Hwkes Lerig Systems: College Algebr Sectio 1.4: Properties of Rdicls
Objectives o Roots d rdicl ottio. o Simplifyig rdicl expressios.
Roots d Rdicl Nottio th Roots d Rdicl Nottio The expressio th expresses the root of i rdicl ottio. The turl umber is clled the idex, is the rdicd d is clled the rdicl sig. By covetio, is usully writte s 2. Rdicl Sig Idex Rdicd
Roots d Rdicl Nottio th Roots d Rdicl Nottio Cse 1: is eve turl umber. If is o-egtive rel umber d is eve turl umber, is the o-egtive rel umber b with the property tht b. Tht is b b. I this cse, ote tht d.
Roots d Rdicl Nottio th Roots d Rdicl Nottio (cot.) Cse 2: is odd turl umber. If is y rel umber d is odd turl umber, is the rel umber b (whose sig will be the sme s the sig of ) with the property tht b. Agi, b b, d.
Note: HAWKES LEARNING SYSTEMS is defied oly whe is o- is ot rel umber. o Whe is eve, egtive. Ex. Roots d Rdicl Nottio o Whe is odd, is defied for ll rel umbers. Ex. = -2 o We prevet y mbiguity i the meig of whe is eve d is o-egtive by defiig to be the o-egtive umber whose power is. Ex: 4 2, NOT 2.
Exmple: Roots d Rdicl Nottio Simplify the followig rdicls.. b. c. d. 3 27 3 becuse 3 3 27. 4 81 is ot rel umber, s o rel umber rised to the fourth power is -81. Note: 0 0 0 0 for y turl umber. 0 0 Note: 1 1 1 1 for y turl umber. 1 1
Exmple: Roots d Rdicl Nottio Simplify the followig rdicls.. 3 125 216 5 6 becuse 3 5 125 6 216 b. 6 5 6 6 15625 5 becuse 5 15625 ( 5) 6 6 I geerl, if is eve turl umber, for y rel umber. Remember, though, tht if is odd turl umber.
Simplifyig Rdicl Expressios Simplified Rdicl Form A rdicl expressio is i simplified form whe: ( Use s referece, ) 1. The rdicd cotis o fctor with expoet greter th or equl to the idex of the rdicl (expoets i ). x 2. The rdicd cotis o frctios ( ). y x 3. The deomitor cotis o rdicl ( ). 4. The gretest commo fctor of the idex d y expoet occurrig i the rdicd is 1. Tht is, the idex d y expoet i the rdicd hve o commo fctor other th 1 ( GCF(y expoet i, )=1 ).
Simplifyig Rdicl Expressios I the followig properties, d b my be tke to represet costts, vribles, or more complicted lgebric expressios. The letters d m represet turl umbers. Property 1. Product Rule b b 2. Quotiet Rule b b 3. m m
Exmple: Simplify Rdicl Expressios Simplify. ) b) c) d)
Cutio! Simplifyig Rdicl Expressios Oe commo error is to rewrite b s b These two expressios re ot equl! To covice yourself of this, observe the followig: = = 5, but = 3 + 4 = 7
Simplifyig Rdicl Expressios Rtiolizig Deomitors Cse 1: Deomitor is sigle term cotiig root. If the deomitor is sigle term cotiig fctor of we will tke dvtge of the fct tht m d eve. m m m m is or, depedig o whether is odd or
Simplifyig Rdicl Expressios Rtiolizig Deomitors Cse 1: Deomitor is sigle term cotiig root. (cot.) Of course, we cot multiply the deomitor by fctor of m without multiplyig the umertor by the sme fctor, s this would chge the expressio. So we must m multiply the frctio by m.
Exmple: Rtiolize the Deomitor Rtiolize the Deomitor. Presume vribles re positive. ) b)
Simplifyig Rdicl Expressios Rtiolizig Deomitors Cse 2: Deomitor cosists of two terms, oe or both of which re squre roots. Let A + B represet the deomitor of the frctio uder cosidertio, where t lest oe of A d B is squre root term. We will tke dvtge of the fct tht 2 2 2 2 A B A B A A B B A B A AB AB B A B Note tht the expoets of 2 i the ed result egte the squre root (or roots) iitilly i the deomitor.
Simplifyig Rdicl Expressios Rtiolizig Deomitors Cse 2: Deomitor cosists of two terms, oe or both of which re squre roots. (cot.) Oce gi, remember tht we cot multiply the deomitor by A B uless we multiply the umertor by this sme fctor. A B Thus, multiply the frctio by. A B The fctor A B is clled the cojugte rdicl expressio of A + B.
Exmple: Rtiolize the Deomitor Rtiolize the Deomitor. Presume vribles re positive. ) b)
Exmple: Rtiolize the Numertor Rtiolize the Numertor. Presume vribles re positive.