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

Transcription

1 128 Sect Simplifyig Rdicl Expressios Cocept #1 Multiplictio d Divisio Properties of Rdicls We c use our properties of expoets to estlish two properties of rdicls: () 1/ 1/ 1/ & ( Multiplictio d Divisio Properties of Rdicls Let d e rel umers such tht 0. The d 1) Multiplictio Property of Rdicls: 2) Divisio Property of Rdicls: ) 1/ 1/ 1/ re rel umers d We c use these properties forwrds d ckwrds to simplify rdicls. Simplify the followig: Ex. 1 Ex. 1 Ex. 1c Ex. 1d Solutio: ) ) c) 5 d) 10 I workig with rdicl expressios where the rdicd is ot perfect power of the idex, 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 idex. For exmple, is simplified sice d the power of 2 d the power of 3 re 1 which is less th the idex. But, is ot simplified sice d the power of 2 is equl to the 129 idex. I fct, 2. Secod, the rdicd hs o frctios. For, istce, is simplified sice the rdicd hs o frctios, wheres is ot simplified sice the rdicd hs frctio. But, s we will see i lter sectio, we c rewrite s. Third, there c e o rdicls i the deomitor. A expressio like is simplified, ut is ot sice there is rdicl i the deomitor. To fix this prolem, s we will see lter, we c multiply top d ottom y : 1. Lstly, the expoets i the rdicd d the idex hve 1 s the oly commo fctor. A prolem like is simplified sice the oly commo fctor of 2 d 9 is 1. But, is ot simplified sice 3 d 9 hve commo fctor of 3. I fct, x 3/9 x 1/3. Let s summrize these coditios. 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 idex. 2) The rdicd hs o frctios. 3) There c e o rdicls i the deomitor. 4) The expoets i the rdicd d the idex hve 1 s the oly commo fctor. Cocept #2 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 130, 40 hs fctors of 8 d 5. The fctor of 8 is perfect cue, so 2. Also, if the idex is less th the power of fctor of the rdicd, we c divide tht power y the idex to figure out how to split the fctors. The quotiet rised to the power of the idex 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 x 9, divide 9 y 2: The perfect power will e (x 4 ) 2 d the fctor tht is ot perfect power is x 1. Thus, x 4. Let s try some dditiol exmples: Simplify the followig: Ex. 2 Ex. 2 Ex. 2c 3 Ex. 2d Solutio: ) Sice R 1 d R 1, the (pply the multiplictio property) (simplify) 5x 3 y ) Sice R 1 d R 2 d , the (pply the mult. property) (simplify) 3 2

4 131 c) Sice 4 4 1, R 1, , d , the 3 3 (pply the mult. prop.) 3 (simplify) 3 2xy 2 z 3 But, sice 2 d y 2 0, the they c come out of the solute vlue: 3 2xy 2 z 3 3 2y 2 xz 3 6y 2 xz 3. d) Sice 9 3 3, 6 3 2, d , the (pply the mult. prop.) (simplify) 2x 3 2 Cocept #3 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: Ex. 3 Ex. 3 Ex. 3c Ex. 3d Ex. 3e Solutio: ) (simplify) (ut 6 3 2) (simplify) r 2

5 132 ) (use the divisio property) (reduce). c) (use the multiplictio property) (simplify) (But c d d re > 0 from the directios) (reduce). d) (use the quotiet rule) (reduce) (simplify) e) The idices re ot the sme so we cot use our properties. But, 1/3 + 1/4 1/5 x x 20/ /60 12/60 x 23/60.