule 7 9 8y Siplify. c) Use the quotient rule to siplify the epression inside the prentheses efore using the power of quotient rule. 4 7 6 Use the quotient rule first. 7 8 Power of quotient rule helpful hint Sury of Rules The rules for eponents re surized in the following o. Note tht the rules of eponents show how eponents ehve with respect to ultipliction nd division only. We studied the ore coplicted prole of using eponents with ddition nd sutrction in Section 5.4 when we lerned rules for ( ) nd ( ). The following rules hold for nonzero rel nuers nd nd nonnegtive integers nd n.. 0 Definition of zero eponent. n n Product rule. n n for n, n n for n Quotient rule 4. ( ) n n Power rule 5. () n n n Power of product rule 6. n n n Rules for Nonnegtive Integrl Eponents Power of quotient rule WARM-UPS True or flse? Assue tht ll vriles represent nonzero rel nuers. A stteent involving vriles is to e rked true only if it is n identity. Eplin your nswer.. 0. 5 8 4. 6 5 4. () 4 4 5. (q ) 5 q 8 6. ( ) 7 6 7. ( ) 4 4 8. 4 6w 9. 4 9 w 5 0. w y 6 9 4y 8
58 (5 6) Chpter 5 Polynoils nd Eponents 5.6 EXERCISES Reding nd Writing After reding this section, write out the nswers to these questions. Use coplete sentences.. Wht is the product rule for eponents?. Wht is the quotient rule for eponents?. Why ust the ses e the se in these rules? 4. Wht is the power rule for eponents? Siplify. See Eple 4. 5. 4 4 7. 4 9. y 4y 4. 6 4 y z 9 6 y 4 z 6. y 8. 9r 5 t 40. 8 4 y zy 4. 0rs 9 t 4 rs t 7 5. Wht is the power of product rule? 6. Wht is the power of quotient rule? For ll eercises in this section, ssue tht the vriles represent nonzero rel nuers. Siplify the eponentil epressions. See Eple. 7. 5 8. 6 7 9. (u 8 )(u ) 0. (r 4 )(6r ). 4 6 () 0. y y 6 ( y) 0. 7 4. 9 t 8 4 6t 5. 5 7 5 6 8 6. 8 y 5y 9 0 4 y 7. 5 8. 0 Siplify. See Eple. 9. ( ) 0. (y ) 4. ( ) 5. (y ) 6 y 5. ( 5 t ) 4 4. ( 4 r ) 5 ( t ) ( r ) 5 5 ( ) 5y ( y ) 5. 6 4 6. ( ) 5 0y ( y ) 6 Siplify. See Eple. 7. (y ) 8. (wy ) 6 9. (t 5 ) 0. (r ). ( y 5 ). (y z ) 4 c 5 ). ( 4 c 4. ( 5 c ) ( 4 c) Siplify ech epression. Your nswer should e n integer or frction. Do not use clcultor. 4. 6 44. (5 ) 45. ( 6) 46. 5 47. 48. 4 49. ( ) 50. ( 4) 5. 5 5. 4 5. 5 54. 0 55. 4 56. 0 0 4 57. 5 58. Siplify ech epression. 59. 4 5 7 60. y (y) 6. (5 4 ) 6. (4z ) 6. y 5 z 9yz 7 64. 4 5 9 65. 4 9 9u v 66. 5 0 5 8 4 u v 5 67. (t )( t) 4 68. () ( ) 4 69. 4 7. 8 4 5 4c 5 4 7 5 8 y 7. 5 6 6 y 8 70. y 5 y 7. 0 5 5 c 5 5 4 5 74. 5 yz 5 yz Solve ech prole. 75. Long-ter investing. Sheil invested P dollrs t nnul rte r for 0 yers. At the end of 0 yers her investent ws worth P( r) 0 dollrs. She then reinvested this oney for nother 5 yers t nnul rte r. At the end of the second tie period her investent
5.7 Negtive Eponents nd Scientific Nottion (5 7) 59 ws worth P( r) 0 ( r) 5 dollrs. Which lw of eponents cn e used to siplify the lst epression? Siplify it. 76. CD rollover. Ronnie invested P dollrs in -yer CD with n nnul rte of return of r. After the CD rolled over two ties, its vlue ws P(( r) ). Which lw of eponents cn e used to siplify the epression? Siplify it. GETTING MORE INVOLVED 77. Writing. When we squre product, we squre ech fctor in the product. For eple, () 9. Eplin why we cnnot squre su y siply squring ech ter of the su. 78. Writing. Eplin why we define 0 to e. Eplin why 0. 5.7 NEGATIVE EXPONENTS AND SCIENTIFIC NOTATION In this section Negtive Integrl Eponents Rules for Integrl Eponents Converting fro Scientific Nottion Converting to Scientific Nottion Coputtions with Scientific Nottion E X A M P L E clcultor close-up You cn evlute epressions with negtive eponents on clcultor s shown here. We defined eponentil epressions with positive integrl eponents in Chpter nd lerned the rules for positive integrl eponents in Section 5.6. In this section you will first study negtive eponents nd then see how positive nd negtive integrl eponents re used in scientific nottion. Negtive Integrl Eponents If is nonzero, the reciprocl of is written s. For eple, the reciprocl of is written s. To write the reciprocl of n eponentil epression in sipler wy, we use negtive eponent. So. In generl we hve the following definition. Since n nd n re reciprocls, their product is. Using negtive eponent for the reciprocl llows us to get this result with the product rule for eponents: n n nn 0 Siplifying epressions with negtive eponents Siplify. ) 5 ) () 5 c) ) 5 5 ) () 5 ( ) 5 Negtive Integrl Eponents If is nonzero rel nuer nd n is positive integer, then c) n n. Definition of negtive eponent (If n is positive, n is negtive.) 8 9 8 9 9 8