Surds, Indices, and Logarithms Radical

Transcription

1 MAT 6 Surds, Idices, d Logrithms Rdicl Defiitio of the Rdicl For ll rel, y > 0, d ll itegers > 0, y if d oly if y where is the ide is the rdicl is the rdicd. Surds A umber which c be epressed s frctio of itegers (ssumig the deomitor is ever 0) is clled rtiol umber. Emples of rtiol umbers re, d. A umber which cot be epressed s frctio of two itegers is clled irrtiol umber. Emples of irrtiol umbers re, 7 d π. A irrtiol umber ivolvig root is clled surd. Surds occur frequetly i trigoometry, clculus d coordite geometry. Usully, the ect vlue of surd cot be determied but pproimte vlue of it c be foud by usig clcultors or mthemticl tbles. I this chpter, mes the positive squre root of while mes the egtive squre root of. Geerl Rules of Surds Multiplictio of surds b b For emple 6 6 (ii) 6 8

2 MAT 6 Divisio of surds b b For emple (ii) b These rules re useful for simplifyig two or more surds of for combiig them ito oe sigle surd. Note, however, tht d 6 6 which c be esily checked by clcultor; d, therefore, i geerl + b + b d b b. Emple + (+ ) (ii) Emple Simplify + 7 (ii) (9 + 0) 7 (ii) ( + + ) Try This Simplify 7 (ii)

3 MAT 6 Rtioliztio of the Deomitor Whe frctio hs surd i its deomitor,, it is usul to elimite the surd i the deomitor. I fct, the writig of surds i the deomitors of frctios should be voided. The process of removig this surd is clled rtiolizig of the deomitor. m + d m re specilly relted surds kow s cojugte surds. The product of cojugte surds is lwys rtiol umber. ( m + )( m ) ( m) ( ) m For emple ( 9 + )( 9 ) ( 9) ( ) 9 ( 7 + )( 7 ) ( 7) ( ) 7 Emple Emple Simplify. Simplify ( 7 ) 7 7 Emple Simplify, without usig tbles or clcultors, the vlue of +. + (+ ) ( ) ( )( + ) ( + )( ) ( + ) + ( ) 9 6 7

4 MAT 6 Try This Simplify (ii) Aswers to Try This.. (ii) 7 (ii)

5 MAT 6 Idices If positive iteger is multiplied by itself three times, we get clled the bse d, the ide or power. Thus I geerl,, i.e. mes the th power of.. Here is mes the th power of, where is y positive ide of the positive iteger. Rules of Idices There re severl importt rules to remember whe delig with idices. If, b, m d re positive itegers, the () () m m + m m 8 m () ( ) m 6 ( ) m m m () b ( b) 6 () b b m m m (6) 0 0 (7) (8) 8 8 m m m (9) ( ) 8 8 ( 8)

6 MAT 6 Emple Evlute (ii) 8 6 (iv) 8 (ii) ( 6) (iv) ( ) 8 Try This Evlute ech of the followig without usig clcultor (v) 7 (ii) (vi) (vii) 9 (iv) 7 (viii) 8 Emple Simplify (ii) ( b ) ( ) (ii) ( b ) 8 b b ( ) Try This Simplify ech of the followig, givig your swer i ide form: (ii) 6 ( b ) 6

7 MAT 6 Solvig Epoetil Equtios Emple Solve the followig epoetil equtios + (ii) 0. (ii) Try This Solve the followig equtios: 8 (ii) 8 (iv) (v) + 7 (vi) Emple Solve the equtio Let y 8y + 8y + y y + 7 y 0 (8y )( y + ) 0 y or 8 Whe y 8 whe y 8 (idmissible) 7

8 MAT 6 Try This Solve the equtio Emple If y 9 7 d y, clculte the vlues of d y. 8 y 9 7 () y () 8 From (): From (): y ( ) y + y + y () y ( ) y y y () () () : 6y 6 y Substitute y ito (): + () d y. Try This + y Solve the simulteous equtios, y 8 8

9 MAT 6 Aswers to Try This. (ii) (iv) 7 (v) 7 (vi) (vii) 8 (viii) 6. 9 (ii) 6 b. (ii) (iv) 0 (v) 9 (vi). or., y 9

10 MAT 6 Logrithms Defiitio: For y umber y such tht y ( > 0 d ), the logrithm of y to the bse is defied to be d is deoted by log y. Thus if y, the log y For emple, log 8 0 log 00 Note: The logrithm of to y bse is 0, i.e. log 0. The logrithm of umber to bse of the sme umber is, i.e. log. The logrithm of egtive umber is udefied. Emple Fid the vlue of log 6 (ii) log9 log (iv) log Let log 6 (ii) Let log Let log (iv) Let log

11 MAT 6 Lws of Logrithms () log m log m + log log + log log0 m () log log m log log + log log p () log m p log m log log 0 0 Emple Without usig tbles, evlute log0 + log0 70 log0 + log0. log 0 + log 0 70 log0 + log0 log0 70 log 00 0 log log 0 Try This Simplify log log 0 + log. Chgig the Bse of Logrithms Logrithms to bse 0 such s log0 d log 0( + ) re clled commo logrithms. A ltertive form of writig log0 is lg.commo logrithms c be evluted usig scietific clcultor. Logrithms to bse e such s loge d log e re clled Nturl logrithms or Npieri logrithms. Nturl logrithms re usully writte i ltertive form, for emple, loge is writte s l. (Note: e ) If, b, d c re positive umbers d, the log logc b b. log c

12 MAT 6 Emple Fid the vlue of log 6. log0 6.0 log6.7 log Try This Fid the vlue of log. Solvig Logrithmic Equtios Emple Solve the equtio 8. 8 Tkig logrithms to bse 0 o both sides, log0 log0 8 log0 log0 8 log0 8. log Try This Solve the equtio + 0. Emple Give tht log0 + log0 p, clculte the vlue of p without usig tbles or clcultors. log + log p p log ( ) p 0 00 p p p ± Sice p cot be becuse log 0( ) is ot defied, p.

13 MAT 6 Try This + Solve the equtio log. 7 Emple 6 Solve the equtio log 0( + ) log0 log 0( ). log ( + ) log log ( ) log ( ) log log ( ) ( + )( ) log0 ( + )( ) ( + 6)( ) 0 6 or Sice cot be egtive,. Try This Solve the equtio log + log ( ). Aswers to Try This. log or.