Mathematics Form 3 PDF by Penerbitan Pelangi Sdn Bhd 2019

DUAL LANGUAGE PROGRAMME FORM  RUKUN NEGARA Bahawasanya Negara Kita Malaysia mendukung cita-cita hendak; Mencapai perpaduan yang lebih erat dalam kalangan seluruh masyarakatnya; Memelihara satu...

DUAL LANGUAGE PROGRAMME FORM  RUKUN NEGARA Bahawasanya Negara Kita Malaysia mendukung cita-cita hendak; Mencapai perpaduan yang lebih erat dalam kalangan seluruh masyarakatnya; Memelihara satu cara hidup demokrasi; Mencipta satu masyarakat yang adil di mana kemakmuran negara akan dapat dinikmati bersama secara adil dan saksama; Menjamin satu cara yang liberal terhadap tradisi-tradisi kebudayaannya yang kaya dan pelbagai corak; Membina satu masyarakat progresif yang akan menggunakan sains dan teknologi moden; MAKA KAMI, rakyat Malaysia, berikrar akan menumpukan seluruh tenaga dan usaha kami untuk mencapai cita-cita tersebut berdasarkan prinsip-prinsip yang berikut: KEPERCAYAAN KEPADA TUHAN KESETIAAN KEPADA RAJA DAN NEGARA KELUHURAN DAN PERLEMBAGAAN KEDAULATAN UNDANG-UNDANG KESOPANAN DAN KESUSILAAN (Sumber : Jabatan Penerangan, Kementerian Komunikasi dan Multimedia Malaysia) KURIKULUM STANDARD SEKOLAH MENENGAH MATHEMATICS FORM 3 Authors Chiu Kam Choon Vincent De Selva A/L Santhanasamy Punithah Krishnan Raja Devi Raja Gopal Translator Yew Chian-Hauo Editors Premah A/P Rasamanie Muhammad Amirullah Bin Miswan Lai Boon Sing Designers Lim Fay Lee Nur Syahidah Mohd Sharif Illustrators Asparizal Mohamed Sudin Mohammad Kamal B Ahmad Penerbitan Pelangi Sdn Bhd. 2019  KEMENTERIAN PENDIDIKAN MALAYSIA Book Series No: 0166 ACKNOWLEDGEMENTS KPM2019 ISBN 978-983-00-9651-3 First Published 2019 The publishing of this textbook involves © Ministry of Education Malaysia cooperation from various parties. Our wholehearted appreciation and gratitude goes All rights reserved. No part of this book may out to all involved parties: be reproduced, stored in a retrieval system, or transmitted in any form or by any means, ‡ &RPPLWWHH PHPEHUV RI Penambahbaikan either electronic, mechanical, photocopying, Pruf Muka Surat, Textbook Division, recording or otherwise, without the prior Ministry of Education, Malaysia. permission of Director General of Education ‡ &RPPLWWHH PHPEHUV RI Penyemakan Malaysia, Ministry of Education Malaysia. Pembetulan Pruf Muka Surat, Textbook Negotiations are subject to an estimation of Division, Ministry of Education, Malaysia. royalty or an honorarium. ‡ &RPPLWWHH PHPEHUV RI Penyemakan Published for the Ministry of Education Naskhah Sedia Kamera, Textbook Division, Malaysia by: Ministry of Education, Malaysia. PENERBITAN PELANGI SDN. BHD. 66, Jalan Pingai, Taman Pelangi, ‡ 2൶FHUV LQ 7H[WERRN 'LYLVLRQ DQG WKH 80400 Johor Bahru, Curriculum Development Division, Johor Darul Takzim. Ministry of Education, Malaysia. ‡ &KDLUSHUVRQ DQG PHPEHUV RI WKH 4XDOLW\ Layout and Typesetting: Control Panel. PENERBITAN PELANGI SDN. BHD. Font type: Times New Roman ‡ (QJOLVK/DQJXDJH7HDFKLQJ&HQWUH (/7&  Font size: 11 point Teacher Education Division, Ministry of Education Malaysia. Printed by: THE COMERCIAL PRESS SDN. BHD. ‡ (GLWRULDO 7HDP DQG 3URGXFWLRQ 7HDP Lot 8, Jalan P10/10, especially the illustrators and designers. Kawasan Perusahaan Bangi, ‡ (YHU\RQHZKRKDVEHHQGLUHFWO\RULQGLUHFWO\ Bandar Baru Bangi, involved in the successful publication of 43650 Bangi, this book. Selangor Darul Ehsan. ii Introduction v Symbols and Formulae vii CHAPTER Indices 1 1 1.1 Index Notation 2 1.2 Law of Indices 6 CHAPTER Standard Form 30  2  6LJQL¿FDQW)LJXUHV  2.2 Standard Form 37 CHAPTER Consumer Mathematics: Savings and Investments, 3 Credit and Debt 3.1 Savings and Investments 50 52 3.2 Credit and Debt Management 73 CHAPTER Scale Drawings 86 4 4.1 Scale Drawings 88 CHAPTER Trigonometric Ratios 106 5 5.1 Sine, Cosine and Tangent of Acute Angles in Right-angled Triangles 108 iii  CHAPTER Angles and Tangents of Circles 128 6 6.1 Angle at the Circumference and Central Angle Subtended by an Arc 130   &\FOLF4XDGULODWHUDOV  6.3 Tangents to Circles 150 6.4 Angles and Tangents of Circles 160 CHAPTER Plans and Elevations 168 7 7.1 Orthogonal Projections 170 7.2 Plans and Elevations 182 CHAPTER Loci in Two Dimensions 198 8 8.1 Loci 200 8.2 Loci in Two Dimensions 204 CHAPTER Straight Lines 224 9 9.1 Straight Lines 226 Answers 252 Glossary 262 References 263 Index 264 iv Introduction This Form 3 Mathematics Textbook is prepared based on Kurikulum Standard Sekolah Menengah (KSSM). This book contains 9 chapters arranged systematically based on Form 3 Mathematics Dokumen Standard Kurikulum dan Pentaksiran (DSKP). At the beginning of each chapter, pupils are introduced to materials related to daily life to stimulate their thinking about the content. The Learning Standard and word lists are included to provide a visual summary of the chapter’s content. Special features of this book are: Description What will you learn? Contains learning standards that pupils need to achieve in each chapter. Why Why do do you you learn learn this this chapter? chapter? $SSOLFDWLRQVRINQRZOHGJHLQUHODWHGFDUHHU¿HOGV Exploring Exp Exp plor g Era loring lori ing Era Era Er Historical background or origin of the content. WORD B A N K Word list contained in each chapter. Brainstorming Helps pupils to understand the basic mathematical concept via individual, pair or group activities. Individual In pairs In groups Provides additional information about the content BULLETIN learned. Questions that test pupils’ ability to understand QU I Z basic concepts in each chapter. Additional facts that pupils need to be reminded of REMINDER and common mistakes to be avoided. TIPS Exposes pupils to additional knowledge that they need to know. Challenging tasks for enhancement of critical and SMART MIND creative thinking skills. v  Description SMART Exposes pupils to the use of technology in mathematics. Develops pupils’ mathematical communication DISCUSSION CORNER skills. FLASHBACK Helps pupils to recall what they have learnt. SMART FINGER 1,234567.89 7 8 4 5 1 2 AC 0 9 ÷ 6 x 3 -. + 6KRZVKRZWRXVHVFLHQWL¿FFDOFXODWRUV P R O J E C T Enables pupils to carry out and present project work. Assesses pupils’ understanding on the concepts they MIND TEST have learnt. 4XHVWLRQVWRHQKDQFHSXSLOV¶KLJKHURUGHUWKLQNLQJ skills. Dynamic Challenge 3URYLGHV GLYHUVL¿HG WDVNV ZKLFK LQFRUSRUDWH WKH elements of LOTS, HOTS, TIMSS and PISA. (QDEOHV SXSLOV WR VFDQ D 45 &RGH XVLQJ D PRELOH device to access further information. Covers applicable concepts of digital tool calculators, EXPLORING MATHEMATICS KDQGVRQDFWLYLWLHVDQGJDPHVWKDWDLPWRH൵HFWLYHO\ enhance pupils’ understanding. CONCEPT MAP Overall chapter summary. SELF-REFLECT Pupils self-assess their achievement. Checking Answers Checking answers using alternative methods. Activities with elements of Science, Technology, S T EM Engineering and Mathematics. vi Symbols and Formulae SYMBOLS ¥ root  is more than or equal to ʌ SL  is less than a:b ratio of a to b  is less than or equal to A × 10n standard form where 6 WULDQJOH 1  A  10 and nLVDQLQWHJHU  DQJOH = is equal to $ GHJUHH §  is approximately equal to ' minute   is not equal to '' second  is more than FORMULAE am × an = am + n sin θ tan θ = ——– am ÷ an = am – n cos θ (am)n = amn 3\WKDJRUDVWKHRUHP: a0 = 1 1 c2 = a2 + b2 a–n = — an c b 1 b2 = c2 – a2 a—n = n¥ a m 1 — 1 a a2 = c2 – b2 a—n = (am) n = (a—n )m m Distance between a—n = n¥ am = (n¥ a )m ¥ x2 – x1)2 + (y2 – y1)2 two points I = Prt MV = P(1 + —) r nt n ( x1 + x2 y1 + y2 Midpoint = ²²²²²² 2 2 ) A = P + Prt opposite side vertical distance sin θ = ——————– *UDGLHQWm = ———————— horizontal distance opposite side hypotenuse e us y2 – y1 ten adjacent side cos θ = ——————– m = ——— po hypotenuse x2 – x1 hy ș opposite side tan θ = ——————– adjacent y-intercept m = – ————— adjacent side side x-intercept Download a free QR Code scanner application to your mobile device. Scan QR Code or visit the website http://bukutekskssm.my/Mathematics/F3/Index.html to download ¿OHVIRUEUDLQVWRUPLQJ7KHQVDYHWKHGRZQORDGHG¿OHIRUR൷LQHXVH Note: Students can download the free GeoGebra and Geometer’s Sketchpad *63 VRIWZDUHWRRSHQUHODWHG¿OHV http://bukutekskssm. my/Mathematics/F3/ Index.html vii CHAPTER Indices 1 What Whatt will illl you wil you learn? learn? lear le ? 1.1 Index Notation 1.2 Law of Indices Why Why do do you you learn learn this this chapter? chapter? ‡ :ULWLQJDQXPEHULQLQGH[QRWDWLRQHQDEOHVWKH number stated in a simple and easily understood form. Various operations of mathematics that involve numbers in index notation can be performed by using laws of indices. ‡ &RQFHSWRILQGH[LVXVHGLQWKHILHOGVRIVFLHQFH engineering, accounting, finance, astronomy, computer and so on. K enyir Lake, located in the district of Hulu Terengganu, in Terengganu, is the biggest man-made lake in Southeast Asia. Kenyir Lake is a world famous tourist destination known for its unique natural beauty. Kenyir Lake is an important water catchment area. Kenyir Lake, which was built in the year 1985, supplies water to Sultan Mahmud Power Station. The estimated catchment area at the main dam is 2 600 km2 with a reservoir volume of 13 600 million cubic metres. During rainy season, the volume of water in the catchment area will increase sharply. What action should be taken to address this situation?  Exploring Expl Ex plor ing Era oriin ing Era Era Er Index notation is an important element in the development of mathematics and computer programming. The use of positive indices was introduced by Rene Descartes (1637), a well-known French mathematician. Sir Isaac Newton, a well-known British mathematician, GHYHORSHGWKH¿HOGRILQGH[QRWDWLRQDQGLQWURGXFHG negative indices and fractional indices. http://bukutekskssm.my/Mathematics/F3/ ([SORULQJ(UD&KDSWHUSGI WORD B A N K ‡ EDVH ‡ DVDV ‡ IDFWRU ‡ IDNWRU ‡ LQGH[ ‡ LQGHNV ‡ IUDFWLRQDOLQGH[ ‡ LQGHNVSHFDKDQ ‡ SRZHU ‡ NXDVD ‡ URRW ‡ SXQFDNXDVD ‡ LQGH[QRWDWLRQ ‡ WDWDWDQGDLQGHNV 1  1.1 Index Notation 1 CHAPTER What is repeated multiplication in index form? LEARNING STANDARD The development of technology not only makes most of our daily Represent repeated WDVNV HDVLHU LW DOVR VDYHV H[SHQVHV LQ YDULRXV ¿HOGV )RU LQVWDQFH multiplication in index form the use of memory cards in digital cameras enable users to store and describe its meaning. photographs in a large number and to delete or edit unsuitable photographs before printing. DISCUSSION CORNER Discuss the value of the capacity of a pen drive. BULLETIN 7KHQXFOHDU¿VVLRQRI uranium U-320 follows the pattern 30, 31, 32, … In the early stage, memory cards were made with a capacity of 4MB. The capacity increases over time to meet the demands of users. Do you know that the capacity of memory cards is calculated using a special form that is 2n?  ,Q)RUP\RXKDYHOHDUQWWKDW3 = 4 × 4 × 4. The number 43 is written in index notation, 4 is the base and 3 is the index or exponent. The number is read as ‘4 to the power of 3’. Hence, a number in index notation or in index form can be written as; an Index Base You have also learnt that 42 = 4 × 4 and 43 îî)RUH[DPSOH 4×4=42 The value of index is 2 Repeated two times The value of index is the same as the number of times 4 is multiplied repeatedly. 4×4×4=43 The value of index is 3 Repeated three times The value of index is the same as the number of times 4 is multiplied repeatedly. Example 1 Write the following repeated multiplications in index form an. REMINDER (a) 5 × 5 × 5 × 5 × 5 × 5 (b) 0.3 × 0.3 × 0.3 × 0.3 25  2 × 5 43  4 × 3          an  a × n (c) (–2) × (–2) × (–2) (d) — × — × — × — × — 4 4 4 4 4 (e) m × m × m × m × m × m × m (f) n × n × n × n × n × n × n × n 2  Chapter 1 Indices Solution: (a) 5 × 5 × 5 × 5 × 5 × 5 = 56 (b) 0.3 × 0.3 × 0.3 × 0.3 = (0.3)4 1 repeated six times repeated four times CHAPTER           5 (c) (–2) × (–2) × (–2) = (–2)3 (d) — × — × — × — × — = — 4 4 4 4 4 4 ( ) repeated three times UHSHDWHG¿YHWLPHV (e) m × m × m × m × m × m × m = m7 (f) n × n × n × n × n × n × n × n = n8 repeated seven times repeated eight times )URPWKHVROXWLRQLQ([DPSOHLWLVIRXQGWKDWWKHYDOXHRILQGH[LQDQLQGH[IRUPLVWKHVDPHDV the number of times the base is multiplied repeatedly. In general, an = a × a × a × … × a ; a n factors MIND TEST 1.1a 1. Complete the following table with base or index for the given numbers or algebraic terms. 53 (– 4)7 Base Index 5  7 (—2 ) m6 (– —37 ) 4  — 2 n0 (0.2)9 6 n 9   2 x20 ( ) 2— 3 x 4 2 8 8 2. State the following repeated multiplications in index form an. (a) 6 × 6 × 6 × 6 × 6 × 6 (b) 0.5 × 0.5 × 0.5 × 0.5 × 0.5 × 0.5 × 0.5       (c) — × — × — × — (d) (–m) × (–m) × (–m) × (–m) × (–m) 2 2 2 2                   H  ²î²î² 3 3 3    ( ) ( ) ( ) ( ) ( ) ( ) I  – – × – – × – – × – – × – – × – – n n n n n n 3. Convert the numbers or algebraic terms in index form into repeated multiplications.    3      5    3 (a) (–3) (b) (2.5)4 (c) — 3 (d) – 2 — 4 ( ) ( ) (g) (— m)   6      8 (e) k (f) (–p)7 (h) (3n)5 3  How do you convert a number into a number in index LEARNING form? STANDARD 1 A number can be written in index form if a suitable base is selected. You Rewrite a number in index form and vice versa. CHAPTER can use repeated division method or repeated multiplication method to convert a number into a number in index form. Example 2 FLASHBACK Write 64 in index form using base of 2, base of 4 and base of 8. 4 × 4 × 4 = 43 Solution: Repeated Division Method (a) Base of 2 (b) Base of 4 (c) Base of 8 ‡LVGLYLGHGUHSHDWHGO\ ‡LVGLYLGHG ‡LVGLYLGHG by 2. repeatedly by 4. repeatedly by 8. 2 ) 64 4 ) 64 2 ) 32 8 ) 64 n=3    n=2    8) 8 n=6 4) 4   2) 8   2) 4 Hence, 64 = 82 2) 2 Hence, 64 = 43   The division is continued until LVREWDLQHG Hence, 64 = 26 Repeated Multiplication Method (a) Base of 2 (b) Base of 4 (c) Base of 8 2×2×2×2×2×2 4×4×4 8 × 8 = 64 4  Hence, 64 = 82 8 64  DISCUSSION CORNER Hence, 64 = 43 32 Which method is easier to convert a number into 64 a number in index form? Is it the repeated division or repeated multiplication Hence, 64 = 26 method? Discuss. 4  Chapter 1 Indices Example 3 32 2 Write ——– in index form using base of —. 1    CHAPTER Solution: Repeated Division Method Repeated Multiplication Method 2 2 2 2 2 2 ) 32    —×—×—×—×— 5 5 5 5 5    5 ) 625 2) 8 4 n=5 n=5    —– 2) 4 25 5) 25 2) 2 5) 5 8 —–       5 —– 32 2 625 Hence, ——– = —   ( ) 32 ——–  32 2 5 Hence, ——– = —  ( ) MIND TEST 1.1b 1. Write each of the following numbers in index form using the stated base in brackets. 64 4  D      [base of 3]   E    [base of 5]   (c) —–  [ base of — 5 ]           (d) 0.00032 [base of 0.2] H  ± [base of (– 4)] (f) —  [ base of – — 4 ( )] How do you determine the value of the number in index form , an? The value of an FDQ EH GHWHUPLQHG E\ UHSHDWHG PXOWLSOLFDWLRQ PHWKRG RU XVLQJ D VFLHQWL¿F calculator. Example 4 Calculate the values of the given numbers in index form. QU I Z (a) 25 (b) (0.6)3 (m)4 = 16 What are the possible 2×2×2×2×2 0.6 × 0.6 × 0.6 values of m? 4 × 2 0.36 × 0.6    î         î  3  32 Hence, 25 = 32 Hence, 0.63  5  Example 5 SMART FINGER 1,234567.89 7 8 4 5 1 2 AC 0 9 ÷ 6 x 3 -. + REMINDER (a) 54 = 625 5 ^ 4 = Negative or fractional base 1 must be placed within (b) (–7)3 = –343 CHAPTER ( (–) 7 ) ^ 3 = brackets when using a calculator to calculate    4  ( ) (c) — = —–     ( 2 ab/c 3 ) ^ 4 = values of given numbers. 3 2 64 (d) (²) = —– ( 1 ab/c 3 ab/c 5 ) ^ 2 = DISCUSSION CORNER 5 25 Calculate questions (c), (e) (– 0.5)6  ( (–) 0. 5 ) ^ 6 = (d) and (e) in Example 5 without using brackets. Are the answers the same? Discuss. MIND TEST 1.1c 1. Calculate the value of each of the following numbers in index form. (a) 94 (b) (– 4)5 (c) (2.5)3 (d) (– 3.2)3    5    4    2    3 (e) — 8 ( ) (f) – — 6 ( ) (g) ² 3 ( ) (h) – 2 — 3 ( ) 1.2 Law of Indices What is the relationship between multiplication of LEARNING numbers in index form with the same base and repeated STANDARD multiplication? Relate the multiplication of numbers in index Brainstorming 1 In pairs form with the same base, to repeated Aim: To identify the relationship between multiplication of multiplications, and hence numbers in index form with the same base and repeated make generalisation. multiplication. Steps: 1. Study example (a) and complete examples (b) and (c). 2. Discuss with your friend and state three other examples. 3. ([KLELWWKUHHH[DPSOHVLQWKHPDWKHPDWLFVFRUQHUIRURWKHUJURXSVWRJLYHIHHGEDFN Multiplication of Repeated multiplication numbers in index form 3 factors 4 factors 7 factors (overall) (a) 23 × 24 (2 × 2 × 2) × (2 × 2 × 2 × 2) = 2 × 2 × 2 × 2 × 2 × 2 × 2 = 27 23 × 24 = 2 7 7=3+4 23 × 24 = 2 3 + 4 2 factors 3 factors 5 factors (overall) (b) 32 × 33 (3 × 3) × (3 × 3 × 3) = 3 × 3 × 3 × 3 × 3 = 35 32 × 33 = 3 32 × 33 = 3 6  Chapter 1 Indices Multiplication of Repeated multiplication numbers in index form 1 4 factors 2 factors 6 factors (overall) (c) 54 × 52 CHAPTER (5 × 5 × 5 × 5) × (5 × 5) = 5 × 5 × 5 × 5 × 5 × 5 = 56 54 × 52 = 5 54 × 52 = 5 Discussion: What is your conclusion regarding the relationship between multiplication of numbers in index form and repeated multiplication? )URP%UDLQVWRUPLQJLWLVIRXQGWKDW 23 × 24 = 23 + 4 DISCUSSION CORNER 32 × 33 = 32 + 3 Given, 54 × 52 = 54 + 2 am × an = bm × bn. Is a = b? Discuss. In general, am × an = a m + n Example 6 Simplify each of the following.  2  3        (a) 7 × 7 (b) (0.2)2 × (0.2)4 × (0.2)5 (c) 2k2 × 4k3 (d) 3m4 × —m5îm 6 Solution: (a) 72 × 73 (b) (0.2)2 × (0.2)4 × (0.2)5 REMINDER = 72 + 3 = (0.2)2 + 4 + 5 = 75 = (0.2) a = a1   2 3   4  5 (c) 2k × 4k (d) 3m × —m îm 6 = (2 × 4)(k2 × k3)     î²î  m4 × m5 × m) Operation of 6 WKHFRHI¿FLHQWV = 8k2 + 3 = 6m SMART MIND = 8k5 = 6m a b 8 If m × m = m , such that a > 0 and b > 0, what are the possible MIND TEST 1.2a values of a and b? 1. Simplify each of the following. (a) 32 × 3 × 34 (b) (– 0.4)4 × (– 0.4)3 × (– 0.4) ( ) ( ) ( ) 4 4 3 4 (c) — × — × — 7 7 7 5 ( 2 2 5 ) ( 2 3 5 2 (d) ±² × ±² × ±² 5 ) ( )5   4m2 × — (e)  m3 × (– 3)m4  n 6 × — (f)  n2 × —  n3 × n 2 25 4    4   2    5  (g) –x × — x × — x (h) – — y × (– 6)y3 × — y4 4 5 2 3 7  How do you simplify a number or an algebraic term in index TIPS form with different bases? Group the numbers or 1 algebraic terms with the VDPHEDVH¿UVW7KHQDGG CHAPTER the indices for the terms Example 7 with the same base. Simplify each of the following. (a) m3 × n2 × m4 × n5 (b) (0.3)2 × (0.2)2 × 0.3 × (0.2)5 × (0.3)3  2  3   4  (c) p × m × p4 × n3 × m4 × n2 (d) –m × 2n5 × 3m × — n2 4 Solution: (a) m3 × n2 × m4 × n5 (b) (0.3)2 × (0.2)2 × 0.3 × (0.2)5 × (0.3)3 = m3 × m4 × n2 × n5 Group the terms = (0.3)2 × (0.3) × (0.3)3 × (0.2)2 × (0.2)5 with the same base. = m3 + 4 × n2 + 5 = (0.3)  × (0.2)(2 + 5) = m7 × n7 Add the indices for terms = (0.3)6 × (0.2)7 7 7 with the same base. =m n  2  3   4  (c) p × m × p4 × n3 × m4 × n2 (d) –m × 2n5 × 3m × —n2 4 = m3 × m4 × n3 × n2 × p2 × p4      ±îîî² m4 × m× n5 × n2 = m3 + 4 × n3 + 2 × p2 + 4 4 3  5 + 2 = m7 n5 p6 = – —m n 2 REMINDER 3 5 7 –an  ±a)n = – —m n 2 Example: ±2 ± 2 MIND TEST 1.2b ± 1. State in the simplest index form. (a) 54 × 93 × 5 × 92 (b) (0.4)2î  3î  î  5î      5 3   4     F  x × y × — x × — y (d) –2k5 × p6 × — p5 × 3k 2 3 4 What is the relationship between division of numbers in LEARNING index form with the same base and repeated multiplication? STANDARD Relate the division of Brainstorming 2 In pairs numbers in index form with the same base, to repeated multiplications, Aim: To identify the relationship between division of numbers in and hence make index form with the same base and repeated multiplication. generalisation. Steps: 1. Study example (a) and complete examples (b) and (c). 2. Discuss with your friend and state three other examples. 3. 3UHVHQW\RXU¿QGLQJV 8  Chapter 1 Indices Division of numbers Repeated multiplication in index form 1 5 factors (a) 45 ÷ 42 CHAPTER 45 4 × 4 × 4 × 4 × 4 = 4 × 4 × 4 = 43 —2 = —————–––– 4 4×4 3 factors (Remainder) 2 factors 45 ÷ =4342 3=5–2 45 ÷ 42 = 4 5–2 6 factors (b) 26 ÷ 22 26 2 × 2 × 2 × 2 × 2 × 2 = 2 × 2 × 2 × 2 = 24 —2 = —————––––—– 2 2×2 4 factors (Remainder) 2 factors 26 ÷ =2 22 26 ÷ 22 = 2 5 factors (c) (–3)5 ÷ (–3)3 (–3)5 (–3) × (–3) × (–3) × (–3) × (–3) = (–3) × (–3) = (–3)2 ——3 = —————––––—–––——–– (–3) (–3) × (–3)× (–3) 2 factors (Remainder) 3 factors (–3)5 ÷ (–3)3 = (–3) (–3)5 ÷ (–3)3 = (–3) Discussion What is the relationship between division of numbers in index form and repeated multiplication? )URP%UDLQVWRUPLQJLWLVIRXQGWKDW SMART MIND 45 ÷ = 42 45 – 2 Given ma ±b = m7 and 26 ÷ 22 = 26 – 2 0 < a < 10. If a > b, state the possible values (–3)5 ÷ (–3)3 = (–3)5 – 3 of a and b. In general, am ÷ an = a m – n Example 8 Simplify each of the following. (a) 54 ÷ 52 (b) (–3)4 ÷ (–3)2 ÷ (–3) (c) m4n3 ÷ m2n (d) 25x2y3 ÷ 5xy (e) m ÷ 4m5 ÷ m2 I  ±p8 ÷ 2p5 ÷ 4p2 Solution: (a) 54 ÷ 52 (b) (–3)4 ÷ (–3)2 ÷ (–3) (c) m4n3 ÷ m2n = 54 – 2 = (–3)4 ÷ (–3)2 ÷ (–3)  = m4n3 ÷ m2n = 52 = (–3)±±  = m4 – 2 n± = (–3)  = m2 n2 = –3 9  (d) 25x2y3 ÷ 5xy H  m ÷ 4m5 ÷ m2  I  ±p8 ÷ 2p5 ÷ 4p2     2 3         ± 8 = 25x y ÷ 5xy = — (m ÷ m5 ÷ m2) = —– (p ÷ p5) ÷ 4p2 4 2 1 25 = 3(m±) ÷ m2 = –8p8–5 ÷ 4p2 = — x± y± CHAPTER 5 Operation of the = 3m5 – 2 = –8p3 ÷ 4p2 = 5xy2 FRHI¿FLHQWV 8 (p3 ÷ p2) = 5xy2 = 3m3 =–— 4 = –2p3 – 2 = –2p = –2p MIND TEST 1.2c 1. Simplify each of the following. (a) 45 ÷ 44 (b) 7 ÷ 76 ÷ 72 m8n6 (c) —— m4n 27x4 y5 (d) ——–– (e) m7 ÷ m2 ÷ m4 (f) –25h4 ÷ 5h2 ÷ h 9x3y2 2. Copy and complete each of the following equations. (a) 8 ÷ 84 ÷ 83 = 8 (b) m4n ÷m n5 = m2n m n4 × m n2 27x3y6 × xy (c) —————— = m 5n (d) —————– = 3x y5 m7n x2y3 2x × 3y 3. If ——— = 6, determine the value of x + y. 2 4 × 32 What is the relationship between a number in index form LEARNING raised to a power and repeated multiplication? STANDARD Relate the numbers in Brainstorming 3 In pairs index form raised to a power, to repeated multiplication, and hence Aim: To identify the relationship between a number in index form make generalisation. raised to a power and repeated multiplication. Steps: 1. Study example (a) and complete examples (b) and (c). 2. Discuss with your friend and state three other examples. 3. 3UHVHQW\RXU¿QGLQJV Index form raised Repeated multiplication in index form Conclusion to a power (a) (32)4 4 factors 32 × 32 × 32 × 32 (32)4 = 32(4) = 32 + 2 + 2 + 2 4 times 2 is added 4 times = 38 = 32(4) 10  Chapter 1 Indices Index form raised Repeated multiplication in index form Conclusion to a power 1 3 factors (b) (54)3 CHAPTER 54 × 54 × 54 (54)3 = 5 = 54 + 4 + 4 3 times 4 is added 3 times =5 = 54(3) 6 factors (c) (43)6 43 × × 43 × 43 × 43 × 43 43 (43)6 = 4 = 43 + 3 + 3 + 3 + 3 + 3 6 times 3 is added 6 times =4 = 43(6) Discussion: What is your conclusion regarding the index form raised to a power and repeated multiplication in index form? The conclusion in Brainstorming 3 can be checked using the following method. Example (a) Example (b) Example (c) (32)4 = 32 × 32 × 32 × 32 (54)3 = 54 × 54 × 54 (43)6 = 43 × 43 × 43 × 43 × 43 × 43 = 32 + 2 + 2 + 2 = 54 + 4 + 4 = 43 + 3 + 3 + 3 + 3 + 3 = 38 = 5 = 4 3 = 32 × 4 2(4) 5 = 54 × 3 4(3) 4 = 43 × 6 3(6) = 38 = 5 = 4 )URP%UDLQVWRUPLQJLWLVIRXQGWKDW SMART MIND (32)4 = 32(4) Given, (54)3 = 54(3) mrt = 312 (43)6 = 43(6) What are the possible values of m, r and t if r > t ? In general, (am)n = amn Example 9 1. Simplify each of the following. (a) (34)2 (b) (h3) (c) ((–y)6)3 2. Determine whether the following equations are true or false. (a) (42)3 = (43)2 (b) (23)4 = (22)6 (c) (32)6 = (272)4 11  Solution: 1. (a) (34)2 (b) (h3) (c) ((–y)6)3 = (–y)6(3) 1 = 34(2) = h  = (–y) CHAPTER = 38 = h30 2. (a) (42)3 = (43)2 (b) (23)4 = (22)6 (c) (32)6 = (272)4 left right left right left right /HIW  /HIW /HIW (42)3 = 42(3) = 46 (23)4 = 23(4) = 2  (32)6 = 32(6) = 3 Same Same 5LJKW 5LJKW 5LJKW (43)2 = 43(2) = 46 (22)6 = 22(6) = 2 (272)4 = (33(2))4 Not the same Hence, (42)3= (43)2 Hence, (23)4 = (22)6 = 36(4) is true. is true. = 324 Hence, (32)6 = (272)4 is false. MIND TEST 1.2d 1. Use law of indices to simplify each of the following statements.  D  5)2 (b) (3)2 (c) (72)3 (d) ((– 4)3)7 (e) (k8)3 (f) (g2) (g) ((–m)4)3 (h) ((–c)7)3 2. Determine whether the following equations are true or false. (a) (24)5 = (22) (b) (33)7 = (272)4 (c) (52)5  2)3 (d) – (72)4 = (– 492)3 How do you use law of indices to perform operations of multiplication and division? (am × bn)q = (am)q × (bn)q (ambn)q = amq bnq = amq × bnq (am ÷ bn)q a m q a mq = (am)q ÷ (bn)q (—– b n) = —– b nq = amq ÷ bnq Example 10 1. Simplify each of the following. (a) (73 × 54)3 (b) (24 × 53î2)5 (c) (p2q3r)4 (d) (5m4n3)2 25 4 2x3 4 (3m2n3)3 (2x3y4)4 × (3xy2)3 ( ) (e) —2 3 (f) (—– 3y ) 7 (g) ———– 6m3n (h) ——————— 36xy 12  Chapter 1 Indices Solution: FLASHBACK (a) (73 × 54)3 (b) (24 × 53î2)5 am × an = am + n 1 = 73(3) × 54(3) = 24(5) × 53(5)î2(5) am ÷ an = am – n CHAPTER = 79 × 5 = 220 × 5î (am)n = amn (c) (p2q3r)4 (d) (5m4n3)2 QU I Z = p2(4) q3(4)r  = 52m4(2)n3(2) mm = 256. = p8qr4 = 25m8n6 What is the value of m? 25 4 2x 3 4 ( ) (e) —2 3 (f) (—– 3y ) 7 DISCUSSION CORNER 25(4) 24 x3(4) Why is 1n = 1 for all = —– = —––– 32(4) 34y7(4) values of n? 220 x Discuss. = —– = —––– 38 y28 (3m2n3)3 (2x3y4)4 × (3xy2)3 (g) ———– (h) ——————— 6m3n 36xy 33m2(3)n3(3) 24x3(4)y4(4) × 33x  y2(3) = ———— = ————————––– 6m3n      36xy 27m6n9 xy × 27x3y6 = ——— = ——————— 6m3n      36xy     6 – 3 ±      î ± ± =—m 2 n ( = ———– x 36 ) y     3 8     x5 y =—m n 2 MIND TEST 1.2e 1. Simplify each of the following. (a) (2 × 34)2 E  3 × 95)3 F  3 ÷ 76)2 (d) (53 × 34)5 –3a5 6 2a5 3 (e) (m3n4p2)5 (f) (2w2 x 3)4 (g) ——– b4 ( ) ( ) (h) —–– 3b4 2. Simplify each of the following.   3 × 42 2 33 × (62)3 4 2 34 2 ((– 4)6)2 × (–52)3   ( (a) ——–— 2 ) (b) ———— 64 (c) (—– 6 )3 ÷ —– 6 3 (d) ——————— (– 4)6 × (–5)2 x2y6 × x3 (h3k2)4 (m5 n7)3 (b2d4)3 (e) ———— (f) ——— (g) ———– (h) ——— xy2 (hk)2 (m2n3)2 (b2d3)2 3. Simplify each of the following. (2m2n4)3 × (3mn4)2 (5xy4)2 × 6xy 24d 3e 5 × (3d 3e 4)2 (a) ———————– (b) —————— (c) ——————––    m7n   x4y6 (d 5e 6) × (6de 2)3 13  1 ; D0? LEARNING How do you verify a0 = 1 and a–n = — an STANDARD 1 Verify that a0 =  Brainstorming 4 CHAPTER In pairs 1 and a–n = –– n ; a. a Aim: To determine the value of a number or an algebraic term with a zero index. Steps: 1. Study and complete the following table. 2. What is your conclusion regarding zero index? Division in Solution Conclusion index form from the Law of indices Repeated multiplication solution 2×2×2 (a) 23 ÷ 23 23 – 3 = 20 ———––   20  2×2×2 m×m×m×m×m (d) m5 ÷ m5 m 5 – 5 = m0 ²²²²²²²±±±  m0  m×m×m×m×m (c) 54 ÷ 54 (d) (–7)2 ÷ (–7)2 (e) n6 ÷ n6 Discussion: 1. Are your answers similar to the answers of the other groups? 2. What is your conclusion regarding zero index? )URP%UDLQVWRUPLQJLWLVIRXQGWKDW 20  m0  7KHUHIRUHDQXPEHURUDQDOJHEUDLFWHUPZLWKD]HURLQGH[ZLOOJLYHDYDOXHRI In general, a0 a 1 How do you verify a–n = ––– ? an Brainstorming 5 In groups      Aim: To verify a–n = —n. a Steps: 1. Study and complete the following table. 14  Chapter 1 Indices Solution Conclusion Division in index form from the 1 Law of indices Repeated multiplication solution CHAPTER îî     (a) 23 ÷ 25 23 – 5 = 2–2 —————–––– = –––– = –– 2 –2 = ––2 2 × 2 × 2 × 2 × 2 2 × 2 22 2  (b) m2 ÷ m5 m2 – 5 = m–3 —m ×— m m –––——————— = ——––––– = ––3 m –3 = ––– m×m×m×m×m m×m×m m m3 (c) 32 ÷ 36 (d) (– 4)3 ÷ (– 4)7 (e) p4 ÷ p8 Discussion 1. Are your answers similar to the anwers of the other groups? 2. What is your conclusion? Scan the QR Code or visit http://bukutekskssm.my/ )URP%UDLQVWRUPLQJLWLVIRXQGWKDW Mathematics/F3/Chapter1 AlternativeMethod.mp4  –2   to watch a video that 2 =— 22 describes alternative   –3  method to verify a± = —. m = —3 an m BULLETIN  In general, a–n = ––n ; a  0 Negative index is a a number or an algebraic Example 11 term that has an index of a negative value. 1. State each of the following terms in positive index form.    –2      TIPS (a) a (b) x – 4 (c) ––– 8–5 1 Ƈ a–n ±n±         a (d) ––– (e) 2m –3 (f) — n – 8 1 y –9 5 Ƈ an  ±±± –n a –n 2 ( ) (g) –– 3 ± (h) (––xy ) –7 ( ) ( ba ) a Ƈ ±± b = ±± n 2. State each of the following in negative index form. REMINDER      1 (a) —4 (b) —5 (c) 75 2a –n²± 3 m 2an  8 SMART MIND (d) n20 (e) (––45 ) (f) (––mn ) 4 ± 3. Simplify each of the following. ( ) ±²   =x y (24)2 × (35)3 (4xy2)2 × x5y What are the values of x (a) 32 × 34 ÷ 38 (b) ————— (c) ————— and y? (28 × 36)2 (2x3y)5 15  Solution: 1 1 1 1 1. (a) a–2 = –– (b) x – 4 = ––4 (c) ––– = 85 (d) ––– = y9 a2 8–5 y –9 1 x CHAPTER 2 3 3 –10 10 –7 7 (e) 2m–3 = —3 m (f) — n– 8 = —–8 5 5n ( ) = (––32 ) 2 (g) –– 3 (h) (––xy ) = (––yx ) 1 1 1 1 2. (a) — = 3– 4 (b) —5 = m–5 (c) 75 = — (d) n20 = —– 34 m 7–5 n–20 4 8 5 –8 m 15 n –15 ( ) ( ) (e) –– = –– 5 4 n( ) ( ) (f) –– = –– m TIPS (24)2 × (35)3 (4xy2)2 × x5y 3. (a) 32 × 34 ÷ 38 (b) ————— (c) ————— (28 × 36)2 (2x3y)5 y0 = 1 = 32 + 4 – 8 28 × 315 42x2y4 × x5y1 y1 = y = 3–2 = ———– = ————— 1 216 × 312 25x15y5 = —2 = 28 – 16 × 315 – 12 3 16 = — x2 + 5 – 15 y4 + 1 – 5 = 2– 8 × 33 32 33 1 = —8 = — x– 8 y0 2 2 1 = —–8 2x MIND TEST 1.2f 1. State each of the following terms in positive index form. 1 (a) 5–3 (b) 8– 4 (c) x– 8 (d) y–16 (e) —– a– 4 1 2 –4 (f) —–– 20–2 (g) 3n– 4 (h) –5n– 6 (i) — m–5 7 (j) (– —83 )m 2 –12 –14 –10 –4 –5 ( ) (k) — 5 (l) (– —37 ) x ( ) (m) — y 2x ( ) (n) —– 3y 1 (o) (—) 2x 2. State each of the following terms in negative index form. (a) —1 (b) — 1 (c) —1 1 (d) — (e) 102 5 4 83 m 7 n9 4 9 10 (f) (– 4)3 (g) m12 (h) n16 (i) — 7 ( ) (j) (—yx ) 3. Simplify each of the following. (42)3 × 45 (23 × 32)3 (52)5 (a) ———– 6 2 (b) ———–– (c) ———––– (4 ) (2 × 34)5 (2 )–2 × (54)2 3 3m2n4 × (mn3)–2 (2m2n2)–3 × (3mn2)4 (4m2n4)2 (d) ——————– (e) ——————–––– (f) ——————––– 9m3n5 (9m3n)2 (2m n)5 × (3m4n)2 –2 16  Chapter 1 Indices How do you determine and state the relationship between LEARNING fractional indices and roots and powers? STANDARD 1 1 — Determine and state the Relationship between n¥a and a n relationship between CHAPTER fractional indices and In Form 1, you have learnt about square and square root as well as cube roots and powers. and cube root. Determine the value of x for (a) x2 = 9 (b) x3 = 64 TIPS Solution: Ƈ 9 = 32 Ƈ 64 = 43 (a) x2 = 9 Square roots are used (b) x3 = 64 to eliminate squares.   ¥x2 ¥2 3¥x3 = 3¥3 Cube roots are used to eliminate cubes. x =3 x =4 Did you know that the values of x in examples (a) and (b) above can be determined by raising the index to the power of its reciprocal? (a) x2 = 9 The reciprocal (b) x3 = 64 BULLETIN 1 1. of 2 is — x2 ( ) = 9 x3 ( ) = 64( ) 1 — — 1 — 1 — 1 2 2 2 3 3 — is the reciprocal of a. The reciprocal a x1 = 43( 3 ) 1 x1 = 32 (—2 ) 1 — 1 of 3 is —. 3 x =3 x =4 From the two methods to determine the values of x in the examples SMART MIND above, it is found that: What is the solution for 2 –12 ¥x = x 1 ¥– 4 ? Discuss. 3 – ¥x = x 3 1 –n ¥a = a ; D0 In general, n Example 12 1 — 1. Convert each of the following terms into the form a n. (a) 2¥ E  3¥± F  5¥m (d) 7¥n 2. Convert each of the following terms into the form n¥a. 1 1 1 1 — — — — (a) 125 5 (b) 256 8 (c) (–1 000) 3 (d) n 12 3. Calculate the value of each of the following terms. 1 1 — — (a) 5¥± E  6¥ F  3 G  ± 5 Solution: 1 1 1 1 — — — — 1. (a) 2¥36 = 36 2 (b) 3¥±  ± 3 F  5¥m = m 5 (d) 7¥n = n 7 1 1 – 1 1 — — — 2. (a) 125 5 = 5¥125 (b) 2568 = 8¥256 (c) (–1 000) 3 = 3¥(–1 000) (d) n12 = 12¥n 17  1– 1 (c) 512 3 = 83(–3) (d) (–243) 5 = (–3)5(—5 ) 1 — 1 — 1 — 1 3. (a) 5¥–32 = (–32) 5 (b) 6 ¥729 = 729 6 = (–3)1 = 36 (—6 ) 1 = (–2)5(—5 ) = 81 1 1 = –3 CHAPTER = (–2)1 = 31 =8 = –2 =3 TIPS VI@ ,IOHYHORIDFFXUDF\LVWRWKHQHDUHVWKXQGUHG (iii) 15 000 [4 s.f.] ,IOHYHORIDF