Name: Period: Date: 2.1 Rules of Exponents

Transcription

1 SM NOTES Ne: Period: Dte:.1 Rules of Epoets The followig properties re true for ll rel ubers d b d ll itegers d, provided tht o deoitors re 0 d tht 0 0 is ot cosidered. 1 s epoet: = e.g.) 7 = 7, π = π, ( 10) 1 = 10 0 s epoet: = 1, 7 = 1, = = 1 e.g.) ( ) 0 8 The Product Rule: = + e.g.) = = + 7 The Quotiet Rule: = e.g.) = = The Power Rule: ( ) ( )( ) 10 = e.g.) ( ) = = k = k = 16k Risig product to power: ( ) b = b e.g.) ( ) Risig quotiet to power: = b b e.g.) p p p = = q q ( q ) 6 Negtive epoets: 1 = e.g.) 1 7, 7 = = 1 = e.g.) 1 9 b bc =, = 9 c d d b b = = b e.g.) v v v = = = v 8 To siplif epressio cotiig powers es to rewrite the epressio without pretheses or egtive epoets.

2 Eples: Siplif the followig epressios. ) b) ( b)( b ) c) 7 r r 9 d) p p f) 6 7 g) ( ) h) i) 1 1 j) 6 k) l) 1 ) ( ) ) o) ( ) 7 p) ( ) 7 q) ( ) r) 1 s) ( ) t) z

3 Dte:. Siplifig Rdicls Squre Root: The squre root of uber is vlue tht, whe ultiplied b itself, gives the uber. Rdicl Sig: Rdicd: The uber uder the. Perfect squres: whole uber which is the squre of other whole uber List of coo perfect squres: Perfect cubes: whole uber which is the cube of other whole uber List of coo perfect cubes: Eples: Siplif ech of the followig: ) 11 b) 81 c) 9 d) 1 z Ide: is the ide Eples: Siplif ech epressio, if possible. ) 1 b) 81 c) 6 d) 8

4 Steps To Siplif Rdicl Epressio with Ide Usig Fctor Tree: 1. Use fctor tree to fid the prie fctoriztio of the uber iside the rdicl.. Deterie the ide of the rdicl. Look for groups equl i size to the ide of the se prie uber.. Move ech group of ubers or vribles fro iside the rdicl to outside the rdicl.. Siplif the epressios both iside d outside the rdicl b ultiplig. Eples: Siplif ech epressio. ) 1 b) 0 c) 7 d) 0 00 f)

5 g) 7 0 h) t 7 u 9 i) 0 6 j) k) 7 1 z l) pr p q r

6 Dte: The Product Rule for Rdicls: For rel ubers d b, b = b.. Multiplig, Addig, d Subtrctig Rdicls Cutio: The product rule does t work if ou re trig to ultipl the eve roots of egtive ubers, becuse those roots re ot rel ubers. For eple, Cutio: The product ol pplies whe the rdicls hve the se ide: Eples: Multipl. ) 7 b) 8 c) 7 1 d) ( ) 8 f) ( ) 11 g) 9 h) 10 6 Questio: C ou dd d subtrct rdicls the se w ou ultipl d divide the? e.g.) Sice b = b, does + b = + b? NO!!!!!!!!!! Do t ke the followig istkes: ( + ) + Like Rdicls: Rdicls with the se ide d the se rdicd. Eples: Deterie whether the followig re like rdicls. If the re ot, epli wh ot. ) d b) d c) d

7 Addig d Subtrctig Rdicls: 1. Siplif ech rdicl copletel.. Cobie like rdicls. Whe ou dd or subtrct rdicls, ou c ol cobie rdicls tht hve the se ide d the se rdicd. The rdicl itself (the root) does ot chge. You sipl dd or subtrct the coefficiets. Eples: ) 7 b) c) d) f) 16 + Multiplig Rdicl Epressios: Use the Product Propert. Use the Distributive Propert d FOIL to ultipl rdicl epressios with ore th oe ter. Eples: Multipl. ) ( + 0) b) ( 6 ) c) ( 6)( 7+ 1) d) ( )( + ) ( 1) f) ( + )( )

8 Dte:. Rtiol Epoets If is positive iteger greter th 1 d is rel uber the 1 =. «The deoitor of the epoet tells ou wht tpe of root to tke. Eples: Write equivlet epressio usig rdicl ottio d, if possible, siplif. ) 1 b) 1 6 c) ( z ) d) ( 6 ) 1 1 f) ( ) 1 Eples: Write equivlet epressio usig epoetil ottio. ) 7 b) b 7 c) z d) z z Positive Rtiol Epoets If d re positive itegers (where 1) d eists, the ( ) = =. e.g.) ( ) 8 = 8 = = or 8 = 8 = 6 = Eples: Write equivlet epressio usig rdicl ottio d siplif. ) t 6 b) 9 c) 6 d) ( ) Eples: Write equivlet epressio usig epoetil ottio. ) b) 7 9 c) ( ) 6 d) 6 ( )

9 Negtive Rtiol Epoets 1 For rtiol uber d, ozero rel uber, =. «The sig of the bse is ot ffected b the sig of the epoet. Eples: Write equivlet epressio usig positive epoets d, if possible, siplif. ) 1 9 b) ( ) c) 7 Lws of Epoets: The lws of epoets ppl to rtiol epoets s well s iteger epoets. Eples: Use the lws of epoets to siplif. 7 1 ) b) c) ( 19 ) d) 1 f) 7 67 z z g) 16 1 ( 1 h) ) To Siplif Rdicl Epressios: 1. Covert rdicl epressios to epoetil epressios.. Use rithetic d the lws of epoets to siplif.. Covert bck to rdicl ottio s eeded. Eples: Use rtiol epoets to siplif. Do ot use epoets tht re frctios i the fil swer. ) 8 9 z b) ( ) bc c) d) 6 9 k k 7 f) 8 6 g) h)