Sect Simplifying Radical Expressions. We can use our properties of exponents to establish two properties of radicals:


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1 70 Sect Simplifyig Rdicl Epressios Cocept #1 Multiplictio d Divisio Properties of Rdicls We c use our properties of epoets to estlish two properties of rdicls: () 1/ 1/ 1/ & ( ) 1/ 1/ 1/ Multiplictio d Divisio Properties of Rdicls Let d e rel umers such tht 0. The d 1) Multiplictio Property of Rdicls: ) Divisio Property of Rdicls: re rel umers d We c use these properties forwrds d ckwrds to simplify rdicls. Simplify the followig: E E. 1 E. 1c 1 Solutio: E. 1d ) ) 7 1 c) 1 d) I workig with rdicl epressios where the rdicd is ot perfect power of the ide, it is importt to simplify the rdicl s much s possile. With frctios, it is firly simple to tell whe frctio is completely simplified. Let s thik out the criteri for rdicl to e completely simplified.
2 First, ll fctors of the rdicd hve powers tht re less th the ide. For emple, 6 is simplified sice 6 d the power of d the power of re 1 which is less th the ide. But, 0 is ot simplified sice 0 d the power of is equl to the ide. I fct, 0. Secod, the rdicd hs o frctios. For, istce, hs o frctios, wheres is simplified sice the rdicd is ot simplified sice the rdicd hs frctio. But, s we will see i lter sectio, we c rewrite rdicls i the deomitor. A epressio like. Third, there c e o is simplified, ut ot sice there is rdicl i the deomitor. To fi this prolem, s we will see lter, we c multiply top d ottom y : 1 71 s is. Lstly, the epoets i the rdicd d the ide hve 1 s the oly commo fctor. A prolem like simplified sice the oly commo fctor of d is 1. But, simplified sice d hve commo fctor of. I fct, 1/. Let s summrize these coditios. is is ot / Simplified Form of Rdicl A rdicl is simplified if ll of the followig coditios re met: 1) All fctors of the rdicd hve powers tht re less th the ide. ) The rdicd hs o frctios. ) There c e o rdicls i the deomitor. ) The epoets i the rdicd d the ide hve 1 s the oly commo fctor. Cocept # Simplifyig Rdicls Usig the Multiplictio Property of Rdicls.
3 The key to simplify rdicls is the seprte out the fctors tht re perfect powers from the fctors tht re ot perfect powers. We c use the multiplictio property to do this d the simplify the perfect powers. For istce, i the 0, 0 hs fctors of 8 d. The fctor of 8 is perfect cue, so Also, if the ide is less th the power of fctor of the rdicd, we c divide tht power y the ide to figure out how to split the fctors. The quotiet rised to the power of the ide will e the perfect power. The remider will e the power of the fctor tht is ot perfect power. For istce, i, to figure out how to split up, divide y : 8 1 The perfect power will e ( ) d the fctor tht is ot perfect power is 1. Thus, ( ) ( ). Let s try some dditiol emples: Simplify the followig: E. 7 y E. 8 E. c y z 1 E. d 8 6 Solutio: ) Sice 7 R 1 d 1 R 1, the 7 y ( ) (y) y (pply the multiplictio property) ( ) (y) y y y ) Sice 1 R 1 d 8 R d 7, the 8 ( ) ( ) 7 (pply the mult. property)
4 c) Sice 1, R 1, 1, d 16, the y z 1 (y ) (z ) (y ) (z ) y y 7 (pply the mult. prop.) y z y But, sice d y 0, the they c come out of the solute vlue: y z y y z y 6y z y d) Sice, 6, d 8 16, the 8 6 ( ) ( ) 6 ( ) ( ) (pply the mult. prop.). Cocept # Simplifyig Rdicls Usig the Divisio Property of Rdicls. We will proceed i the sme mer i simplifyig rdicls usig the divisio property. Simplify the followig; Assume the vriles represet positive rel umers: E. E. d r 11 E. Solutio: ) r 7 1 y 7 z 6 y z 6 r 11 E. e r r 6 (r ) r 1 (ut 6 ) E. c 7 7c d
5 7 ) c) d) c d (use the divisio property) (reduce) 7 6(c ) d 7 6 c d c d 7cd. 7 1 y 7 z 6 y z y 7 z 6 y z 6 7 6(c ) d (use the multiplictio property) (But c d d re > 0 from the directios) (reduce) (reduce) 1 1 y 7 ( ) (y ) (use the quotiet rule) y e) The idices re ot the sme so we cot use our properties. But, 1/ 1/ 1/ + 1/ 1/ 1/ 0/60 + 1/60 1/60 /60 60.
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