ALGEBRAIC PRODUCTS AND QUOTIENTS IN INDEX NOTATION

Chpter 8 Indices Contents: A Algeric products nd quotients in index nottion B Index lws C Expnsion lws D Zero nd negtive indices E Scientific nottion (Stndrd form) F Significnt figures

74 INDICES (Chpter 8) OPENING PROBLEM Hilert wnted to find the numer of toms in one kilogrm of pure gold. He ws surprised to find tht some scientists ctully define wht kilogrm mens in terms of gold. They do this ecuse gold is perfectly stle element. Hilert found tht under such definition, kilogrm equls 3 057 443 66 23 38 88 734 962 toms of gold. Things to think out: ² How cn we est pproximte this numer? ² How cn we write this numer to mke it esier to use? ² Cn such form e used to write very smll numers? Some other lrge numers include: ² The numer of grins of snd in lrge ech is out 00 000 000 000 000. ² The weight of se wter throughout the world is out 50 000 000 000 000 000 000 kg. ² Ech drop of wter contins out 2 000 000 000 000 000 000 000 molecules. A ALGEBRAIC PRODUCTS AND QUOTIENTS IN INDEX NOTATION We hve seen tht: If n is positive integer, then n is the product of n fctors of. n = :::::: {z } n fctors Exmple Simplify: m 3 m 2 p 5 p m 3 m 2 =(m m m) (m m) = m 5 p 5 p =(p p p p p) p = p 6 p 5 is the product of five ps.

INDICES (Chpter 8) 75 EXERCISE 8A Simplify: 2 c c c 3 d n 2 n 2 e m m 2 f s s 4 g e 3 e 3 h k 3 k 2 i p 2 p 4 j p 2 p k k 6 k 3 l m 3 m m k 6 k n n n 4 n o e 4 e 3 e p n n n 2 Exmple 2 Simplify: (x 2 ) 2 (5 3 ) 2 (x 2 ) 2 = x 2 x 2 = x x x x = x 4 (5 3 ) 2 =5 3 5 3 =5 5 =25 6 2 Simplify: ( 2 ) 2 (s 3 ) 2 c (g 2 ) 3 d (c) 2 e (m 4 ) 2 f (3) 2 g (2 2 ) 2 h (5 2 ) 2 i (3 2 ) 2 j (7m 2 n) 2 k (8 2 c) 2 l (3x 2 y) 3 m ( ) 2 n ( 2) 2 o ( 3) 3 p 2 ( 5) 2 Exmple 3 Simplify these quotients: x 5 x 6 3 x 5 x = x x x x x x = x 4 6 3 = = 3 3 Simplify: e i 2 k5 k 2 2g 7 3g 4 f j m3 m c t 2 t 2 d y4 y 2 p 5 p 3 g 2n6 n 2 h 6r5 3r 4 8d 8 2d 3 k 0e7 2e 2 l 5h 9 5h 4

76 INDICES (Chpter 8) B INDEX LAWS INVESTIGATION Look for ny ptterns s you complete the following: Copy nd complete: 2 2 2 3 = (2 2) (2 2 2) = 2 5 3 3 3 = = c 3 4 = = From the ove exmples, m n =. 2 Copy nd complete: DISCOVERING INDEX LAWS 25 2 3 = 2 2 2 2 2 =2 2 34 2 2 2 3 = = 5 c 2 = = d x7 x 4 = = m From the ove exmples, n =. 3 Copy nd complete: (2 3 ) 2 = 2 3 2 3 = (2 2 2) (2 2 2) = (3 2 ) 4 = = = From the ove exmples, ( m ) n =. From Investigtion you should hve found these index lws for positive indices: If m nd n re positive integers, then: WORKSHEET ² m n = m +n To multiply numers with the sme se, keep the se nd dd the indices. ² m = n m n, 6= 0 To divide numers with the sme se, keep the se nd sutrct the indices. ² ( m ) n = m n When rising power to power, keep the se nd multiply the indices. Exmple 4 Simplify using the lws of indices: 2 3 2 2 x 4 x 5 When multiplying, keep the se nd dd the indices. 2 3 2 2 =2 3+2 =2 5 =32 x 4 x 5 = x 4+5 = x 9

INDICES (Chpter 8) 77 EXERCISE 8B Simplify using the index lws: 2 2 2 4 3 2 3 2 c 5 3 5 4 d 7 2 7 3 e 2 3 f n 4 n g x 5 x 6 h 2 5 Exmple 5 When dividing, keep Simplify using the index lws: 35 3 3 p7 p 3 the se nd sutrct the indices. 3 5 3 3 =3 5 3 =3 2 =9 p 7 p 3 = p 7 3 = p 4 2 Simplify using the index lws: 25 2 2 34 3 3 c e x7 x 2 f 5 6 5 4 d 07 0 4 y 8 y 2 g 6 5 h 9 4 For power to power, keep the se nd multiply the indices. Exmple 6 Simplify using the index lws: (2 3 ) 2 (x 4 ) 5 (2 3 ) 2 =2 3 2 =2 6 =64 (x 4 ) 5 = x 4 5 = x 20 3 Simplify using the index lws: (2 2 ) 3 (0 4 ) 2 c (3 3 ) 4 d (2 3 ) 5 e (x 3 ) 4 f (x 5 ) 2 g ( 4 ) 5 h ( 3 ) 7 4 Simplify using the index lws: 3 2 7 2 c c 4 c 3 d d 4 d e ( 4 ) 3 f 8 5 g ( 2 ) 5 h 3 n i 4 3 j m 4 m 2 m 3 k ( 2 ) 3 l (g 3 ) 3 g 2

78 INDICES (Chpter 8) Exmple 7 Simplify using the index lws: 2 2 3 2 3 6x4 y 3 3x 2 y 2 2 3 2 3 =2 3 2 2 3 =6 2 2+3 =6 2 5 6x 4 y 3 3x 2 y = 6 3 x4 2 y 3 =2x 2 y 2 5 Simplify using the index lws: 3 e 22 3 3 4 2 2 3 c 0hk 3 4h 4 d 47 2 2 m 5 n 4 m 2 f m 2 n 3 g (m 2 ) 8 h p2 p 7 (p 3 ) 2 C EXPANSION LAWS INVESTIGATION 2 DISCOVERING EXPANSION LAWS Look for ny ptterns s you complete the following: Copy nd complete the following: () 4 = = = () 3 = = = c (2) 5 = = = In generl, () n =. 2 Copy nd complete: ³ 3 = = = ³ 4 = = = In generl, ³ n = for 6= 0. WORKSHEET From Investigtion 2 you should hve found these expnsion lws for positive indices: If n is positive integer, then: ² () n = n n µ n ² = n provided 6= 0. n

INDICES (Chpter 8) 79 Exmple 8 Rise ech Remove the rckets nd simplify: () 5 (2xy) 3 () 5 = 5 5 (2xy) 3 =2 3 x 3 y 3 =8x 3 y 3 fctor to the given power. EXERCISE 8C Remove the rckets of the following nd simplify: () 4 (xy) 3 c (c) 4 d (xyz) 3 e (2x) 5 f (3) 4 g (2m) 3 h (5) 3 Exmple 9 Remove the rckets nd simplify: ³ µ m 4 2 n 3 Rise oth the numertor nd the denomintor to the given power. ³ m n 4 µ 3 2 = 23 3 = m4 n 4 = 8 3 2 Remove the rckets of the following nd simplify: µ 2 x ³ µ 3 4 p c y q d e µ 2 3 ³ µ 4 y 2 f g x 5 3 h ³ c d 5 µ 5 2 Exmple 0 Express the following in simplest form, without rckets: (3 2 ) 2 (4 2 ) 3 (3 2 ) 2 =3 2 ( 2 ) 2 =9 4 (4 2 ) 3 =4 3 ( 2 ) 3 3 =64 6 3

80 INDICES (Chpter 8) 3 Express the following in simplest form, without rckets: (2 3 ) 2 (5n 2 ) 2 c (3x 2 ) 2 d (3 3 ) 2 e (xy 2 ) 2 f (x 2 y) 3 g (2) 2 3 h (3 2 ) 2 i (3 2 ) 3 j (7 3 c) 2 k (2 2 ) 4 l (3 3 ) 3 Exmple Express the following in simplest form, without rckets: ³ µ xy 2 4 2 3 3 3 ³ xy 2 3 = x2 y 2 3 2 = x2 y 2 9 µ 4 2 3 3 = 24 4 3 4 ( 3 ) 4 = 64 8 2 4 Express the following in simplest form, without rckets: µ 2 µ 2 µ 3 3 2m c 2 c n d e µ 2 2 2 µ c 2 3 µ 4 2 2 f g 2d 3 h µ m 2 4 µ 2 3 3 2 4 D ZERO AND NEGATIVE INDICES From our originl definition n = ::::: {z }, 0 hs no mening. n times However, consider the pttern: 2 3 =8 2 2 =4 2 =2 If we continue the pttern, we get: 2 0 = Ech time the power of 2 decreses y, the result is hlved. 2 = 2 = 2 2 2 = 4 = 2 2 In other ses such s 3 or 5, the sme pttern will occur. We therefore define the following lws for zero nd negtive indices:

INDICES (Chpter 8) 8 ZERO INDEX LAW 0 = for ll 6= 0. NEGATIVE INDEX LAW If is ny non-zero numer nd n is n integer, then n = This mens tht n nd n re reciprocls of one nother. In prticulr notice tht =. n Exmple 2 Simplify: 7 0 x 0 c y 4 y 4 d 2+5 0 7 0 = x 0 = c y 4 y 4 = y 4 4 = y 0 = d 2+5 0 =2+ =3 Exmple 3 Simplify: 3 5 2 c 0 4 The negtive index indictes the reciprocl. 3 = 3 = 3 5 2 = 5 2 = 25 c 0 4 = 0 4 = 0 000 EXERCISE 8D Simplify: 3 0 6 0 c 0 0 d 8 0 e y 0 f 0 g 2x 0 h (2x) 0 i x 3 5 x 3 j 5 k 3+2 0 l 5 7 0 2 Simplify, giving nswers in simplest rtionl form: 4 2 c 6 d 8 e 2 2 f 3 2 g 7 2 h 9 2 i 3 3 j 0 5

82 INDICES (Chpter 8) Exmple 4 Simplify, giving nswers in simplest rtionl form: 2 3 3 2 5 c 8 0 8 2 3 3 5 2 c 8 0 8 = 3 2 = 3 2 = 5 2 3 = 52 3 2 = 25 9 = 8 = 7 8 3 Simplify, giving nswers in simplest rtionl form: ( 2 ) ( 4 ) c ( 4 5 ) d ( 5 3 ) e ( 9 ) f 2 0 +2 g ( 2 3 ) 2 h ( 3 4 ) 2 i ( 3 4 ) 3 j 3 0 +3 3 Exmple 5 Write the following without rckets or negtive indices: 8 8() 8 = 8 8() =8 = 8 = 8 4 Write the following without rckets or negtive indices: 2 (2) c 4 d (4) e 3 2 f (3) 2 g (5c) 2 h 5c 2 i xy j (xy) k xy 2 l (xy) 2 m 2 n (2) o 2() p (3n 2 ) 5 Write s powers of 0: 000 000 000 c 0:00 d 0:000 000 0 6 Write s powers of 2, 3 or 5: 8 8 c 9 d 9 e 25 f 25 g 32 h 32 i 8 j 8 k 25 l

INDICES (Chpter 8) 83 E SCIENTIFIC NOTATION (STANDARD FORM) Oserve the pttern: We cn use this pttern to simplify the writing of very lrge nd very smll numers. For exmple, 300 000 =3 00 000 =3 0 5 SCIENTIFIC NOTATION nd 0:0002 = 2 0 000 = 2 0 000 =2 0 4 Scientific nottion or stndrd form involves writing ny given numer s numer etween inclusive nd 0, multiplied y power of 0, i.e., 0 k where 6 <0 nd k is n integer. Exmple 6 0 0 0 0 000 = 0 4 0 0 0 0 000 = 0 3 00=0 2 0 = 0 =0 0 0 =0 00 =0 2 000 =0 3 As we divide y 0, the exponent or power of 0 decreses y one. Write in scientific nottion: 37 600 0:000 86 37 600 = 3:76 0 000 fshift deciml point 4 plces to the =3:76 0 4 left nd 0 000g 0:000 86 = 8:6 0 4 fshift deciml point 4 plces to the =8:6 0 4 right nd 0 000g Notice tht: ² If the originl numer is > 0, the power of 0 is positive (+). ² If the originl numer is <, the power of 0 is negtive ( ). ² If the originl numer is etween nd 0, we write the numer s it is nd multiply it y 0 0, which is relly just.

84 INDICES (Chpter 8) EXERCISE 8E. Write the following s powers of 0: 2 00 000 c 0 d 00 000 e 0: f 0:0 g 0:000 h 00 000 000 Express the following in scientific nottion: 259 259 000 c 2:59 d 0:259 e 0:000 259 f 40:7 g 4070 h 0:0407 i 407 000 j 407 000 000 k 0:000 040 7 3 Express the following in scientific nottion: The distnce from the erth to the sun is 49 500 000 000 m. Bcteri re single cell orgnisms, some of which hve dimeter of 0:0003 mm. c A speck of dust is smller thn 0:00 mm. d e f The proility tht your six numers will e selected for Lotto on Mondy night is 0:000 000 4 62. The centrl temperture of the sun is 5 million degrees Celsius. A single red lood cell lives for out four months nd during this time it will circulte round the ody 300 000 times. Exmple 7 Write s n ordinry numer: 3:2 0 2 5:76 0 5 3:2 0 2 =3:20 00 = 320 5:76 0 5 = 000005:76 0 5 =0:000 057 6 4 Write s n ordinry deciml numer: 4 0 3 5 0 2 c 2: 0 3 d 7:8 0 4 e 3:8 0 5 f 8:6 0 g 4:33 0 7 h 6 0 7 5 Write s n ordinry deciml numer: 4 0 3 5 0 2 c 2: 0 3 d 7:8 0 4 e 3:8 0 5 f 8:6 0 g 4:33 0 7 h 6 0 7 6 Express the following quntities s ordinry deciml numers: The wvelength of light is 9 0 7 m. The estimted world popultion for the yer 2000 ws 6:30 0 9. c The dimeter of our glxy, the Milky Wy, is 0 5 light yers. d The smllest viruses re 0 5 mm in size. e The mss of ee s wing is 0 7 kg.

INDICES (Chpter 8) 85 Exmple 8 Simplify the following, giving your nswer in scientific nottion: (5 0 4 ) (4 0 5 ) (8 0 5 ) (2 0 3 ) (5 0 4 ) (4 0 5 ) =5 4 0 4 0 5 =20 0 4+5 =2 0 0 9 =2 0 0 (8 0 5 ) (2 0 3 ) = 8 05 2 0 3 = 8 2 05 3 =4 0 2 7 Simplify the following, giving your nswer in scientific nottion: (3 0 3 ) (2 0 2 ) (5 0 2 ) (7 0 5 ) c (5 0 4 ) (6 0 3 ) d (3 0 3 ) 2 e (5 0 4 ) 2 f (8 0 2 ) 2 g (8 0 4 ) (4 0 2 ) h (8 0 5 ) (2 0 3 ) SCIENTIFIC NOTATION ON A CALCULATOR Scientific clcultors cn disply very lrge nd very smll numers using scientific nottion. If you perform the opertion 2 300 000 400 000 on your clcultor, it will disply 9.2 or 9.2E or something similr. This ctully represents 9:2 0. Likewise, if you perform 0:0024 0 000 000 your clcultor will disply 2.4 0 or 2.4E 0 or something similr. This ctully represents 2:4 0 0. Numers which re lredy represented in scientific nottion cn e entered into the clcultor using the EXP or EE key. For exmple, 4:022 0 4 cn e entered s: 4:022 EXP 4 or 4:022 EE 4 nd will pper s 4.022 04 or 4.022E4 on the disply. Likewise, 5:446 0 cn e entered s 5:446 EXP + / or 5:446 EE ( ) nd will pper s 5.446 or 5.446E. Exmple 9 Use your clcultor to find: (:42 0 4 ) (2:56 0 8 ) (4:75 0 4 ) (2:5 0 7 ) :42 EXP 4 2:56 EXP 8 = Answer: 3:6352 0 2 4:75 EXP 4 + / 2:5 EXP 7 = Answer: :9 0

86 INDICES (Chpter 8) EXERCISE 8E.2 Write ech of the following s it would pper on the disply of clcultor in scientific nottion: 220 000 0:000 046 4 c :26 0 4 d 2:464 0 0 e 2 400 000 f 0:000 007 3 2 Clculte ech of the following, giving your nswer in scientific nottion. The deciml prt should e written correct to 2 deciml plces. 0:06 0:002 4000 426 760 42 000 c 627 000 74 000 d 320 600 5 400 e 0:004 28 20 000 f 0:026 0:00 42 0:08 3 Find, in scientific nottion with deciml prt correct to 2 plces: (3:42 0 5 ) (4:8 0 4 ) (6:42 0 2 ) 2 c 3:6 0 0 6 0 7 d (9:8 0 4 ) (7:2 0 6 ) e 3:8 0 5 f (:2 0 3 ) 3 4 If missile trvels t 5400 km per hour, how fr will it trvel in: dy week c 2 yers? Give your nswers in scientific nottion with deciml prt correct to 2 plces. Assume tht yer ¼ 365:25 dys. 5 Light trvels t speed of 3 0 8 metres per second. How fr will light trvel in: minute dy c yer? Give your nswers in scientific nottion with deciml prt correct to 2 plces. Assume tht yer ¼ 365:25 dys. F SIGNIFICANT FIGURES There re mny occsions when it is sensile to give n pproximte nswer to n rithmetic clcultion. For exmple, if we wnt to know how mny people ttended footll mtch, figure of 32 000 would e cceptle even though the exct numer ws 3 964. Likewise, if trffic survey showed tht 852 crs crried 4376 people, it would not e sensile to give the verge numer of pssengers per cr s 2:362 850 972. An pproximte nswer of 2:4 is more pproprite. There is clerly need to shorten or round off some numers which hve more figures in them thn re required. We round off to certin numer of deciml plces, or else to certin numer of significnt figures.

INDICES (Chpter 8) 87 THE PROCEDURE FOR ROUNDING OFF TO SIGNIFICANT FIGURES To round off to n significnt figures, look t the (n +)th digit from the left: ² if it is 0,, 2, 3 or 4, do not chnge the nth figure, ² if it is 5, 6, 7, 8 or 9, increse the nth figure y. Delete ll figures fter the nth figure, replcing y 0 s if necessry. Converting to scientific nottion provides us with sfe method of rounding off. Exmple 20 Write 278 463 correct to 3 significnt figures 0:007 658 4 correct to 3 significnt figures. 278 463 = 2:784 63 0 5 ¼ 2:78 0 5 ¼ 278 000 0:007 658 4 = 7:6584 0 3 ¼ 7:66 0 3 ¼ 0:007 66 f4th figure is 4, so3rd stys s n 8g f4th figure is 8 nd so 3rd goes up y g EXERCISE 8F Write correct to 2 significnt figures: 567 6 342 c 70:7 d 3:00 e 0:76 f 49:6 g 3:046 h 760 i 0:040 9 j 45 600 2 Write correct to 3 significnt figures: 43 620 0 076 c 0:6 d 0:036 82 e 0:38 6 f 0:79 6 g 0:63 h 0:063 7 i 8:997 j 256 800 3 Write correct to 4 significnt figures: A r over digit indictes recurring deciml. 0: 6 = 0: 666 666 :::: 0: 63 = 0: 636 363 :::::: 28:039 2 0:005 362 c 23 683:9 d 42 366 709 e 0:038 792 f 0:006 377 9 g 0:000 899 95 h 43:076 32 4 Consider the Opening Prolem on pge 74. Write the numer of toms in kilogrm of gold in stndrd form with deciml prt correct to 3 significnt figures. RUSSIAN PEASANT MULTIPLICATION LINKS click here Ares of interction: Humn ingenuity, Approches to lerning

88 INDICES (Chpter 8) REVIEW SET 8A Simplify using index lws: x 8 x 3 ( 4 ) 3 c 2 Express s powers of 3: 3 2 3 3 3 3 3 7 3 Write in scientific nottion: 3762 0:000 04 3 c 8:62 4 Simplify: 6 0 +6 +6 2 2 +( 3 ) c ( 5 8 ) 2 5 Simplify, giving your nswer in scientific nottion: (4 0 4 ) 3 8:4 0 4 :2 0 3 c (2 0 4 ) (8 0 7 ) µ 2 6 Simplify: 2x 3 4x 0 2 3 7 Round these correct to 3 significnt figures: 2:6583 0:000 620 435 c 22 7 8 Write 34:7 0 7 in scientific nottion. A plnet trvels 5:782 0 8 km in its orit ech dy. How fr will it trvel in 800 yers? Use yer ¼ 365:25 dys. c A spce crft trvels 3:45 0 8 km in 5 0 3 hours. Find its speed. 2 8 3 REVIEW SET 8B Simplify the following: 2 3 (2x 3 ) 5 c 6c 3 7c 5 2 Write s powers of 7: 7 2 7 3 3 Write in scientific nottion: 0:003 5 43 200 c 0:8904 4 Simplify: ( 3 2 ) 2 3 0 3 c ( ) 0 5 Simplify, giving your nswers in scientific nottion: (8 0 3 ) 2 6 Simplify: 4d 3 5d 8 4:5 04 9 0 2 c (4 0 3 ) (9 0 7 ) µ 2 4x y 3 7 Round these correct to 3 significnt figures: 58:04999 0:008 255 c 3 8 Write 32 0 7 in scientific nottion. On verge nk receives 4:578 0 7 per week. How much will it receive in 20 yers? Use yer ¼ 365:25 dys. c A su-tomic prticle trvels distnce of 5 0 5 cm in 8 0 6 sec. Find its speed.