We will begin this chapter with a quick refresher of what an exponent is.

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1 .1 Exoets We will egi this chter with quick refresher of wht exoet is. Recll: So, exoet is how we rereset reeted ultilictio. We wt to tke closer look t the exoet. We will egi with wht the roerties re for the exoets. I the followig tle, we will give the roerty, d the sile exle to illustrte the roerty. Proerties of Exoets Proerty Exle 1. ; 5. ; 6 x x x. ; 6 1. ; 5. ; ; 1 ; ; ; 1 1,000,000 0 My of these roerties re cler if you sily write out the exoets d the roerties will ecoe cler. We use these roerties to silify exressios which coti exoets. Wht tht es is, we wt exressio cotiig exoets to hve o vriles or uer of the se se, o egtive exoets d o exoets which c e worked out. Bsiclly, if soethig c e doe. It ust e doe. Oe other thig to lwys reeer whe workig with these roerties of exoets, the roerties re fro exoets to other exoets. So the oertios we do, eed to e doe to the exoets of ech uer or vrile. Let s strt with soe sile exles. -ties se exoet

2 Exle 1: Silify... c. d. Solutio:. I this exle, we do t relly hve y thigs we c do. It looks like ll we c do is use roerty to rig the exoet fro the outside of the rethesis to the iside. Also, reeer tht eve though there is o exoet of the y, everythig hs ivisile exoet of 1. Puttig this together we get. This tie, we hve to otice tht we hve two exressios ultilied together. So, we eed to use roerty 1 to get together the vriles of the se se. Be creful here, eve though we re ultilyig the together, we re suosed to dd the exoets. This gives us c. Here, the oly thig tht eeds to e ddressed is tht we hve egtive exoet. We re ot llowed to hve egtive exoets. So we use roerty 6 to ke the exoet ositive. The ide ehid roerty 6 is tht wheever you hve egtive exoet, you sily eed to ove the vlue cross the frctio r to ke it ositive exoet. This ide works either wy. This es, if soethig o the deoitor hs egtive exoet, you ove it to the uertor. If soethig o the uertor hs egtive exoet, you ove it to the deoitor. I this cse, we eed to ove the x - to the deoitor (which is curretly 1) to get the sig to chge. We hve d. Lstly, we just eed to use roerty to silify. Proerty c cuse soe otetil issues. The ide is, you eed to sutrct the exoets d you hve to kee i id two thigs: its lwys to exoet - otto exoet d the swer lwys lds o to. If it hes to e tht you get egtive exoet fter tht, the you del with it t tht oit.

3 With this i id we hve Now ove it to the otto to get rid of the egtive exoet. Now let s look t few ore difficult exles ivolvig exoets. Exle : Silify... c. Solutio:. The first thig you hve to kee i id is tht there re severl differet wys you c work these roles. All of the differet directios you could go, will still ed u t the se sot, s log s you use the roerties correctly. I this cse, it sees esier to strt with riig the exoet through fro the outside, d the roceedig fro there. Reeer, whe usig roerty to rig the ower i, you eed to ultily it y ech ower tht is lredy o the iside of the rethesis. Multily the ower i Move the egtive exoet to the otto Silify = 7. This tie, it sees est to ove ll of our egtive exoets roud to ke the ositive efore we eve thik out doig ythig else. Reeer, we oly wt to ove the ites tht hve egtive exoets. If soethig does ot hve egtive exoet, tht is, the exoet is lredy ositive, we eed to leve it where it is. Oce we hve doe tht, the we c decide wht to do ext.

4 Silify d 1 d coie like ses y ddig exoets Reduce the d 8 c. Siilr to rt ove, let s strt y ullig through the exoet tht is o the outside y ultilyig it y ech exoet o the iside. Multily through the - Move the egtive exoet dow d silify Filly, we will tke look t soe very chllegig roles with exoets. Exle : Silify.. [ ]. c. Solutio:. I this role, we hve y differet wys we c strt. We c try to get rid of ll the egtive exoets, we c ull though the exoets fro the outside, we c get se ses together, etc. However, i this cse, it is est to strt with ultilyig the d -1 exoets fro the outside together. It does t see like we re llowed to do this, ut it ctully is sile usge of roerty. The we cotiue s usul [ ] Multily d -1 Brig through the - Move the egtive exoets Coie the y ses y sutrctio d silify 8 d 1

5 . Here we will strt y ovig the lrger egtive exoets to ke the ositive the ove log like we usully do. It looks like Move lrge egtive exoets Brig through the owers fro the outside Move the egtive exoet d silify 6 d Reduce d coie se ses c. Lstly, we will work this exle s we did ll of the others. Pull through the - o otto Move egtive exoets Reduce d coie se ses.1 Exercises Silify ( ) 8. 9.

6 [ ] 8. [ ] 9. * + 0. *

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