[PDF] Oxford Insight Mathematics (Australian Curriculum for NSW) Stage 5.2 5.3 - Year 9 (John Ley).pdf | Plain Text

LEY FULLER JOHN LEY MICHAEL FULLER 9 INSIGHT AUSTRALIAN CURRICULUM FOR NSW OXFORD MATHEMATICS STAGE 5.2 / 5.3

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JOHN LEY MICHAEL FULLER 9 INSIGHT AUSTRALIAN CURRICULUM FOR NSW OXFORD MATHEMATICS STAGE 5.2/5.3 00_LEY_IM_9SB_52_53_22631_SI.indd i 00_LEY_IM_9SB_52_53_22631_SI.indd i 9/09/13 9:39 AM 9/09/13 9:39 AM

1 Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide. Oxford is a registered trademark of Oxford University Press in the UK and in certain other countries. Published in Australia by Oxford University Press 253 Normanby Road, South Melbourne, Victoria 3205, Australia © John Ley, Michael Fuller 2014 The moral rights of the author have been asserted First published 2014 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, by licence, or under terms agreed with the appropriate reprographics rights organisation. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above. Yo u must not circulate this work in any other form and you must impose this same condition on any acquirer. National Library of Australia Cataloguing-in-Publication data Oxford Insight Mathematics 9, stage 5.2/5.3: Australian Curriculum for NSW/ John Ley, Michael Fuller. ISBN 978 019 552263 1 (paperback) For secondary school students. Mathematics--Problems, exercises, etc. Mathematics–Australia–Textbooks. 510 Reproduction and communication for educational purposes The Australian Copyright Act 1968 (the Act) allows a maximum of one chapter or 10% of the pages of this work, whichever is the greater, to be reproduced and/ or communicated by any educational institution for its educational purposes provided that the educational institution (or the body that administers it) has given a remuneration notice to Copyright Agency Limited (CAL) under the Act. For details of the CAL licence for educational institutions contact: Copyright Agency Limited Level 15, 233 Castlereagh Street Sydney NSW 2000 Telephone: (02) 9394 7600 Facsimile: (02) 9394 7601 Email: info@copyright.com.au Edited by Marta Veroni and Anna Beth McCormack Text design by Ana Cosma Proofread by Marta Veroni Technical artwork by Paulene Meyer Typeset by Idczak Enterprises Links to third party websites are provided by Oxford in good faith and for information only. Oxford disclaims any responsibility for the materi\ als contained in any third party website referenced in this work. ACKNOWLEDGEMENTS The author and the publisher wish to thank the following copyright holders for reproduction of their material. Alamy/CBKfoto, p. 85 (bottom) /Justin Kase z05z, p. 216 /Monteverde, p. 327; Brent Parker Jones, pp. 25, 51, 79, 82, 86, 156, 201, 265, 342, 433, 453; Corbis/Gallerystock/Robert Eikelpoth, Chapter 12 Opener /Hulton-Deutsch Collection, p. 372 /Image Source, front cover; Dreamstime. com/John Braid, p. 333 /Michael Rolands, p. 66 /Skripko Ievgen, p. 476 /Svetlana Yudina, p. 388 /Don Farrall, Chapter 4 opener; Getty Images, p. 280 /bbq, p. 278 /Fritz Goro, back cover / Stocktrek Images, p. 58 /Christian Hacker, Chapter 11 Opener /Don Farrall, Chapter 7 Opener, Chapter 14 Opener /Dorling Kindersley, Chapter 13 Opener; iStockphoto.com/Melbye, p. 209 /savas keskiner, p. 85 (top) /Adrian Lindley, p. 239 (top); Lindsay Edwards, p. 34, 180, 184, 221, 304, 338; shutterstock.com/ Gustavo Miguel Fernandes , p. 242 /Chad McDermott, p. 183 /Darrenp, p. 501 /Jeffrey J Coleman, p. 447 /Richard Cavalleri, p. 473 / TonyV3112, p. 65 /Tyler Olson, p. 291 /Philip Date, p. 325 /Stephanie Rousseau, p. 214; Courtesy of Sylvia Wiegand/Mathematical Association of America via Flickr, p. 461; Shutterstock.com, all other images Every effort has been made to trace the original source of copyright material contained in this book. The publisher will be pleased to hear from copyright holders to rectify any errors or omissions. 00_LEY_IM_9SB_52_53_22631_SI.indd ii 9/09/13 9:39 AM

Contents iii C O N T E N T S 1 Review of Year 8 1 A Data ................................................................... 2 B Ratios and rates ................................................ 2 C Congruence ....................................................... 3 D Index laws .......................................................... 4 E Perimeter and area ........................................... 5 F Time ................................................................... 6 G Fractions, decimals and percentages ............... 6 H Circles and cylinders ......................................... 8 I Mean, mode, median and sampling .................. 8 J Pythagoras’ theorem ......................................... 9 K Algebra ............................................................ 10 L Volume and capacity ........................................ 11 M Equations and inequations .............................. 12 N Probability and Venn diagrams ....................... 13 O Coordinate geometry and straight lines ......... 14 2 Indices Number & Algebra 15 A The index laws ................................................. 16 B Applying the index laws ................................... 18 C The zero index ................................................. 22 D Negative indices .............................................. 23 E Negative indices with variables ....................... 26 F Further use of the index laws ......................... 28 G Removing grouping symbols ........................... 30 Language in mathematics .................................... 35 Check your skills .................................................. 35 Review set A ......................................................... 36 Review set B ......................................................... 37 Review set C ......................................................... 38 3 Collecting and analysing data Statistics & Probability 39 A Investigating data ............................................ 40 B The shape of displays ...................................... 43 C Comparing like sets of numerical data ........... 47 D Statistical claims in the media ........................ 51 Investigation 1 Statistical claims made in the media ...................................................... 51 Language in mathematics .................................... 52 Check your skills .................................................. 52 Review set A ......................................................... 54 Review set B ......................................................... 55 Review set C ......................................................... 56 4 Numbers of any magnitude Number & Algebra 57 A Scientific notation ............................................ 58 B Calculations using scientific notation ............. 63 C Units of measurement .................................... 66 Investigation 1 Kilobytes ....................................... 69 D Measurement of time ...................................... 69 E Approximations ............................................... 71 F Significant figures ........................................... 76 G Calculations and rounding numbers............... 80 H Error in measurement .................................... 82 Investigation 2 Accuracy of measures ................... 82 I Calculations involving measurements ............ 87 Language in mathematics .................................... 91 Check your skills .................................................. 91 Review set A ......................................................... 93 Review set B ......................................................... 94 Review set C ......................................................... 95 Cumulative review chapters 2–4 97 5 Financial mathematics Number & Algebra 99 Investigation 1 Earning an income ...................... 100 A Salaries and wages ....................................... 100 B Additional payments ...................................... 103 Investigation 2 Calculating total pay .................... 108 C Casual employment ...................................... 108 D Piecework ...................................................... 111 E Commission ................................................... 112 F Taxable income .............................................. 115 G Calculating tax ............................................... 116 H Simple interest .............................................. 118 I Purchasing goods by cash ............................. 120 J Using credit cards ......................................... 122 K Lay-by ............................................................ 123 L Buying on terms ............................................ 124 Investigation 3 Monthly repayments .................... 127 M Deferred payment ......................................... 128 N Personal loans............................................... 129 O Goods and services tax (extension) ............... 131 Language in mathematics .................................. 133 Check your skills ................................................ 133 Review set A ....................................................... 135 Review set B ....................................................... 136 Review set C ....................................................... 137 CONTENTS 00_LEY_IM_9SB_52_53_22631_SI.indd iii 9/09/13 9:39 AM

iv C O N T E N T S Insight Mathematics 9 stages 5.2/5.3 Australian Curriculum 6 Area, surface area and volume Measurement & Geometry 139 A Review of area ............................................... 140 Investigation 1 Formulas for area ........................ 144 B Areas of special quadrilaterals ..................... 145 C Composite shapes ......................................... 148 D Area applications ........................................... 152 E Areas involving sectors of a circle................. 154 F Surface areas of prisms ................................ 156 G Problems involving surface area ................... 162 H Surface areas of right cylinders .................... 163 I Volumes of prisms and cylinders .................. 165 J Surface areas and volumes of composite solids............................................ 167 K Problems with surface area and volume ...... 168 Language in mathematics .................................. 170 Check your skills ................................................ 171 Review set A ....................................................... 173 Review set B ....................................................... 174 Review set C ....................................................... 175 7 Probability Statistics & Probability 177 A Relative frequency and probability ................ 178 Investigation 1 Probability experiments ............... 178 B Venn diagrams and two-way tables .............. 182 C Probability for multistage events .................. 186 D Sampling with and without replacement ...... 190 E The product rule in probability ...................... 193 F Conditional probability .................................. 200 Language in mathematics .................................. 202 Check your skills ................................................ 203 Review set A ....................................................... 205 Review set B ....................................................... 206 Review set C ....................................................... 207 Review set D ....................................................... 208 Cumulative review chapters 5–7 210 8 Right-angled trigonometry Measurement & Geometry 213 A Review of Pythagoras’ theorem .................... 214 Investigation 1 Ratios of sides ............................. 217 B Defining trigonometric ratios ........................ 218 Investigation 2 Comparing ratios ......................... 221 C Trigonometric ratios of acute angles ............ 222 D Using trigonometry to find sides ................... 226 E Using trigonometry to find angles ................ 232 F Compass bearings ......................................... 235 G Worded problems using trigonometry .......... 239 Language in mathematics .................................. 244 Check your skills ................................................ 245 Review set A ....................................................... 247 Review set B ....................................................... 248 Review set C ....................................................... 249 Review set D ....................................................... 250 9 Similarity Measurement & Geometry 251 A The enlargement transformation ................. 252 Investigation 1 Similar figures ............................. 255 Investigation 2 Enlargements and scale factors ................................................... 256 B Properties of similar figures ......................... 258 C Finding sides in similar figures ..................... 260 D Applications of similar figures ...................... 267 Investigation 3 Methods of producing scale drawings ................................................ 270 Investigation 4 Uses of scale drawings ................ 270 Language in mathematics .................................. 273 Check your skills ................................................ 273 Review set A ....................................................... 275 Review set B ....................................................... 277 Review set C ....................................................... 278 Review set D ....................................................... 280 10 Linear and non-linear relationships Number & Algebra 283 A Midpoint ......................................................... 284 B Distance between two points ........................ 287 Investigation 1 Distance formula ......................... 288 C Slope (gradient) ............................................. 289 Investigation 2 Varying the slope ......................... 291 D Positive and negative gradients .................... 291 Investigation 3 Formula for gradient ................... 296 Investigation 4 The slope of a line ........................ 297 Investigation 5 Relating gradient and the tangent ratio ............................................. 297 E Sketching lines parallel to the axes .............. 298 F Graphing linear relationships ....................... 299 Investigation 6 Graphics calculator .................... 304 G Non-linear relationships ............................... 305 Language in mathematics .................................. 307 Check your skills ................................................ 308 Review set A ....................................................... 310 Review set B ....................................................... 311 Review set C ....................................................... 312 Review set D ....................................................... 313 Cumulative review chapters 8–10 314 11 Proportion and rates Number & Algebra 317 A Rates .............................................................. 318 B Direct and inverse proportion ....................... 319 C Graphs involving direct and inverse proportion ......................................... 321 D Direct linear proportion ................................. 324 E Conversion graphs ........................................ 327 F Modelling direct linear proportion ................ 329 Language in mathematics .................................. 332 Check your skills ................................................ 332 00_LEY_IM_9SB_52_53_22631_SI.indd iv 9/09/13 9:39 AM

Contents v C O N T E N T S Review set A ....................................................... 333 Review set B ....................................................... 334 Review set C ....................................................... 335 Review set D ....................................................... 336 12 Equations, inequalities and simultaneous equations Number & Algebra 337 A Linear equations review ................................ 338 B Pronumerals on both sides ........................... 339 Investigation 1 Using a spreadsheet ...................... 343 C Equations with fractions ............................... 344 Investigation 2 Equation solver .............................. 348 D Practical equations ....................................... 349 E Substitution into formulas ............................ 351 Investigation 3 Algebraic solutions ........................ 353 F Inequalities .................................................... 354 Investigation 4 Further inequalities ....................... 356 G Solving inequalities ....................................... 358 H Simultaneous equations ............................... 360 I Graphical solutions ....................................... 363 Investigation 5 Graphics calculator .......................365 J Solution by substitution ................................. 366 K Solution by elimination .................................. 367 L Problem solving with two unknowns ............ 369 Investigation 6 Simultaneous equations solutions ..371 Language in mathematics .................................. 372 Check your skills ................................................ 373 Review set A ....................................................... 375 Review set B ....................................................... 376 Review set C ....................................................... 377 Review set D ....................................................... 378 13 Further trigonometry Measurement & Geometry 379 A Right-angled triangle trigonometry (review) ... 380 Investigation 1 Complementary angles ................ 383 B Angles greater than 90° ................................ 384 Investigation 2 Special right-angled triangles ..... 386 C Exact values of trigonometric functions ....... 386 Investigation 3 Trigonometry and the number plane ................................................. 389 D Area of a triangle ........................................... 389 E Sine rule ........................................................ 391 F Cosine rule .................................................... 394 G Miscellaneous questions ............................... 397 H Practical problems ........................................ 398 Language in mathematics .................................. 403 Check your skills ................................................ 403 Review set A ....................................................... 405 Review set B ....................................................... 406 Review set C ....................................................... 407 Review set D ....................................................... 408 Cumulative review chapters 11–13 410 14 Surds and indices Number & Algebra 413 A Square root of a number ............................... 414 B Recurring decimals ....................................... 415 C Real number system ..................................... 418 D Properties of surds ........................................ 421 E Addition and subtraction of surds ................. 424 F Multiplication of surds .................................. 426 G Rationalising the denominator ...................... 429 H Fractional indices .......................................... 432 I Some properties of real numbers ................. 436 Language in mathematics .................................. 437 Check your skills ................................................ 437 Review set A ....................................................... 439 Review set B ....................................................... 440 Review set C ....................................................... 441 15 Surface area and volume Measurement & Geometry 443 Investigation 1 Volumes of pyramids and cones .......................................................... 444 A Volumes of pyramids and cones ................... 444 B Volume of a sphere ........................................ 449 C Further problems involving volumes ............ 450 D Surface areas of pyramids ............................ 452 E Surface areas of cones and spheres ............. 456 F Further problems involving surface area...... 459 Language in mathematics .................................. 461 Check your skills ................................................ 462 Review set A ....................................................... 464 Review set B ....................................................... 464 Review set C ....................................................... 465 Review set D ....................................................... 466 16 Functions and logarithms Number & Algebra 467 A Graphs in practical situations ....................... 468 B Functions ....................................................... 470 C Function notation .......................................... 474 D Domain and range of a relation .................... 477 E Inverse functions ........................................... 478 Investigation 1 Inverse functions ........................ 479 F Conditions for an inverse function ................ 481 Investigation 2 Conditions for an inverse function .............................................. 482 G Quadratic functions and inverses ................. 484 H Indices ........................................................... 488 I Logarithms .................................................... 489 J Laws of logarithms ........................................ 491 K Exponential and logarithmic graphs ............. 496 L Practical applications .................................... 498 Language in mathematics .................................. 503 Check your skills ................................................ 503 Review set A ....................................................... 505 Review set B ....................................................... 507 Answers .......................................................................\ .................... 509 Index .......................................................................\ ........................... 558 00_LEY_IM_9SB_52_53_22631_SI.indd v 9/09/13 9:39 AM

Insight Mathematics 9 stages 5.2/5.3 Australian Curriculum vi Insight Mathematics 9 stages 5.2/5.3 Australian Curriculum vi 9 AUSTRALIAN CURRICULUM FOR NSW OXFORD MATHEMATICS STUDENT BOOKS Peerless maths content designed to support deep understanding of mathematical concepts and development of skills. Written for the Mathematics Syllabus for the Australian Curriculum in New South Wales. 00_LEY_IM_9SB_52_53_22631_SI.indd vi 9/09/13 9:39 AM

Contents vii Insight Mathematics 9 stages 5.2/5.3 Australian Curriculum vii PROFESSIONAL SUPPORT • Interactive tutorials and guided examples support independent 24/7 study. • Diagnostic tools aid student understanding. • Teachers can manage class progress, set tests, and plan instruction to meet individual and whole-class needs. Drive student progress through the Mathematics Syllabus for the Australian Curriculum in New South Wales. d t 00_LEY_IM_9SB_52_53_22631_SI.indd vii 9/09/13 9:39 AM

Insight Mathematics 9 stages 5.2/5.3 Australian Curriculum viii S Y L LABUS SYLLABUS GRID Chapter Name Outcomes NSW Syllabus referencesAC references 1 Review of Year 8 2 Indices MA5.1-1WM, MA5.1-3WM, MA5.1-5NA, MA5.2-1WM, MA5.2-3WM, MA5.2-6NA, MA5.2-7NA 5.1 N&A Indices, 5.2 N&A Indices, 5.2 N&A Algebraic techniques (part)ACMNA209, ACMNA212, ACMNA213, ACMNA231 3 Collecting and analysing data MA5.1-1WM, MA5.1-2WM, MA5.1-3WM, MA5.1-12SP, MA5.1-13SP, MA5.2-1WM, MA5.2-3WM, MA5.2-15SP 5.1 S&P Probability ACMSP228, ACMSP253, ACMSP282, ACMSP283 4 Numbers of any magnitude MA5.1-1WM, MA5.1-2WM, MA5.1-3WM, MA5.1-5NA, MA5.1-9MG 5.1 N&A Indices, 5.1 M&G Numbers of any magnitudeACMNA210, ACMNA219 CR 2–4 Cumulative review chapters 2–4 5 Financial mathematics MA5.1-1WM, MA5.1-2WM, MA5.1-3WM, MA5.1-4NA 5.1 N&A Financial mathematics ACMNA211 6 Area, surface area and volume MA5.1-1WM, MA5.1-2WM, MA5.1-8MG, MA5.2-1WM, MA5.2-2WM, MA5.2-11MG, MA5.2-12MG 5.1 M&G Area and surface area, 5.2 M&G Area and surface area, 5.2 M&G Volume ACMMG216, ACMMG217, ACMMG218, ACMMG242 7 Probability MA5.1-1WM, MA5.1-2WM, MA5.1-3WM, MA5.1-13SP, MA5.2-1WM, MA5.2-2WM, MA5.2-3WM, MA5.2-17SP 5.1 S&P Probability, 5.2 S&P Probability ACMSP225, ACMSP226, ACMSP246, ACMSP247 CR 5–7 Cumulative review chapters 5-7 8 Right-angled trigonometry MA5.1-1WM, MA5.1-2WM, MA5.1-3WM, MA5.1-10MG, MA5.2-1WM, MA5.2-2WM, MA5.2-13MG 5.1 M&G Right-angled triangles (trigonometry), 5.2 M&G Right-angled triangles (trigonometry)ACMMG222, ACMMG223, ACMMG224, ACMMG245 9 Similarity MA5.1-1WM, MA5.1-2WM, MA5.1-3WM, MA5.1-11MG 5.1 M&G Properties of geometrical figuresACMMG220 (part), ACMMG221 10 Linear and non-linear relationships MA5.1-1WM, MA5.1-3WM, MA5.1-6NA, MA5.1-7NA, MA5.2-1WM, MA5.2-2WM, MA5.2-3WM, MA5.2-10NA 5.1 N&A Linear relationships, 5.1 N&A Non-linear relationships ACMNA214, ACMNA215, ACMNA239, ACMNA294, ACMNA296 CR 8–10 Cumulative review chapters 8–10 11 Proportion and rates MA5.2-1WM, MA5.2-2WM, MA5.2-5NA 5.2 N&A Ratios and rates ACMNA208 12 Equations, inequalities and simultaneous equations MA5.2-1WM, MA5.2-2WM, MA5.2-3WM, MA5.2-8NA, MA5.2-9NA 5.2 N&A Equations ACMNA215, ACMNA234, ACMNA235, ACMNA236, ACMNA237, ACMNA240 13 Further trigonometry MA5.3-1WM, MA5.3-2WM, MA5.3-3WM, MA5.3-15MG 5.3 M&G Trigonometry and Pythagoras’ theorem (part)ACMMG273, ACMMG275 CR 11–13 Cumulative review chapters 11-13 14 Surds and indices MA5.3-1WM, MA5.3-2WM, MA5.3-3WM, MA5.3-6NA 5.3 N&A Surds and indices ACMNA264 15 Surface area and volume MA5.3-1WM, MA5.3-2WM, MA5.3-3WM, MA5.3-13MG, MA5.3-14MG 5.3 M&G Area and surface area, 5.3 M&G VolumeACMMG271 16 Functions and logarithms MA5.3-1WM, MA5.3-3WM, MA5.3-11NA, MA5.3-12NA 5.3 N&A Functions and other graphs, 5.3 N&A Logarithms ACMNA265, ACMNA270 00_LEY_IM_9SB_52_53_22631_SI.indd viii 9/09/13 9:39 AM

Review of Year 8 This chapter reviews the Year 8 component of the mathematics syllabus and includes outcomes from Number and Algebra, Measurement and Geometry, and Statistics and Probability. You should be able to: 1 ▶ complete data investigations ▶ calculate ratios and rates ▶ identify congruent figures including triangles, stating the conditions ▶ work with numbers and algebraic terms involving indices ▶ calculate the perimeter and area of plane and compound shapes ▶ calculate time using mixed units ▶ operate with fractions, decimals and percentages in worded problems ▶ calculate area and circumference of circles and surface area and volume of cylinders ▶ analyse sample data using mean, mode and median and make inferences ▶ use Pythagoras’ theorem to perform calculations in right-angled triangles ▶ use algebraic techniques to simplify, expand and factorise simple algebraic expressions ▶ calculate volume and capacity ▶ solve linear equations and simple inequations ▶ solve probability problems involving simple events and use Venn diagrams ▶ graph and interpret linear relationships on the number plane. 01_LEY_IM_9SB_52_53_22631_SI.indd 1 01_LEY_IM_9SB_52_53_22631_SI.indd 1 6/09/13 8:07 AM 6/09/13 8:07 AM

REVIEW OF YEAR 8 2 Insight Mathematics 9 stages 5.2/5.3 Australian Curriculum Data Exercise 1A 1 Defi ne the statistical term ‘sample’. 2 Would a census or a sample be used to investigate the number of people who use a particular brand of toothpaste? Why? 3 Describe the sample you would use if you wanted to gather support for improved skateboard facilities at your local park. 4 For the scores 11, 14, 15, 19, 19, 21, fi nd the: a mean b mode c median d range. 5 For the scores in this stem-and-leaf plot, fi nd the: a mean b mode c median d range. 6 The back-to-back stem-and-leaf plot compares the marks gained by classes A and B in their half-yearly Mathematics exam. a Find the mean, mode, median and range for each class. b Which class performed better? Explain your answer. 7 From a school of 800 students, a random sample of 50 students was selected. There were 13 left-handed students in the sample. a What fraction of the sample was left-handed? b Estimate how many students at the school were left-handed. Ratios and rates Exercise 1 B 1 Express the ratio 25 min : 1 1 _ 4 h in simplest form. 2 Which of the following is equivalent to 3 : 5? A 21 : 36 B 45 : 75 3 Find x if 4 __ x = 7 __ 5 . 4 Simplify the following ratios. a 25 cm : 0.6 m b 360 m : 0.5 km A Stem Leaf 2 7 8 8 3 0 0 1 2 3 4 5 6 6 4 1 2 4 4 4 6 8 5 3 5 7 8 6 2 3 Class A Class B Leaf Stem Leaf 2 1 2 8 8 6 4 2 1 3 0 3 5 6 6 5 3 1 0 4 0 2 6 6 8 1 1 0 5 3 6 97 6 7 B 01_LEY_IM_9SB_52_53_22631_SI.indd 2 6/09/13 8:07 AM

Chapter 1 Review of Year 8 3 REVIEW OF YEAR 8 5 The ratio of teachers to students is 2 : 11. Calculate the number of stu\ dents if there are 10 teachers. 6 A scalene triangle has side lengths in the ratio 2 : 5 : 4. a If the shortest side is 12.4 cm, fi nd the lengths of the other two sides. b Calculate the perimeter of the triangle. 7 Ian jogs 3.5 km in 20 minutes. Express this as a rate of km/min. 8 Which is the better buy, A or B? A 1.2 L of Fizz Whiz Cola at $1.05 B 2.5 L of Fizz Whiz Cola at $2.20 9 The scale of a model aeroplane is 1 : 120. If the wingspan of the model \ is 17 cm, calculate the actual wingspan of the real aeroplane. The actual height of a building is 825 m. If a model of the building is \ constructed using a scale of 1 : 1500, calculate the height of the model. Congruence Exercise 1C 1 Which transformation(s) could have been used to produce each pair of congruent fi gures? a b 2 For each pair of triangles, state the congruency test used to show that the triangles are congruent. a b c d 3 For each pair of triangles, state why the triangles are not congruent. a b c 10 C 1 2 1 2 01_LEY_IM_9SB_52_53_22631_SI.indd 3 6/09/13 8:07 AM

REVIEW OF YEAR 8 4 Insight Mathematics 9 stages 5.2/5.3 Australian Curriculum 4 Which triangles in each group are congruent? Give a reason for your answer. a b 5 For each pair of triangles, state why the triangles are congruent. Hence fi nd the values of the pronumerals. a b c Index laws Exercise 1D 1 Write the following in index form. a 7 × 7 × 7 × 7 × 7 b 9 × 9 × 9 × 9 × 9 × 9 × 9 c 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 d 10 × 10 × 10 2 Write the base and index of each number. a 38 b 52 c 84 d 60 3 Write the following in expanded form. a 64 b 73 c 62 d 57 4 Evaluate: a 29 b 36 c 54 d 73 5 Simplify, leaving your answers in index form. a 310 × 3 3 b (72)6 c 410 ÷ 4 5 d 612 ÷ 6 e (25)4 × 2 10 6 Determine whether these calculations are true or false. a 45 × 2 6 = 8 11 b 57 ÷ 5 3 = 1 4 c 47 × 3 4 = 12 11 d 15 8 ÷ 3 2 = 5 6 7 Evaluate: a 62 b 61 c 60 d (72)0 8 Simplify: a a × a × a b 6 × r × r × r × r c x × x × y × y × y × y 9 Expand: a t 2 b 5a 4 c p6 d 15e 5 If a = 3, evaluate: a a2 b 4a 2 c (4a) 2 d 4a 0 1 55° 65° 8 2 65° 55° 8 55° 65° 8 3 50° 6 8 1 8 50° 6 2 50° 6 8 3 35° 95° 50° x y z y z x 35° 45° 15 x y z 50° 10 9 D 10 01_LEY_IM_9SB_52_53_22631_SI.indd 4 6/09/13 8:07 AM

Chapter 1 Review of Year 8 5 REVIEW OF YEAR 8 Perimeter and area Exercise 1E 1 Estimate the width of your classroom. 2 Convert the following lengths to millimetres. a 0.27 m b 0.004 km 3 Convert the following lengths to centimetres. a 0.34 m b 0.07 km 4 Calculate the perimeter of a rectangle with width 11.9 cm and length 26.\ 3 cm. 5 Calculate the perimeter of each shape. All measurements are in centimetres. a b 6 A regular octagon has a perimeter of 1012.16 cm. Calculate the length of eac\ h side. 7 Find the length of each side marked with a pronumeral, then calculate the perimeter. All measurements are in centimetres. 8 By counting squares, fi nd the area of the shape. 9 Find the areas of the following shapes. a b c d Find the areas of the following composite shapes. a b c E 38.2 31.5 20.3 33.4 51.3 11.08 47.26 15.3 3.8 11.74.2 9.6 25.39.9 6.2 y z x w 7.2 19.8 15 mm 24 mm 7 cm 2 m5 m 8 cm 11 cm 10 4 cm10 cm 40 m 50 m 120 m 18 m 18 m 01_LEY_IM_9SB_52_53_22631_SI.indd 5 6/09/13 8:07 AM

REVIEW OF YEAR 8 6 Insight Mathematics 9 stages 5.2/5.3 Australian Curriculum Find the shaded area in each shape. a b Complete these conversions. a 5 cm = _____ mm b 800 cm = _____ m c 640 mm = _____ cm d 11.6 m = _____ cm e 43.8 cm = _____ mm f 8400 cm = _____ m g 8 cm 2 = _____ mm 2 h 7.2 m 2 = _____ cm 2 i 9000 mm 2 = _____ cm 2 Time Exercise 1F 1 How many hours in 2 days? 2 Complete the following conversions. a 240 s = ____ min b 300 min = ____ h 3 Convert 210 min to hours and minutes. 4 Calculate the following. a 3 h 35 min + 5 h 48 min b 3 h 21 min − 1 h 42 min 5 If Sergio caught the bus at 6:35 am, at what time did he arrive at work, given the bus trip took 42 min? 6 High tide is at 5:20 am and low tide is at 9:08 am. Calculate the time diff erence between high and low tide. 7 Convert 2 1 _ 3 h to hours and minutes. 8 Round the digital clock display 03:16:41 to the nearest minute. Express the time in hours and minutes. Fractions, decimals and percentages Exercise 1G 1 Shade 7 __ 10 of this diagram. 2 In a class of 20 students, 1 _ 4 play soccer, 1 _ 5 play netball and the remainder play football. What fraction of the class plays football? 11 4 m 1 m 2 m 7 m2 m 2 m 20 cm 50 cm 12 F G 01_LEY_IM_9SB_52_53_22631_SI.indd 6 6/09/13 8:07 AM

Chapter 1 Review of Year 8 7 REVIEW OF YEAR 8 3 a Convert 146 ___ 11 to a mixed numeral. b Convert 3 5 _ 8 to an improper fraction. 4 a Complete: 155 ____ □ = 31 ___ 20 b Simplify 175 ___ 240 . 5 Arrange in descending order: 4 _ 5 , 8 __ 15 , 2 _ 3 6 a State the reciprocal of 2 2 _ 3 . b Calculate 3 _ 8 of 592 kg. 7 Liam earns $600 per week. He banks 1 _ 5 , spends 2 _ 3 on rent and food, and keeps the remaining money for personal use. a How much does Liam bank each week? b How much does he spend weekly on rent and food? c What fraction of Liam’s weekly wage is for personal use? d How much is kept for personal use? 8 State the value of 2 in 4.0203. 9 Express 8 + 3 __ 10 + 7 ____ 1000 as a decimal. a Write 4.2 as a mixed numeral. b Write 3 3 _ 8 as a decimal. Express 1 __ 6 as a decimal correct to 2 decimal places. a Round 3.854 44 to the nearest hundredth. b Round 3.5217 to the nearest whole number. Simplify the following. a 12.6 − 11.8 + 3.84 b 15.5 ÷ 0.05 + 22.4 c 16.2 ÷ 2 + 5.7 − 1.9 Simplify the following. a (2.1 + 3) × (11.9 − 5.9) b (10.3 − 8.7) + (0.4 × 9) a Ahmed earns $4.60 per hour. How much does he earn if he works for 10 1 _ 2 hours? b Sylvanna won $1 216 320 in a lottery. She decided to share it equally between eight people. How much did each person receive? Shade 75% of this diagram. Write 48 out of 100 as a percentage. a Convert 37% to a fraction. b Convert 57% to a decimal. Express the following as percentages. a 3.8 b 5 _ 8 Convert to percentages and arrange in ascending order: 4 _ 5 , 70%, 0.65 Convert: a 27 ___ 100 to a percentage b 15% to a simplifi ed fraction c 425% to a decimal. a Calculate 15% of $360. b Find 25% of 48 m. Express 13 kg as a percentage of 52 kg. a Increase 100 by 30%. b Decrease 320 by 25%. 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 01_LEY_IM_9SB_52_53_22631_SI.indd 7 6/09/13 8:07 AM

REVIEW OF YEAR 8 8 Insight Mathematics 9 stages 5.2/5.3 Australian Curriculum Circles and cylinders Exercise 1H 1 Name the features of each circle shown in orange. a b c d 2 What fraction of a circle is represented by this sector? 3 Write the formula for the circumference of a circle when given the diameter. 4 Calculate the circumference of a circle with a diameter of 11.4 cm correct to 1 decimal place. 5 Write the formula for the circumference of a circle when given the radius. 6 Calculate the circumference of a circle with a radius of 6.8 cm correct to 2 decimal places. 7 Write the formula for calculating the area of a circle when given the radius. 8 Calculate the area of a circle correct to 1 decimal place, given: a radius = 7 cm b diameter = 3.9 cm 9 Calculate the area of this shape correct to 1 decimal place. Calculate the volume of each cylinder. a b c Mean, mode, median and sampling Exercise 1I 1 Write the outlier in each data set. a 0, 71, 72, 72, 75 b 23, 24, 25, 25, 27, 28, 89 H 108° 15.7 cm 10 A = 0.82 m 2 1.6 m 58 cm 15.3 cm 19 cm 8.5 cm I 01_LEY_IM_9SB_52_53_22631_SI.indd 8 6/09/13 8:07 AM

Chapter 1 Review of Year 8 9 REVIEW OF YEAR 8 2 Consider the three data sets given. A 8, 10, 11, 11, 13, 96 B 7, 7, 8, 9, 11, 12 C 4, 7, 8, 9, 9, 9, 9 In which data set(s) is the following measure not a central value? a mean b mode c median 3 The sexes of 5 students chosen at random from Year 9 are female, male, male, female, male. For this data, fi n d, where possible, the: a mean b mode c median. 4 a The weights, in kilograms, of seven 1-year-old horses of the same breed were 420, 420, 430, 440, 460, 470, 650. For these weights, fi nd the: i mean ii mode iii median. b Which measure would be the most appropriate to represent the weight of 1-year-old horses of this breed? 5 Five samples of 20 students were chosen from all students in a school. The students were asked to state the number of text messages they had sent the day before. The mean number of texts per day for each sample is shown in the table. Using the information given, what is the best estimate of the mean number of texts sent per day by students of this school? Give the answer to the nearest whole number. Sample number 1 2 3 4 5 Mean number of texts 15.2 6.8 9.4 12.8 11.6 Pythagoras’ theorem Exercise 1J 1 Consider the following triangles. i Which side is the hypotenuse? ii Write an expression for Pythagoras’ theorem for the triangle. a b 2 State whether each triangle is right-angled. a b 3 a Find the value of 9 2. b Calculate the value of √ ___ 70 correct to 1 decimal place. 4 Find the length of the hypotenuse in each triangle correct to 1 decimal place. a b J c b a R Q P 7 cm 8 cm 13 cm 24 cm 26 cm 10 cm 5.8 cm 4.7 cm 7.6 cm 15.3 cm 01_LEY_IM_9SB_52_53_22631_SI.indd 9 6/09/13 8:07 AM

REVIEW OF YEAR 8 10 Insight Mathematics 9 stages 5.2/5.3 Australian Curriculum 5 Find the length of the third side of each triangle correct to 1 decimal place. a b 6 Find the value of the pronumeral in each of the following triangles correct to 1 decimal place. a b 7 Calculate the length of the diagonal of a square with side length 36 cm \ correct to 2 decimal places. Algebra Exercise 1K 1 Simplify: a 9x + 5x b 7y − y c 3a 2 + 4a 2 d 9ac − 3ca 2 Simplify: a 5 × 12n b −5 × 2a c 8m × 3 d −5p × −7 3 Simplify: a 10a ÷ 2 b 12m ÷ −3 c abc ___ a d 12m ____ 3 4 Simplify: a 4wx + 2y − 5xw − 5y b 6m + 2m − 8m 5 Simplify: a 4a ___ 7 + a __ 7 b a __ 3 − a __ 5 c q __ 5 × q d 6p ÷ 2p 6 Expand: a a(a − n) b mn(2n − 5) c 4p(3p + 2) d −2p(4y − 2w) 7 Expand and simplify: a 3(5a + 3) − 4(8 − 4a) b 3x(y − 4) + 4y(5x − 2) 8 Factorise: a mn 2 + mn b pq − aq c 4p − 12d d 25f − 15 9 Factorise each by taking out a negative factor. a −3k + 9 b −4p − 12d 9.4 cm x 7.2 cm 21.3 cm 44.2 cm x 12 cm 8 cm x 29.3 cm 10.8 cm p K 01_LEY_IM_9SB_52_53_22631_SI.indd 10 6/09/13 8:07 AM

Chapter 1 Review of Year 8 11 REVIEW OF YEAR 8 Defi ne the following terms. a pronumeral b coeffi cient If Q = 7 and p = −4, evaluate: a 4Q + p b Qp ___ 8 c 6p − 5Q d 3(Q − p) + 7p − 8Q Write an algebraic expression for each. a The product of six and d plus twenty-three b The diff erence between x and seven multiplied by three and the result divided by eight Volume and capacity Exercise 1L 1 a If ABFE is the top face of the rectangular prism, name the bottom face. b Name the front and back faces. c Name the two side faces. 2 a Draw a net of the cube shown. b Use the net to calculate the total surface area of the cube. 3 Calculate the surface area of each prism. a b 4 Construct prisms with the following cross-sections. a b 5 Draw the cross-section of each prism if it is cut along the orange dotted l\ ine shown. a b 10 11 12 L A F E G H C D B 3.8 cm 6.8 cm 15.2 cm 3 cm 5 cm 18.8 cm 12 cm 26 cm 8 cm 01_LEY_IM_9SB_52_53_22631_SI.indd 11 6/09/13 8:07 AM

REVIEW OF YEAR 8 12 Insight Mathematics 9 stages 5.2/5.3 Australian Curriculum 6 Calculate the volume of each solid. a b 7 Calculate the volume of this cylinder to the nearest cm 3. 8 Calculate the volume of each solid. a b 9 Complete the following capacity conversions. a 1 cm 3 = ____ mm 3 b 1 kL = ____ mL c 1 kL = ____ cm 3 d 1 m 3 = ____ L e 1 m 3 = ____ kL f 5.3 kL = ____ cm 3 Equations and inequations Exercise 1M 1 Show each step required to get from the expression 4x + 12 back to x. 2 Solve the following equations. a x + 11 = 17 b x + 9 = −6 c 4x = 36 d −9x = 63 e 3y + 18 = 29 f 5 − 4p = −47 g 4d + 8 = 3d − 12 h 18 + 7c = 32 − 3c i 3(m + 6) = 2(m − 1) j 8(q − 5) = −3(10 + 3q) 3 Solve the following equations. a 4p ___ 5 = 6 b 3x + 12 _______ 7 = 12 4 Is the given value for the pronumeral a solution to the equation? a 5d + 12 = 28; d = 3 b x __ 5 + 7 = 24; x = 3 2 _ 5 5 Write an equation and solve this problem. The sum of a certain number and 23 is 114. What is the number? 6 Solve the following inequations. a x + 9 ⩾ −3 b m __ 7 < 4 7.4 cm A = 427.5 cm 2 A = 133 mm 2 25 mm 15.3 cm 38 cm 23.5 cm 7.8 cm 38.7 cm 10.3 cm 5.8 cm 9.4 cm M 01_LEY_IM_9SB_52_53_22631_SI.indd 12 6/09/13 8:07 AM

Chapter 1 Review of Year 8 13 REVIEW OF YEAR 8 Probability and Venn diagrams Exercise 1N 1 A hat contains 1 red, 1 blue, 1 green and 1 yellow ticket. One ticket is chosen. a List the sample space. b What is the probability of selecting the red ticket? 2 Te n cards with the numbers 1 to 10 written on them are shuffl ed and one card is chosen. a List the sample space. b What is the probability that the card selected has 7 written on it? 3 Complete this table. Fraction Decimal Percentage a 0.7 b 25% c 5 _ 8 4 A bag contains 4 green, 9 red and 7 blue marbles. One marble is selected at random. a How many marbles are in the bag? b How many marbles are red? c What is the probability of selecting a red marble? 5 One card is selected at random from a normal deck of 52 cards. What is: a P(diamond)? b P(red card)? c P(king)? 6 a Write a statement describing a probability of 0. b Estimate a percentage probability for the phrase ‘even chance’. c Write a phrase to describe a probability of about 85%. 7 A die with the numbers 1–6 is rolled once. Describe an event that would be: a certain b impossible c of even chance. 8 A spinner has 5 equal-sized sectors coloured green, yellow, orange, brown and white. It is spun once. What is the probability of getting: a white? b any colour except white? c yellow or orange? d any colour except yellow or orange? 9 a In a group of 29 girls, 15 play netball, 11 play oztag and 8 play both. Draw a Venn diagram to show this. b How many girls: i play netball but not oztag? ii play oztag but not netball? iii play netball or oztag or both? iv play netball or oztag but not both? v play neither netball or oztag? N 01_LEY_IM_9SB_52_53_22631_SI.indd 13 6/09/13 8:07 AM

REVIEW OF YEAR 8 14 Insight Mathematics 9 stages 5.2/5.3 Australian Curriculum Coordinate geometry and straight lines Exercise 1 O 1 Plot these points on a number plane: A(0, −3), B(−2, −3), C(3, − 4), D(−3, 2), E(2, 5) 2 a Plot the points A(−3, 6), B(3, 6) and C(3, 0). b If ABCD is a square, fi nd the coordinates of point D. 3 a Use this pattern of matches to complete the table. Shape number 1 2 3 4 5 Number of matches b Write a rule describing the number of matches required to make each shape. c Using x to represent the shape number and y to represent the number of matches, write a set of coordinate points describing this information. d Graph these points on a number plane. e Mark in the next two points and write their coordinates. 4 Bulk ‘minute steak’ for barbecues is sold for $7.50 per kilogram with a minimum purchase of 2 kg. The following table shows weight versus cost for various quantities of minute steak. Weight (kg) 2 4 6 10 20 Cost ($) 15 30 45 75 150 a Using x to represent the number of kilograms and y to represent the cost in dollars, write a set of coordinate points describing this information. b Graph these points on a number plane and draw a straight line through them. c Use the graph to fi nd how much 16 kg of minute steak would cost. d Use the graph to fi nd how much minute steak could be purchased for $90. 5 Complete the table and draw the graph of y = 2x − 3. x −2 −1 0 1 2 y −5 1 6 The graph on the right shows a straight line. a Use the graph to complete this table of values. x −2 −1 0 1 y b Write the rule describing this straight line. The rule is of the form y = □ x ± △. O Shape 1 Shape 2 Shape 3 Shape 4 Shape 5 7 6 y 9 8 2 1 3 4 5 1 2 3 –3 –2 x 10 –1 01_LEY_IM_9SB_52_53_22631_SI.indd 14 6/09/13 8:07 AM

Indices This chapter deals with indices and the distributive law. After completing this chapter you should be able to: 2 ▶ simplify algebraic products and quotients using the index laws ▶ simplify expressions involving the zero index ▶ evaluate numerical expressions involving negative (integral) indices ▶ simplify algebraic expressions involving negative (integral) indices ▶ apply the index laws to expressions with negative indices ▶ apply the distributive law to the expansion of algebraic expressions. NSW Syllabus references: 5.1 N&A Indices, 5.2 N&A Indices, 5.2 N&A Algebraic techniques (part) Outcomes: MA5.1-1WM, MA5.1-3WM, MA5.1-5NA, MA5.2-1WM, MA5.2-3WM, MA5.2-6NA, MA5.2-7NA NUMBER & ALGEBRA – ACMNA209, ACMNA212, ACMNA213, ACMNA231 This c distr i After be ab ▶ sim qu ▶ sim ze ▶ ev inv 02_LEY_IM_9SB_52_53_22631_SI.indd 15 02_LEY_IM_9SB_52_53_22631_SI.indd 15 6/09/13 8:21 AM 6/09/13 8:21 AM

NUMBER & ALGEBRA 16 Insight Mathematics 9 stages 5.2/5.3 Australian Curriculum The index laws The index laws for numbers were established in Year 7: 1 When multiplying numbers with the same base, add the indices. For example: 3 6 × 3 4 = 3 6 + 4 = 3 10 2 When dividing numbers with the same base, subtract the indices. For example: 3 6 ÷ 3 4 = 3 6 − 4 = 3 2 3 When raising a power of a number to a higher power, multiply the indices. For example: (3 6)4 = 3 6 × 4 = 3 24 If we use letters to represent numbers, the rules can be generalised: a m × a n = a m + n a m ÷ a n = a m − n (a m)n = a mn E XAMPLE 1 Show by writing in expanded form that m 4 × m 3 = m 7. Solve Think Apply m 4 × m 3 = m 7 m4 × m 3 = (m × m × m × m) × (m × m × m) = m × m × m × m × m × m × m = m 7 Expand each term then write the answer in index form. Exercise 2A 1 Show by writing in expanded form that: a m2 × m 4 = m 6 b m6 ÷ m 2 = m 4 c (m 2)4 = m 8 EXAMPLE 2 a Use a calculator to evaluate these expressions when a = 3. i a4 × a 3 ii a7 b Does the value of a 4 × a 3 equal the value of a 7? Solve/Think Apply a i a4 × a 3 = 3 4 × 3 3 = 81 × 27 = 2187 Substitute the value of the variable into each expression and evaluate using a calculator. ii a7 = 3 7 = 2187 b Yes, a 4 × a 3 = a 7. Compare the numerical answers. A 2 3 BaseIndex, power or exponent The plural of index is indices. The power of a number is how many of that number are multiplied together. Expand means to spread out. Here it means by writing with multiplication signs. 02_LEY_IM_9SB_52_53_22631_SI.indd 16 6/09/13 8:21 AM

Chapter 2 Indices 17 NUMBER & ALGEBRA 2 a Use a calculator to evaluate the following when a = 2. i a5 × a 4 ii a 9 b Does the value of a 5 × a 4 equal the value of a 9? 3 a Use a calculator to evaluate the following expressions when m = 5. i m8 ÷ m 2 ii m 6 b Does the value of m 8 ÷ m 2 equal the value of m 6? 4 a Use a calculator to evaluate the following expressions when n = 3. i (n4)2 ii n 8 b Does the value of (n 4)2 equal the value of n 8? E XAMPLE 3 Use the index laws to simplify the following. a y7 × y 3 b y18 ÷ y 17 c (b5)32 Solve Think Apply a y7 × y 3 = y 10 y7 × y 3 = y 7 + 3 = y 10 When multiplying powers with the same base, add the indices. b y18 ÷ y 17 = y 1 = y y 18 ÷ y 17 = y 18 − 17 = y 1 = y When dividing powers with the same base, subtract the indices. c (b5)3 = b 15 (b5)3 = b 5 × 3 = b 15 When raising a power of a number to a higher power, multiply the indices. 5 Use the index laws to simplify the following. a m3 × m 6 b q8 × q 7 c t10 × t 9 d b15 × b × b 4 e v × v 5 × v 7 6 Use the index laws to simplify the following. a a12 ÷ a 10 b x15 ÷ x 5 c w 8 ÷ w 2 d b6 ÷ b 5 e z20 ÷ z 19 7 Use the index laws to simplify the following. a (b4)2 b (h5)3 c (k8)2 d (z10)6 e (n2)4 Index comes from the Latin word 'indicare': to point, disclose, show; as in using your index fi nger. 8 Use the index laws to simplify the following. a m4 × m 2 b x9 ÷ x 6 c (b4)6 d m3 × m 6 × m 4 e (v7)10 f n8 ÷ n 7 g b8 ÷ b h (y5)5 i t 10 × t 20 × t j a12 ÷ a 6 02_LEY_IM_9SB_52_53_22631_SI.indd 17 6/09/13 8:21 AM

NUMBER & ALGEBRA 18 Insight Mathematics 9 stages 5.2/5.3 Australian Curriculum EXAMPLE 4 Explain why the index laws cannot be used to simplify the following. a p3 × q 4 b m6 ÷ n 4 Solve/Think Apply a p3 × q 4 = p × p × p × q × q × q × q = p 3q4 As the bases are not the same, we cannot simplify further. The index laws can only be used if the bases are the same. b m6 ÷ n 4 = m × m × m × m × m × m _______________________ n × n × n × n = m 6 ___ n4 Again, as the bases are not the same, we cannot simplify further. 9 Explain why the index laws cannot be used to simplify the following. a k5 × m 3 b x9 ÷ y 6 Determine whether these statements are true or false. If they are false, rewrite the answer to make them true. a b4 × b 3 = b 7 b m5 × m 2 = m 10 c p4 × p 5 = p 20 d e6 × e 10 = e 16 e a4 × b 5 = ab 9 f z10 ÷ z 2 = z 8 g p12 ÷ p 3 = p 4 h t 8 ÷ t 7 = t i w15 ÷ w 3 = w 5 j p 6 ___ q2 = p 4 __ q k (b7)2 = b 14 l (n10)3 = n 13 Applying the index laws E XAMPLE 1 Simplify the following. a p 5 × p 6 ______ p8 b (a 5)4 ______ a3 × a 2 Solve Think Apply a p 5 × p 6 ______ p8 = p 3 p 5 × p 6 ______ p8 = p 5 + 6 ____ p8 = p 11 ___ p8 = p 11 − 8 = p 3 When multiplying powers with the same base, add the indices. When dividing, subtract the indices. b (a 5)4 ______ a3 × a 2 = a 20 ___ a5 = a 15 (a 5)4 ______ a3 × a 2 = a 5 × 4 ____ a3 + 2 = a 20 ___ a5 = a 20 − 5 = a 15 When raising a power to a higher power, multiply the indices. 10 B 02_LEY_IM_9SB_52_53_22631_SI.indd 18 6/09/13 8:21 AM

Chapter 2 Indices 19 NUMBER & ALGEBRA Exercise 2B 1 Simplify the following. a x5 × x 7 ______ x6 b w3 × w 10 ________ w8 c m8 × m 4 _______ m10 d k10 × k 6 _______ k8 × k 5 e a7 × a 6 ______ a8 × a 2 f y 9 × y 11 _______ y10 × y 8 g z16 × z 2 _______ z10 × z 7 h x 14 ______ x3 × x 4 i k 30 _______ k16 × k 5 j (m 2)3 × m 5 k (a4)5 × (a 3)4 l (t 5)6 ____ t10 m (y 5)5 ____ y20 n a16 × a 6 × a 4 ___________ a12 × a 8 × a o b10 × b 20 × b 30 ____________ (b4)5 E XAMPLE 2 Simplify the following. a 5m 4 × 3m 6 b 2k7 × 4k 3 × 3k 5 Solve Think Apply a 5m 4 × 3m 6 = 15m 10 5m 4 × 3m 6 = 5 × 3 × m 4 × m 6 = 15 × m 4 + 6 = 15m 10 Multiply the numerical coeffi cients and use the index laws to multiply the pronumerals. b 2k7 × 4k 3 × 3k 5 = 24k 15 2k7 × 4k 3 × 3k 5 = 2 × 4 × 3 × k 7 × k 3 × k 5 = 24 × k 7 + 3 + 5 = 24k 15 2 Simplify the following. a 4m 5 × 3m 7 b 5p 4 × 2p 6 c 3t 8 × 6t 4 d 10a 12 × 7a 4 e 4w 9 × 6w 10 f 5b 3 × 6b 2 × b 4 g 3z 6 × 4z 8 × 2z 3 h 2q 5 × 5q 7 × 8q 6 i d 4 × 6d 6 × 3d 8 EXAMPLE 3 Simplify the following. a 12m 8 _____ 3m 6 b 20a 10 _____ 16a 4 Solve Think Apply a 12m 8 _____ 3m 6 = 4m 2 12m 8 _____ 3m 6 = 12 × m 8 _______ 3 × m 6 = 12 ___ 3 × m 8 ___ m6 = 4 × m 2 = 4m 2 Divide the numerical coeffi cients and use the index laws to divide the pronumerals. 02_LEY_IM_9SB_52_53_22631_SI.indd 19 6/09/13 8:21 AM

NUMBER & ALGEBRA 20 Insight Mathematics 9 stages 5.2/5.3 Australian Curriculum EX AMPLE 3 CONTINUED Solve ThinkApply b 20a 10 _____ 16a 4 = 5a 6 ___ 4 20a 10 _____ 16a 4 = 20 × a 10 _______ 16 × a 4 = 20 ___ 16 × a 10 ___ a4 = 5 __ 4 × a 6 = 5a 6 ___ 4 Be careful not to mix up the numerators and denominators. 3 Simplify the following. a 6m 7 ____ 3m 2 b 10a 12 _____ 5a 7 c 12w 10 _____ 4w 8 d 8z12 ____ 6z8 e 16k 9 ____ 12k 3 f 9e 10 ____ 6e6 g 2m 8 ____ 6m 3 h 6a 15 _____ 12a 10 i 9t13 ____ 12t 6 j 15b 11 _____ 20b 6 E XAMPLE 4 Simplify the following. a a3 __ a7 b 10w 2 _____ 8w 4 Solve ThinkApply a a3 __ a7 = 1 __ a4 a 3 __ a7 = 1a × 1a × 1a _________________________ 1a × 1a × 1a × a × a × a × a = 1 __ a4 Or a 3 __ a7 = a 3 ÷ a 3 ______ a7 ÷ a 3 = 1 ____ a7−3 = 1 __ a4 Write in expanded form and cancel or divide both the numerator and the denominator by the numerator. b 10w 2 _____ 8w 4 = 5 ____ 4w 2 10w 2 _____ 8w 4 = 10 ___ 8 × w 2 ___ w4 = 5 __ 4 × 1w × 1w _______________ 1w × 1w × w × w = 5 __ 4 × 1 ___ w2 = 5 ____ 4w 2 Or 10w 2 _____ 8w 4 = 10 ___ 8 × w 2 ÷ w 2 _______ w4 ÷ w 2 = 5 __ 4 × 1 ___ w2 = 5 ____ 4w 2 Divide the numerical coeffi cients. Write the pronumerals in expanded form and cancel, or divide both the numerator and the denominator by the numerator. 02_LEY_IM_9SB_52_53_22631_SI.indd 20 6/09/13 8:21 AM

Chapter 2 Indices 21 NUMBER & ALGEBRA 4 Simplify the following. a m4 ___ m7 b 3k 2 ___ k6 c 4p 3 ___ 2p 5 d 5y 3 ____ 15y 7 e 8z2 ___ 6z7 f 12x 3 _____ 18x 10 E XAMPLE 5 Simplify (2a 3)5. Solve ThinkApply (2a 3)5 = 32a 15 (2a 3)5 = 2a 3 × 2a 3 × 2a 3 × 2a 3 × 2a 3 = 2 × 2 × 2 × 2 × 2 × a 3 × a 3 × a 3 × a 3 × a 3 = 2 5 × (a 3)5 = 32 × a 15 = 32a 15 Expand, separate the numerical coeffi cients and the pronumerals and simplify using the index laws. 5 Simplify the following. a (3a 4)3 b (2m 3)6 c (7p 5)2 d (10k 2)4 e (5t 11)3 f (x3y2)5 g (m 4n6)3 h (p7q3)4 i (a4b10)2 j (2x 5y2)3 k (3x 7y4)3 l (5p 2q4)3 E XAMPLE 6 Simplify the following. a 5m 2n4 × 3m 5n8 b 12x 10y8 ______ 8x6y2 Solve Think Apply a 5m 2n4 × 3m 5n8 = 15m 7n12 5m 2n4 × 3m 5n8 = 5 × 3 × m 2 × m 5 × n 4 × n 8 = 15 × m 7 × n 12 = 15m 7n12 Separate the terms and group together the numerical coeffi cients and the like terms. Use the index laws to simplify, where appropriate. b 12x 10 y8 ______ 8x6y2 = 3x 4y6 _____ 2 12x 10 y8 ______ 8x6y2 = 12 ___ 8 × x 10 ___ x6 × y 8 __ y2 = 3 __ 2 × x 4 × y 6 = 3x 4y6 _____ 2 6 Simplify the following. a 4a 3b2 × 2a 5b3 b 5m 6n7 × 2m 4n c 3p 5q8 × 4p 6q7 d 10x 4y3 × 3x 6y e 2w 10z12 × 6w 4z5 f 5a 2b3c4 × 7ab 3c2 g 6a 5b6 _____ 4a 3b2 h 15x 10y9 ______ 5x6y2 i 2k 7m12 ______ 10k 3m6 j 9a 11b6 _____ 12a 8b k 12m 7n8 ______ 15m 6n8 l 5a 2b3c4 ______ 7ab 3c6 Remember to add indices when multiplying and subtract them when dividing, 02_LEY_IM_9SB_52_53_22631_SI.indd 21 6/09/13 8:21 AM

NUMBER & ALGEBRA 22 Insight Mathematics 9 stages 5.2/5.3 Australian Curriculum The zero index Any number raised to the power zero is equal to 1. In general, a 0 = 1 EXAMPLE 1 a Use the index laws to simplify a 5 ÷ a 5. b Hence show that a 0 = 1. Solve Think Apply a a5 ÷ a 5 = a 0 a5 ÷ a 5 = a 5 − 5 = a 0 Use the index laws to divide one term by the other. b a5 ÷ a 5 = 1 Hence a 0 = 1. Any number divided by itself = 1.Hence any number raised to the power zero is equal to 1. Exercise 2C 1 a Use the index laws to simplify a 4 ÷ a 4. b Hence show that a 0 = 1. EXAMPLE 2 Evaluate the following. a x0 b (3x) 0 c 3x0 Solve Think/Apply a x0 = 1Any number raised to the power zero is equal to 1. b (3x) 0 = 1 3x = 3 × x is a number. Any number raised to the power zero is equal to 1. c 3x0 = 33x 0 = 3 × x 0 The 3 is not to the power zero; only the x is to the zero power. = 3 × 1 = 3 2 Evaluate the following. a y0 b (3y) 0 c 3y0 d 4k0 e 9t0 f (6z) 0 g (10m) 0 h 10m 0 i 8b 0 j (7q) 0 k 3m 0 + 1 l 9e0 − 3 m 6p 0 + 7 n 3a 0 + 2b 0 o 6x0 − 4y 0 C 0 1 1 = 0 5 = 0 127 = 0 a = 02_LEY_IM_9SB_52_53_22631_SI.indd 22 6/09/13 8:21 AM

Chapter 2 Indices 23 NUMBER & ALGEBRA Negative indices EXAMPLE 1 Complete the table to fi nd the meaning of 3 −1, 3 −2, 3 −3. 33 32 31 30 3−1 3−2 3−3 27 9 3 Solve Think/Apply 33 32 31 30 3−1 3−2 3−3 27 9 3 1 1 __ 3 1 __ 9 = 1 __ 32 1 ___ 27 = 1 __ 33 Each number in the second row can be found by multiplying the number before it by 1 _ 3 . Exercise 2D 1 Multiply the numbers in the second row by 1 _ 2 to complete the table. Hence fi nd the meaning of 2 −1, 2 −2, 2 −3. 23 22 21 20 2−1 2−2 2−3 8 8 × 1 _ 2 = __ __ × 1 _ 2 = __ Hence 2 −1 = 1 ___ □ = 1 ___ 2□ 2 −2 = 1 ___ □ = 1 ___ 2□ 2 −3 = 1 ___ □ = 1 ___ 2□ 2 Multiply the numbers in the second row by 1 __ 10 to complete the table and fi nd the meaning of 10 −1, 10 −2, 10 −3. 10 3 10 2 10 1 10 0 10 −1 10 −2 10 −3 1000 Hence 10 −1 = 1 ___ □ = 1 ____ 10 □ 10 −2 = 1 ___ □ = 1 ____ 10 □ 10 −3 = 1 ___ □ = 1 ____ 10 □ E XAMPLE 2 a Use the index laws to simplify 3 4 ÷ 3 6. b Write in expanded form and show that 3 4 ÷ 3 6 = 1 __ 32 . c Hence show that 3 −2 = 1 __ 32 . Solve/Think Apply a 34 ÷ 3 6 = 3 4 − 6 = 3 −2 Simplify using the index laws and by writing in expanded form and cancelling. In general: 3 −n = 1 __ 3n b 34 ÷ 3 6 = 13 × 13 × 13 × 13 ______________________ 13 × 13 × 13 × 13 × 3 × 3 = 1 __ 32 c From parts a and b, 3 −2 = 1 __ 32 . D Multiplying a number by 1 _ 3 is the same as dividing it by 3. 02_LEY_IM_9SB_52_53_22631_SI.indd 23 6/09/13 8:21 AM

NUMBER & ALGEBRA 24 Insight Mathematics 9 stages 5.2/5.3 Australian Curriculum 3 a Use the index laws to simplify 5 3 ÷ 5 7. b By writing in expanded form, show that 5 3 ÷ 5 7 = 1 __ 54 . c Hence show that 5 −4 = 1 __ 54 . 4 Write the following with positive indices. a 3−1 b 4−3 c 2−5 d 8−2 e 5−4 f 12 −1 g 9−2 h 6−1 i 7−3 j 3−6 k 2−8 l 5−1 m 10 −5 n 5−10 o 4−15 E XAMPLE 3 Write the following as simplifi ed fractions or mixed numerals. a 5−2 b 3−5 Solve Think Apply a 5−2 = 1 ___ 25 5 −2 = 1 __ 52 = 1 ___ 25 Write with a positive index then evaluate using a calculator if necessary. b 3−5 = 1 ____ 243 3 −5 = 1 __ 35 = 1 ____ 243 5 Write the following as simplifi ed fractions or mixed numerals. a 3−2 b 2−5 c 4−3 d 5−4 e 2−10 f 6−3 g 9−2 h 3−4 i 5−5 j 2−9 k 7−3 l 4−4 m 3−6 n ( 2 _ 5 ) −1 o ( 1 3 _ 4 ) −1 EXAMPLE 4 Write the following with negative indices. a 1 __ 3 b 1 __ 32 c 1 __ 38 Solve/Think Apply a 1 __ 3 = 1 __ 31 = 3 −1 3−n = 1 __ 3n is equivalent to 1 __ 3n = 3 −n. b 1 __ 32 = 3 −2 c 1 __ 38 = 3 −8 6 Write the following with negative indices. a 1 __ 2 b 1 __ 22 c 1 __ 28 d 1 __ 25 e 1 __ 23 f 1 __ 5 g 1 __ 72 h 1 __ 43 i 1 __ 34 j 1 __ 56 k 1 ___ 310 l 1 __ 6 m 1 __ 75 n 1 __ 49 o 1 ___ 10 02_LEY_IM_9SB_52_53_22631_SI.indd 24 6/09/13 8:21 AM

25 NUMBER & ALGEBRA EXAMPLE 5 Write 1 ___ 5−3 with a positive index. Solve/Think Apply 1 ___ 5−3 = 1 ___ 1 __ 53 = 1 × 5 3 __ 1 = 5 3 Or 1 ___ 5−3 = 5 0 ___ 5−3 = 5 0 − (−3) = 5 3 Write 5 −3 with a positive index and divide the fractions. Or write 1 as 5 0 and divide using the index laws. 7 Write the following with positive indices. a 1 ___ 3−4 b 1 ___ 2−7 c 1 ___ 7−2 d 1 ___ 6−1 e 1 ___ 4−5 E XAMPLE 6 Evaluate ( 3 __ 7 )−1. Solve/Think Apply ( 3 __ 7 )−1 = 1 ___ 3 __ 7 = 1 × 7 __ 3 = 7 __ 3 or 2 1 _ 3 Write ( 3 __ 7 )−1 with a positive index and divide the fractions. To divide by a fraction, invert the fraction (turn it upside down) and multiply. 02_LEY_IM_9SB_52_53_22631_SI.indd 25 6/09/13 8:21 AM

NUMBER & ALGEBRA 26 Insight Mathematics 9 stages 5.2/5.3 Australian Curriculum 8 Evaluate the following. a ( 2 _ 3 )−1 b ( 3 _ 4 )−1 c ( 7 _ 8 )−1 d ( 1 _ 5 )−1 e ( 1 __ 10 )−1 f (1 1 _ 2 )−1 g (2 3 _ 4 )−1 9 Using the results of questions 7 and 8, simplify ( a __ b )−1. Negative indices with variables A negative index means to invert the number (turn it upside-down). In general, a −n = 1 __ an E XAMPLE 1 a Use the index laws to simplify a 4 __ a5 . b Expand and simplify a 4 __ a5 . c Hence show that a −1 = 1 __ a . Solve/Think Apply a a4 __ a5 = a 4 – 5 = a −1 Simplify using the index laws and by writing in expanded form and cancelling. b a4 __ a5 = 1a × 1a × 1a × 1a ___________________ 1a × 1a × 1a × 1a × a = 1 __ a c Since a 4 __ a5 = a −1 and a 4 __ a5 = 1 __ a , then a −1 = 1 __ a . Exercise 2E 1 a Use the index laws to simplify a 2 __ a5 . b Expand and simplify a 2 __ a5 . c Hence show that a −3 = 1 __ a3 . 2 a Use the index laws to simplify a 2 __ a6 . b Expand and simplify a 2 __ a6 . c Hence show that a −4 = 1 __ a4 . 3 a Use the index laws to simplify a __ a6 . b Expand and simplify a __ a6 . c Hence show that a −5 = 1 __ a5 . 4 Complete: From questions 1 to 3 it can be seen that, in general, a −n = ____. E bles 02_LEY_IM_9SB_52_53_22631_SI.indd 26 6/09/13 8:21 AM

Chapter 2 Indices 27 NUMBER & ALGEBRA EXAMPLE 2 Write the following with positive indices. a k−9 b m−15 Solve/Think Apply a k−9 = 1 __ k 9 In general a −n = 1 __ an . b m−15 = 1 ___ m15 5 Write the following with positive indices. a y−2 b k−1 c m−3 d x−6 e t −10 E XAMPLE 3 Write the following with negative indices. a 1 __ a5 b 1 __ y7 Solve/Think Apply a 1 __ a5 = a −5 a−n = 1 __ an is equivalent to 1 __ an = a −n. b 1 __ y7 = y −7 6 Write the following with negative indices. a 1 __ a8 b 1 __ k2 c 1 ___ x11 d 1 ___ n14 e 1 ___ z20 E XAMPLE 4 Write the following with positive indices. a 3m −2 b (3m) −2 Solve Think Apply a 3m −2 = 3 ___ m2 3m −2 = 3 × m −2 = 3 __ 1 × 1 ___ m2 = 3 ___ m2 First express the term as a product, then write it with a positive index. b (3m) −2 = 1 ____ 9m 2 (3m) −2 = 1 _____ (3m) 2 = 1 ____ 9m 2 First write the term with a positive index, then simplify. 7 Write the following with positive indices. a 3k−1 b (3k) −1 c 2y−5 d (2y) −5 e 3t−4 f (3t) −4 02_LEY_IM_9SB_52_53_22631_SI.indd 27 6/09/13 8:21 AM

NUMBER & ALGEBRA 28 Insight Mathematics 9 stages 5.2/5.3 Australian Curriculum EXAMPLE 5 Write the following with negative indices. a 1 ___ m4 b 3 ___ m4 c 1 ____ 3m 4 Solve/Think Apply a 1 ___ m4 = m −4 Write as a product, then use 1 __ an = a −n. b 3 ___ m4 = 3 × 1 ___ m4 = 3 × m −4 = 3m −4 c 1 ____ 3m 4 = 1 __ 3 × 1 ___ m4 = 1 __ 3 × m −4 = 1 _ 3 m−4 or m −4 ____ 3 8 Write the following with negative indices. a i 1 __ p5 ii 7 __ p5 iii 1 ___ 7p 5 b i 1 ___ m10 ii 6 ___ m10 iii 1 ____ 6m 10 c i 1 __ y7 ii 4 __ y7 iii 1 ___ 4y7 Further use of the index laws E XAMPLE 1 Use the index laws to simplify the following. a m−6 × m 2 b q −2 ÷ q −7 c (x −3)5 Solve Think Apply a m−6 × m 2 = m −4 m−6 × m 2 = m −6 + 2 = m −4 When multiplying add the indices. b q−2 ÷ q −7 = q 5 q−2 ÷ q −7 = q −2 – (−7) = q −2 + 7 = q 5 When dividing subtract the indices. c (x−3)5 = x −15 (x−3)5 = x −3 × 5 = x −15 When raising a power to another power, multiply the indices. Exercise 2F 1 Use the index laws to simplify the following. a a−5 × a −2 b y−3 × y 7 c e5 × e −7 d n4 × n −3 e b−6 ÷ b 2 f w3 ÷ w −2 g z−2 ÷ z −4 h k−6 ÷ k −2 i (y−2)4 j (t 5)−4 F 02_LEY_IM_9SB_52_53_22631_SI.indd 28 6/09/13 8:21 AM

Chapter 2 Indices 29 NUMBER & ALGEBRA EXAMPLE 2 Simplify the following. a 5m −3 × 6m 7 b 4y7 ÷ 5y −2 c (5y −2)3 Solve Think Apply a 5m −3 × 6m 7 = 30m 4 5m −3 × 6m 7 = 5 × 6 × m −3 × m 7 = 30 × m −3 + 7 = 30 × m 4 = 30m 4 Separate the terms and group together the numerical coeffi cients and the like terms. Use the index laws to simplify, where appropriate. b 4y7 ÷ 5y −2 = 4 _ 5 y9 or 4y 9 ___ 5 4y 7 ÷ 5y −2 = 4y 7 ____ 5y−2 = 4 __ 5 × y 7 ___ y−2 = 4 __ 5 × y 7 − (−2) = 4 __ 5 × y 9 = 4 _ 5 y9 or 4y 9 ___ 5 c (5y −2)3 = 125y −6 (5y −2)3 = 5y −2 × 5y −2 × 5y −2 = 5 3 × (y −2)3 = 125y −6 2 Simplify the following. a 10a 5 × 9a −3 b 6b −5 × 3b −2 c 3v−6 × 2v 2 d 8y 5 ÷ 2y −1 e 6p −4 ÷ 2p 2 f 3k −4 ÷ 8k −2 g (5z −4)3 h (2m −3)5 i (3w −6)2 j 4n −3 × 3n −4 ÷ 6n −5 k (3x) −2 l (5m 2) −3 EXAMPLE 3 State whether the following are true or false. a m3 ÷ m 5 = m 2 b 3y0 = 1 c 6k 4 ÷ 2k 4 = 3 d 2p −3 = 1 ___ 2p 3 e x−3 = −3x Solve Think Apply a Statement is false. m 3 ÷ m 5 = m 3 – 5 = m −2 Use the index laws, the results for the zero index and negative indices. b Statement is false. 3y 0 = 3 × y 0 = 3 × 1 = 3 c Statement is true. 6k 4 ÷ 2k 4 = 6 __ 2 × k 4 __ k4 = 3 × k 4 − 4 = 3 × k 0 = 3 × 1 = 3 d Statement is false. 2p −3 = 2 × p −3 = 2 × 1 __ p3 = 2 __ p3 e Statement is false. x −3 = 1 __ x3 02_LEY_IM_9SB_52_53_22631_SI.indd 29 6/09/13 8:21 AM

NUMBER & ALGEBRA 30 Insight Mathematics 9 stages 5.2/5.3 Australian Curriculum 3 State whether the following are true or false. a 6m 0 = 1 b a4 ÷ a 7 = a 3 c 8t 9 ÷ 2t 9 = 4 d 3c−2 = 1 ___ 3c2 e 4k0 = 4 f b ÷ b 6 = b 5 g 5x6 ÷ x 6 = 5x h 4y−3 = 4 __ y3 i (2p −1)3 = 8 __ p3 j x−2 = −2x k x−4 = −4 ___ x l (2x −1)−2 = x 2 __ 4 E XAMPLE 4 By substituting a = 5, show that a −2 ≠ −2a. Solve Think Apply If a = 5, a −2 = 5 −2 = 1 ___ 25 and −2a = −10 Hence a −2 ≠ −2a. 5 −2 = 1 __ 52 and −2a = −2 × a Evaluate each expression by substituting the value of the variable. 4 By substituting a = 3, show the following. a a2 ≠ 2a b a3 ≠ 3a c a−2 ≠ −2 ___ a d a−3 ≠ −3 ___ a e a2 × a ≠ a 2 + a f a2 + a 2 ≠ a 4 g a2 − a 2 ≠ a 0 h 5a 2 × 3a ≠ 5a 2 + 3a Removing grouping symbols E XAMPLE 1 Expand the following. a 3(x + 5) b 4(3y − z) Solve Think Apply a 3(x + 5) = 3x + 15 3(x + 5) = (x + 5) + (x + 5) + (x + 5) = x + x + x + 5 + 5 + 5 = 3 × x + 3 × 5 = 3x + 15 Use a × (b + c) = a × b + a × c. b 4(3y − z) = 12y − 4z 4(3y − z) = (3y − z) + (3y − z) + (3y − z) + (3y − z) = 3y + 3y + 3y + 3y − z − z − z − z = 4 × 3y + 4 × (−z) = 12y + (−4z) = 12y − 4z G Expand means to write the expression without the grouping symbols. Adding −4z is the same as subtracting 4z. 02_LEY_IM_9SB_52_53_22631_SI.indd 30 6/09/13 8:21 AM

Chapter 2 Indices 31 NUMBER & ALGEBRA From Example 1 we can see that, in general: a × (b + c) = a × b + a × c To remove grouping symbols, multiply each term inside them by the term (number and/or pronumeral) at the front. This is known as the distributive law . EXAMPLE 2 Use the distributive law to expand the following. a 5(2y + 3) b 7(3y − 4w) Solve Think Apply a 5(2y + 3) = 10y + 15 5(2y + 3) = 5 × 2y + 5 × 3 = 10y + 15 Use a(b + c) = a × b + a × c. b 7(3y − 4w) = 21y − 28w 7(3y − 4w) = 7 × 3y − 7 × 4w = 21y − 28w Exercise 2G 1 Use the distributive law to expand the following. a 3(2w + 5) b 6(3z − 2) c 5(4a + 3b) d 2(4x – 3y) e 10(z 2 + 6) f 7(ab − 2a 2) g 4(m 2 + n 2) h 2(m 3 − 3mn) i 5(4b + 2a + 3) j 3(5x − 3y − 2z) EXAMPLE 3 Expand the following. a 3w(2y + 4z) b 2a(3a − 4b) c 4m 2(m 3 + 2m 5) Solve Think Apply a 3w(2y + 4z) = 6wy + 12wz3w(2y + 4z) = 3w × 2y + 3w × 4z = 6wy + 12wz Use a(b + c) = a × b + a × c and the index laws. b 2a(3a − 4b) = 6a 2 − 8ab 2a(3a − 4b) = 2a × 3a − 2a × 4b = 6a 2 − 8ab c 4m 2(m 3 + 2m 5) = 4m 5 + 8m 7 4m 2(m 3 + 2m 5) = 4m 2 × m 3 + 4m 2 × 2m 5 = 4m 5 + 8m 7 2 Expand the following. a 3a(2b + 4c) b 4x(3x − 2y) c 10k(6k − 4m) d m(m 2 + 2) e 6x(2y − 5x 2) f 3k2(2k 2 + 5) g a3(5a 2 − 2) h 2p 5(p2 + 3p 3) 02_LEY_IM_9SB_52_53_22631_SI.indd 31 6/09/13 8:21 AM

NUMBER & ALGEBRA 32 Insight Mathematics 9 stages 5.2/5.3 Australian Curriculum EXAMPLE 4 Expand the following. a −3(2w + 5) b −2(4a – 3b) c −(4m + 3n) Solve Think Apply a −3(2w + 5) = −6w − 15−3(2w + 5) = −3 × 2w + (−3) × 5 = −6w + (−15) = −6w − 15 Use a(b + c) = a × b + a × c. b −2(4a − 3b) = −8a + 6b −2(4a − 3b) = −2 × 4a + (−2) × (−3b) = −8a + 6b c −(4m + 3n) = −4m − 3n −(4m + 3n) = −1 × 4m + (−1) × 3n = −4m + (−3n) = −4m − 3n 3 Expand the following. a −2(y + 3) b −5(a + 2) c −3(w + 4) d −4(m − 7) e −(t + 3) f −(b + 6) g −3(2k + 5) h −2(4m − 5) i −(7w + 3) j −(4x − 1) k −(4 − 3x) l −(7x − 5y) EXAMPLE 5 Expand the following. a −3a(5a 2 + 2ab ) b −n 3(2n 4 − 5n 2p) Solve Think Apply a −3a(5a 2 + 2ab) = −15a 3 − 6a 2b −3a(5a 2 + 2ab ) = −3a × 5a 2 + (−3a) × 2ab = −15a 3 + (−6a 2b) = −15a 3 − 6a 2b Use a(b + c) = a × b + a × c with the index laws. b −n 3(2n 4 − 5n 2p) = − 2n 7 + 5n 5p −n 3(2n 4 − 5n 2p) = −n 3 × 2n 4 + (−n 3) × (−5n 2p) = −2n 7 + 5n 5p 4 Expand the following. a −2a(3a 2 + 2ab ) b −4x(2x 2 − 3xy) c −3p 2(3p 2 + 4pq) d −y 3(4y 2 − 3xy) e −3m 4(2m 2 + 5mn) f −y 2(y3 − 4) g −5x 2(2x 3 − 3xy) h −t(mt 2 + t) i −3x(2x − 4y) 02_LEY_IM_9SB_52_53_22631_SI.indd 32 6/09/13 8:21 AM

Chapter 2 Indices 33 NUMBER & ALGEBRA EXAMPLE 6 Expand and simplify by collecting like terms. a 3(a + 2) + 7 b 3 + 2(3n − 5) Solve ThinkApply a 3(a + 2) + 7 = 3a + 6 + 7 = 3a + 13 3(a + 2) + 7 = 3 × a + 3 × 2 + 7 = 3a + 6 + 7 = 3a + 13 Use a(b + c) = a × b + a × c. b 3 + 2(3n − 5) = 3 + 6n − 10 = 6n − 7 3 + 2(3n − 5) = 3 + 2 × 3n − 2 × 5 = 3 + 6n − 10 = 6n − 7 Do the multiplication using the distributive law before the addition. 5 Expand and simplify the following. a 4(a + 3) + 6 b 2(3b − 12) + 12 c 3(4w + 2) − 7 d 5(2y − 3) − 2 e 6(3z − 1) + 4 f 10 + 2(4x + 3) g 12 + 2(3b − 5) h 13 + 4(y + 5) i 4 + 3(2w − 4) E XAMPLE 7 Expand and simplify by collecting like terms. a 5 − 2(4y − 3) b −4(3x + 1) − 6 Solve ThinkApply a 5 − 2(4y − 3) = 5 − 8y + 6 = −8y + 11 5 − 2(4y − 3) = 5 − [2(4y − 3)] = 5 − (2 × 4y − 2 × 3) = 5 − (8y − 6) = 5 − 8y −(−6) = 5 − 8y + 6 = −8y + 11 Do the multiplication using the distributive law before the subtraction. b −4(3x + 1) − 6 = −12x − 4 − 6 = −12x − 10 −4(3x + 1) − 6 = −4 × 3x + (−4) × 1 − 6 = −12x − 4 − 6 = −12x − 10 Use a(b + c) = a × b + a × c. 6 Expand and simplify the following. a 12 − 2(a + 5) b 8 − 3(y − 2) c 9 − 4(b + 3) d 7 − 2(v − 6) e 20 − 3(2w + 5) f 2 − 5(3t − 4) g 4 − 3(5x + 2) h 10 − 2(3k − 1 i 5 − 3(3 + 4z) Like terms are terms that have the same power(s) of the same base(s). Fo r example, x 2 and 3x 2 are like terms, but x 2 and 3x are not. 02_LEY_IM_9SB_52_53_22631_SI.indd 33 6/09/13 8:21 AM

NUMBER & ALGEBRA 34 Insight Mathematics 9 stages 5.2/5.3 Australian Curriculum EXAMPLE 8 Expand and simplify the following. a 3(x + 2) + 2(x − 4) b 4(2m − 3) + 3(m − 2) Solve/Think Apply a 3(x + 2) + 2(x − 4) = 3x+ 6 + 2x − 8 = 3x+ 2x + 6 − 8 = 5x − 2 Expand using the distributive law and then collect like terms. b 4(2m − 3) + 3(m − 2) = 8m − 12 + 3m − 6 = 8m + 3m − 12 − 6 = 11m − 18 7 Expand and simplify the following. a 5(2k + 3) + 3(k − 2) b 2(6m + 7) + 3(m − 1) c 4(2p − 1) + 2(3p + 5) d 3(3a + 2) + 4(a − 3) e 2(5x − 3) + 5(3x − 1) f 3(4y − 2) + (2y + 7) g (6v − 1) + 3(2v − 5) h 4(3x + 2y) + 2(5x − 3y) i 7(2a − 3b) + 3(3a + 4b) EXAMPLE 9 Expand and simplify the following. a 2(3p + 4q) − 4(2p − 3q) b 3(4m − 1) − (m + 4) Solve/Think Apply a 2(3p + 4q) − 4(2p − 3q) = 6p + 8q − 8p + 12q = 6p − 8p + 8q + 12q = −2p + 20q Expand using the distributive law and then collect like terms. b 3(4m − 1) − (m + 4) = 12m − 3 − m − 4 = 12m − m − 3 − 4 = 11m − 7 8 Expand and simplify the following. a 3(2k + 5) − 2(k + 3) b 5(w + 4) − 3(w − 2) c 2(6t + 1) − 3(t + 4) d 3(5z − 1) − (2z + 5) e 2(a + 5) − 4(a − 1) f 5(d − 3) − 3(2d + 1) g 4(3x + y) − (2x − 7y) h 3(2a − 3b) − (2a + 3b) i 2(q − 5) − 4(q − 5) 02_LEY_IM_9SB_52_53_22631_SI.indd 34 6/09/13 8:21 AM

Chapter 2 Indices 35 NUMBER & ALGEBRA 1 Write the following in words. a 35 b x2 c 70 d x−3 2 Write expressions for: a seven squared b y cubed c 6 to the power 5 d m to the fourth 3 Replace the missing vowels to make words that mean the same as ‘power’. a _nd_x b _xp_n_nt 4 When writing numbers in index notation, what name is given to: a the number that is being repeated? b how many of this number are multiplied together? 5 Explain the following. a m3 × m 2 ≠ m 6 b (x3)2 ≠ x 5 c y0 = 1 d a−2 ≠ −2a e 3(k + 5) ≠ 3k + 5 f 5 + 4(n + 2) ≠ 9(n + 2) 6 Explain the diff erence between 3p 0 and (3p) 0. Terms base distributive evaluate expand exponent expression grouping index law indices invert like terms negative notation power simplify symbol variable zero 1 Which of the following does not simplify to a 8? A a16 ÷ a 8 B (a2)4 C a7 × a D a16 ÷ a 2 2 5m 3n2 × 4m 4n3 = A 20m 12n16 B 20m 7n5 C 9m 12n6 D 9m 7n5 3 4a 2b2 × 2a 5b10 ____________ 6a 4b8 = A 4a 6b12 _____ 3 B 7a 6b12 C 4a 3b4 _____ 3 D 7a 3b4 4 (3m 7)3 = A 3m 21 B 3m 10 C 27m 21 D 27m 10 5 x5 ___ 4x8 = A 4 __ x3 B 1 ___ 4x3 C 4x3 D x3 __ 4 6 5m 0 − 1 = A −1 B 0 C 4 D 5 Language in mathematics Check your skills 02_LEY_IM_9SB_52_53_22631_SI.indd 35 6/09/13 8:21 AM

NUMBER & ALGEBRA 36 Insight Mathematics 9 stages 5.2/5.3 Australian Curriculum 7 (5x) 0 = A 0 B 1 C 5 D 5x 8 3−4 is the same as: A 1 __ 43 B 1 __ 34 C −12 D 3 __ 4 9 Which of the following statements is not correct? A (1 1 _ 2 )−1 = 2 __ 3 B 1 ___ 2−3 = 2 3 C 5p −2 = 5 __ p2 D 25 ÷ 2 −3 = 1 8 (5y) −3 simplifi es to: A y 3 ____ 125 B 125 ____ y3 C 1 _____ 125y 3 D 5 __ y3 5p 3q−7 × 2p −2q−3 = A 7pq −10 B 10pq −10 C 7p 5q−4 D 10p −6q21 12k −2m3 ÷ 4k −5m2 = A 3k−7m5 B k3 ___ 3m C k3m ____ 3 D 3k3m −5(4 − 3t) = A −20 − 15t B −20 + 15t C −20 − 8t D −20 + 8t 3x 2(7x − 2y) = A 21x 3 − 6x 2y B 21x 3 + 6x 2y C 10x 3 – 5x 2y D 21x 3 + 5x 2y 7 − 3(2x − 5) = A 8x − 20 B 12 − 6x C 2 − 6x D 22 − 6x 3(2a + 3) − 4(a − 2) = A 2a + 17 B 2a + 1 C 10a + 17 D 10a + 1 If you have any diffi culty with these questions, refer to the examples and questions in the sections listed in the table. Question 1 2–5 6, 7 8 9, 10 11, 12 13–16 Section A B C D E FG 2A Review set 1 Simplify: a y10 × y 7 b k11 ÷ k 5 c (p7)2 d t7 × t 8 ______ t3 × t 4 e (5m 4)3 f 3a 5b3 × 2ab 6 g 5m 10n8 ____________ 10m 3n × m 5n2 2 Simplify, writing the answers with positive indices. a a4 __ a9 b 2a 4 ___ a9 c a4 ___ 2a 9 d 6a 4 ____ 15a 9 10 11 12 13 14 15 16 02_LEY_IM_9SB_52_53_22631_SI.indd 36 6/09/13 8:21 AM

Chapter 2 Indices 37 NUMBER & ALGEBRA 3 Evaluate: a v0 b 5v0 c (5v) 0 d 2v0 + 1 4 Evaluate: a 1 ___ 2−3 b ( 3 _ 4 )−1 c ( 9 _ 4 )−1 5 Write the following with positive indices. a z−3 b 2z−3 c (2z) −3 6 Write the following with negative indices. a 1 __ k7 b 2 __ k7 c 1 ___ 2k7 7 Simplify: a y−3 × y 5 b e6 ÷ e −2 c (n−4)5 d 6b −2 × 3b 7 e 4k−5 ÷ 2k −3 8 State whether the following are true or false. a 4q 0 = 4 b a5 ÷ a 7 = a 2 c 6m 5 ÷ 3m 5 = 2m d 4b −2 = 4 __ b2 e n2 × n = n 2 + n 9 Expand: a 5(2v − 4w) b a3(2a 2 + 4a) c −3(4x + 5) Expand and simplify: a 4(m – 2) + 3(2m + 5) b 3(3a − b) − (2a − b) c x(2 − x) − 2(x − 2) 2B Review set 1 Simplify: a m14 × m 6 b t25 ÷ t 5 c (z6)4 d (b 5)6 ______ b7 × b 4 e (2m 7)4 f 4p 3q7 × 5p 4q g 6a 7b4 × 3a 3b5 ____________ 24a 9b 2 Simplify, writing the answers with positive indices. a m7 ___ m10 b 3m 7 ____ m10 c m 7 ____ 3m 10 d 10m 7 _____ 15m 10 3 Evaluate: a s0 b 4s0 c (4s) 0 d 4s0 − 1 4 Evaluate: a 1 ___ 3−2 b (1 1 _ 4 )−1 c ( 6 __ 11 )−1 5 Write the meaning of the following. a e−4 b 3e−4 c (3e) −4 6 Write the following with negative indices. a 1 __ p4 b 5 __ p4 c 1 ___ 5p 4 7 Simplify: a k−6 × k 2 b m−4 ÷ m −1 c (n3)−5 d 5n −3 × 4n 8 e 2a −5 ÷ 4a −8 10 02_LEY_IM_9SB_52_53_22631_SI.indd 37 6/09/13 8:21 AM

NUMBER & ALGEBRA 38 Insight Mathematics 9 stages 5.2/5.3 Australian Curriculum 8 State whether the following are true or false. a 3w 0 = 1 b b6 ÷ b 9 = b −3 c a7 ÷ a 7 = a d 2t −2 = 1 ___ 2t 2 e p−2 = −2p 9 Expand: a −10(4p + 3) b m2(3m 5 – m 3) c −a(2a – 5) Expand and simplify: a 3(2q + 6) − 2(q − 7) b 2a(2a − 5) − (3a + 1) c m(6 − t) + t (m − 6) 2C Review set 1 Simplify: a p6 × p 8 b y16 ÷ y 8 c (t 5)6 d c12 × c 8 _______ (c4)5 e (3v 7)3 f 3x8y9 × 6x 2y5 g 5m 5n3 × 3m 10n4 _____________ 2m 6n2 × 5m 7n4 2 Simplify, writing the answers with positive indices. a k4 __ k6 b 8k 4 ___ k6 c k4 ___ 8k6 d 6k 4 ___ 8k6 3 Evaluate: a a0 b 7a 0 c (7a) 0 d 7a 0 + 5 4 Evaluate: a 1 ___ 3−3 b ( 7 _ 8 )−1 c (1 1 _ 5 )−1 5 Write the following with positive indices. a b−5 b 3b −5 c (3b) −5 6 Write the following with negative indices. a 1 __ t 3 b 2 __ t 3 c 1 ___ 2t 3 7 Simplify: a d−5 × d −3 b n−2 ÷ n −3 c (k−2)−3 d 5a −6 × 3a 3 e 6m 3 ÷ 9m −4 8 State whether the following are true or false. a 7h 0 = 7 b a ÷ a 5 = a −5 c 2p 5 ÷ 6p 5 = 1 _ 3 p d 3s−4 = 3 __ s4 e n2 + n 2 = n 4 9 Expand: a 4(5w + 2x) b −k 4(2k 3 − 4k) c −(4s − 7) Expand and simplify: a 2(5t − 1) + 3(2 − 3t) b 10(x − 2y) − 5(2x − y) c 3(m + 2) − 8(3m − 2) 10 10 02_LEY_IM_9SB_52_53_22631_SI.indd 38 6/09/13 8:21 AM

Collecting and analysing data ▶ identify eve r yday questions and issues involving numerical and categorical variables ▶ describe data us ing the terms ‘symmetric’, ‘skewed’ and ‘bimodal’ ▶ construct back-to-back stem-and-leaf plo ts and histograms ▶ construct side -by-side histograms and dot plots ▶ compare data d isplays using the mean, median and range ▶ evaluate s tatistical reports in the media. This chapter deals with single variable data analysis. After completing this chapter you should be able to: 3 NSW Syllabus references: 5.1 S&P Probability Outcomes: MA5.1-1WM, MA5.1-2WM, MA5.1-3WM, MA5.1-12SP, MA5.1-13SP, MA5.2-1WM, MA5.2-3WM, MA5.2-15SP STATISTICS & PROBABILITY – ACMSP228, ACMSP253, ACMSP282, ACMSP283 03_LEY_IM_9SB_52_53_22631_SI.indd 39 03_LEY_IM_9SB_52_53_22631_SI.indd 39 6/09/13 8:13 AM 6/09/13 8:13 AM

STATISTICS & PROBABILITY 40 Insight Mathematics 9 stages 5.2/5.3 Australian Curriculum Investigating data Exercise 3A Table 1 shows the monthly and annual rainfall for Sydney (Observatory Hill) from 2002 to 2011. Measurements are to the nearest millimetre. Ta b l e 1: Rainfall for Sydney (mm) Year J F M A M J J A S O N D Annual 2002 98 348 45 68 93 28 24 20 22 6 32 75 860 2003 14 59 132 192 349 76 58 43 6 103 109 60 1200 2004 51 129 101 33 8 39 44 153 60 234 67 76 995 2005 68 125 154 33 48 79 63 2 51 43 125 25 816 2006 121 51 40 10 40 177 140 86 192 17 45 74 994 2007 45 108 65 180 10 511 67 152 41 27 170 123 1499 2008 57 258 63 147 3 127 90 44 99 67 73 54 1083 2009 25 128 61 153 126 130 53 6 16 180 13 67 956 2010 36 239 51 30 168 147 115 27 42 85 130 83 1154 2011 54 18 192 206 136 94 282 52 72 37 148 78 1369 1 In this time period, which year had the: a highest annual rainfall? b lowest annual rainfall? 2 How much rain fell in: a January 2006? b May 2007? c November 2011? A 3 Which month had the highest rainfall in: a 2004? b 2010? 4 Which month had the lowest rainfall in: a 2003? b 2007? 5 Which year had the wettest: a January? b June? c December? 6 Which year had the driest: a February? b May? c November? 7 Considering winter to be the months June, July and August, which year had the: a wettest winter? b driest winter? 03_LEY_IM_9SB_52_53_22631_SI.indd 40 6/09/13 8:13 AM

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