Next we encountered the exponent equaled 1, so we take a leap of faith and generalize that for any x (that s not zero),

Transcription

1 79 CH 0 MORE EXPONENTS Itroductio T his chpter is cotiutio of the epoet ides we ve used m times efore. Our gol is to comie epressios with epoets i them. First, quick review of epoets: 0 0 () () 0 ( ) 0 Review of Strge Epoets We ve worked with se etesivel, lzig wht hppes whe we ppl vrious epoets to it. First we oserved tht. It s resole to coclude tht if is umer, Net we ecoutered the epoet 0. 0 equled, so we tke lep of fith d geerlize tht for (tht s ot zero), 0 [0 0 hs o meig util clculus] The cme the strge egtive epoet. We sw, for emple, tht 0. It s lso the cse tht for o-zero se, 0 Ch 0 More Epoets

2 0 where,,,... Zero rised to egtive power will e delt with i the homework. Homework. Evlute ech power: c. (.) d. (7) e. f. 9 0 g. (.) 0 h. 0 i. j. k. l.. Fid the vlue of the epressio 0 / 7 si d 0. i e l t. Assumig ll vriles re ot zero, simplif ech epressio:.. 0 c. d. z e. (c + ) 0 f. g. 0 h. () i. t j. () 0 + k. () 0 l. 0 m. 0. ( + ) 0 o. + 0 p. 0 q. (c d) 0 r. 0 0 s. w t. z u. v. u w.. w. 7 z. k Hit: Ectl wht is eig rised to the 0 power?. Prove tht 0 is udefied. Ch 0 More Epoets

3 Comiig Thigs with Epoets EXAMPLE : Simplif ech epoetil epressio usig the stretch-d-squish techique. Stretch-d-squish mes to epd the ses i the prolem usig the defiitio of epoets, the do some kid of simplifig usig previous techiques, d the squish it ck to epoet form. A. ( ) ( ) B. C. 7 7 D. () () ()()() (defiitio of cuig) (pretheses ot ecessr) ()() (rerrge the fctors) (rewrite with epoets) Ch 0 More Epoets

4 E. (uwz) (uwz) (uwz)(uwz) (uu)(ww)(zz) u w z F. G. H. ( ) ( ) ( )( ) ( )( ) EXAMPLE : Simplif ech epressio: A. ()() 7 B ??? Note: We do t hve simple product of s (due to the plus sig), so we c t write the sum usig sigle epoet. Besides, d 7 re ulike terms, d therefore cot e dded. However we look t it, this prolem cot e simplified. O test ou c write our swer either s + 7 or As is. C. + These re like terms, so the dd up. Ch 0 More Epoets

5 D. u w 7 cot e simplified, sice (uuu)(wwwwwww) is just wht it is, fctors of u multiplied 7 fctors of w. E. 7 ()() F. (c )(0c ) ()(0)(ccc)(ccccc) 0(cccccccc) 0c G. (qt) (qt)(qt)(qt) ()(qqq)(ttt) q t H. ( ) ( )( ) (the is ot eig squred) ()()()() (stretch) ()() (rerrge fctors) (squish) I. ( ) ( ) ( ) ( ) ( ) (stretch) ()()()()()()()()()()()() (stretch) ()() (rerrge) (squish) J. g h g g g g gggg gggg h h h h hhh hhh g h Ch 0 More Epoets

6 Homework Use the stretch-d-squish techique (where pproprite) to simplif ech epressio:... c. z z d. e. u + u f. w 9 + w 9 g. v v h. c c c. z z 7 d (uv). (c) c. () d. (mpq) e. (jk) f. () g. ( + ) 0. True/Flse: ( + ) + Check it out usig umers for d. 9.. e. F H F H I K. I u vk f. F w H I z F f I HG g K J 0 g. K c. F I HG K J d. F H I K h. ( ) 0.. ( ). ( ) c. ( ) d. (0 ) e. ( ) f. ( 0 ) g. ( ) 0 h. (m 0 ) 0.. ()(). ( )() c. ( ) d. ( )( ) e. + f. 9 + g. 0q 0q h. ()() i. ( )( ) j. (g )(g ) k. (9 ) (9 ) l. t + t Ch 0 More Epoets

7 .. ( ). ( ) c. () d. ( ) e. (z ) f. ( ) g. (m ) h. (m ) i. (cd ) j. ( ) k. m.. o. c d u w l. p. ( ) Review Prolems. Evlute ech epressio:.. 9 c. d. e. 0 f. 0 g. 0 h. 0 i. 0 7 j. 0 k. 0 0 l. 0 + m ( ) o. (0 ) p. q. 0 0 r. 0 s. + 0 t. + u v. (0 ) w. ( ). (0 9 ) 0. ( ) 0 z. (7 0 ). Simplif ech epressio: c. d. w e. () f. ( + ) 0 g. h. + Ch 0 More Epoets

8 i. j. 7 k. + l. m. q.. 0 o. () p. (z) r. ( ) s. (u ) t. (w ) u. (A ) 0 v. (Q 0 ) 9 w. ( ). ( ). 0 9 z Simplif ech epressio:. ( )(). ( w)(w ) c. ( ) d. ( ) e. k g. ( )(0 ) h. j k. + + l. 0 f. k i Ch 0 More Epoets

9 7 Solutios c.. d. 7 e. g. h. i. j. k. f. l c. g. h. i. t d. z e. f. j. k. l. m.. o. + p. q. r. 0 s.. w 7 t. z. z k u. v. u w.. w., which is udefied ()(). () c. z d. e. As is f. w 9 g. As is h. 0.. c.. 7 z zzzzzzz zzzzzzzz z zzzzzzzz zzzzzzzz z d. As is 7.. (uv) (uv)(uv)(uv) uuuvvv u v. c c. d. m p q e. j k f. g. Ch 0 More Epoets

10 . The sttemet is flse; pick some umers for d, plug them ito ech side of the sttemet, d ou ll see wh. [See Chpter.] 9.. d. e.. w z u v f. g. h. 0 + c. 0.. ( ) ( ) ( ) ( ). 79 c. d. 0 e. f. ( 0 ) g. h.... c. 0 d. e. As is f. 7 g. 0 h. i. j. g 7 k. 0 l. As is... c. d. 7 9 e. 7z f. g. m h. 9m i. c d j. k. m.. 9 o. c d u w... c. d. e. f. 000 g.,000,000 h. 0 l. p. 09 i. j. k. l. 0,000,000 m. 0. o.,000,000,000,000 0,000 Ch 0 More Epoets

11 9 p. q. 0,000,000 r. 00 s. 0 t. u v. w.,000,000.. z c. d. w e. f. g. h. + i. j. 7 k. l. 0 m.. o. p. z q. r. s. u t. w 9 u. v. w z.... w c. d. e. i. k 9 0 f. As is g. j. k. 0 7 h. l. 0 To d Beod! Do some reserch to determie the meig of / 9. Ch 0 More Epoets

12 90 Nothig c stop the m with the right metl ttitude from chievig his gol; othig o erth c help the m with the wrog metl ttitude. Thoms Jefferso Ch 0 More Epoets