EXPONENTS AND RADICALS


 Amy Ford
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1 Expoets d Rdicls MODULE  EXPONENTS AND RADICALS We hve lert bout ultiplictio of two or ore rel ubers i the erlier lesso. You c very esily write the followig, d Thik of the situtio whe is to be ultiplied ties. How difficult is it to write?... ties? This difficulty c be overcoe by the itroductio of expoetil ottio. I this lesso, we shll expli the eig of this ottio, stte d prove the lws of expoets d ler to pply these. We shll lso ler to express rel ubers s product of powers of prie ubers. I the ext prt of this lesso, we shll give eig to the uber /q s qth root of. We shll itroduce you to rdicls, idex, rdicd etc. Agi, we shll ler the lws of rdicls d fid the siplest for of rdicl. We shll ler the eig of the ter rtiolisig fctor d rtiolise the deoitors of give rdicls. OBJECTIVES After studyig this lesso, you will be ble to write repeted ultiplictio i expoetil ottio d vicevers; idetify the bse d expoet of uber writte i expoetil ottio; express turl uber s product of powers of prie ubers uiquely; stte the lws of expoets; expli the eig of 0, p d q ; siplify expressios ivolvig expoets, usig lws of expoets; Mthetics Secodry Course
2 MODULE  Expoets d Rdicls idetify rdicls fro give set of irrtiol ubers; idetify idex d rdicd of surd; stte the lws of rdicls (or surds); express give surd i siplest for; clssify siilr d osiilr surds; reduce surds of differet orders to those of the se order; perfor the four fudetl opertios o surds; rrge the give surds i scedig/descedig order of gitude; fid rtiolisig fctor of give surd; rtiolise the deoitor of give surd of the for d b x x where x d y re turl ubers d d b re itegers; siplify expressios ivolvig surds. EXPECTED BACKGROUND KNOWLEDGE Prie ubers Four fudetl opertios o ubers Rtiol ubers Order reltio i ubers. y,. EXPONENTIAL NOTATION Cosider the followig products: (i) I (i), is ultiplied twice d hece is writte s. I, is ultiplied three ties d so is writte s. I, is ultiplied five ties, so is writte s. is red s rised to the power or secod power of. Here, is clled bse d is clled expoet (or idex) Siilrly, is red s rised to the power or third power of. Here, is clled the bse d is clled expoet. Siilrly, is red s rised to the power or Fifth power of. Agi is bse d is the expoet (or idex). 0 Mthetics Secodry Course
3 Expoets d Rdicls Fro the bove, we sy tht The ottio for writig the product of uber by itself severl ties is clled the Expoetil Nottio or Expoetil For. Thus,... 0 ties 0 d ( ) ( )... 0 ties ( ) 0 I 0, is the bse d expoet is 0. I ( ) 0, bse is d expoet is 0. Siilrly, expoetil ottio c be used to write precisely the product of rtioil uber by itself uber of ties. MODULE  Thus, d... ties...0 ties 0 I geerl, if is rtiol uber, ultiplied by itself ties, it is writte s. Here gi, is clled the bse d is clled the expoet Let us tke soe exples to illustrte the bove discussio: Exple.: Evlute ech of the followig: () i Solutio: (i) ( ) ( ) Exple.: Write the followig i expoetil for: (i) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) () Solutio: (i) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Mthetics Secodry Course
4 MODULE  Expoets d Rdicls Exple.: Express ech of the followig i expoetil ottio d write the bse d expoet i ech cse. (i) 0 Solutio: (i) 0 Altertively 0 () () Bse, expoet Here, bse d expoet Here, bse d expoet Here, bse d expoet Exple.: Siplify the followig: Solutio: Siilrly Exple.: Write the reciprocl of ech of the followig d express the i expoetil for: (i) Mthetics Secodry Course
5 Expoets d Rdicls Solutio: (i) MODULE  Reciprocl of Reciprocl of ( ) Reciprocl of Fro the bove exple, we c sy tht if q p is y ozero rtiol uber d is y positive iteger, the the reciprocl of p q q is. p CHECK YOUR PROGRESS.. Write the followig i expoetil for: (i) ( ) ( ) ( ) ( )... 0 ties... 0 ties. Write the bse d expoet i ech of the followig: Mthetics Secodry Course
6 MODULE  Expoets d Rdicls (i) ( ) (). Evlute ech of the followig (i). Siplify the followig: (i). Fid the reciprocl of ech of the followig: (i) ( ). PRIME FACTORISATION Recll tht y coposite uber c be expressed s product of prie ubers. Let us tke the coposite ubers, 0 d. (i) 0 We c see tht y turl uber, other th, c be expressed s product of powers of prie ubers i uique er, prt fro the order of occurrece of fctors. Let us cosider soe exples Exple.: Express 00 i expoetil for. Solutio: Mthetics Secodry Course
7 Expoets d Rdicls 00 Exple.: Express i expoetil for. Solutio: 0 0 MODULE  CHECK YOUR PROGRESS.. Express ech of the followig s product of powers of pries, i.e, i expoetil for: (i). Express ech of the followig i expoetil for: (i) (iv) 0 (v). LAWS OF EXPONENTS Cosider the followig (i) ( ) ( ) ( ) ( ) ( ) [( ) ( )] [( ) ( ) ( ) ( )] [ ( ) ( ) ( ) ( ) ( ) ( )] ( ) ( ) Mthetics Secodry Course
8 Expoets d Rdicls MODULE  Mthetics Secodry Course (iv) ( ) ( ) Fro the bove exples, we observe tht Lw : If is y ozero rtiol uber d d re two positive itegers, the Exple.: Evlute. Solutio: Here, d. Exple.: Fid the vlue of Solutio: As before, 0 0 Now study the followig: (i) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( )
9 Expoets d Rdicls MODULE  Mthetics Secodry Course Fro the bove, we c see tht Lw : If is y ozero rtiol uber d d re positive itegers ( > ), the Exple.0: Fid the vlue of. Solutio: I Lw, < >, the ( ) Lw : Whe > Exple.: Fid the vlue of Solutio: Here, d. Let us cosider the followig: (i) ( )
10 Expoets d Rdicls MODULE  Mthetics Secodry Course 0 Fro the bove two cses, we c ifer the followig: Lw : If is y ozero rtiol uber d d re two positive itegers, the ( ) Let us cosider exple. Exple.: Fid the vlue of Solutio:.. Zero Expoet Recll tht, if >, if > Let us cosider the cse, whe 0 0 Thus, we hve other iportt lw of expoets,. Lw : If is y rtiol uber other th zero, the o. Exple.:Fid the vlue of (i) 0 0 Solutio: (i) Usig 0, we get 0
11 Expoets d Rdicls MODULE  Mthetics Secodry Course Agi usig 0, we get 0. CHECK YOUR PROGRESS.. Siplify d express the result i expoetil for: (i) () (). Siplify d express the result i expoetil for: (i) ( ) ( ). Siplify d express the result i expoetil for: (i) ( ) (iv) 0 (v) 0. Which of the followig stteets re true? (i) (iv) (v) 0 0 (vi) (vii) 0
12 MODULE  Expoets d Rdicls. NEGATIVE INTEGERS AS EXPONENTS i) We kow tht the reciprocl of is. We write it s d red it s rised to power. ii) The reciprocl of ( ) is power.. We write it s ( ) d red it s ( ) rised to the iii) The reciprocl of. We write it s d red it s rised to the power ( ). Fro the bove ll, we get If is y ozero rtiol uber d is y positive iteger, the the reciprocl of i. e. is writte s d is red s rised to the power ( ). Therefore, Let us cosider exple. Exple.: Rewrite ech of the followig with positive expoet: Solutio: (i) (i) ( ) Fro the bove exple, we get the followig result: If q p is y ozero rtiol uber d is y positive iteger, the p q q p q p. 0 Mthetics Secodry Course
13 Expoets d Rdicls MODULE  Mthetics Secodry Course. LAWS OF EXPONENTS FOR INTEGRAL EXPONENTS After givig eig to egtive itegers s expoets of ozero rtiol ubers, we c see tht lws of expoets hold good for egtive expoets lso. For exple. (iv) (i) Thus, fro the bove results, we fid tht lws to hold good for egtive expoets lso. For y ozero rtiol ubers d b d y itegers d,.. if > if >. ( ). ( b) b CHECK YOUR PROGRESS.. Express s rtiol uber of the for q p :
14 MODULE  Expoets d Rdicls. Express s power of rtiol uber with positive expoet: (i). Express s power of rtiol uber with egtive idex: (i) [ ] ( ). Siplify: (i). Which of the followig stteets re true? (i) ( ) b (b) (iv) b b (v) o. MEANING OF p/q You hve see tht for ll itegrl vlues of d, Wht is the ethod of defiig /q, if is positive rtiol uber d q is turl uber. Cosider the ultiplictio q q q q q q... q ties...q ties q q q Mthetics Secodry Course
15 Expoets d Rdicls MODULE  I other words, the qth power of q or i other words q is the qth root of d is writte s q. For exple, or is the fourth root of d is writte s, Let us ow defie rtiol powers of If is positive rel uber, p is iteger d q is turl uber, the p q q We c see tht p q p p q p q p q p p p p...q ties. q q q q q... q ties p p q q p p/q is the qth root of p Cosequetly, is the cube root of. Let us ow write the lws of expoets for rtiol expoets: (i) ( ) (iv) (b) b (v) b b Let us cosider soe exples to verify the bove lws: Exple.: Fid the vlue of (i) ( ) ( ) Mthetics Secodry Course /
16 Expoets d Rdicls MODULE  Mthetics Secodry Course Solutio: (i) ( ) ( ) ) ( ( ) ( ) ) ( CHECK YOUR PROGRESS.. Siplify ech of the followig: (i) ( ). Siplify ech of the followig: (i) ( ) ( ). SURDS We hve red i first lesso tht ubers of the type d, re ll irrtiol ubers. We shll ow study irrtiol ubers of prticulr type clled rdicls or surds.
17 Expoets d Rdicls MODULE  A surd is defied s positive irrtiol uber of the type x, where it is ot possible to fid exctly the th root of x, where x is positive rtiol uber. The uber x is surd if d oly if (i) it is irrtiol uber it is root of the positive rtiol uber.. Soe Teriology I the surd x, the sybol is clled rdicl sig. The idex is clled the order of the surd d x is clled the rdicd. Note: i) Whe order of the surd is ot etioed, it is tke s. For exple, order of ( ) is. ii) is ot surd s its vlue c be deteried s which is rtiol. iii), lthough irrtiol uber, is ot surd becuse it is the squre root of irrtiol uber.. PURE AND MIXED SURD i) A surd, with rtiol fctor is oly, other fctor beig rrtiol is clled pure surd. For exple, d 0 re pure surds. ii) A surd, hvig rtiol fctor other th logwith the irrtiol fctor, is clled ixed surd. For exple, d re ixed surds.. ORDER OF A SURD I the surd, is clled the coefficiet of the surd, is the order of the surd d is the rdicd. Let us cosider soe exples: Exple.: Stte which of the followig re surds? (i) (iv) Mthetics Secodry Course
18 MODULE  Expoets d Rdicls Solutio: (i), which is rtiol uber. is ot surd. is irrtiol uber. is surd., which is irrtiol is surd. (iv) is irrtiol. is surd, d (iv) re surds. Exple.: Fid idex d rdicd i ech of the followig: (i) (iv) Solutio: (i) idex is d rdicd is. idex is d rdicd is. idex is d rdicd is. (iv) idex is d rdicd is. Exple.: Idetify pure d ixed surds fro the followig: (i) Solutio: (i) is pure surd. is ixed surd. is ixed surd..0 LAWS OF RADICALS Give below re Lws of Rdicls: (without proof): (i) [ ] Mthetics Secodry Course
19 Expoets d Rdicls MODULE  b b b b where d b re positive rtiol ubers d is positive iteger. Let us tke soe exples to illustrte. Exple.: Which of the followig re surds d which re ot? Use lws of rdicls to scerti. (i) 0 0 (iv) Solutio: (i) which is rtiol uber. 0 is ot surd , which is irrtiol is surd. (iv) It is ot surd., which is irrtiol is surd. CHECK YOUR PROGRESS.. For ech of the followig, write idex d the rdicd: (i) Mthetics Secodry Course
20 MODULE  Expoets d Rdicls. Stte which of the followig re surds: (i) (iv) (v). Idetify pure d ixed surds out of the followig: (i) (iv). LAWS OF SURDS Recll tht the surds c be expressed s ubers with frctiol expoets. Therefore, lws of idices studied i this lesso before, re pplicble to the lso. Let us recll the here: (i) x. y xy or x.y ( xy) x y x y or x y x y x x x or x x x x x or x (iv) ( ) (v) p p p p ( ) ( ) x x x p p or x x x Here, x d y re positive rtiol ubers d, d p re positive itegers. Let us illustrte these lws by exples: (i) ( ) x () () Mthetics Secodry Course
21 Expoets d Rdicls (iv) ( ) MODULE  Thus, we see tht the bove lws of surds re verified. A iportt poit: The order of surd c be chged by ultiplyig the idex of the surd d idex of the rdicd by the se positive uber. For exple d. SIMILAR (OR LIKE) SURDS Two surds re sid to be siilr, if they c be reduced to the se irrtiol fctor, without cosidertio for coefficiet. For exple, d re siilr surds. Agi cosider d. Now d re expressed s d. Thus, they re siilr surds.. SIMPLEST (LOWEST) FORM OF A SURD A surd is sid to be i its siplest for, if it hs ) sllest possible idex of the sig b) o frctio uder rdicl sig c) o fctor of the for, where is positive iteger, uder the rdicl sig of idex. For exple, Let us tke soe exples. Exple.0: Express ech of the followig s pure surd i the siplest for: (i) Mthetics Secodry Course
22 MODULE  Expoets d Rdicls Solutio: (i), which is pure surd., which is pure surd., which is pure surd. Exple.: Express s ixed surd i the siplest for: (i) 0 0 Solutio: (i), which is ixed surd. 0, which is ixed surd. 0, which is ixed surd. CHECK YOUR PROGRESS.. Stte which of the followig re pirs of siilr surds: (i),, 0,. Express s pure surd: (i). Express s ixed surd i the siplest for: (i) 0. FOUR FUNDAMENTAL OPERATIONS ON SURDS.. Additio d Subtrctio of Surds As i rtiol ubers, surds re dded d subtrcted i the se wy. 0 Mthetics Secodry Course
23 Expoets d Rdicls MODULE  For exple, ( ) d [ ] For ddig d subtrctig surds, we first chge the to siilr surds d the perfor the opertios. For exple i) 0 ( ) ii) ( ) Exple.: Siplify ech of the followig: (i) Solutio: (i) 0 Exple.: Show tht 0 0 Solutio: 0 Mthetics Secodry Course
24 MODULE  Expoets d Rdicls [ ] 0 0 RHS Exple.: Siplify: 000 Solutio: Required expressio 0 ( 0 ) CHECK YOUR PROGRESS. Siplify ech of the followig: Mthetics Secodry Course
25 Expoets d Rdicls MODULE .. Multiplictio d Divisio i Surds Two surds c be ultiplied or divided if they re of the se order. We hve red tht the order of surd c be chged by ultiplyig or dividig the idex of the surd d idex of the rdicd by the se positive uber. Before ultiplyig or dividig, we chge the to the surds of the se order. Let us tke soe exples: [ d re of se order] Let us ultiply d 0 d Let us cosider exple: Exple.:(i) Multiply d 0. Divide by. Solutio: (i) Exple.: Siplify d express the result i siplest for: 0 Mthetics Secodry Course
26 MODULE  Expoets d Rdicls Solutio: 0 0 Give expressio 0 0. COMPARISON OF SURDS To copre two surds, we first chge the to surds of the se order d the copre their rdicds log with their coefficiets. Let us tke soe exples: Exple.: Which is greter or? Solutio: > > > Exple.: Arrge i scedig order:, d. Solutio: LCM of,, d is. Now < < < < Mthetics Secodry Course
27 Expoets d Rdicls MODULE  CHECK YOUR PROGRESS.. Multipliy d.. Multipliy d.. Divide by.. Divide by 0.. Which is greter or?. Which i sller: 0 or?. Arrge i scedig order:,,. Arrge i descedig order:,,. RATIONALISATION OF SURDS Cosider the products: (i) I ech of the bove three ultiplictios, we see tht o ultiplyig two surds, we get the result s rtiol uber. I such cses, ech surd is clled the rtiolisig fctor of the other surd. (i) is rtiolisig fctor of d vicevers. is rtiolisig fctor of d vicevers. is rtiolisig fctor of d vicevers. Mthetics Secodry Course
28 MODULE  Expoets d Rdicls I other words, the process of covertig surds to rtiol ubers is clled rtiolistio d two ubers which o ultiplictio give the rtiol uber is clled the rtiolistio fctor of the other. For exple, the rtiolisig fctor of x is x, of is. Note: (i) The qutities x y d x y re clled cojugte surds. Their su d product re lwys rtiol. Rtiolistio is usully doe of the deoitor of expressio ivolvig irrtiol surds. Let us cosider soe exples. Exple.: Fid the rtiolisig fctors of d. Solutio: Rtiolisig fctor is.. Rtiolisig fctor is. Exple.0: Rtiolise the deoitor of Solutio: ( )( ) ( )( ). ( ) 0 0 Exple.: Rtiolise the deoitor of Solutio: ( )( ) ( )( ). Mthetics Secodry Course
29 Expoets d Rdicls MODULE  Mthetics Secodry Course Exple.: Rtiolise the deoitor of. Solutio: ( ) ( ) [ ]( ) [ ] ( ) Exple.: If, b fid the vlues of d b. Solutio: b b, CHECK YOUR PROGRESS.0. Fid the rtiolisig fctor of ech of the followig: (i) xy y x. Siplify by rtiolisig the deoitor of ech of the followig: (i) (iv)
30 MODULE  Expoets d Rdicls. Siplify:. Rtiolise the deoitor of. If. Fid.. If x y, fid x d y. LET US SUM UP... ties is the expoetil for, where is the bse d is the expoet. Lws of expoet re: (i) (b) b (iv) b b (vi) o (vii) (v) ( ) p q q p A irrtiol uber x is clled surd, if x is rtiol uber d th root of x is ot rtiol uber. I x, is clled idex d x is clled rdicd. A surd with rtiol coefficiet (other th ) is clled ixed surd. The order of the surd is the uber tht idictes the root. The order of x is Lws of rdicls ( > 0, b > 0) b b b (i) [ ] b Mthetics Secodry Course
31 Expoets d Rdicls MODULE  Opertios o surds x y ( xy) ; x x x ; x y x y ( x ) x ; x x or ( x ) x x ( x ) Surds re siilr if they hve the se irrtiol fctor. Siilr surds c be dded d subtrcted. Orders of surds c be chged by ultiplyig idex of the surds d idex of the rdicd by the se positive uber. Surds of the se order re ultiplied d divided. To copre surds, we chge surds to surds of the se order. The they re copred by their rdicds logwith coefficiets. If the product of two surds is rtiol, ech is clled the rtiolisig fctor of the other. x y is clled rtiolisig fctor of x y d vicevers. TERMINAL EXERCISE. Express the followig i expoetil for: (i). Siplify the followig: (i). Siplify d express the result i expoetil for: (i) ( ) ( ) ( ) 0 Mthetics Secodry Course
32 MODULE  Expoets d Rdicls 0. Siplify ech of the followig: (i) o o o ( o o ) ( o o ). Siplify the followig: 0 (i) ( ) ( ) ( ) ( ). Fid x so tht. Fid x so tht x x. Express s product of pries d write the swers of ech of the followig i expoetil for: (i) The str sirus is bout. 0 k fro the erth. Assuig tht the light trvels t.0 0 k per secod, fid how log light fro sirus tkes to rech erth. 0. Stte which of the followig re surds: (i). Express s pure surd: (i) (iv). Express s ixed surd i siplest for: (i) 0 0. Which of the followig re pirs of siilr surds? (i),,, 0 0 Mthetics Secodry Course
33 Expoets d Rdicls.Siplify ech of the followig: (i) MODULE  0. Which is greter? (i) or or. Arrge i descedig order: (i),,,,. Arrge i scedig order:,, 0. Siplify by rtiolisig the deoitor: (i). Siplify ech of the followig by rtiolisig the deoitor: (i) 0. If b, fid the vlues of d b, where d b re rtiol ubers.. If x, fid the vlue of x. x. ANSWERS TO CHECK YOUR PROGRESS. (i) ( ) 0 0 Mthetics Secodry Course
34 MODULE  Expoets d Rdicls. Bse Expoet (i). (i) 0. (i). (i).. (i). (i) (iv) (v) ( ).. (i) (). (i) ( ). (i) (iv) (v). True: (i),, (vii) Flse:, (iv), (v), (vi) Mthetics Secodry Course
35 Expoets d Rdicls. MODULE .. (i). (i). (i). True:,, (iv) (i). (i).. (i),,,.,(iv). Pure: (i), (iv) Mixed:,.. (i),. (i). (i)..... Mthetics Secodry Course
36 MODULE  Expoets d Rdicls ,,.,,.0. (i) x y. (i).. [ ]. (iv). 0 ANSWERS TO TERMINAL EXERCISE. (i). (i) 0. (i). (i) zero zero. (i) () Mthetics Secodry Course
37 Expoets d Rdicls. x. x.. 0 secods 0.,, (iv) MODULE . (i) (i) 0. (i),. (i) zero. (i). (i),,,,., 0,. (i) ( ) ( ). (i) , b. Mthetics Secodry Course
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