LAWS OF INDICES M.K. HOME TUITION. Mathematics Revision Guides Level: GCSE Higher Tier

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1 Mthetics Revisio Guides Lws of Idices Pge of 7 Author: Mrk Kudlowski M.K. HOME TUITION Mthetics Revisio Guides Level: GCSE Higher Tier LAWS OF INDICES Versio:. Dte: 0--0

2 Mthetics Revisio Guides Lws of Idices Pge of 7 Author: Mrk Kudlowski LAWS OF INDICES. Whe we ultiply uer y itself, we re sid to squre it, or rise it to the power of. Thus we write 6 6 = 6 = 6, i.e six squred Whe we ultiply uer y itself twice, we re cuig it, or risig it to the power of. Therefore we write =, or =, i.e cued. Higher powers lso exist, thus = = 6 ( to the fourth ). I the expressio, the is the se d the is the idex (plurl: idices). The sic lws of idices re s follows, pplicle to ll positive uers. Multiplictio d divisio: The Multiplictio Lw : Additio of idices correspods to ultiplictio of ctul uers. Exples ():, or 8 =. This lw lso holds for frctios, s do the divisio lw d the powers of powers lw :, or The Divisio Lw : Sutrctio of idices correspods to divisio of ctul uers. Exple ():, or. Brckets ( Powers of powers ): Whe we ultiply idices, we tke "power of power". 6 Exple ():, or 9 = 79.

3 Mthetics Revisio Guides Lws of Idices Pge of 7 Author: Mrk Kudlowski Here re soe power tles: Powers of 0: Power of 0 6 Nuer Powers of : Power of Nuer Usig powers of is ore coveiet for illustrtio, ecuse the uers i questio do ot ecoe ugely lrge. Exples (): Show how you would use the Powers of tle to work out: i) 6 ; ii) 08 ; iii) 8 6 dd these powers get result Power of Nuer Redig upwrds, we see tht = d 6 = 6. Addig the idices gives, or 6 = sutrct this fro this 8 Power of Nuer get result Redig upwrds, we see tht 08 = d 8 = 7. Sutrctig idices gives, or 08 = 6. 8 doule this get result Power of Nuer Redig upwrds, we see tht =. Doulig the idex gives 0, or = 0. So fr, we hve restricted ourselves to rithetic ivolvig positive whole uer powers of uers, ut we c lso hve zero, frctiol d egtive powers s well. The ext sectio shows how to uderstd such expressios s -, 8 0 d 6 ½.

4 Mthetics Revisio Guides Lws of Idices Pge of 7 Author: Mrk Kudlowski Lookig ck t the Powers of tle, we c see how the etries i the tle re douled every tie the power of, ely the idex, is icresed y. Coversely, whe the idex is decresed y, the correspodig etry i the tle is hlved. Power of Nuer Negtive Idices - Reciprocls: Ay uer rised to egtive power is the reciprocl of the se uer rised to the correspodig positive power. Exples () :, or 8. Avoid the coo error: - is ot -8. or 6 Here we covert the expressio with egtive power to expressio with positive power y reversig the sides of the origil frctio. Zero idex: 0 or Ay positive uer rised to the zero power is equl to. This c e deostrted y the ultiplictio lw: 0 0, ut s + 0 is siply, 0. Addig zero to uer leves it uchged; so does ultiplyig y. Powers of : The uer rised to y power is just. Ay positive uer rised to the power is equl to the uer itself. Exples (6): ; 7 7

5 Mthetics Revisio Guides Lws of Idices Pge of 7 Author: Mrk Kudlowski This leves us with hvig to fid eig to expressios like 6 ½. We will show the process of squrig 8 to oti 6. 8 doule this get result Power of Nuer To squre uer, we doule the idex. The iverse of squrig is to tke squre roots, d to do so, we hlve the idex. 6 ½ hlve this Power of Nuer get result Hece Frctiol Idices - Roots: This lst result c e geerlised: Ay positive uer rised to the reciprocl of idex is equivlet to the th root of tht uer. Exples(7): 6. 6 Recll the powers of powers rule: (6 ) Whe we cue uer, we triple the idex; whe we tke cue roots, we divide the idex y.

6 Mthetics Revisio Guides Lws of Idices Pge 6 of 7 Author: Mrk Kudlowski Hrder Frctiol Idices. Frctiol idices of the for or re less esy to ge, ut they ll use the stdrd lws ove. is the se s rised to the power of ; Exples (8): 6 = ) 6 ( - ely or 6. We tke the cue root of 6, ely, d the squre to oti the fil swer, We tke the squre root, followed y cuig the iteredite result. is the se s the reciprocl of rised to the power of, s the rcket d reciprocl lws ove show. Exples (9): ; (We tke the reciprocl, followed y the squre root.) (Tke the reciprocl, the the fourth root, d filly tke the cue.) (Tke reciprocl, the cue root, d filly squre.)

7 Mthetics Revisio Guides Lws of Idices Pge 7 of 7 Author: Mrk Kudlowski More o rithetic usig idices. The idex lws c lso e used to siplify lgeric expressios.. Exples (0): Evlute the followig, givig swers i their siplest for : i) ; ii) ; iii) ( ) ; iv) 6 ; v) (7 ) - i ) = 6 (Note tht uers re ultiplied, ut idices dded! ) ii) = (Nuers divided, ut idices sutrcted) iii) ( ) = 6 6 (The uer hs ee cued, ut power of hs ee tripled!) iv) 6 = (Hlve the powers of d, ut tke squre root of ) v) (7 ) - = (Tke reciprocl of squre of 7) Still other questios ight hve ixture of ses: Exples (): Siplify i) prie uer. 9 ii) 6 8 d iii), levig the swer s power of The first step whe delig with this type of questio is to reduce ll ters i the expressio to the se se. For exple, is defiitely ot 8! I i) we use 9 = to rewrite the expressio s 7. I ii), we use = : 7. I iii), we use =, 8 = d 6 = :