Numbers and indices. 1.1 Fractions. GCSE C Example 1. Handy hint. Key point

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1 GCSE C Emple 7 Work out 9 Give your nswer in its simplest form Numers n inies Reiprote mens invert or turn upsie own The reiprol of is 9 9 Mke sure you only invert the frtion you re iviing y 7 You multiply y the reiprol of Cnel the ommon ftor : Bring the numertors together: Bring the enomintors together: So 7 Common ftors reue the size of the numers you re working with Without nelltion, this lultion is The rules for frtion rithmeti re: Frtions Wht you shoul lrey know: how to, sutrt, multiply n ivie y frtions how to onvert etween mie numers n top-hevy frtions In this setion you will lern: how to sustitute frtion into formul You must mke the enomintors equl efore you n or sutrt frtions In AS Mths, you will e epete to work onfiently with frtions They re often neee when simplifying, or fining the vlue of, n epression You will generlly fin it esier to work with top-hevy frtions rther thn mie numers For emple, it is not esy to work out the vlue of without lultor However, y writing s, you n esily work out this nswer () Moving on to AS Level Revisiting GCSE In GCSE Mths you performe lultions with frtions The numers involve my hve een top-hevy (or improper) frtions, suh s 7 or mie numers suh s You shoul e onfient in onverting mie numers into top-hevy frtions AS Level Emple Using the formul y, fin the vlue of y when You sustitute into the formul For emple, Use ommon enomintor of : 0 A the numertors: So NUMBERS AND INDICES Write s the top-hevy frtion Formul: y Cnel the ommon ftor : Simplify the prout: A the numertors: So y when Common error oes not equl To voi this type of error, write s NUMBERS AND INDICES

2 AS Level Emple Using the formul y, fin the vlue of y when You n write s, or simply s So the formul is y You sustitute into this formul Formul: y The oeffiient of is You n seprte the oeffiient from the -term You nee to know the rules of surs: The rules of surs re:, (where is not zero) These rules llow you to rek own sur For emple, It is not possile to rek own ny further using integers hs een epresse in simplifie sur form or et form 0 9 Sutrt the frtions y using ommon enomintor : Sutrt the frtions: So y when The epression n e written s, where n re ny numers,? 0 Tking it further You will nee to e onfient in using frtions s you progress through your ourse In prtiulr, you use them when working with stright-line equtions Surs Wht you shoul lrey know: how to re-write sur suh s how to rtionlise the enomintor of frtion suh s 0 In this setion you will lern: how to use the rules of surs to simplify more omple epressions how to rtionlise more omple enomintors in frtions how to lulte with numers n epressions involving surs the mening of et form Revisiting GCSE A sur is numer of the form where is not squre numer For emple, is sur A eiml pproimtion for is 7 ut you n never etly esrie sur using eimls However, 9 is not sur euse you n write own its et vlue, You shoul e fmilir with the squre numers:,, 9, 6,, 6 You n ssume 9 mens the positive squre root of 9 GCSE A Emple Epress in the form k where k is n integer Simplify y rtionlising the enomintor You nee to re-write using ftors of Re-write using ftors of hs lots of ftors Look for the gretest squre numer whih is ftor of (ie 6) 6 6 So, 6 (so k 6) The enomintor of is the sur You nee to fin frtion with the sme vlue s whih oes not hve sur in its enomintor Sine, you multiply the numertor n enomintor of this frtion y 7 Common error is NOT the sme s hek this on lultor Use squre numer whih is ftor of (ie ) Before nswering sur question, mke list of the first few squre numers (ie,, 9, 6, ) You n like surs You must multiply top n ottom y for ny numer > 0 Et form is the sme s giving n nswer in sur form 6 NUMBERS AND INDICES NUMBERS AND INDICES 7

3 Moving on to AS Level In AS Mths you will nee to e le to simplify epressions involving surs Also, you sometimes nee to give nswers in et form, rther thn using eimls AS Level Emple Show tht ( 7)( 7) is n integer, stting its vlue 7 Hene epress 7 in the form 7 where n re integers to e stte You nee to epn the rkets n then simplify the numers ( 7)( 7) , whih is n integer You nee to rtionlise the enomintor of the frtion 7 7 This n e one y multiplying the numertor n enomintor y ( 7) 7 ( 7 ) ( 7 ) So, 7 ( 7 ) ( 7 ) Comine numers n like surs: 7 7 Compre this nswer to the form 7 given in the question: 7 looks like 7 where, These re some efinitions for negtive n frtionl inies: for ny numer > 0 for ny numer 0 for ny numer, n for ny non-zero numer Prt oes not tell you how to strt Write 7 s 7 Use Prt is Hene question this mens you shoul mke use of prt You shoul use the result of prt to simplify the enomintor You must write own the vlue of n the vlue of, s irete y the question is the ue root of This is the numer suh tht = If n re integers, then ( )( ) is lso n integer, n equls N To rtionlise the enomintor of frtion of the form, multiply top n ottom y ( ) N To rtionlise the enomintor of frtion of the form, multiply top n ottom y ( ) Tking it further Surs pper in mny res of AS Mths, suh s solving qurti equtions n lulting istnes etween points Inies Wht you shoul lrey know: how to simplify epressions suh s ( ) how to lulte vlues suh s or In this setion you will lern: how to work out the vlue of more omplite epressions using the rules of inies Revisiting GCSE In GCSE Mths you woul hve use vrious rules of inies to simplify n epression For emple, in the epression you n the inies together so tht simplifies to This works euse ( ) ( ) Here re some results involving inies: m n m+n n m n for not zero ( m ) n ( n ) m mn n n () n n n n for not zero GCSE B Emple 6 Simplify (p q ) You n write rule s () n n n So (p q ) (p ) (q ) Use rule on eh rket: (p ) (q ) Simplify the inies: p 6 q So (p q ) simplifies to p 6 q You hve lso met the efinition of negtive n frtionl inies For emple, mens n so hs vlue Similrly, mens the positive squre root of, n so hs vlue Rule mens ( m ) n n ( n ) m re equl to eh other n re lso equl to mn Rule n e etene to three or more terms NUMBERS AND INDICES NUMBERS AND INDICES 9

4 GCSE A Emple 7 Fin the vlue of You pply the efinitions to eh epression 9 So 9 You must split the frtion up into two prts: Seprte the numers from the -terms: Use rules of inies: So Compre with n :,, n Common error is not Compring terms, = so = 9 Moving on to AS Level Here re some generl efinitions whih you nee to know for AS Mths 6 n n where n is ny integer 7 n n where n is ny positive integer In AS Mths you will e epete to evlute epressions involving inies, possily without the use of lultor m n ( n ) m n m When lulting m n it is usully esier to use the result ( n ) m rther thn n m Tking it further Inies pper in mny AS Mths topis, ut in prtiulr you will nee to e onfient in using them when stuying ifferentition n integrtion AS Level Emple Fin the vlue of 6 Simplify Epress in the form n, stting the vlue of the onstnts, n n Using rule 6 you n write 6 s 6 Now onentrte on the enomintor You n write 6 s 6 y using rule Now use rule 7: 6 ( 6) So 6 6 () 6 You n write s whih simplifies to 6 Now epress the enomintor s power of You n see tht the numers n epressions re more omple t this level You nee to omine severl rules of inies to nswer these questions Del with the negtive sign on the ine first Using rule, you oul lso work out 6 s (6 ), ut this woul men hving to fin 6 Although =, you shoul voi writing n ine using mie numer So 6 6 Use rule : 6 6 So simplifies to 6 0 NUMBERS AND INDICES NUMBERS AND INDICES

5 Prtie: Frtions Numers n inies Work out these Give eh nswer in its simplest form 6 S D Work out these Give eh nswer in its simplest form 0 Convert these mie numers into top-hevy frtions 7 S D Simplify these Give eh nswer s tophevy frtion in its simplest form 7 ( ) ( ) ( ) ( 6) ( 9) By onverting mie numers to top-hevy frtions, fin the vlue of these Give eh nswer s frtion in its simplest form 6 ( 9 ) 6 6 Use these formule to fin the vlue of y for the given vlue of Give eh nswer for y s frtion in its simplest terms y, y, y, y, e y, f y 6, 7 A urve hs eqution y Fin the y-oorinte of the point on this urve where The formul for onverting the temperture C egrees Celsius into the temperture F egrees Fhrenheit is F 9C Wht is C in egrees Fhrenheit? The minimum overnight temperture in villge ws 7 C Wht ws this temperture in egrees Fhrenheit? 9 The formul for onverting the istne K kilometres into the istne M miles is M K How fr in miles is the istne 9 km? How fr in km is the istne miles? PRACTICE: NUMBERS AND INDICES 7

6 0 Solve these equtions Give eh nswer s frtion in its simplest form 7 A smll retngulr gren lwn hs length metres n with w metres The lwn hs re m w --- Show tht w Fin the perimeter of the lwn in metres Give your nswer s mie numer Surs Simplify these 0 ( ) e f 6 7 Epress these in simplifie sur form 0 7 By simplifying eh sur, fin the vlue 0 of 7 Simplify these epressions 6 e Epress these frtions in the form, where n re integers 6 Simplify these frtions Show tht 6 is n integer, stting its vlue Given tht D fin the vlue of D when i,, ii,, Give eh nswer in simplifie sur form Eplin why D oes not hve rel vlue when n 0 9 Epress these in simplifie sur form ABC is right-ngle tringle AB, AC C A + Fin the re of this tringle Show tht BC 6 Hene fin the et perimeter of this tringle Give your nswer in the form, where n re integers to e stte B Fin the gretest of these numers You my use lultor if you wish,, Show tht these three numers re sies of right-ngle tringle Fin the re of this tringle, giving your nswer in the form, where n re integers to e stte PQR is right-ngle tringle PQ, PR P + + Epn n then simplify Show tht QR R Q Using your nswer to prt, fin the et length QR Show tht the re n perimeter of this tringle re numerilly equl Inies Epress eh of these in the form n where n is n integer ( ) ( ) Epress these s frtions in their simplest terms Evlute these Evlute these Where pproprite, give nswers s frtions in their simplest form 7 e 6 f g h 6 By writing 6 s power of, or otherwise, solve the eqution 6 6 Solve these equtions Epress these terms in the form n where is rel numer e f g 6 h 0 Determine whether eh of these sttements is true or flse Use rules of inies to prove those whih you think re true For those sttements tht you think re flse, give n emple to show tht it is inorret n n n n n n mn m n n n e ( n ) n for ll numers n positive integers n for ll numers n n positive integers n for ll numers n positive integers m n n for ll non-zero numers n positive integers n for ll numers n positive integers n f 76 PRACTICE: NUMBERS AND INDICES PRACTICE: NUMBERS AND INDICES 77

7 9 Evlute these epression using the given vlue of Give nswers s top-hevy frtions where pproprite when when 9 6 when 6 when 6 e when f when 9 0 Epress in the form n, where, n n re onstnts Epress in the form, where, n re onstnts Epress these s sums of powers of ( )( ) ( ) 6 ( ) A urve C hs eqution y ( )( ) where 0 Epress y in the form, where, n re onstnts Eplin why, s inreses, the vlue of y pprohes 6 Is there point on this urve with y-oorinte 6? 7 PRACTICE: NUMBERS AND INDICES