Indices and Surds. Learning Outcomes

Transcription

1 6 Indices nd Surds Wht gret spce sver! Chpter Contents 6:0 Indices nd the index lws NS, PAS Investigtion: Exploring index nottion Chllenge: Fmily trees 6:0 Negtive indices NS, PAS Investigtion: Zero nd negtive indices 6:0 Frctionl indices NS, PAS Fun Spot: Why is room full of mrried people lwys empty? Investigtion: Resoning with frctionl indices 6:0 Scientific (or stndrd) nottion NS Investigtion: Multiplying nd dividing by powers of 0 6:0 Scientific nottion nd the clcultor NS Investigtion: Using scientific nottion 6:06 The rel number system NS Reding Mths: Proof tht is irrtionl Chllenge: f-stops nd 6:07 Surds NS 6:08 Addition nd subtrction of surds NS 6:09 Multipliction nd division of surds NS Investigtion: Itertion to find squre roots 6:0 Binomil products NS 6: Rtionlising the denomintor NS Fun Spot: Wht do Eskimos sing t birthdy prties? Chllenge: Rtionlising binomil denomintors Mths Terms, Dignostic Test, Revision Assignment, Working Mthemticlly Lerning Outcomes PAS Applies the index lws to simplify lgebric expressions. PAS Simplifies, expnds nd fctorises lgebric expressions involving frctions nd NS NS negtive nd frctionl indices. Applies index lws to simplify nd evlute rithmetic expressions nd uses scientific nottion to write smll nd lrge numbers. Performs opertions with surds nd indices. Working Mthemticlly Stges. Questioning, Applying Strtegies, Communicting, Resoning, Reflecting. 67

2 6:0 Indices nd the Outcomes NS, PAS Index Lws = 6 is clled the bse. is clled the index. = 6 6 is clled the bsic numerl. is the 'index form' (bse ) of 6 x n = x x x... x x (where n is positive integer) n fctors For: x is the bse x n n is the index. is clled power of. Multipliction using indices = ( ) ( ) x x = (x x x x x) (x x x) = 6 [= + ] = x 8 [= x + ] Lw When multiplying terms, dd the indices: x m x n = x m + n Division using indices = x x x x x x - = - x x x = [= ] = x [= x ] Lw When dividing terms, subtrct the indices: x m x n = x m n Powers of indices ( ) = (x ) = x x x x = + [Using Lw ] = x [Using Lw ] = 6 [= ] = x 0 [= x ] Lw For powers of power, multiply the indices: (x m ) n = x mn You should lern these lws. If we simplify the division x n x n, using the second lw bove: x n x n = x n n = x 0 But ny expression divided by itself must equl. x n x n = Therefore x 0 must be equl to. Lw x 0 = x 0 = 68 NEW SIGNPOST MATHEMATICS 9 STAGE..

3 worked exmples Simplify: b c ( ) Simplify: b x x c 6m n mn Simplify: x 7 x b c 0 b 0b Simplify: ( ) b ( ) c (p ) (p ) Simplify: 7 0 b 8x 6x c (y ) (y 6 ) Solutions = = 6 b = Using the clcultor to evlute PRESS x y = = 7 9 c ( ) = = 6 Using index lw : = + = 7 b x x = x + = x c 6m n mn = 6 m + n + = 6m n Using index lw : x 7 x = x 7 b = - = x = - = = Another nme for n INDEX is n EXPONENT. Remember the x y button. Enter the bse x first, press x y then enter the index y. Note: n = n b ( ) = ( ) c 0 b 0b 0 = b 0b 0 = - b 0 b = b Using index lw : ( ) = c (p ) (p ) = p p 8 = 8 = 8 = p = 8 Using index lw : 7 0 = b 8x 6x 8x = - c (y ) (y 6 ) = ( y ) - 6x ( y 6 ) With prctice, mny of = x 6 y = x 0 = - the steps in the bove 6 y solutions cn be left out. = = y 0 = = CHAPTER 6 INDICES AND SURDS 69

4 6 7 8 Exercise 6:0 Write ech expression in index form. b c d e f g x x h i n n n j m m m m m k p p p p p p l y y m n t t t t o x x x x x Rewrite in expnded form. b c 6 d 0 e 7 f g h x i y j m k n 7 l p Determine the bsic numerl for: b c 7 d e 0 f 6 g 8 h 7 i 8 j 9 k l m 6 n 7 o 9 p 7 Simplify these products, writing nswers in index form. 0 0 b 0 0 c 0 0 d e f 7 7 g h i Simplify these quotients, writing nswers in index form. 0 0 b c 0 0 d 8 e 7 7 f 0 g h i 8 Simplify these powers, writing nswers in index form. (0 ) b (0 ) c (0 6 ) d ( ) e ( ) f ( 7 ) g ( ) h ( ) i (7 ) j ( ) k (7 ) l ( ) Simplify: x x b y y c m m d m m e p p f g y y h x x i m m j y y k m 6 l x Simplify: x 6 x b x 6 x c x 6 x d m m e y y f m m g y 6 y 6 h x x i y 8 y j 6m 6 k y 8 l 0x Foundtion Worksheet 6:0 The index lws NS Evlute: b c Use your clcultor to evlute: 6 b c 8 This sure is powerful stuff! 70 NEW SIGNPOST MATHEMATICS 9 STAGE..

5 9 0 Simplify: (x ) b (y ) c ( ) d (m ) 0 e (x 0 ) f ( 7 ) 0 g (y ) h ( 6 ) i (x ) j (x) k (x ) l (m ) Simplify: 8x x b c m 6 m d 8x x e f m 6 m g 0y y h 6m m i 8 j 0y y k 6m m l 8 m x 6x n 9 7 o 8y 6 6y p x 9 - q 8y r - 6 6x 6y s 0x t - u Simplify: 6 0 b 6( ) 0 c (6 ) 0 d b 0 e x 0 y f m n 0 g (m ) h (n ) i (p ) j x y x k b b l xy x m x y xy n b b o m n mn p (x y ) q (bc) r ( pq ) s x y xy t b 7b u b v w x c c 0c y x x + 0x z x(x + ) (x ) Simplify: x x x b c y y xy d x 8x e 7y 9y f 00x 0x g (x ) x h ( ) i (y 7 ) y j ( ) k (m ) m 0 l n 8 (n ) m (y ) (y ) n ( ) ( ) o (b ) (b ) p (x x 7 ) x 9 q ( ) 0 r 7p 7 q ( p q) s x x 7 ( x t ) x x u ( xy) 0x 6x x ( x) Expnd nd simplify: x (x ) b ( ) c ( ) d x(x + y) e m(7 m ) f y(y xy) g ( + ) h x(x x) i m (n m ) j x(x x + 7) k x (x + 7x ) l y(y 7y ) m x (x 7) x n y(y + ) y o x(x 7x + ) (x x ) Simplify: x x + b y y + c ( x ) ( x ) d e x + e x e e x + e x f (e x + ) e x + CHAPTER 6 INDICES AND SURDS 7

6 investigtion 6:0 Investigtion 6:0 Exploring index nottion Serching for ptterns is prt of mthemtics nd being ble to explin concepts is importnt if our ides re to be shred. Find pirs of terms tht cn be multiplied to give 7. Explin the reltionship between the members of ny pir. Find pirs of terms tht cn be divided to give x. Explin the difference between the members of ny pir. Explin the difference between 7x 0 nd (7x) 0. Explin why 9 6. List ll the pirs of expressions tht could be multiplied together to give 0xy. (Use only whole numbers.) Fold sheet of A pper in hlf s mny times s you cn. How does the number of regions increse with ech new fold. Write formul for the number of regions (R) for n folds. As rewrd for service, mn sked his king for the mount of rice tht would be needed to plce one grin of rice on the first squre of chessbord, on the second, on the third, 8 on the fourth, doubling ech time, until the 6th squre is considered. How mny grins of rice would be needed for the nd squre? chllenge Chllenge 6:0 Fmily trees A prt of Aln s fmily tree is drwn below. 6:0 Robert McSeveny Srh McNughton Stephen Newby Ruby Dnn Thoms McSeveny Edn Newby Aln McSeveny How mny gret-grndprents would Aln hve hd? Estimte the number of genertions you would need to go bck to before more thn one million boxes would be required to show tht genertion. Would this men tht in tht genertion there would be over different ncestors? Estimte Aln s totl number of ncestors in the previous 0 genertions. It is much hrder to estimte the number of descendnts Aln will hve in ny one genertion s lot of ssumptions will need to be mde. Will ll of his five children mrry? Will there be wrs, diseses or popultion control in the future? Estimte how mny genertions would be needed before Aln hs totl number of descendnts in excess of NEW SIGNPOST MATHEMATICS 9 STAGE..

7 6:0 Negtive Indices Outcomes NS, PAS All the indices seen so fr hve been positive integers or zero. If we hd, the nswer, ccording to the second index lw, should be, ie =. But this could lso be written in this wy: = - = - = So: = Wht hppens if the index is negtive? Also = 8 - = = In generl, the mening of negtive index cn be x m = -, (x 0) summrised by the rules: x m x m is the reciprocl of x m, since x m x m = Exmples x = x x x = x 0 = worked exmples Simplify the following: b c x 7 x d 6x x e ( ) f ( ) Evlute, using the clcultor: b ( ) Solutions = b = c x 7 x = x 7 + ( ) d 6x x = x = x = x = = = 9 x e ( ) = f ( ) = = - = Note: 6 7 = Since x -m is the nd reciprocl of x m. = 6 7 = - = = - = - = 6 = continued CHAPTER 6 INDICES AND SURDS 7

8 x y The key cn lso be used for negtive indices by entering y s negtive number. Exmine the following: Press: x y + / nswer: 0, ie or = 8 b Press: = x y + / nswer: 9, ie ( ) = 6 Exercise 6:0 Write down the vlue of ech of the following. b c d 6 e f 0 g h 0 i Write ech with negtive index. - b c d e f g h Foundtion Worksheet 6:0 Negtive indices NS Write down the vlue of: - b - c - Write with negtive index. b c i - j - k - l Write true or flse for: 0 = 0 b 8 = c = d () = 6 9 e () = f = g < h 8 = ( ) 8 6 Simplify, leving nswers s powers of ten. 0 0 b 0 0 c d 0 0 e 0 0 ( 0 f ) g h ( 0 ) Write ech without negtive index. b x c m d y e x f y g x h m 6 i x j k 0y l 6q Rewrite ech using negtive index to void hving frction. b c d x x x x e 0 7 f g h - y m x i x m x j k - l - y y 8 b NEW SIGNPOST MATHEMATICS 9 STAGE..

9 Write ech s n integer, frction or mixed number. b c d - 0 e f g h - 0 Rewrite ech expression with positive index. x b c x d m e (x + ) f ( + ) g (6x) h (x + ) Evlute the following, using your clcultor. Leve your nswers in deciml form. b c d 8 e 6 f g (0 ) h (0 ) i (0 0) j ( ) k (0 ) l (0 6) Simplify, writing your nswers without negtive indices. x x b c m m d n n e f 6x x g h m m i x x j k y y l m m Simplify, writing your nswers without negtive indices. m m b x x c y 6 y 8 d x x e f y y g y y h x x i 6x x j 0 7 k l 8n 9n Simplify, writing your nswers without negtive indices. ( ) b (x ) c (y ) d (m ) e (x ) f (x) g (x ) h (7x ) i (bc) j ( b c ) k ( b) l ( b) If x =, y = nd z =, evlute: x + y b (xy) c (xz) d x y z Simplify: x x b y y c e x + e x d (e x ) e (x ) The formul for the volume of sphere is: V = - πr where π nd r is the rdius of the sphere. CHAPTER 6 INDICES AND SURDS 7

10 investigtion 6:0 Investigtion 6:0 Zero nd negtive indices Wht does it men to hve zero or negtive index? Complete these tbles writing nswers less thn s frctions. Divide ech nswer by 0 to reduce the power. Power of Answer As the power of 0 decreses, does the nswer decrese? 0 0 = = = = = =... Divide ech nswer by to reduce the power. Power of 0 Answer 6 As the power of decreses, does the nswer decrese? 0 = -m = m = = = = 0 0 =..... =... Use the tbles bove to write true or flse for: 0 = b 0 = c - = 0 d 0 = e - = f 0 0 = 0 g 0 = h = i 0 - = NEW SIGNPOST MATHEMATICS 9 STAGE..

11 6:0 Frctionl Indices Outcomes NS, PAS Complete: = 7 = 9 = 8 = =... 9 =... 8 =... =... Consider the following: = = 8 If n n =, If 8 n 8 n = 8, wht is the vlue of n? wht is the vlue of n? 7 x x = x 8 y y = y If x n x n = x, If y n y n = y, wht is the vlue of n? wht is the vlue of n? 9 = 0 x x x = x If n n n =, If x n x n x n = x, wht is the vlue of n? wht is the vlue of n? prep quiz 6:0 Wht is the mening of frctionl index? The mening is shown in the exmples below = 9 = 9 = = 9 = 9 9 multiplied by itself gives 9 nd 9 multiplied by itself gives 9. So 9 is the squre root of 9. 9 = 9 Tht's net! = + Tht mens tht = is the squre = root of. = Now = The number tht multiplies itself to give So = (ie ) is the squre root of. Similrly: = 8 = 8 = 8 So 8 = 8, (the cube root of 8) 8 = + + = = 8 Two is the cube root of 8. Since ( x) = x, x = x CHAPTER 6 INDICES AND SURDS 77

12 x x = x, x = x, xn = nth root of x is the number tht, when used three times in product, gives x. Simplify the following: worked exmples b 7 c x x d ( 9m 6 ) e 8 f Evlute using your clcultor: b c d 96 Solutions 6 = b 7 = 7 c x x = x = = = x d ( 9m 6 ) = 9 m 6 e 8 = 8 f 9 = 9 = 9 m = ( 8) = ( 9) = 7m = = = = - 7 Note from exmples e nd f the rule: 9 + x p - q q x p q = or ( x) p (f) is pretty tricky! b = 96 Using the squre root key 96 = 96 To evlute Press: x /y = Answer: = For roots higher thn squre root ( ) [or cube root ( x key], the x /y or ), if your clcultor hs key cn be used. You my need to use the inverse button. c To evlute 6 d To evlute Press: 6 x /y + / = Press x /y x y + / = Answer: 6 = 0 Answer: = Exercise 6:0 Write ech of the following using squre root sign. b 0 c 6 d e f 7 6 Use frctionl index to write: b c d 7 Foundtion Worksheet 6:0 Frctionl indices NS Simplify: 6 b 6 c Evlute: b 6 c 78 NEW SIGNPOST MATHEMATICS 9 STAGE..

13 Find the vlue of the following: b c d e f g h i j k l Assuming tht ll pronumerls used re positive, simplify: x x b c m m d 6x x e y y f 9n n g ( x ) h ( y 6 ) i ( 6 ) j ( b ) k ( 9x y 6 ) l ( 8x y ) 6 Evlute: ( x ) b = x b 9 b c 8 9 d e f 6 g h i j k l 6 6 Evlute using your clcultor, leving nswers s deciml numerls: b c d e f ( b) = b 79 g h i 8000 j ( 0 ) k ( 0 ) l ( 0 0) 7 If =, b = 8 nd c = 9, evlute the following: b - + b ( b) c c d ( c) e b + f ( b) g ( b) 6 h ( bc) 8 Use the fct tht = q x p, to simplify: ( 7 ) b ( x 6 y ) c ( 8m 9 ) d e f y6 b x p q x As = x, x x stnds for the positive squre root of x. Note x = x. 6:0 Who wnts to be millionire? Chllenge worksheet 6:0 Algebric expressions nd indices CHAPTER 6 INDICES AND SURDS 79

14 fun spot 6:0 Fun Spot 6:0 Why is room full of mrried people lwys empty? Work out the nswer to ech prt nd put the letter for tht prt in the box tht is bove the correct nswer. Write in index form: E 0 0 E E yyyy Find the vlue of: E E E A 0 A Find the vlue of x in: I x = 6 I x = 9 Write s bsic numerl: I 7 0 I Evlute: O x 0 O U y 0 N N 0 N ( ) N 6 B 7 C 7 H To fill jr in 6 minutes, Jn doubled the number of penuts in the jr every minute. After how mny minutes ws the jr hlf full? Simplify: S x x S x x T x R 0x 0 T x x T x 0 x S (x ) L 60x x b S R x 0 x P x x - G - b y x 0 x 7 x x x 9 x 8 x 8 x x investigtion 6:0 Investigtion 6:0 Resoning with frctionl indices Write x, x, x, x, x,... s expressions with frctionl indices nd describe the pttern tht emerges. Find the vlue of b if (x b ) = x Explin why 8 = = = ( ). Find some vlues tht x, p nd q could tke if x p q =. 80 NEW SIGNPOST MATHEMATICS 9 STAGE..

15 6:0 Scientific (or Stndrd) Outcome NS Nottion Investigtion 6.0 Multiplying nd dividing by powers of 0 Use the x y button on your clcultor to nswer these questions. Look for connection between questions nd nswers nd then fill in the rules t the end of the investigtion. Exercise 8 0 b 8 0 c 8 0 d 0 0 e 0 0 f 0 0 g 6 0 h 6 0 i j 6 0 k 6 0 l 6 0 investigtion 6:0 To multiply by 0 n move the deciml point plces to the. 8 0 b 8 0 c 8 0 d e f To divide by 0 n move the deciml point plces to the. The investigtion bove should hve reminded you tht: when we multiply deciml by 0, 00 or 000, we move the deciml point, or plces to the right when we divide deciml by 0, 00 or 000, we move the deciml point, or plces to the left. When expressing numbers in scientific (or stndrd) nottion ech number is written s the product of number between nd 0, nd power of 0. This number is written in scientific nottion (or stndrd form). 6 0 The first prt is between nd 0. The second prt is power of 0. Scientific nottion is useful when writing very lrge or very smll numbers. Scientific nottion is sometimes clled stndrd nottion or stndrd form. Numbers greter thn = 97 0 To write 970 in stndrd form: put deciml point fter the first digit count the number of plces you hve to move the deciml point to the left from its originl position. This will be the power needed. To multiply. 9 by 0, we move the deciml point plces to the right - which gives 9 0. CHAPTER 6 INDICES AND SURDS 8

16 worked exmples Express the following in scientific nottion. b c Write the following s bsic numerl. 0 b 0 c Solutions = 00 = 0 b = = 6 0 c = = = 0 00 = 0 If end zeros re significnt, write them in your nswer. eg (to nerest 00) = We hve moved the deciml point 7 plces from its originl position. b 0 = 00 = c = = To multiply by 0 7, move the deciml point 7 plces right. Numbers less thn = 97 0 To write in scientific nottion: put deciml point fter the first non-zero digit count the number of plces you hve moved the deciml point to the right from its originl position. This will show the negtive number needed s the power of Multiplying by 0 is the sme s dividing by 0 so we would move the deciml point plces left to get is the sme s worked exmples Express ech number in scientific nottion. 0 0 b c 0 00 Write the bsic numerl for: 9 0 b c 00 0 Short-cut method: 0 0 How mny plces must we move the deciml point for scientific nottion? Answer = Is 0 0 bigger or smller thn? Answer = smller So the power of 0 is. 0 0 = 0 8 NEW SIGNPOST MATHEMATICS 9 STAGE..

17 Solutions 0 0 = 00 b = c 0 00 = 000 = 0 = 97 0 = = b = = 0 09 = c 00 0 = = Explin the difference between 0 nd. b Explin the difference between 0 nd. c How mny seconds re in yers? d Hve you lived 8 0 hours? e Order the following, from smllest to lrgest f g Exercise 6:0 Write the thickness of sheet of pper in scientific nottion if 00 sheets of pper hve thickness of 8 cm. Estimte the thickness of the cover of this book. Write your estimte in scientific nottion. Write the bsic numerl for: 0 b 0 c 0 d e f g 7 0 h 7 0 i 7 0 Express in scientific nottion. (Assume tht finl zeros re not significnt.) 70 b 600 c 000 d 700 e f g 6 h i 90 j 970 k 6 00 l m 97 n 69 o 976 p q r s t u Express in scientific nottion b c 0 9 d 0 08 e f g 0 h 0 00 i j 0 6 k 0 00 l m n o Foundtion Worksheet 6:0 Scientific nottion NS Evlute: 6 0 b 6 0 c 6 0 Write in scientific nottion. 0 b 00 c 000 If you re stuck with this exercise, think bck to Investigtion 6:0... CHAPTER 6 INDICES AND SURDS 8

18 Write the bsic numerl for: 0 b 9 0 c 7 0 d 9 0 e f g 07 0 h 0 0 i 8 0 j 9 0 k 9 0 l 9 0 m 76 0 n 6 0 o 07 0 p 7 0 q r 0 6 s t 6 0 u v 9 0 w 0 0 x :0 Scientific Nottion nd Outcome NS the Clcultor prep quiz 6:0 Write in scientific nottion: Rewrite s bsic numerls: On clcultor: 7 0 is shown s is shown s.8 06 This is the clcultor s wy of showing scientific nottion. To enter scientific nottion, press: 7 Exp, to enter 7 0, nd Some clcultors re clled Scientific Clcultors becuse they cn give nswers in scientific (or stndrd) nottion. 8 Exp 6, to enter / To convert clcultor nswers into deciml form: 6 0 First prt Second prt Locte the deciml point in the first prt of the number (the prt betwen nd 0). Look t the sign of the second prt. This tells you in which direction to move the deciml point. If it is negtive the point moves to the left. If it is positive the point moves to the right. Look t the size of the second prt. This tells you how mny plces the deciml point hs to be moved. Move the deciml point to its new position, filling in ny gps, where necessry, with zeros. worked exmples Use clcultor to find the nswers for: (7 000) (86) (8 0 6 ) + ( ) 0 6 A clcultor will give n nswer in scientific nottion if the number is too lrge or smll to fit on the screen. 8 NEW SIGNPOST MATHEMATICS 9 STAGE..

19 Solutions (7 000) (86) The nswers to nd re too long to fit on the screen. = 967 = = = = (8 0 6 ) + ( ) 0 6 Press: 8 Exp Exp 7 Press: Exp 6 + / = = = 0 00 Note: Not ll clcultors work the sme wy. Exercise 6:0 Enter ech of these on your clcultor using the Exp key, nd copy the clcultor redout. 6 0 b 0 c 9 0 Rewrite these clcultor redouts in scientific nottion using powers of b.6 09 c.7 d. 0 e f.67 0 g h.0 i Explin why clcultor redout of different vlue to.. 0 hs Plce the nine numbers in question in order of size from smllest to lrgest. Give the nswers to these in scientific nottion, correct to significnt figures. 8 b c ( ) d e (6 87) f (0 00) (0 008) g h ( ) hs significnt figures, s four figures re used in the deciml prt. Use index lws to check the size of your nswer. Use clcultor to nswer correct to significnt figures, then use the index lws to check your nswer. 8 ( 0 ) b (8 0 ) c ( 0 8 ) ( ) d ( 8 0 ) e 68 ( 8 0 ) f (9 0 ) + (6 8 0 ) g h i b An Americn reported tht the dimeter of the sun is pproximtely miles. Write this in kilometres, using scientific nottion written correct to four significnt figures. There re 609 km in mile. If the sun s dimeter is 09 times tht of the erth, wht is the erth s dimeter, correct to three significnt figures? The distnce to the sun vries from km in Jnury to 0 8 km in July. This is becuse the erth s orbit is n ellipse. Wht is the difference between these distnces? CHAPTER 6 INDICES AND SURDS 8

20 c d e If we use the verge distnce to the sun ( km), how long would it tke light trvelling t m/s to rech the erth? (Answer correct to the nerest minute.) The mss of the erth is pproximtely 6 0 tonnes. The sun s mss is bout 00 times greter thn the mss of the erth. Wht is the mss of the sun correct to one significnt figure? We belong to the glxy known s the Milky Wy. It contins bout 0 strs. If the sun is tken to hve verge mss [see prt d], wht is the totl mss, correct to significnt figure, of the strs in the Milky Wy? investigtion 6:0 Investigtion 6:0 Using scientific nottion The speed of light is m/s. Use reference books nd your clcultor to complete this tble for five strs of your choice (eg Veg, Dog Str, Pole Str, Sirius). Nme of str The sun Distnce from the erth 0 8 km Time tken for light to trvel to the erth 0 6 is million. 0 9 is billion. Alph Centuri 0 km Order the distnces of the five strs from the erth, from smllest to lrgest. Reserch nnotechnology, which involves the use of very smll mchine prts. Prts re often mesured in micrometres. Mke comprisons between the sizes of components. Distnces in stronomy re mesured in light yers, which is the distnce tht light trvels in yer. A light yer is pproximtely km. 86 NEW SIGNPOST MATHEMATICS 9 STAGE..

21 6:06 The Rel Number System Outcome NS The rel number system is mde up of two groups of numbers: rtionl nd irrtionl numbers. Rtionl numbers Any number tht cn be written s frction, where nd b re whole numbers nd b 0, is b rtionl number. These include integers, frctions, mixed numbers, terminting decimls nd recurring decimls. 7 eg, 6,, 0 07, 0, 8 8 These exmples cn ll be written s frctions , -,,,, Note: An integer is rtionl number whose denomintor is. Irrtionl numbers It follows tht irrtionl numbers cnnot be written s frction, where nd b re whole numbers. We hve b met few numbers like this in our study of the circle nd Pythgors theorem. eg π,,, + The clcultor cn only give pproximtions for these numbers. The decimls continue without terminting or repeting , 6..., , Irrtionl numbers on the number line Although irrtionl numbers cnnot be given n exct deciml vlue, their positions cn still be plotted on the number line. As cn be seen from exercises on Pythgors theorem, number such s cn correspond to the length of side of tringle, nd so this length cn be shown on number line. Exmine the digrm on the right. If the length of the hypotenuse of this tringle is trnsferred, using compsses, to the number line s shown, we hve the position of on the number line. (This grees with the 0 deciml pproximtion from the clcultor of 6.) The previous construction cn be extended to give the position of other squre roots on the number line. Another irrtionl number you hve met before is π. You should know tht it hs n pproximte vlue of, so it would lie on the number line in the position shown. 0 6 π 0 CHAPTER 6 INDICES AND SURDS 87

22 Exercise 6:06 For ech number write rtionl or irrtionl. (A clcultor might help you to decide.) b c 0 d e 6 f π g h i j 0 99 k - l 9 m 6 n o 70 p q r + s t + 9 u v 9 w x 7+ y z 0 Tht looks irrtionl. Use your clcultor to find n pproximtion correct to deciml plce for the following. Also use these vlues to show the position of ech number on the number line. b c d 6 e 7 f 8 g 0 h i 0 j π Between which two consecutive integers does ech number below lie? b 8 c d 78 e 9 f g 80 h 0 i 90 j 90 Arrnge ech set of numbers below in order, from smllest to lrgest.,, b 8,, π c 0,, d 7, 0, 0, 6 e π,,, f 6, 6, 6, 0 g 8, 6, 7 9, 60 h 98, 0, 0, 0 i, π, 9, j 0,,, k 0, 90,, 0 l 600, 60,, This digrm shows nother construction for locting squre roots on the number line. Use set squre to drw the tringles on grph pper, then use compsses to drw the rcs on the number line. Extend your digrm to show 6. Check the ccurcy of your constructions with your clcultor NEW SIGNPOST MATHEMATICS 9 STAGE..

23 6 7 To show multiples of squre root on the number line we cn use pir of compsses to mrk off equl intervls. Repet the instructions in question to find the position of on the number line. Then use pir of compsses to mrk the position of nd. 0 b Drw digrm nd show the position of, nd on number line. The position of π on the number line cn be shown by doing the following. Use the dimeter of twenty cent coin to mrk off units on number line. Why is it so? 0 Then mrk point on the circumference of the coin, lign it with zero on the number line, then roll it long the line crefully until the mrk meets the number line gin. This will show the position of π on the number line. 0 Reding mths 6:06 Proof tht Let us suppose tht is rtionl nd, therefore, cn be written s frction in p the form where p nd q re positive q integers with no common fctor. (This ssumption is essentil.) p so = q p q then = (squring both sides) nd q = p is irrtionl These numbers re bsurd. They cn t be written s n exct deciml. reding mths 6:06 CHAPTER 6 INDICES AND SURDS 89

24 This lst step implies tht p must be divisible by (since is prime). Therefore must divide into p exctly. p cn be expressed in the form k for some integer k. q = (k) q = k q = k Perhps tht s why they clled them SURDS. Now, s for p bove, it cn be rgued from this lst step tht q must be divisible by. But p nd q were sid to hve no common fctor, hence contrdiction exists. So our originl ssumption ws wrong. p Therefore p nd q cnnot be found so tht =. Hence must be irrtionl. q Try to use the method bove to prove tht these re irrtionl. chllenge 6:06 Chllenge 6:06 f-stops nd Professionl photogrphers hve cmers tht cn lter shutter time nd perture settings using wht re clled f-stops The f-stops,, 8 nd 6 re ccurte. More ccurte redings for the rest re given below the scle. Find the pttern in the ccurte f-stops. * Try squring ech ccurte f-stop number. * Try dividing ech f-stop number by the one before it. Try to discover how f-stops re used. 90 NEW SIGNPOST MATHEMATICS 9 STAGE..

25 6:07 Surds Outcome NS Find the vlue of: ( 6) 9 9 prep quiz 6:07 Surds re numericl expressions tht involve irrtionl roots. They re irrtionl numbers. So, 7, + nd 0 re ll surds. Surds obey the following rules, which re suggested by Prep Quiz 6:07. Rule xy = x y worked exmples 00 = 7 = 9 7 = 7 = = = = 0 (which is true) = Rule x y = x y Note: x mens the positive squre root of x when x > 0. x = 0 when x = 0. worked exmples = - = 0 = 0 = = 6 ie = = = (which is true) Rule ( x) = x Note: For x to exist, x cnnot be negtive. worked exmples ( ) = () ( 7) = 7 ( ) = ( ) = = 9 = 8 CHAPTER 6 INDICES AND SURDS 9

26 A surd is in its simplest form when the number under the squre root sign is s smll s possible. To simplify surd we mke use of Rule by expressing the squre root s the product of two smller squre roots, one being the root of squre number. Exmine the exmples below. worked exmples Simplify the following surds. 8 = 9 7 = 8 = 6 = = = = = = 0 Exercise 6:07 Simplify: b c 7 6 d 6 7 e 0 f g h i j 7 k 0 l 7 m 6 n o 77 p 7 q r - s - t - 0 Foundtion Worksheet 6:07 Surds NS Simplify: 6 b 7 Simplify: 8 b 8 Squre ech of the surds below (Rule ). 6 b 9 c d 00 e f 8 g h 7 i j k l m 7 n 7 o 9 p 6 q 0 0 r 9 0 s 6 0 t Simplify ech of these surds. 8 b 0 c d 0 e f g h i 8 j 90 k 6 l 6 m n o 08 p 0 q 99 r 60 s 96 t 76 u 68 v 6 w 00 x 6 9 NEW SIGNPOST MATHEMATICS 9 STAGE..

27 Simplify ech surd nd then, tking = nd = 7, give deciml pproximtion correct to deciml plce. 8 b 7 c 8 d e f 8 g 0 h 6 Write ech surd below in its simplest form. b 8 c 0 d 8 e 0 f 7 g 0 7 h 6 i j k l m 7 6 n 7 o p 90 q 6 00 r 98 s 9 08 t 68 6 Simplified surds cn be written s n entire root by reversing the bove process. For exmple, = 6 = 8. Express the following s entire squre roots. b c d 6 e f g 7 h i 6 j 6 k 0 l 7 m 6 7 n 0 o 7 p 0 q 9 r 8 s 7 9 t 6:08 Addition nd Subtrction Outcome NS of Surds Simplify: 0 0 Evlute: (to deciml plce) (to deciml plce) (to deciml plce) prep quiz 6:08 As cn be seen from the Prep Quiz, if x nd y re two positive numbers, x+ y does not equl x+ y nd Surds re x y does not equl x y irrtionl. For exmple: + 8 does not equl. 0 7 does not equl. CHAPTER 6 INDICES AND SURDS 9

28 Only like surds cn be dded or subtrcted. For exmple: + = 6 nd = However, we cn only tell whether surds re like or unlike if ech is expressed in its simplest form. Exmine the following exmples. I think I m beginning to like this. When dding or subtrcting surds write ech surd in its simplest form dd or subtrct like surds only. worked exmples Simplify ech of the following = 9 = = + = + = = = ( ) + ( ) = ( ) + ( ) = + = 0 + = 6 = 0 + Remember how in lgebr only like terms could be dded or subtrcted. eg x + x = 8x 7 6 = Exercise 6:08 Foundtion Worksheet 6:08 Addition nd subtrction of surds NS Simplify. Simplify: + b + 7 c 6+ 6 b d 0 7 e 9 6 f g 7+ 7 h + i b j k l + m 0 n + 7 o Collect like surds Simplify by collecting like surds b c d e f + 6 g h i + 6 j NEW SIGNPOST MATHEMATICS 9 STAGE..

29 Simplify completely: 8+ b + c + 8 d + 0 e 7 + f 6+ g 8 h 0 i j 8 + k 0 + l 7 8 m 7 n o 0+ p 8+ 6 q 0 r 7 8 Simplify: b + 0 c d e + 0 f g h :09 Multipliction nd Outcome NS Division of Surds worked exmples Simplify the following ( ) Solutions x y = x y 7 = 7 =, ( = ) = = = = 96 = 8 6 = 8 = 8 = = = 60 8 = - 6 ( ) = = = = 6 6 = - At this point we could cncel like this: = = 6 = 6 Using the surd rules, these re esy! x y = xy CHAPTER 6 INDICES AND SURDS 9

30 Exercise 6:09 Simplify these products: b 7 c d e 7 f 0 7 g 8 h i 0 j 6 k 0 l 0 m n o 7 p q r s 0 t 6 8 u v x x w 8x x x x Foundtion Worksheet 6:09 Multipliction nd division of surds NS Simplify: 7 b Simplify: 0 b 0 6 Simplify: 0 b c 6 d 7 e 8 f g h 6 i 0 j k 0 0 l 7 m 0 n 9 6 o 0 p 0 q 7 r 8 0 s x t u 0p p Simplify fully: b - 0 c d 6 - e - f g h - 0 i Expnd nd simplify: ( + ) b ( + ) c 7 ( 7 ) d ( ) e ( ) f 0( ) g ( + ) h ( + ) i ( ) j 6( + 6) k 7( 7 ) l 7 ( ) m ( + ) n ( ) o 6( 6 ) p ( + ) q x( x + ) r y( y + x) 6:09 Golden rtio investigtions 96 NEW SIGNPOST MATHEMATICS 9 STAGE..

31 Investigtion 6:09 Itertion to find squre roots Itertion is the repetition of process. We cn use simple process to find squre roots. Exmple Find correct to deciml plces without using clcultor. Step Estimte. Let E = 6 (We wnt E to be close to.) Step Divide by your estimte. 6 = 87 Since 6 87 =, the correct nswer must lie between 6 nd 87. Step Averge these two numbers to get better estimte = Use 77 s the new estimte nd repet the steps bove (iterte). E = = To iterte, repet the process over nd over gin. To iterte, repet the process over nd over gin. Cn you find other uses for itertion? investigtion 6:09 If we use 706 s our next estimte we get better pproximtion (ie 70). Since lies between 706 nd 70 then = 7 correct to deciml plces. Use itertion to find, correct to deciml plces: b c 70 d 0 Investigte finding the squre root of number n, using itertion of the formul: x New estimte = + n - x where x is your lst estimte nd we wish to find n. CHAPTER 6 INDICES AND SURDS 97

32 6:0 Binomil Products Outcome NS prep quiz 6:0 Simplify the following: 8 ( ) ( ) Expnd nd simplify where possible: 8 ( 7 ) 9 0( 0 + ) 0 ( ) see :06 I remember! In Chpter, you sw how to expnd binomil product. ( + b)(c + d) = (c + d) + b(c + d) = c + d + bc + bd The sme procedure is used if some of the terms re not pronumerls, but surds. Exmine the following exmples. worked exmples see :07A see :07B Expnd nd simplify: ( + ) ( ) b ( ) ( + ) ( + ) b ( 7 ) ( ) ( + ) b ( 7) ( + 7) Solutions ( + ) ( ) b ( ) ( + ) = ( ) + ( ) = ( + ) ( + ) = ( ) + = = = 7 0 These re = perfect ( + ) b ( 7 ) squres. = ( ) + + ( ) = ( 7) 7 + ( ) = = 7 + Remember! = = 0 ( + b) = + b + b ( b) = b + b These give the difference ( ) ( + ) = ( ) ( ) of two squres. = = b ( 7) ( + 7) = ( ) ( 7) Remember! ( + b)( b) = b = 0 7 = 98 NEW SIGNPOST MATHEMATICS 9 STAGE..

33 Exercise 6:0 Expnd nd simplify the following: ( + ) ( + ) b ( + ) ( ) b c ( 7 ) ( 7 ) d ( + ) ( + ) e ( 7 ) ( ) f ( 0 + ) ( + ) g ( + ) ( + ) h ( ) ( ) i ( 6 ) ( 6) j ( + ) ( + ) k ( + ) ( + ) l ( + ) ( + ) m ( 7) ( 7 ) n ( ) ( + ) o ( 7 ) ( 7) p ( + 7) ( 7) q ( ) ( 7+ 0) r ( + 7 ) ( 7 ) s ( 9 ) ( + ) t ( 7+ ) ( 7 7) u ( 6 + ) ( 7 ) v ( x + ) ( x + ) w ( m + n) ( m+ n) x ( b) ( + b) Foundtion Worksheet 6:0 Binomil products surds NS Expnd nd simplify: (x + )(x + ) ( + ) ( + ) Simplify: (x + ) ( + ) b ( ) c ( + ) d ( + ) e ( ) f ( + 0) g ( + ) h ( ) i ( + ) j ( + 7) k ( + 0) l ( 7 ) m ( + ) n ( 7 ) o ( 0 0 ) p ( x+ y) q ( m + ) r ( p q) ( + ) ( ) b ( + ) ( ) c ( 0 7) ( 0 + 7) d ( ) ( + ) e ( + ) ( ) f ( 7 ) ( 7+ ) g ( 0 8) ( 0 + 8) h ( + 7) ( 7) i ( ) ( + ) j ( 6 ) ( 6+ ) k ( 7+ ) ( 7 ) Importnt notice! l ( ) ( + ) The two binomils in ech prt of m ( ) ( + ) question re sid to be conjugte surds. n ( + ) ( ) Note tht when binomil surd is o ( x+ y) ( x y) multiplied by its conjugte, the nswer is lwys rtionl number. p ( + b) ( b) b ( + ) CHAPTER 6 INDICES AND SURDS 99

34 6: Rtionlising the Outcome NS Denomintor prep quiz 6: Simplify the following: ( + ) ( ) 7 ( ) ( + ) 8 ( ) ( + ) 9 ( ) ( + ) 0 ( ) ( + ) If frction hs surd (ie n irrtionl number) in its denomintor, we generlly rewrite the frction with rtionl denomintor by using the method shown below. Solutions worked exmples Rewrite with rtionl denomintors: For these frctions, we multiply top nd bottom by the squre root in the denomintor. = - = - = - = - = = = - = Note: = - ( + ) Multiplying by is the = sme s multiplying by. + = - = Exercise 6: Rtionlise the denomintor for ech of the following: b c d - e 0 f g 0 h - i j k - 0 l m n - o - p - q - r s t - u v w 7 x NEW SIGNPOST MATHEMATICS 9 STAGE..

35 Evlute ech frction correct to significnt figures (using your clcultor). Then rtionlise the denomintor nd evlute the frction gin. Compre this nswer with your first clcultion. 7 b c - d - 7 Rtionlise ech denomintor, then express s single frction. + b c + d e f - - g h + i Fun Spot 6: Wht do Eskimos sing t birthdy prties? Answer ech question nd put the letter for tht question in the box bove the correct nswer. x A = E = E x x L = 0 8 E 0 O 7 Simplify: x x - = 86 0 x = 0 60 = fun spot 6: L x x O x x Y x x E x y L x x O x 6 x F ( x ) G x 0 D 0 E 69 F 7 7 G ( ) H + I 0 J 6 L 7 7 N O R S 6 T + Solve: W x + = Y - x = Z x = 7 6x 0 x 8 x x 7 x = 8 x 0 8x 6xy x = 9 7 x = 6 x = 8 6 x = x = x = x = x = 0 CHAPTER 6 INDICES AND SURDS 0

36 chllenge 6: Chllenge 6: Rtionlising binomil denomintors Exmples Rtionlise the denomintors for ech expression. - + For these frctions, we multiply top nd bottom by the conjugte of the denomintor. + - Solutions + - = - = ( + ) = = - + = = + - = = = ( + ) Note: The product of binomil surd nd its conjugte is lwys rtionl. Now try these exercises! Express with rtionl denomintor. b - c d e 0 f - g - + h i - j - k l m + + n o + p Rtionlise ech denomintor, then express s single frction. - + b - c NEW SIGNPOST MATHEMATICS 9 STAGE..

37 Mths terms 6 bse The term which is operted on by the index. eg for x n, x is the bse for, is the bse. conjugte The binomils tht multiply to give the difference of two squres re the conjugte of ech other. eg ( b) nd ( + b) ( + ) nd ( ) These re conjugte pirs. exponent Another term for power or index. Equtions which involve power re clled exponentil equtions. eg x = 7 frctionl indices Another wy of writing the root of number or term. x = x, x = x, x index A number indicting how mny of bse term need to be multiplied together. eg for x n, n is the index x n = x x x x x x x x n fctors The plurl of index is indices. irrtionl numbers Numbers tht cnnot be expressed in the form where nd b re integers. b They cnnot be given n exct deciml vlue. eg π,, + n = n x negtive indices Indicte the reciprocl of term. eg x =, x n = x x n ie =, = = power Another term for n index or exponent. rtionl numbers Numbers tht cn be written in the form where nd b re integers (b 0). b They cn be expressed s terminting or repeting deciml. eg integers, frctions, percentges rel numbers The combintion of rtionl nd irrtionl numbers. scientific (stndrd) nottion A useful wy to write very big or very smll numbers. Numbers re written s the product of number between nd 0 nd power of 0. eg = = surds Numericl expressions tht involve irrtionl roots. eg,, 7+ zero index A term or number with zero index is equl to. eg x 0 =, 0 = mths 6 terms Mths terms 6 CHAPTER 6 INDICES AND SURDS 0

38 dignostic test 6 Dignostic Test 6: Indices nd Surds These questions reflect the importnt skills introduced in this chpter. Errors mde will indicte res of wekness. Ech wekness should be treted by going bck to the section listed. These questions cn be used to ssess outcomes PAS, PAS, NS, NS. Express in index form: b c m m m Evlute: b c 0 Simplify: b x x c 6m n mn Simplify: x 7 x b c 0 b 0b Simplify: ( ) b (x ) c ( ) 6 Simplify: 7 0 b p 0 c 8x 6x 7 Simplify: b c ( ) 8 Simplify, writing nswers without negtive indices: x 7 x b 6x x c (x ) 9 Simplify: b 7 c 0 If x > 0 nd m > 0, simplify: x x b ( 9m 6 ) c Express in scientific nottion: b c Write s bsic numerl: 0 b 0 c Express in scientific nottion: 0 0 b c 0 00 Write the bsic numerl for: 9 0 b c 00 0 Simplify, giving nswers in scientific nottion: ( 0 8 ) b (8 0 6 ) + ( ) c 96 0 d ( 8x ) Section 6:0 6:0 6:0 6:0 6:0 6:0 6:0 6:0 6:0 6:0 6:0 6:0 6:0 6:0 6:0 0 NEW SIGNPOST MATHEMATICS 9 STAGE..

39 6 For ech, write rtionl or irrtionl. 0 b 0 c d 6 7 Evlute correct to deciml plces: b c d 7 8 Simplify ech surd. 0 b 7 c 8 d 7 9 Express ech of the following s n entire squre root. b c 7 d 0 Simplify completely: + b 6 c 8 d 7 + Simplify: 6 b c d 8 Simplify: b 8 c d 0 0 Expnd nd simplify: c ( + ) ( + ) b ( + ) ( ) ( 7+ ) d ( ) ( + ) Rtionlise the denomintor of: + b c d - 6:06 6:06 6:07 6:07 6:08 6:09 6:09 6:0 6: Cn you use your clcultor to find the vlue of 00? Wht is the lrgest power of tht cn be clculted using your clcultor? CHAPTER 6 INDICES AND SURDS 0

40 ssignment 6A Chpter 6 Revision Assignment Simplify, writing the nswers in index form: b c b b d b b e f 6 g 7m m h y 6 y i 0 b 0 b j 7 k ( ) l (x ) m ( ) n m 7 (m ) o x - 6x Express in simplest form: (x ) 0 b 6x 0 c (x ) d (0 ) e (x ) 8x Write ech of the following in stndrd form (scientific nottion). 600 b c d Write ech of the following s bsic numerl. 8 0 b 67 0 c 0 d 06 0 Use your clcultor to evlute: 0 b c 6 6 d 7 6 Find the vlue of n, if: n = 8 b n = c 0 n = Simplify nd evlute: b c ( ) 8 Simplify: m 7 m - 6 b ( ) ( ) m 0 8x c 7 9x 6x 6 6x 9 Noting tht x x = ( ), evlute without using clcultor: b 8 c 9 d p Simplify: x x b 0x x c ( 6m n 6 ) Simplify ech expression. 0 b c 8 8 d 7 e 0 f g ( + ) h ( 7+ ) i ( + 6) ( + ) j m n k ( m + n) l ( m + n) ( m n) Rtionlise the denomintor of ech expression: - b - c + d - - Extension Rtionlise the denomintor of ech expression. c Simplify ech expression, writing your nswer with rtionl denomintor. c - + b d - b NEW SIGNPOST MATHEMATICS 9 STAGE..

41 Chpter 6 Working Mthemticlly Use ID Crd 7 on pge xxii to identify: b 8 c 7 d 8 e 9 f 0 g h i j Use ID Crd 6 on pge xxi to identify numbers to. How mny digonls cn be drwn from one vertex of regulr hexgon? How mny vertices hs hexgon? b Ech digonl joins two vertices nd digonl cnnot be drwn from vertex to the two djcent vertices or to itself. The number of digonls of hexgon 66 ( ) is -. How mny digonls hs: i regulr octgon? ii regulr decgon? iii regulr polygon tht hs 0 sides? Tom ws given $.6... cheque for n mount... $6.? between $ nd $. The bnk teller mde mistke nd exchnged dollrs nd cents on the cheque. Tom took the money without exmining it nd gve cents to his son. He now found tht he hd twice the vlue of the originl cheque. If he hd no money before entering the bnk, wht ws the mount of the cheque? ssignment 6B In the decibel scle, for mesuring noise, 0 decibels is noise tht is brely udible. A noise 0 times s intense is 0 decibels, nd so on up to 0 decibels, which is the threshold of pin. Study the tble nd nswer the questions below. Noise Reltive intensity Decibels Minimum of udible sound 0 Soft wind on leves 0 0 Whisper t metre 0 0 Bush quiet 0 0 b c d e If ordinry converstion hs reltive intensity of 0 6, wht is its loudness in decibels? If lwn mower hs reltive intensity of 0, wht is its loudness in decibels? By how mny times is the reltive intensity of the mower greter thn tht of converstion? By how mny times is the reltive intensity of hevy trffic (loudness 80 db) greter thn tht of bush quiet? From the bove it would pper tht hevy trffic (80 db) is four times s noisy s whisper t metre (0 db). However, rise of 0 db corresponds to doubling in the subjective loudness to the humn er. How much louder to the humn er is: i the verge office (0 db) thn bush quiet (0 db)? ii hevy trffic (80 db) thn whisper t metre (0 db)? iii rock group (0 db) thn business office (60 db)? Index lws Negtive indices Frctionl indices Simplifying surds Opertions with surds CHAPTER 6 INDICES AND SURDS 07