Math 153: Lecture Notes For Chapter 1

Transcription

1 Mth : Lecture Notes For Chpter Sectio.: Rel Nubers Additio d subtrctios : Se Sigs: Add Eples: = - - = - Diff. Sigs: Subtrct d put the sig of the uber with lrger bsolute vlue Eples: - = - = - Multiplictio d Divisios : Se Sigs: (positive swer) Eples: ().() = = = (-)(-) = (-) / (-) = = = Diff. Sigs: - (egtive swer) Eples: (-).() = - = = ()(-) = - (-) / () = - = = Book, Eer. - : If < 0 d > 0, deterie the sig of the rel uber:. ) b) c) d) ( ) Book, Eer. - : Replce the sbol? with either <, >, or = to ke the resultig stteet true.. ) -? - b) π? 0. c)?. )? 0. b)? 0. c)?. Book, Eer. - : Epress the stteet s iequlit.. ) b is positive. b) s is o positive. c) w is greter th or equl to -. d) c is betwee d e) p is ot greter th -. f) The egtive of is ot less th -. g) The quotiet of r d s is t lest h) The reciprocl of f is t ost. i) The bsolute vlue of is less th.

2 Book, Eer. - : Rewrite the uber without usig the bsolute vlue sbol, d siplif the result. 0. ) b) c). ) ( ) b) / c) Book, Eer. - : The give ubers re coordites of poits A, B, d C, respectivel, o coordite lie. Fid the distce. ) d(a, B) b) d(b, C) c) d(c, B) d) d(a, C). -, -,., -, - Book, Eer. - : The two give ubers re coordites of poits A d B, respectivel, o coordite lie. Epress the idicted stteet s iequlit ivolvig the bsolute vlue sbol. 0., ; d(a, B) is greter th., ; d(a, B) is t ost. -, ; d(a, B) is ot less th Book, Eer. - : Rewrite the epressio without usig the bsolute vlue sbol, d siplif the result.. if >. if > - 0. b if > b. Book, Eer. - 0: Replce the sbol? with either = or to ke the resultig stteet true for ll rel ubers, b, c, d d, wheever the epressios re defied. b c.? b c.. ( - b) - c? - (b - c) c b d? b c d Book, Eer. - : Approite the rel-uber epressio to four decil plces.... ) b) π.. Book, Eer. - : Approite the rel-uber epressio. Epress the swer i scietific ottio ccurte to four sigifict figures.. b) (. 0 ) (. 0 ) Covert to Scietific for: ),00,000,000,000 b) Covert to Decil for: ). 0 b). 0

3 Sectio.: Epoets d Rdicls Prt I: Iteger Epoets Rule Forul Eple I) Product Rule = = II)Divisio Rule = = = III) Power Rule ( ) =. ( ) = 0 Other Rules ( b ) = b = b b 0 ( ) = = Note : A ter to the power of zero =, ecept zero to the power of zero. Note : Whe ou ove ter fro uertor to deoitor or vice vers, chge the sig of the epoet ol. Eples: = is correct, but = is wrog, wh? Note : (-) to the power of odd is = - ; ( ) = (-) to the power of eve = ; ( ) = Eples: fid the swer of the followig: ) ( ) b) ( ) c) ( ) d) ( ) e) f) ( ) Note : it is helpful to siplif ll the w whe ou c Eples: = ( ) = ; = ( ) = Book, Eer. - 0: Epress the uber i the for /b, where d b re itegers... Book, Eer. - : Siplif. 0. ( z )( z )( ) 0. ( r s) (r s ).. (r) (r ) 0. ( )( ).. c d ( ) b b

4 Prt II: Rtiol Epoet: = / / ( ) or = the ide or the degree of the root, ( = the squre root, = the third root ) : the rdicl sig : the rdicd or uder the root Rewrite without rtiol epoets: ) ) = ( ) ) = ( ) ( ) = ( ) Rewrite with rtiol epoets: ) ( ) = ) ( ) = = ) ( ) Rewrite ech eple i differet for: ) = = ( ) ( ) = ) = = ( ) = ( ) = ( ) Hit: Write =, = or ± = ±, do ot write: = ± Hit: The followig tble should be es to reeber d to use i this chpter Mi Nuber Relted Nubers Epoetil Reltio Mi Nuber Relted Nubers Epoetil Reltio = = = ; - = (-) = ; - = (-) = = = ; - = (-) = = = = = = ; - = (-) = 0 = = Eples: = ( ) = ; = ( ) = = ( ) = ; = ( ) =

5 Hits o Siplifictios, Divisio d Multiplictio of Rdicls These re iportt hits to reeber i siplifig: Hit : The power uder the root ust lws be less th the degree of the root (ide). If it is higher, the divide b the degree of the root or ide. Eple: = Hit : If the power uder the root is lrger th the ide but cot be divided b the ide, the brek the power to ubers, oe of the is the lrgest uber tht c be divided b the ide Eple: 0 = = Hit : Siplif both the power uder the root d the ide of the root if both c be divided b the se uber Eple: = (ll powers c be divided b ) Hit : Alws write the uber s prie uber if it is ot Eple: = = Eple: = ( ) = ( ) = = = Hit : Do ot hve ore th oe root i ultiplictio or divisio. Eple: (the coo deoitor betwee d is ) 0 = = = = Eple: (the coo deoitor betwee d is ) = = = = For the followig eples, Use rtiol epoets to write sigle rdicl epressio for:

6 Multiplictio d Divisio of Rtioles: (ust hve se ide, se degree of root). b = b d = b b For the followig eples, Siplif the followig without usig clcultor: Rtiolizig the Deoitor es Reove the rdicl fro the deoitor To reove squre root, ultipl uertor d deoitor b the se squre root. Eple: = = To reove roots tht re ot squre, follow the followig eples d see wht is eeded to be doe to reove the roots: If the deoitor is: Multipl Nu d De b: Result... You ultipl b root with Se Degree, d ke the power of ech eleet uder the ew root equl to uber tht c be dded to the origil power (of the se eleet). The result should be uber tht c be divided b the degree of the root. For the followig eples, Rtiolize the deoitor (reove the rdicl fro the deoitor):...

7 Eve Root or = eve uber If is eve, the the rdicd or the epressio uder the root ust be positive. Eple: =, but = No RelNuber. If = eve uber d uder the root cotis epressio with vrible to eve power d the vrible (letters) represet rel uber Use Absolute Vlue = Eples: ) = ; b) c) = ( ) = ( ) For the followig eples, Siplif the followig without usig clcultor... Book, Eer. - 0: Siplif the epressio, d rtiolize the deoitor whe pproprite.. r s Book, Eer. - : Siplif the epressio ssuig d be egtive.. ( ) Book, Eer. - 0: Replce the sbol? with either = or to ke the resultig stteet true.. ( )? Book, Eer. - : Approite the rel-uber epressio to four decil plces.. () ( ) (b) ( )

8 Sectio.: Algebric Epressio Fid the ledig ter, ledig coefficiet d the degree of the poloil for: ) - - ) - Siplif ) (b - c bc) (b - c - bc) ) (0 - - ) - (- -) Note : Whe ou dd or subtrct poloils, the ust hve se vrible d se power Note : To get the opposite of poloil, ultipl b - Multiplictio Book, Eer. - : Epre ss s poloil. 0. ( u )( u ) u( u ). ( )( )( ) b b b. 0. ( ) b *. ( )( ) 0. ( ) Fctorig A. Fctor ) ) - - B. Fctorig b c whe = : ) ) - 0 ) C. Fctorig b c whe : ) - - ) ) 0 - ) -

9 D. Usig: - b = ( - b)( b): ) - ) b - c ) - ) - ) - ) - E. Fctor b Groupig: ) z w wz ) b b - - ) - - ) - b ) - 00 ) - - F. Cubes: ( b ) = ( b) ( - b b ) ( - b ) = ( - b) ( b b ) ) ) ) - ) - b ) Book, Eer. - 00: Fctor the poloil.. r s r s r s c c

10 0 Sectio.: Frctiol Epressio A. Fid the doi of: ) ) ) B. Siplif the epressio: ) ) 0 ) ) ) ) C. Fid the LCD of: ) d - ) ; d D. Siplif the epressio: ) ) ) ) E. Siplif the epressio: ) ) ) )

11 F. Multipl d Siplif Copletel, (ppl ( -b)( b) = - b whe it is possible): ) ( ) ) ( 0 ) ) ( )( ) ) ( )( ) ) ( )( ) ) ( )( ). G. Rtiolize the deoitor: ). ). Book, Eer. - 0: Rtiolize the uertor. b c h.. b c h Book, Eer. - : Epress s su of ters of the for r, where r is rtiol uber: ( ). Book, Eer. - : Epress s quotie t:. / / Book, Eer. - : Siplif the epressio:. ( ) ( )( ) / ( ) / ()( ) (). ( ) ( ) ()( [( ) ] ) () 0. ( / ) () ( )( / [( ) ] / ) ( ) Solve proble s bous. The swer is: / / ( ) ( )