Chpter Wht you will lern A Irrtionl numers n surs (0A) B Aing n sutrting surs (0A) C Multiplying n iviing surs (0A) D Binomil prouts (0A) E Rtionlising the enomintor (0A) F Review of inex lws (Consoliting) G Negtive inies H Sientifi nottion (Consoliting) I Rtionl inies (0A) J Exponentil growth n ey K Compoun interest L Compring interest using tehnology M Exponentil equtions (0A) Inies n surs Austrlin urriulum NUMBER AND ALGEBRA Ptterns n lger Simplify lgeri prouts n quotients using inex lws. Money n finnil mthemtis Connet the ompoun interest formul to repete pplitions of simple interest using pproprite igitl tehnologies. Rel numers (0A) Define rtionl n irrtionl numers n perform opertions with surs n frtionl inies. Liner n non-liner reltionships (0A) Solve simple exponentil equtions. Unorrete r smple pges Cmrige University Press Greenwoo et l., 04 ISBN 98--0-6890- Ph 0 86 400
Investment returns Does n verge investment return of % soun goo to you? Given this return is ompoune nnully, $0 000 investment woul grow to over $40 000 fter 0 yers. This is lulte y multiplying the investment totl y. (to return the originl mount plus the %) for eh yer. Using inies, the totl investment vlue fter n yers woul e given y Vlue = 0 000. n. This is n exmple of n exponentil reltion tht uses inies to onnet vriles. We n use suh rule to introue the set of speil numers lle surs. If, for exmple, you wnte to lulte the verge investment return tht elivers $00 000 fter 0 yers from $0 000 investment, then you woul nee to lulte x in the eqution 00 000 = 0 000 + This gives x = ( 0 ) x 0. 00 0 00.9, representing n nnul return of.9%. Inies n surs re ommonple in the worl of numers, espeilly where money is involve! Online resoures Chpter pre-test Vieos of ll worke exmples Intertive wigets Intertive wlkthroughs Downlole HOTsheets Aess to HOTmths Austrlin Curriulum ourses Unorrete r smple pges Cmrige University Press Greenwoo et l., 04 ISBN 98--0-6890- Ph 0 86 400
66 Chpter Inies n surs A Irrtionl numers n surs 0A Key ies You will rell tht when using Pythgors theorem to fin unknown lengths in right-ngle tringles, mny nswers expresse in ext form re surs. The length of the hypotenuse in this tringle, for exmple, is, whih is sur. A sur is numer tht uses root sign ( ), sometimes lle ril sign. They re irrtionl numers, mening tht they nnot e expresse s frtion in the form, where n re integers n π 0. Surs, together with other irrtionl numers suh s pi, n ll rtionl numers (frtions) mke up the entire set of rel numers, whih n e illustrte s point on numer line. (rtionl) Let s strt: Construting surs (rtionl) 0 (irrtionl) π (irrtionl) 4 Someone sks you: How o you onstrut line tht is 0 m long? Use these steps to nswer this question. First, rw line segment AB tht is m in length. Construt segment BC so tht BC = m n AB BC. You my wish to use set squre or pir of ompsses. Now onnet point A n point C n mesure the length of the segment. Use Pythgors theorem to hek the length of AC. Use this ie to onstrut line segments with the following lengths. You my nee more thn one tringle for prts to f. 0 e 6 f A m All rel numers n e lote s point on numer line. Rel numers inlue: rtionl numers (i.e. numers tht n e expresse s frtions) For exmple:, 4 9,,.6,., 0.9 The eiml representtion of rtionl numer is either terminting or reurring eiml. irrtionl numers (i.e. numers tht nnot e expresse s frtions) For exmple:,,, p, p The eiml representtion of n irrtionl numer is n infinite non-reurring eiml. C B m Unorrete r smple pges Cmrige University Press Greenwoo et l., 04 ISBN 98--0-6890- Ph 0 86 400
Numer n Alger 6 Surs re irrtionl numers tht use root sign ( ). For exmple:,, 00, + These numers re not surs: 4 (= ), (= ), 4 6 (= ). The nth root of numer x is written n x. If n x = y then y n = x. For exmple: = sine =. The following rules pply to surs. ( x) = x n x = x when x 0. xy = x y when x 0 n y 0. x x y = when x 0 n y > 0. y When ftor of numer is perfet squre we ll tht ftor squre ftor. Exmples of perfet squres re:, 4, 9, 6,, 6, 49, 64, 8, 00,... When simplifying surs, look for squre ftors of the numer uner the root sign n then use =. Exmple Defining n loting surs Express eh numer s eiml n eie if they re rtionl or irrtionl. Then lote ll the numers on the sme numer line. % SOLUTION =.0080... is irrtionl. % = 00 =. % is rtionl. = 0.48 is rtionl. EXPLANATION Use lultor to express s eiml. The eiml oes not terminte n there is no reurring pttern. % is frtion n n e expresse using terminting eiml.. 0 is n infinitely reurring eiml. Use the eiml equivlents to lote eh numer on the rel numer line. Key ies Unorrete r smple pges Cmrige University Press Greenwoo et l., 04 ISBN 98--0-6890- Ph 0 86 400
68 Chpter Inies n surs Exmple Simplifying surs Simplify the following. 40 00 6 SOLUTION = 6 = 6 = 4 00 = 00 = 00 = 0 = 0 40 = 4 0 6 6 = 4 0 6 = 0 0 6 = 0 9 = 9 = = 9 EXPLANATION 9 When simplifying, hoose the highest squre ftor of (i.e. 6 rther thn 4) s there is less work to o to rrive t the sme nswer. Compre with = 4 8 = 8 = 4 = = 4 Selet the pproprite ftors of 00 y fining its highest squre ftor: 00. Use x y = x y n simplify. Selet the pproprite ftors of 40. The highest squre ftor is 4. Cnel n simplify. x x Use y =. y Exmple Expressing s single squre root of positive integer Express these surs s squre root of positive integer. SOLUTION Then selet the ftors of tht inlue squre numer n simplify. EXPLANATION = 4 = 0 = 49 = 98 Write s 4 n then omine the two surs using x y = xy. Write s 49 n omine. Unorrete r smple pges Cmrige University Press Greenwoo et l., 04 ISBN 98--0-6890- Ph 0 86 400
Numer n Alger 69 Exerise A, (½) (½) Exmple Exmple Exmple, Choose the orret wor(s) from the wors given in re to mke the sentene true. A numer tht nnot e expresse s frtion is rtionl/irrtionl numer. A sur is n irrtionl numer tht uses root/squre symol. The eiml representtion of sur is terminting/reurring/non-reurring eiml. is sur/rtionl numer. Write own the highest squre ftor of these numers. For exmple, the highest squre ftor of 4 is 9. 0 8 4 e 48 f 96 g h 08 Express eh numer s eiml n eie if it is rtionl or irrtionl. Then lote ll the numers on the sme numer line. 8% 4% e f g 4 Deie if these numers re surs. 44 9 e f g h + 4 6 Simplify the following surs. 4 4 48 e f 00 g 98 h 90 i 8 j 60 k 6 l 80 Simplify the following. 8 0 4 48 6 e 98 4 4 8 f g h i 4 4 80 99 j k l 4 0 8 m 44 n 00 o 8(½) 98 8(½) h p p 68 8(½) UNDERSTANDING FLUENCY q 6 0 r 4 0 s 08 8 t 4 4 Unorrete r smple pges Cmrige University Press Greenwoo et l., 04 ISBN 98--0-6890- Ph 0 86 400
0 Chpter Inies n surs A Exmple Exmple 8 Simplify the following. 8 e i 9 0 9 f j 49 44 4 g k 8 6 4 h l 8 0 6 6 Express these surs s squre root of positive integer. 4 e f 6 g 8 h 0 i 9 0 j k l 9 Simplify y serhing for the highest squre ftor. 6 8 80 88 0 Determine the ext sie length, in simplest form, of squre with the given re. m 0 m 40 mm Determine the ext rius n imeter of irle, in simplest form, with the given re. 4 p m 4 p m 8 p m Use Pythgors theorem to fin the unknown length in these tringles, in simplest form. 6 m mm m m mm 4 m m m e 0 mm 9, 0 9, 0, (½) mm f 4 m 0 m, (½) Riky uses the following working to simplify. Show how Riky oul hve simplifie using fewer steps. = 9 8 = 8 = 4 = = 6, 4 4, FLUENCY PROBLEM-SOLVING REASONING Unorrete r smple pges Cmrige University Press Greenwoo et l., 04 ISBN 98--0-6890- Ph 0 86 400
Numer n Alger 4 List ll the ftors of 40 tht re perfet squres. Now simplify 40 using the highest of these ftors. Use Pythgors theorem to onstrut line segment with the given lengths. You n use only ruler n set squre or ompsses. Do not use lultor. 0 m 9 m 6 m m Proving tht is irrtionl 6 We will prove tht is irrtionl y the metho lle proof y ontrition. Your jo is to follow n unerstn the proof, then opy it out n try explining it to frien or teher. Before we strt, we first nee to show tht if perfet squre is even then is even. We o this y showing tht if is even then is even n if is o then is o. If is even then = k, where k is n integer. So = (k) = 4k = k, whih must e even. If is even then is even. If is o then = k +, where k is n integer. So = (k + ) = 4k + 4k + = (k + k) +, whih must e o. Now, to prove is irrtionl let s suppose tht is inste rtionl n n e written in the form in simplest form, where n re integers ( π 0) n t lest one of or is o. = So = (squring oth sies) = is even n, from prt ove, must e even. If is even then = k, where k is n integer. If = Then (k) = 4k = k = is even n therefore is even. This is ontrition euse t lest one of or must e o. (Rell tht in simplest 6 REASONING ENRICHMENT A form will hve t lest one of or eing o.) Therefore, the ssumption tht n e written in the form must e inorret n so is irrtionl. Unorrete r smple pges Cmrige University Press Greenwoo et l., 04 ISBN 98--0-6890- Ph 0 86 400
Chpter Inies n surs B Aing n sutrting surs 0A Key ies We n pply our knowlege of like terms in lger to help simplify expressions involving the ition n sutrtion of surs. Rell tht x n x re like terms, so x + x = 0x. The pronumerl x represents ny numer. When x = then + = 0, n when x = then + = 0. Multiples of the sme sur re lle like surs n n e ollete (i.e. ounte) in the sme wy s we ollet like terms in lger. Let s strt: Cn + 8 e simplifie? The term sur n e tre k to the gret Persin mthemtiin Al-Khwrizmi, who ws orn here in the nient ity of Khiv in the 9th entury. To nswer this question, first isuss these points. Are n 8 like surs? How n 8 e simplifie? Now eie whether + 8 n e simplifie. Disuss why nnot e simplifie. Like surs re multiples of the sme sur. For exmple:,, =, = 0 Like surs n e e n sutrte. Simplify ll surs efore ttempting to or sutrt them. Exmple 4 Aing n sutrting surs Simplify the following. + 4 4 6 + 6 + SOLUTION EXPLANATION + 4 = 6 Collet the like surs y ing the oeffiients: + 4 = 6. 4 6 + 6 + = 6 + Collet like surs involving 6: 4 6 6 = 6 = 6 Then ollet those terms with. Unorrete r smple pges Cmrige University Press Greenwoo et l., 04 ISBN 98--0-6890- Ph 0 86 400
Numer n Alger Exmple Simplifying surs to or sutrt Exmple 4 Simplify these surs. 8 0 + 6 4 SOLUTION 8 = 4 = = 0 + 6 4 = 4 + 6 9 Exerise B = 6 + 8 = 4 EXPLANATION First, look to simplify surs: 8 hs highest squre ftor of 4 n n e simplifie to. Then sutrt like surs. Simplify the surs n then ollet like surs. Note tht 4 = 4 = 6. (½),, 4 4 Deie if the following pirs of numers re like surs.,,, 4 6, 6 e, f, g, h, Rell your si skills in lger to simplify these expressions. x x 8y y x x e 4 + f 8x + x g 4t t + t Simplify the sur 48. Hene, simplify the following. i + 48 ii 48 iii 48 h y + y 0y 4 Simplify the sur 00. Hene, simplify the following. i 00 ii 00 + 4 iii 00 0 (½) (½) Simplify the following. + 4 8 e + 4 f 6 g 4 0 + 0 0 h 6 4 + i + j 8 k + 4 l 0 0 0 0 (½) UNDERSTANDING FLUENCY Unorrete r smple pges Cmrige University Press Greenwoo et l., 04 ISBN 98--0-6890- Ph 0 86 400
4 Chpter Inies n surs B Exmple 4 Exmple Exmple 6 Simplify the following. + + 6 + 4 6 + 4 + + e + + f + 6 6 g 4 0 + 0 h 4 + + 8 Simplify the following. 8 8 + + 0 e 4 8 f + g 44 + h 8 8 i 4 + 4 j 4 k + 98 l 800 4 00 Simplify the following. + 0 + 98 6 4 + 96 + 8 80 4 + 6 e 0 96 6 + f + 0 + 80 8 + + g 8 h 6 08 + i 49 + 88 44 8 j 00 + + 4 9 Simplify these surs tht involve frtions. Rememer to use the LCD (lowest ommon enomintor). + 4 + 6 4 e f + 6 4 g 9 0 8 0 h + 0 i 6 8 0 Fin the perimeter of these retngles n tringles, in simplest form. m m 8 m m 8 m m e m 8(½) 8(½), 0(½) 0 m f m 8 0(½) FLUENCY PROBLEM-SOLVING m m Unorrete r smple pges Cmrige University Press Greenwoo et l., 04 ISBN 98--0-6890- Ph 0 86 400
Numer n Alger, (½) Explin why n 0 n e thought of s like surs. Explin why n 8 n e thought of s like surs. Prove tht eh of the following simplifies to zero y showing ll steps. 08 + 6 + 4 6 6 + 8 8 + e 0 + 4 f 0 + 6 + 8 Prove tht the surs in these expressions nnot e e or sutrte. 8 4 8 + 0 0 4 40 + 00 + e 00 f 80 4 Simplifying oth surs n frtions 4 To simplify the following, you will nee to simplify surs n omine using ommon enomintor. 8 4 + 0 6 4 98 4 6 e 9 4 f 8 4 4 08 g h 4 + i 0 48 + 4 96 600 j k 80 l 6 4 (½), 4 REASONING ENRICHMENT B Unorrete r smple pges Cmrige University Press Greenwoo et l., 04 ISBN 98--0-6890- Ph 0 86 400
6 Chpter Inies n surs C Multiplying n iviing surs 0A Key ies When simplifying surs suh s 8, we write 8 = 9 = 9 =, where we use the ft tht xy = x y. This n e use in reverse to simplify the prout of two surs. A similr proess is use for ivision. Let s strt: Exploring prouts n quotients When ing n sutrting surs we n omine like surs only. Do you think this is true for multiplying n iviing surs? Use lultor to fin eiml pproximtion for n for. Use lultor to fin eiml pproximtion for 0 n for. Wht o you notie out the results from ove? Try other pirs of surs to see if your oservtions re onsistent. When multiplying surs, use the following result. x y = xy More generlly: x y = xy When iviing surs, use the following result. x x = More generlly: x y y y = x y Use the istriutive lw to expn rkets. ( + ) = + Exmple 6 Simplifying prout of two surs Simplify the following. SOLUTION = = 6 = = 6 4 = 6 9 = 6 9 = 8 EXPLANATION Use x y = xy. Use x y = xy. Then simplify the sur 4, whih hs highest squre ftor of 9, using 9 =. Alterntively, using = : = = = 8 Unorrete r smple pges Cmrige University Press Greenwoo et l., 04 ISBN 98--0-6890- Ph 0 86 400
Numer n Alger Exmple Simplifying surs using ivision Simplify these surs. 0 8 SOLUTION EXPLANATION 0 0 = = 8 = 8 = 4 6 Exmple 8 Using the istriutive lw Use x x y = y. Use x y = x y. Use the istriutive lw to expn the following n then simplify the surs where neessry. ( 6) 6( 0 4 6) SOLUTION ( 6) = 8 = 9 = 6( 0 4 6) = 6 60 6 Exerise C Copy n omplete. = - - - = 6 4 6 = EXPLANATION Expn the rkets = n 6 = 8. Then simplify 8. = = - - - - - - Expn the rkets n simplify the surs. Rell tht 6 6 = 6. (½) 4 4 = (½) - - - UNDERSTANDING 6 = 6 - - - = - - - = - - - = - - - Unorrete r smple pges Cmrige University Press Greenwoo et l., 04 ISBN 98--0-6890- Ph 0 86 400
8 Chpter Inies n surs C Exmple 6 Exmple Exmple 6 Exmple Exmple 8 Simplify the following. e ( ) f 6 g 6 ( ) ( h ) i 0 Simplify the following. 0 8 ( ) 0 ( 6) e f 0 g 40 8 h 6 i 0 0 4 Simplify the following. 0 e f 9 9 g 4 h i 8 j 0 k 8 l 0 6 Simplify the following. 4 4 6 0 0 e 6 ( 8) f g 4 h 4 6 i 0 ( ) j ( 4) k 4 8 l 9 4 Simplify the following. 6 4 8 e 6 9 4 6(½) f 4 0 8 6 0 8 4 Use the istriutive lw to expn the following n simplify the surs where neessry. ( + ) ( ) ( + ) ( + ) e ( ) f 4 ( 0) g ( 6 + 0) h 6( ) i ( + 4) j 6 ( 8) k 8( 0) l ( 6 + ) 4 (½) 4 (½) UNDERSTANDING FLUENCY Unorrete r smple pges Cmrige University Press Greenwoo et l., 04 ISBN 98--0-6890- Ph 0 86 400
Numer n Alger 9 8 Simplify the following. ( ) ( ) ( ) ( ) 8 e 8( 6 + ) f ( + ) 40 g 44 ( ) h 4 ( 4) i ( 6 ) 0 9 Determine the unknown sie of the following right-ngle tringles. 8 6 6 4 0 The perimeter of squre is m. Fin its re. Fin the length of igonl of squre tht hs n re of m. 9 8(½), 9 8(½), 9, 0 Use x y = xy to prove the following results. 6 6 = 6 8 8 = 8 ( ) = 8 oul e simplifie in two wys, s shown. Metho A 8 = 4 9 = = = 6 6, Metho B 8 = 8 = 6 = 6 6 = 6 6 Desrie the first step in metho A. Why is it useful to simplify surs efore multiplying, s in metho A? Multiply y first simplifying eh sur. i 8 ii 4 0 iii 0 4 iv 4 v 8 48 vi 08 ( ) 4 ( vii 8) viii 98 00 ix 80, PROBLEM-SOLVING REASONING C Pythgorens re ttriute to the isovery of irrtionl numers. Unorrete r smple pges Cmrige University Press Greenwoo et l., 04 ISBN 98--0-6890- Ph 0 86 400
80 Chpter Inies n surs C oul e simplifie in two wys. Metho A = = 4 = Metho B = = Choose metho to simplify these surs. Compre your metho with tht of nother stuent. 0 6 8 Higher powers e 4 4 Look t this exmple efore simplifying the following. ( ) = ( ) = = 8 = 4 ( ) ( ) ( ) ( ) 4 e ( ) 4 f ( ) g ( ) h ( ) i ( ) 4 ( ) j 4 ( ) ( ) 4 k l 4 4 ( ) ( ) ( ) m ( ) 4 n 4 9 ( ) ( ) ( ) o ( ) p 4 4 ( ) 4 ( ) ( ) q ( 8) r 0 9 ( ) f 4 REASONING ENRICHMENT Unorrete r smple pges Cmrige University Press Greenwoo et l., 04 ISBN 98--0-6890- Ph 0 86 400
Numer n Alger 8 D Binomil prouts 0A In the previous setion we use the istriutive lw to expn surs of the form ( + ). Now we will exten this to expn n simplify the prout of two inomil terms, inluing perfet squres n the ifferene of perfet squres. We expn inomil prouts in the sme wy tht we expn the prout of inomil expressions in lger, suh s (x + )(x ). Let s strt: Show the missing steps You re tol tht the following three equtions re ll true. Provie ll the missing steps to show how to otin the right-hn sie from eh given left-hn sie. ( )( + ) = ( + ) = + ( + )( ) = 0 Use the istriutive lw to expn inomil prouts n simplify y olleting terms where possile. ( + )(+ ) = + + + Expning squres ( + ) = ( + )( + ) = + + + = + + ( ) = ( )( ) = + = + Forming ifferene of squres ( + )( ) = + = Key ies The istriutive lw is use in most res of mthemtis. Unorrete r smple pges Cmrige University Press Greenwoo et l., 04 ISBN 98--0-6890- Ph 0 86 400
8 Chpter Inies n surs Exmple 9 Expning inomil prouts Expn n simplify. (4 + )( ) ( )( + 4) SOLUTION (4 + )( ) = 4 8 + ( )( + 4) = = 6 + 8 4 = 0 + 4 = 6 + Exmple 0 Expning perfet squres Expn n simplify. ( ) ( + ) SOLUTION ( ) = ( )( ) = 4 + = 4 Alterntively: ( ) = 4 + ( + ) = 4 = ( + )( + ) = 9 + 6 + 6 + = 8 + 0 6 + = 9 + 0 6 Alterntively: ( + ) = 9 + 6 + = 8 + 0 6 + = 9 + 0 6 EXPLANATION Use ( + )( + ) = + + + n note tht =. Simplify y olleting like surs. Use the istriutive lw n ollet like surs. Rell = EXPLANATION = 6 = 0 Note tht ( ) = ( )( ). Expn n simplify using the istriutive lw. Rell tht =. Use ( ) = +. Use the istriutive lw to expn n simplify. Use ( ) = +. ( ) = = 9 = 8 Unorrete r smple pges Cmrige University Press Greenwoo et l., 04 ISBN 98--0-6890- Ph 0 86 400
Numer n Alger 8 Exmple Expning to form ifferene of perfet squres Expn n simplify. ( + )( ) ( )( + ) SOLUTION ( + )( ) = 4 + = Alterntively: ( + )( ) = ( ) = 4 = ( )( + ) = 49 + 6 6 = 98 = 9 Alterntively: ( )( + ) = ( ) ( ) Exerise D = 49 = 9 EXPLANATION Expn using the istriutive lw n nel the two mile terms. Use ( + )( ) =. Use the istriutive lw n then nel the two mile terms. Use ( )( + ) =. (½) Simplify the following. ( ) e ( ) f ( ) g ( ) h ( ) i (9 ) Simplify the following. + 6 6 8 e 4 f 48 Use the istriutive lw ( + )( + ) = + + + to expn n simplify these lgeri expressions. (x + )(x + ) (x )(x + ) (x + 4)(x ) (x + )(x ) e (x + )(x ) f (6x + )(x 4) g (x + 4)(x 4) h (x )(x + ) i (x 6)(x + 6) j (x + ) k (x ) l (x ) (½) UNDERSTANDING Unorrete r smple pges Cmrige University Press Greenwoo et l., 04 ISBN 98--0-6890- Ph 0 86 400
84 Chpter Inies n surs D 4 (½) 4 (½) 4 (½) Exmple 9 Exmple 9 Exmple 0 Exmple Exmple 0 Exmple 4 6 8 9 Expn n simplify. ( + )( ) (4 + )( ) ( 6 + )( 6 ) ( )( + ) e ( + )(4 ) f ( )( + ) g ( )(4 ) h (4 0)( 0) i ( 4)( 4) Expn n simplify. ( )( + ) (4 + )( ) (6 )( + ) ( 6 + 4)( 6) e ( 0 )( 0 ) f ( )( 4 ) g (4 )( ) h ( )( 4 ) i (4 )( 4 ) Expn n simplify these perfet squres. ( ) ( 6) (4 + ) ( + ) e ( + ) f ( ) g ( + ) h ( ) i ( 0 ) j ( + 9) k ( + ) l ( 9) Expn n simplify these ifferenes of perfet squres. ( )( + ) ( 6)( + 6) (4 + )(4 ) ( )( + ) e ( 8 )( 8 + ) f ( 0 4)( 0 + 4) g ( + )( ) h ( + )( ) i ( )( + ) 8(½), 0(½) 8 0(½) Expn n simplify these perfet squres. ( + ) ( + 4 ) ( 6 + ) (8 4 ) e (6 ) f ( 6) g ( + 6) h ( 0 + ) i ( 8) Expn n simplify these ifferenes of perfet squres. ( )( + ) ( )( + ) (4 + )(4 ) ( )( + ) e (4 6)(4 + 6) f (4 0 6)(4 0 + 6) g ( 8 0 )( 8 + 0 ) h ( 4 6)( + 4 6) i ( 0 + 4 )( 0 4 ) j ( + 6)( 6) 8 0(½) FLUENCY PROBLEM-SOLVING Unorrete r smple pges Cmrige University Press Greenwoo et l., 04 ISBN 98--0-6890- Ph 0 86 400
Numer n Alger 8 0 Fin the re of these retngles n tringles, in expne n simplifie form. e + m mm m mm m Show tht the following simplify to n integer. ( + )( ) ( )( + ) ( )( + ) f m + 6 m + m 6 mm m, (½) Use your knowlege of the simplifition of surs to fully simplify the following. ( )( 4 ) ( 6 + )( 0 ) ( + )( + ) (4 + )( 4 ) e ( + 4)( 6 ) f ( )( 0 + ) Is it possile for ( + ) to simplify to n integer if t lest one of or is sur? If your nswer is yes, give n exmple. Expnsion hllenge 4 Fully expn n simplify these surs. ( ) + ( + ) ( ) + ( + ) ( 4 )( + 4 ) ( ) 0 ( ) e ( 6) + ( + ) f ( ) ( ) g ( )( + ) ( 6 ) h ( ) + ( 6 + ) (½), 4 PROBLEM-SOLVING REASONING ENRICHMENT D Unorrete r smple pges Cmrige University Press Greenwoo et l., 04 ISBN 98--0-6890- Ph 0 86 400
86 Chpter Inies n surs E Rtionlising the enomintor 0A Key ies As you know, it is esier to or sutrt frtions when the frtions re expresse with the sme enomintor. In similr wy, it is esier to work with surs suh s n when they re expresse using whole numer in the enomintor. The proess tht removes sur from the enomintor is lle rtionlising the enomintor euse the enomintor is eing onverte from n irrtionl numer to rtionl numer. Let s strt: Wht o I multiply y? When trying to rtionlise the enomintor in sur like, you must multiply the sur y hosen numer so tht the enomintor is onverte to whole numer. First, eie wht eh of the following is equivlent to. i ii iii Rell tht x x = x n simplify the following. i ii iii 4 Now, eie wht you n multiply y so tht: the vlue of oes not hnge, n the enomintor eomes whole numer. Repet this for: i ii Rtionlising enomintor involves multiplying y numer equivlent to, whih hnges the enomintor to whole numer. x = x y = x y y y y y Exmple Rtionlising the enomintor Rtionlise the enomintor in the following. SOLUTION 6 EXPLANATION = = Choose the pproprite frtion equivlent to to multiply y. In this se, hoose sine =. Unorrete r smple pges Cmrige University Press Greenwoo et l., 04 ISBN 98--0-6890- Ph 0 86 400
Numer n Alger 8 Exmple = = 0 6 = 6 = 0 = 4 0 = = = Exerise E Simplify. 6 6 8 e f Choose the pproprite frtion. In this se, use sine =. Rell = = 0. Choose the pproprite frtion; i.e.. = = 0 Simplify the sur n nel. Expn using the istriutive lw: ( ) = = (½) 4 4 g 4 h 9 Write the missing numer. = = 0 0 = = 0 e 4 = f = g = h = = i Use lultor to fin eiml pproximtion to eh numer in the following pirs of numers. Wht o you notie?,,, 4 Rtionlise the enomintors. 4 (½) 4 (½) 4 4 (½) UNDERSTANDING FLUENCY e f 8 g h Unorrete r smple pges Cmrige University Press Greenwoo et l., 04 ISBN 98--0-6890- Ph 0 86 400
88 Chpter Inies n surs E Exmple Exmple Exmple Rewrite eh of the following in the form n then rtionlise the enomintors. 6 6 0 e f g h 6 8 Rtionlise the enomintors. 4 6 Rtionlise the enomintors. e 4 Rtionlise the enomintors. e i + + 6 0 f f j e 0 + 0 4 6 g g k 9 6 + 6 + 0 f h h l 4 0 90 0 + 0 0 + 9 Determine the ext vlue of the re of the following shpes. Express your nswers using rtionl enomintor. m mm m m mm m 6 9 8(½), 9 mm 8(½), 9, 0(½) 0 Simplify the following y first rtionlising enomintors n then using ommon enomintor. + + FLUENCY PROBLEM-SOLVING g e h + 4 0 6 + 4 f i + 4 6 Unorrete r smple pges Cmrige University Press Greenwoo et l., 04 ISBN 98--0-6890- Ph 0 86 400
Numer n Alger 89 x Explin why multiplying numer y oes not hnge its vlue. x Rtionlise the enomintors n simplify the following. + 6 + e g 4 + 0 h + 6 f i + 6 + 4 To explore how to simplify numer suh s 4, first nswer these questions. Simplify: i (4 )(4 + ) ii ( )( + ) iii ( )( + ) Wht o you notie out eh question n nswer in prt ove? Now eie wht to multiply 4 y to rtionlise the enomintor. Rtionlise the enomintor in these expressions. 4 i ii iii Binomil enomintors, (½) 4 iv 6 6 (½), 4 Rtionlise the enomintors in the following y forming ifferene of two perfet squres. For exmple: = + + e i + ( ) = ( + )( ) = = f j 4 6 6 + g k 4 0 4 + h 4 l + 4 REASONING ENRICHMENT E m 6 6 n o + p + q r + s + t Unorrete r smple pges Cmrige University Press Greenwoo et l., 04 ISBN 98--0-6890- Ph 0 86 400
90 Chpter Inies n surs F Review of inex lws CONSOLIDATING Key ies From your work in Yer 9 you will rell tht powers (i.e numers with inies) n e use to represent repete multiplition of the sme ftor. For exmple, = n x x x x = x 4. The five si inex lws n the zero power will e revise in this setion. Let s strt: Rell the lws Try to rell how to simplify eh expression n use wors to esrie the inex lw use. x 4 x ( ) () ( ) 4 x (4x ) 0 Rell tht = n =. The inex lws Lw : m n = m + n Retin the se n the inies. Lw : m n = m n = m n Retin the se n sutrt the inies. Lw : ( m ) n = m n Retin the se n multiply the inies. Lw 4: ( ) m = m m Distriute the inex numer ross the ses. ( ) m Lw : = m m Distriute the inex numer ross the ses. The zero power: 0 = Any numer (exept 0) to the power of zero is equl to. Exmple Using inex lw Simplify the following using the first inex lw. x x 4 4 SOLUTION EXPLANATION x x 4 = x 9 There is ommon se of x, so the inies. 4 = 4 Multiply oeffiients n inies for eh se n. Rell tht =. Unorrete r smple pges Cmrige University Press Greenwoo et l., 04 ISBN 98--0-6890- Ph 0 86 400
Numer n Alger 9 Exmple 4 Using inex lw Simplify the following using the seon inex lw. m m 4x y (8xy ) SOLUTION EXPLANATION m m = m Sutrt the inies when iviing terms with the sme se. 4x y (8xy ) = 4x y 8xy = xy Exmple Comining inex lws Simplify the following using the inex lws. ( ) 4 (y ) SOLUTION First, express s frtion. Divie the oeffiients n sutrt the inies of x n y (i.e. x y ). ( x ) y z EXPLANATION (xy ) 4x 4 y ( ) 4 = Use inex lw n multiply the inies. (y ) = y 8x y Use inex lw 4 n multiply the inies for = 8y eh se n y. Note: =. ( x ) = x 6 y z y 6 z = x6 y 6 z (xy ) 4x 4 y 8x = x y 6 4x 4 y y 8x y = x y 8 8x y = x y Rise the oeffiients to the power n multiply the inies of eh se y. Remove rkets first y multiplying inies for eh se. Simplify the numertor using inex lw. Simplify the frtion using inex lw, sutrting inies of the sme se. Unorrete r smple pges Cmrige University Press Greenwoo et l., 04 ISBN 98--0-6890- Ph 0 86 400
9 Chpter Inies n surs Exmple 6 Using the zero power Exmple Evlute, using the zero power. 4 0 p 0 + (p) 0 SOLUTION 4 0 = 4 = 4 p 0 + (p) 0 = + Exerise F = EXPLANATION Any numer to the power of zero is equl to. Note: (p) 0 is not the sme s p 0. Simplify, using inex form. 8 8 8 x x x e y y y y f 4 Copy n omplete this tle. x 4 0 x = 8 (½),, Copy n omplete. x = - - - - - - x = x x x - - - - - - - - - - - - = --- = x--- ( ) = - - - - - - - - - - - - (x) 0 x 0 = - - - - - - - - = --- = 4 Simplify, using the first inex lw. 4 x x m m e s 4 s f t 8 t 8 g p p h 4 4 4 (½) i 4 (½) s s j x y xy k l v w 6v w m x 4 xy 0y 4 n rs r 4 s r s o 4m 6 n mn mn 4 (½) UNDERSTANDING FLUENCY Unorrete r smple pges Cmrige University Press Greenwoo et l., 04 ISBN 98--0-6890- Ph 0 86 400
Numer n Alger 9 F Exmple 4 Exmple Exmple 6 Exmple 6 8 Simplify, using the seon inex lw. x x 6 q 9 q 6 g j j 6 e h y 8 y m m 9 f i 8 x y x j r s (r s) k 6p 4 q (q p ) l 6m x (8m x 4 ) m p 4 4 n q 8st 4 t x 4 y 9x y o r v 8v 8x y 6x y Simplify using the thir, fourth n fifth inex lws. (x ) (t ) 4( ) (y ) e (4t ) f (u ) g (r ) h (p 4 ) 4 ( ) ( x ) ( x y ) ( u 4 w ) 4 i j y 4 k z 4 l v ( f ) ( ) ( t ) ( 4p q ) 4 m n g pq o g 4 p r Evlute the following using the zero power. 8x 0 t 0 (z) 0 (0 ) 0 e (g h ) 0 f 8x 0 g 4 0 9 h x 0 4(y) 0 Use pproprite inex lws to simplify the following. x 6 x x x y (xy) xy e g x 4 n x n (xn) m w m w m 4 w 9x y 6x y xy 6 f h 8(½), 9 8(½), 9 x y x y 4 x y r 4 s r 4 s r 4 s 4x y x y 4x 4 y 6 8 4 i 6 j (m n 4 ) mn k ( ) () ( 4f g ) f g 4 ( ( fg ) ) l 8 0(½) FLUENCY PROBLEM-SOLVING m o 4m n (m n) 6m n () ( ) 4 4 n p (y z) yz (yz) (m ) (mn 4 ) 0 (6n ) ( n) m 4 Unorrete r smple pges Cmrige University Press Greenwoo et l., 04 ISBN 98--0-6890- Ph 0 86 400
94 Chpter Inies n surs F 9 Simplify: ( ) () ( ) 4 4 0 Simplify: ( (x ) ) Evlute without the use of lultor. e 4 f 8 8 6 6 6 4 When Billy uses lultor to rise to the power 4 he gets 6 when the nswer is tully 6. Wht hs he one wrong? ( ( ) ) g (½) (½), 9 8 9 6 4 ( ( ) ) h 4 0 4 Fin the vlue of in these equtions in whih the inex is unknown. = 8 = 8 + = 4 ( ) = e ( ) = 6 f ( 4) = Inies in equtions 4 If x 4 =, fin the vlue of: x 8 x 4 x 6 x 4 x 8 Fin the vlue(s) of x. x 4 = 6 x = 6 x = 6 x = 6 6 Fin the possile pirs of positive integers for x n y when: x y = 6 x y = 64 x y = 8 x y =, 4 6 PROBLEM-SOLVING REASONING ENRICHMENT Unorrete r smple pges Cmrige University Press Greenwoo et l., 04 ISBN 98--0-6890- Ph 0 86 400
Numer n Alger 9 G Negtive inies The stuy of inies n e extene to inlue negtive powers. Using the seon inex lw n the ft tht 0 =, we n estlish rules for negtive powers. 0 n = 0 n (inex lw ) lso 0 n = n (s 0 = ) = n = n Therefore: n = n Also: Therefore: n = n = n = n = n n = n. Let s strt: The isppering nk lne Due to fees, n initil nk lne of $64 is hlve every month. Blne ($) 64 6 8 4 Negtive inies n e use to esrie very smll numers, suh s the mss of grin of sn. Positive inies only 6 Positive n negtive inies 6 4 Copy n omplete the tle n ontinue eh pttern. Disuss the ifferenes in the wy inies re use t the en of the rows. Wht woul e wy of writing using positive inies? 6 Wht woul e wy of writing 6 using negtive inies? Key m = m For exmple, = = 8. 4 8 ies m = m For exmple, = = 8. Unorrete r smple pges Cmrige University Press Greenwoo et l., 04 ISBN 98--0-6890- Ph 0 86 400
96 Chpter Inies n surs Exmple Writing expressions using positive inies Express eh of the following using positive inies. 4 x 4 y SOLUTION x EXPLANATION 4 = 4 Use n = n. x 4 y = y x 4 = x x = x ( ) = 6 = 9 = 9 Exmple 8 Simplifying more omplex expressions ( ) x is the only se with negtive power. x 4 y = y x 4 Use n = n n note tht x = x. Use inex lws, 4 n to pply the power to eh se in the rkets. Apply inex lw to sutrt the inies of. Use n = n to express with positive inex. Alterntively, first simplify n then rise to the power of. Simplify the following n express your nswers using positive inies. (p q) 4 ( p ) ( m ) ( m p q n ) q r n 4 r SOLUTION (p q) 4 p q ( p ) q = p 8 q 4 p q p6 q 9 = p q 4 p q 6 = p q 0 = q0 p EXPLANATION Del with rkets first y multiplying the power to eh of the inies within the rkets. Use inex lws n to omine inies of like ses. Simplify eh numertor n enomintor first: p 8 + 6 = p n q + ( 9) = q 6. Then p ( ) q 4 ( 6) = p q 0. Use n = n to express p with positive inex. Unorrete r smple pges Cmrige University Press Greenwoo et l., 04 ISBN 98--0-6890- Ph 0 86 400
Numer n Alger 9 Exmple, Exmple ( m r n 4 Exerise G ) ( m n r ) = m 9 r 6 n m 4 n 6 r = 8m9 r 6 n r m 4 n 6 = 8m r 4 n 6 = 8m n 6 r 4 Write the next three terms in these ptterns.,,, 0,,,, x, x, x 0,,, x, x 0,,, Multiply the rket power to eh of the inies within the rket. Multiply y the reiprol of the ivisor. Use inex lws n to omine inies of like ses. Write the nswer with positive powers. Rell tht 4 =. Similrly, write these using positive inies. 9 6 Write eh rule for negtive inies y ompleting eh sttement. = = 4 Express the following using positive inies. x 4 m 4 y e f 4m n g 0x y z h x 4 y z i p q r j e 4 f Express the following using positive inies. x y 4 m e f n 4 k g 8 u v 6 w 4 m 4 4 l h h g 6 Use inex lws n to simplify the following. Write your nswers using positive inies. x x 4 9 y 6 y e 4 f 4x x 4 g m 4 ( m ) h 6 i m x x s s j n 0 4f f k o 4 (½) 6 6 4 8(½) l p 4 4 t 4 8t 4 8(½) UNDERSTANDING FLUENCY Unorrete r smple pges Cmrige University Press Greenwoo et l., 04 ISBN 98--0-6890- Ph 0 86 400
98 Chpter Inies n surs G Exmple Exmple 8 Express the following with positive inies. ( x ) 4 ( m ) x 4m (x ) 4( ) e (t 4 ) f (x ) g (x ) 4 h 8(x ) i (4y ) j (h ) 4 k (j ) 4 l (t ) 8 Express the following in simplest form with positive inies. 9 x y x y 4 4 6 y 4 y e 4 ( ) f p q (p q ) i m p q r 4 pq 4 r f g f g j n x y 6xy r s 4 r s g k o 4 4m n 4 m n w x 6w x Simplify the following n express your nswers with positive inies. h l p 6 4 6 m n mn r 4 s 6 9rs ( ) ( 4 ) (p ) 4 (p q) (x y ) (xy 4 ) g 4 ( ) ( 4 ) e h (rs ) 4 r s 4 (r s) s ( m 4 n ) ( m n ) r r f i 4(x y 4 ) x y xy4 x y (x y 4 ) (xy ) (xy) (x y 4 ) 0 Evlute without the use of lultor. 4 ( 4 ) e 0 ( ) 6 f (4 ) 4(4 ) ( ) ( ) i j 4 The with of hir on spier is pproximtely m. How mny entimetres is this, orret to four eiml ples? g k 9(½) (4 ) 4 4 9 0(½) h l 4 (0 4 ) (0 ) 9 0(½), FLUENCY PROBLEM-SOLVING Unorrete r smple pges Cmrige University Press Greenwoo et l., 04 ISBN 98--0-6890- Ph 0 86 400
Numer n Alger 99 Simplify these numers. ( ) ( ) i ii ( ) Wht is when expresse in simplest form? A stuent simplifies x n writes x =. Explin the error me. x 4 Evlute the following y omining frtions. + + 6 ( ) ( ) ( ) x Prove tht = x giving resons. Simple equtions with negtive inies 6 Fin the vlue of x. e ( ) ( 4 ) x = 4 x = ( ) x x = = 4 4 e ( ) x = 9 4 f, ( ) x = 8 g x = 8 h x = 8 i x = j x = k x = l 0 9 x = 000 m o ( ) x + = 64 4 ( ) x + = 6 8 n p ( ) x = 4 f iii ( ) x = 4 4 ( ) x y ( ) ( ) 0 4 ( ) ( ) 4, 6 REASONING ENRICHMENT G Unorrete r smple pges Cmrige University Press Greenwoo et l., 04 ISBN 98--0-6890- Ph 0 86 400
00 Chpter Inies n surs H Sientifi nottion CONSOLIDATING Key ies Sientifi nottion is useful when working with very lrge or very smll numers. Comine with the use of signifint figures, numers n e written own with n pproprite egree of ury n without the nee to write ll the zeros tht efine the position of the eiml ple. The pproximte istne etween the Erth n the Sun is 0 million kilometres or. 0 8 km when written in sientifi nottion using two signifint figures. Negtive inies n e use for very smll numers, suh s 0.00008 g =.8 0 g. Let s strt: Amzing fts lrge n smll Think of n ojet, ple or living thing tht is ssoite with very lrge or smll numer. Give three exmples using very lrge numers. Give three exmples using very smll numers. Cn you rememer how to write these numers using sientifi nottion? How re signifint figures use when writing numers with sientifi nottion? A numer written using sientifi nottion is of the form 0 m, where < 0 or 0 < n m is n integer. Lrge numers: 4 800 000 =.48 0 9 00 000 000 = 9.0 0 9 Smll numers: 0.000 =.0 0 0.00000 =. 0 6 Signifint figures re ounte from left to right, strting t the first non-zero igit. When using sientifi nottion the igit to the left of the eiml point is the first signifint figure. For exmple: 0 90 000 =.09 0 shows four signifint figures. The 0 n, EE or Exp keys n e use on lultors to enter numers using sientifi nottion; e.g..e 4 mens. 0 4. Unorrete r smple pges Cmrige University Press Greenwoo et l., 04 ISBN 98--0-6890- Ph 0 86 400
Numer n Alger 0 Exmple 9 Converting from sientifi nottion to si numerl Write these numers s si numerl..06 0. 0 SOLUTION EXPLANATION.06 0 = 0 600 Move the eiml point ples to the right.. 0 = 0.000000 Move the eiml point ples to the left. Exmple 0 Converting to sientifi nottion using signifint figures Write these numers in sientifi nottion, using three signifint figures. 8 00 0.0040 SOLUTION EXPLANATION 8 00 =. 0 6 Ple the eiml point fter the first non-zero igit. The igit following the thir igit is t lest, so roun up. 0.0040 = 4.0 0 Roun own in this se, ut retin the zero to show the vlue of the thir signifint figure. Exerise H How mny signifint figures re showing in these numers?. 0.90 0 4.8 0.0 0 e 46 f 9 g 0.000 h 0.000040 Write these numers s powers of 0. (½) 000 0 000 000 0.00000 000 Convert to numers using sientifi nottion. 4 000 000 90 00 0 00 e 0.0008 f 0.000 g 0.0000 h 0.00040 4 (½) (½) 4 8(½) 4 8(½) UNDERSTANDING Exmple 9 4 Write these numers s si numerl.. 0.49 0 4.0 0 8. 0 6 e.9 0 4 f 8.00 0 g.0 0 4 h 9.99 0 6 i.0 0 8 j. 0 4 k. 0 9 l. 0 FLUENCY Unorrete r smple pges Cmrige University Press Greenwoo et l., 04 ISBN 98--0-6890- Ph 0 86 400
0 Chpter Inies n surs H Exmple 9 Exmple 0 Exmple 0 Write these numers s si numerl. 4. 0. 0.08 0 4.8 0 e 9. 0 f.6 0 g.00 0 4 h 6. 0 6 i 9.8 0 j.4 0 0 k.8 0 l 8. 0 6 Write these numers in sientifi nottion, using three signifint figures. 64 644 0 48 4 8 e 0 089 f 4 9 86 g 4 h 6 088 i 0 48 j 909 k 4 68 l 9 86 00 00 Write these numers in sientifi nottion, using three signifint figures. 0.004 0.084 0.000 0.00866 e 0.00008 f 0.96 g 0.00000640 h 0.000089 i 0.000998 j 0.0000000084 k 0.000000009886 l 0.0004998 8 Write in sientifi nottion, using the numer of signifint figures given in the rkets. 900 () 0 9 () 0 80 00 () 48 () e 0.009 () f 0.00000 () g 0.0000098 () h 0.08 (4) i 0.0000004 () 9 0(½) 9 Write the following numeril fts using sientifi nottion. 9 0(½) The re of Austrli is out 00 000 km. The numer of stones use to uil the Pyrmi of Khufu is out 00 000. The gretest istne of Pluto from the Sun is out 400 000 000 km. A humn hir is out 0.0 m wie. e The mss of neutron is out 0.000000000000000000000000006 kg. f The mss of teri ell is out 0.0000000000009 g. 0(½), FLUENCY PROBLEM-SOLVING Unorrete r smple pges Cmrige University Press Greenwoo et l., 04 ISBN 98--0-6890- Ph 0 86 400
Numer n Alger 0 0 Use lultor to evlute the following, giving the nswers in sientifi nottion using three signifint figures. (.) (.04) 4 (.8 0 ). (8 0 ) e (9. 0 ) (8.6 0 8 ) f (. 0 4 ) (9 0 ) g. 0 6 h p (. 0 ) i. 0 4 The spee of light is pproximtely 0 km/s n the verge istne etween Pluto n the Sun is out.9 0 9 km. How long oes it tke for light from the Sun to reh Pluto? Answer orret to the nerest minute. Explin why 8 0 is not written using sientifi nottion., (½) 4(½), Write the following using sientifi nottion. 0 94 0 6004 0 9 0 6 e 0. 0 4 f 0.00 0 g 0.0 0 9 h 0.06 0 8 i 0.4 0 j 0.00 0 k 0. 0 6 l 964 0 4 4 Comine your knowlege of inex lws with sientifi nottion to evlute the following n express using sientifi nottion. ( 0 ) ( 0 ) (8 0 4 ) ( 0 ) e ( 0 ) f (4 0 ) g (. 0 ) h (8 0 8 ) i ( 0 ) ( 0 4 ) j ( 0 ) (4. 0 ) k ( 0 8 ) ( 0 ) l (8 0 ) (9 0 ) m (40 0 4 ) ( 0 ) n ( 0 8 ) (0 0 4 ) o ( 0 ) (0 0 ) Rewrite 0 4 with positive inex n use this to explin why, when expressing 0 4 s si numerl, the eiml point is move four ples to the left. E = m 6 E = m is formul erive y Alert Einstein (89 9). The formul reltes the energy (E joules) of n ojet to its mss (m kg), where is the spee of light (pproximtely 0 8 m/s). Use E = m to nswer these questions, using sientifi nottion. Fin the energy, in joules, ontine insie n ojet with these given msses. i 0 kg ii 6 000 kg iii 0.0 kg iv 0.0000 kg 6 PROBLEM-SOLVING REASONING ENRICHMENT H Fin the mss, in kilogrms, of n ojet tht ontins these given mounts of energy. Give your nswer using three signifint figures. i 0 J ii.8 0 6 J iii 8. 0 4 J iv. 0 J The mss of the Erth is out 6 0 4 kg. How muh energy oes this onvert to? Unorrete r smple pges Cmrige University Press Greenwoo et l., 04 ISBN 98--0-6890- Ph 0 86 400
04 Chpter Inies n surs Progress quiz 8pt A 0A 8pt A 0A 8pt A 0A 8pt B 0A 8pt C 0A 8pt C 0A 8pt D 0A 8pt D 0A 8pt F 8pt G 4 6 8 9 0 Express eh numer s eiml n eie if it is rtionl or irrtionl. Then lote ll the numers on the sme numer line. 0 p % Simplify the following. 98 8 Express 8 s the squre root of positive integer. 6 Simplify the following. + 4 + + 48 4 4 0 + Simplify the following. ( 4) 4 Use the istriutive lw to expn ( 6 + 4) n simplify the surs where neessry. Expn n simplify. ( 4)( + ) ( ) ( )( + ) Rtionlise the enomintors. 6 Simplify, using inex lws. 4x y xy h 6 h m 9 n 4 (0m n) ( p e ( ) f (m ) 4 q ) g rt h () 0 + m 0 Simplify the following where possile n express your nswers using positive inies. x 4 4 m ( 4k ) e k f ( ) g 6 m 4 m 0 h 8pt H i Write these numers s si numerl..0 0 6 9.06 0 ii Write these numers in sientifi nottion, using three signifint figures. 0 0.00004 8pt G Simplify the following n express your nswers using positive inies. ( ) ( 4 ) ( x ) ( x ) 6 Unorrete r smple pges Cmrige University Press Greenwoo et l., 04 ISBN 98--0-6890- Ph 0 86 400
Numer n Alger 0 I Rtionl inies 0A The squre n ue roots of numers, suh s 8 = 9 n 64 = 4, n e written using frtionl powers. The following shows tht 9 = 9 n 8 = 8. Consier: 9 9 = n 9 9 = 9 + Also: = 9 = 9 9 = 9 8 8 8 = n 8 8 8 = 8 + + = 8 = 8 8 = 8 A rtionl inex is n inex tht n e expresse s frtion. Let s strt: Mking the onnetion For eh prt elow use your knowlege of inex lws n si surs to simplify the numers. Then isuss the onnetion tht n e me etween numers tht hve sign n numers tht hve frtionl powers. n n ( ( ) n ) ( ( 64) n 64 ) n = n n is the nth root of. For exmple: = or, =, 0 = 0 m ( n = ) m n = ( n ) m or m n = ( m ) n = n m For exmple: 8 ( = 8 ) or 8 = (8 ) = ( 8) = 64 = = 64 = 4 = 4 Key ies In most ses, the inex lws pply to rtionl inies (i.e. frtionl inies) just s they o for inies tht re integers. Unorrete r smple pges Cmrige University Press Greenwoo et l., 04 ISBN 98--0-6890- Ph 0 86 400
06 Chpter Inies n surs Exmple Writing in inex form Express the following in inex form. x 4 x 0 0 SOLUTION = x = (x ) = x 4 x = (x ) 4 = x 4 0 0 = 0 0 = 0 Exmple Writing in sur form Express the following in sur form. SOLUTION EXPLANATION mens the squre root or. Note: n = n. Rewrite to simplify: =. s power, then pply inex lws 4 mens to the power of 4. Apply inex lw to multiply inies. Rewrite the squre root s power n then inies for the ommon se 0. Rell 0 = 0, so + =. An lterntive nswer is 00 0 = 000. EXPLANATION = n = n = ( ) Use inex lw wherey =. = ( ) = Alterntively: = ( ) = is the sme s. Unorrete r smple pges Cmrige University Press Greenwoo et l., 04 ISBN 98--0-6890- Ph 0 86 400
Numer n Alger 0 Exmple Evluting numers with frtionl inies Evlute the following without lultor. 6 6 4 SOLUTION 6 = 6 = 4 6 4 = 4 6 = = EXPLANATION 6 mens 6. 6 4 mens 4 6 n 4 = 6. = mens n =. = Exerise I Copy n omplete eh sttement. = 8 n 8 = = 9 n 9 = Rewrite, using positive inies. Rell tht m = m. (½) = n = = n = e = 8 n 8 = f 0 = 00 000 n 00 000 = 0 Evlute: 9 6 e 8 f g h i 4 6 4 j 8 k l 64 00 000 Using lultor, enter n evlute eh pir of numers in their given form. Roun your nswer to two eiml ples., 0, 0 00, 00 UNDERSTANDING 4 (½) 4 (½) 4 (½) Exmple, 4 Express the following in inex form. 9 e f 4t g x 0t h 4 8 8m 4 FLUENCY Unorrete r smple pges Cmrige University Press Greenwoo et l., 04 ISBN 98--0-6890- Ph 0 86 400
08 Chpter Inies n surs I Exmple, Exmple Exmple 6 Express the following in inex form. x 6 n 4 y p r e 4 f 4 g h g h i 4 4 Express the following in sur form. 8 6 0 e f g h 4 Evlute without using lultor. 6 64 49 e 6 4 f g 9 h i 8 4 j 000 k 400 l 0 000 4 8 Evlute without using lultor. 8 6 6 4 e 6 4 f g 64 h j 4 k 9 9 Use inex lws to simplify the following. i l 0 00 m m x x 4 4 4 ( s )4 ( e y ) ( f t ) 0 g 4 h 4 0 Simplify the following. s 4 t 6 e (x ) f ( ) g 8 9(½) 4 6t 8 ( t 4 ) 8 9(½) h t ( m ) 0 i (6 8 ) j (6m 6 n ) k (x 0 y ) l (4r 9 t 6 ) 8x ( ) ( 0 x 4 ) 4 m n o 49 x 0 p 0.0 8 0(½) FLUENCY PROBLEM-SOLVING Unorrete r smple pges Cmrige University Press Greenwoo et l., 04 ISBN 98--0-6890- Ph 0 86 400
Numer n Alger 09 As shown elow, 6 4 n e evlute in two wys. Metho A 6 4 = (6 ) 4 = ( 048 6) 4 = 4 048 6 = Metho B 6 ( 4 = 6 4 ) = ( 4 6) = = If 6 4 is to e evlute without lultor, whih metho ove woul e preferle? Use your preferre metho to evlute the following without lultor. i 8 ii 6 iii 6 4 iv 4 ( ) ( ) ( )4 v 4 4 vi vii viii 9 000 Explin why 6 64 is not sur. Does it exist? We know tht when y = x, where x < 0, y is not rel numer. This is euse the squre of y nnot e negtive; i.e. y π x sine y is positive n x is negtive. But we know tht ( ) = 8 so 8 =. Evlute: i ii 000 Deie if these re rel numers. i ii If y = n x n x < 0, for wht vlues of n is y rel numer? iii iv 8 iii 6 4 iv The squre root of negtive vlue is not rel numer. = i n involves rnh of mthemtis lle omplex numers., REASONING ENRICHMENT I Unorrete r smple pges Cmrige University Press Greenwoo et l., 04 ISBN 98--0-6890- Ph 0 86 400
0 Chpter Inies n surs J Key ies Exponentil growth n ey The popultion of ountry inresing y % per yer n n investment inresing, on verge, y % per yer re exmples of exponentil growth. When n investment grows exponentilly, the inrese per yer is not onstnt. The nnul inrese is lulte on the vlue of the investment t tht time, n this hnges from yer to yer euse of the e investment returns. The more money you hve investe, the more interest you will mke in yer. In the sme wy, popultion n grow exponentilly. A growth of % in lrge popultion represents mny more ies orn in yer thn % of smll popultion. Popultion growth n e moelle using exponentil equtions. Exponentil reltions will e stuie in more etil in Chpter. Here we will fous on exponentil growth in reltion to ompoun interest n other simple situtions involving growth n ey. In the finnil worl, it is importnt to unerstn how ompoun interest works n how investments n grow n ey exponentilly. Let s strt: A ompoun rule Imgine you hve n ntique r vlue t $00 000 n you hope tht it will inrese in vlue t 0% p.. The 0% inrese is to e e to the vlue of the r eh yer. Disuss how to lulte the vlue of the r fter yer. Disuss how to lulte the vlue of the r fter yers. Complete this tle. Yer 0 Vlue ($) 00 000 00 000. 00 000. = = = Disuss how inies n e use to lulte the vlue of the r fter the seon yer. Disuss how inies n e use to lulte the vlue of the r fter the tenth yer. Wht might e the rule onneting the vlue of the r ($A) n the time n yers? Repet the steps ove if the vlue of the r ereses y 0% p.. Per nnum (p..) mens per yer. Exponentil growth n ey n e moelle y the rule A = k t, where A is the mount, k is the initil mount n t is the time. When >, exponentil growth ours. When 0 < <, exponentil ey ours. For growth rte of r% p.., the se is lulte using = + r 00. For ey rte of r% p.., the se is lulte using = r 00. Unorrete r smple pges Cmrige University Press Greenwoo et l., 04 ISBN 98--0-6890- Ph 0 86 400
Numer n Alger ( ) n The si exponentil formul n e summrise s A = A 0 ± r. 00 Key ies The susript zero is often use to inite the initil vlue of quntity (e.g. P 0 is initil popultion). Exmple 4 Writing exponentil rules Form exponentil rules for the following situtions. John hs pinting tht is vlue t $00 000 n whih is expete to inrese in vlue y 4% per nnum. A ity s initil popultion of 0 000 is eresing y % per yer. SOLUTION Let A = the vlue of the pinting t ny time n = the numer of yers the pinting is kept A 0 = initil vlue r = 4 A 0 = 00 000 ( ) n A = 00 000 + 4 00 A = 00 000(.4) n Let P = the popultion t ny time n = the numer of yers the popultion ereses P 0 = strting popultion r = P 0 = 0 000 ( ) n P = 0 000 00 P = 0 000(0.88) n EXPLANATION Define your vriles. ( ) n A = A 0 ± r. 00 Sustitute r = 4 n A 0 = 00 000 n use + sine we hve growth. Define your vriles. ( ) n P = P 0 ± r. 00 Sustitute r = n P 0 = 0 000 n use sine we hve ey. Unorrete r smple pges Cmrige University Press Greenwoo et l., 04 ISBN 98--0-6890- Ph 0 86 400
Chpter Inies n surs Exmple Applying exponentil rules House pries re rising t 9% per yer n Zoe s flt is urrently vlue t $600 000. Determine rule for the vlue of Zoe s flt ($V) in n yers time. Wht will e the vlue of her flt: i next yer? ii in yers time? Use tril n error to fin when Zoe s flt will e vlue t $900 000, to one eiml ple. SOLUTION Let V = vlue of Zoe s flt t ny time V 0 = strting vlue $600 000 n = numer of yers from now r = 9 V = V 0 (.09) n V = 600 000(.09) n i When n =, V = 600 000(.09) = 64 000 Zoe s flt woul e vlue t $64 000 next yer. ii When n =, V = 600 000(.09) = 0.40 In yers time Zoe s flt will e vlue t out $ 0. n 4 4.6 4.8 4. V 846 949 9 4 89 894 90 99 899 6 Zoe s flt will e vlue t $900 000 in out 4. yers time. Exerise J EXPLANATION Define ( your vriles. ) n V = V 0 ± r. 00 Use + sine we hve growth. Sustitute n = for next yer. For yers, sustitute n =. Try vlue of n in the rule. If V is too low, inrese your n vlue. If V is too high, erese your n vlue. Continue this proess until you get lose to 900 000. An ntique ring is purhse for $000 n is expete to grow in vlue y % per yer. Roun your nswers to the nerest ent. e Fin the inrese in vlue in the first yer. Fin the vlue of the ring t the en of the first yer. Fin the inrese in vlue in the seon yer. Fin the inrese in vlue in the thir yer. Fin the vlue of the ring t the en of the fifth yer. UNDERSTANDING Unorrete r smple pges Cmrige University Press Greenwoo et l., 04 ISBN 98--0-6890- Ph 0 86 400
Numer n Alger Exmple 4 Exmple The mss of limestone kg rok expose to the wether is eresing t rte of % per nnum. Fin the mss of the rok t the en of the first yer. Copy n omplete the rule for the mss of the rok (M kg) fter t yers. M = ( ) t t = Use your rule to lulte the mss of the rok fter yers, orret to two eiml ples. Deie if the following represent exponentil growth or exponentil ey. A = 000. t A = 00.8 t A = 0 0.9 t ( ) t ( ) t P = 0 000 0.8 t e P = P 0 + f T = T 00 0 00 4 Define vriles n form exponentil rules for the following situtions. A flt is purhse for $00 000 n is expete to grow in vlue y % per nnum. A house initilly vlue t $0 000 is losing vlue t % per nnum. The vlue of r, ought for $4 00, is eresing t % per nnum. A popultion, whih is initilly 00, is inresing t % per yer. e f g h A tnk with 00 litres of wter is leking t rte of 0% of the wter in the tnk every hour. A humn ell of re 0.0 m oules its re every minute. An oil spill, initilly overing n re of squre metres, is inresing t % per minute. A sustne of mss 0 g is eying t rte of 8% per hour. The vlue of house purhse for $00 000 is expete to grow y 0% per yer. Let $A e the vlue of the house fter t yers. Copy n omplete the rule onneting A n t. A = 00 000 t Use your rule to fin the expete vlue of the house fter the following numer of yers. Roun your nswer to the nerest ent. i yers ii 0 yers iii 0 yers 4 6 4 Use tril n error to estimte when the house will e worth $ million. Roun your nswer to one eiml ple. 8 UNDERSTANDING FLUENCY J Unorrete r smple pges Cmrige University Press Greenwoo et l., 04 ISBN 98--0-6890- Ph 0 86 400
4 Chpter Inies n surs J 6 A shre portfolio, initilly worth $00 000, is reue y % p.. over numer of yers. Let $A e the shre portfolio vlue fter t yers. Copy n omplete the rule onneting A n t. A = 0.8 t Use your rule to fin the vlue of the shres fter the following numer of yers. Roun your nswer to the nerest ent. i yers ii yers iii yers Use tril n error to estimte when the shre portfolio will e vlue t $80 000. Roun your nswer to one eiml ple. A wter tnk ontining 000 L hs smll hole tht reues the mount of wter y 6% per hour. Determine rule for the volume of wter (V) left fter t hours. Clulte (to the nerest litre) the mount of wter left in the tnk fter: i hours ii hours How muh wter is left fter two ys? Roun your nswer to two eiml ples. Using tril n error, etermine when the tnk hols less thn 00 L of wter, to one eiml ple. 8 Megn invests $0 000 in supernnution sheme tht hs n nnul return of %. Determine the rule for the vlue of her investment (V) fter n yers. How muh will Megn s investment e worth in: i 4 yers? ii 0 yers? Fin the pproximte time efore her investment is worth $00 000. Roun your nswer to two eiml ples. 9 A ertin type of teri grows oring to the eqution N = 000(.6) t, where N is the numer of ells present fter t hours. 9, 0 9, 0 How mny teri re there t the strt? Determine the numer of ells (roun to the whole numer) present fter: i 0 hours ii hours iii 4.6 hours If 0 000 000 teri re neee to mke rop of serum, etermine how long you will hve to wit to mke rop (to the nerest minute). 0, FLUENCY PROBLEM-SOLVING Unorrete r smple pges Cmrige University Press Greenwoo et l., 04 ISBN 98--0-6890- Ph 0 86 400