Section 3.6: Rational Exponents

Transcription

1 CHAPTER Sectio.6: Rtiol Epoets Sectio.6: Rtiol Epoets Objectives: Covert betwee rdicl ottio d epoetil ottio. Siplif epressios with rtiol epoets usig the properties of epoets. Multipl d divide rdicl epressios with differet idices. We defie rtiol epoets s follows: DEFINITION OF RATIONAL EXPONENTS: ( ) d The deoitor of rtiol epoet is the se s the ide of our rdicl while the uertor serves s epoet. Either for of the defiitio c be used but we tpicll use the first for s it will ivolve sller ubers. Notice whe the uertor of the epoet is, the specil cse of the defiitio: ( ) th roots follows fro CONVERTING BETWEEN EXPONENTIAL AND RADICAL NOTATION We c use this defiitio to chge rdicl epressio ito epoetil epressio. Eple. Rewrite with rtiol epoets. ( ) 6 6 ( ) ( ) Ide is deoitor, epoet is uertor ( ) ( ) ( ) Negtive epoets fro reciprocls Pge 6

2 CHAPTER Sectio.6: Rtiol Epoets We c lso chge rtiol epoet ito rdicl epressio b usig the deoitor s the ide. Eple. Rewrite usig rdicl ottio. ( ) ( ) ( ) Epoet is uertor; ide is deoitor ( ) ( ) 9 9 ( ) Negtive epoet es reciprocls The bilit to chge betwee epoetil epressios d rdicl epressios llows us to evlute epressios we hd o es of evlutig previousl. Eple. Use rdicl ottio to rewrite d evlute. 6 Chge to rdicl fort; uertor is epoet, deoitor is ide ( 6) Evlute rdicl () Evlute epoet 6 Our Aswer Eple. Use rdicl ottio to rewrite d evlute. Negtive epoet is reciprocl Chge to rdicl fort; uertor is epoet, deoitor is ide Evlute rdicl ( ) Evlute epoet () Our Aswer 8 Pge 6

3 CHAPTER Sectio.6: Rtiol Epoets SIMPLIFY EXPRESSIONS WITH RATIONAL EXPONENTS The lrgest dvtge of beig ble to chge rdicl epressio ito epoetil epressio is we re ow llowed to use ll our epoet properties to siplif. The followig tble reviews ll of our epoet properties. PROPERTIES OF EXPONENTS ( b) b b b ( ) 0 b b Whe ddig d subtrctig with frctios we eed to hve coo deoitor. Whe ultiplig we ol eed to ultipl the uertors together d deoitors together. The followig eples show severl differet probles, usig differet properties to siplif the rtiol epoets. Eple. Siplif. 6 b b Need coo deoitor for s (6) d for b s (0) b b Add epoets o s d b s 6 0 b Our Aswer Eple 6. Siplif. Multipl ech epoet b ; 0 reduce frctios Our Aswer Pge 6

4 CHAPTER Sectio.6: Rtiol Epoets Eple. Siplif. 0 Need coo deoitor for s () to subtrct epoets 0 Subtrct epoets o i deoitor, 0 Negtive epoet oves dow to deoitor Our Aswer MULTIPLY AND DIVIDE RADICAL EXPRESSIONS WITH DIFFERENT INDICES We will use rtiol epoets to ultipl or divide rdicl epressios hvig differet idices. We will covert ech rdicl epressio to its equivlet epoetil epressio. The, we will ppl the pproprite epoet propert. For our swer, we will covert the epoetil epressio to its equivlet rdicl epressio. Our swer will the be writte s sigle rdicl epressio. Eple 8. Multipl, writig the epressio usig sigle rdicl. Rewrite rdicl epressios usig rtiol epoets Need coo deoitor of 0 to dd epoets 0 0 Add epoets 0 Rewrite s rdicl epressio 0 Our Aswer Pge 6

5 CHAPTER Sectio.6: Rtiol Epoets Eple 9. Divide, writig the epressio usig sigle rdicl. Rewrite rdicl epressios usig rtiol epoets Need coo deoitor of to subtrct epoets 0 6 Subtrct epoets Rewrite s rdicl epressio Our Aswer It is iportt to reeber tht s we siplif with rtiol epoets, we re usig the ect se properties we used whe siplifig iteger epoets. The ol differece is we eed to follow our rules for frctios s well. It be worth reviewig our otes o epoet properties to be sure ou re cofortble with usig the properties. Pge 6

6 CHAPTER Sectio.6: Rtiol Epoets Prctice Eercises Sectio.6: Rtiol Epoets Write ech epressio i rdicl for. ) ) ( ) ) r (0 ) ) b (6 ) Write ech epressio i epoetil for. ) ( 6) ( ) ) 6) v 8) Evlute. 9) 8 ) 0) ) ) 6 00 ) ) 6) 8 Siplif. Your swer should coti ol positive epoets. ) 0) ( ) 0 8) v 9) v ( b) ( ) 0 0 ) ) u v ( u ) The Prctice Eercises re cotiued o the et pge. Pge 66

7 CHAPTER Sectio.6: Rtiol Epoets Prctice Eercises: Sectio.6 (cotiued) Siplif. Your swer should coti ol positive epoets. ) b b b 8) ) 9) ( ) 0 ) 0 0) ( ) 6) b b b ) ( ) ) ( ) ) 0 ( ) Perfor the idicted opertio, writig the epressio usig sigle rdicl. ) ) 6 Pge 6

8 CHAPTER Sectio.6: Rtiol Epoets ANSWERS to Prctice Eercises Sectio.6: Rtiol Epoets ) ( ) ) ( ) ) ( 0r) ) ( 6b) ) 6 ) 6) v 8) ( ) 9) 0) ) 8 ) 000 ) ) 8 ) 6) ) 8) v 9) b 0) ) v ) u The Aswers to Prctice Eercises re cotiued o the et pge. Pge 68

9 CHAPTER Sectio.6: Rtiol Epoets ANSWERS to Prctice Eercises: Sectio.6 (cotiued) ) ) b 6 8) 9) ) 0) 6) ) b ) ) 0 ) ) 0 Pge 69

10 CHAPTER Sectio.6: Rtiol Epoets Pge 0