Maths in Focus Ext 1 Preliminary-4

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maths

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Margaret Grove

maths

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Enquiries should be made to the publisher via www.mcgraw-hill.com.au National Library of Australia Cataloguing-in-Publication Data

Author: Grove, Margaret.

Title: Maths in focus: mathematics extension preliminary course/Margaret Grove.

Edition: 2nd ed.

ISBN: 9780070278585 (pbk.)

Target Audience: For secondary school age.

Subjects: Mathematics–Problems, exercises, etc. Mathematics–Textbooks. Dewey Number: 510.76

Published in Australia by McGraw-Hill Australia Pty Ltd

Level 2, 82 Waterloo Road, North Ryde NSW 2113

Publisher: Eiko Bron

Managing Editor: Kathryn Fairfax Production Editor: Natalie Crouch Editorial Assistant: Ivy Chung Art Director: Astred Hicks

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PREFACE ix

ACKNOWLEDGEMENTS ix

CREDITS ix

FEATURES OF THIS BOOK ix

SYLLABUS MATRIX x

STUDY SKILLS xi

Chapter 1: Basic Arithmetic 2

INTRODUCTION 3

REAL NUMBERS 3

DIRECTED NUMBERS 9

FRACTIONS, DECIMALS AND PERCENTAGES 12

POWERS AND ROOTS 19

ABSOLUTE VALUE 37

TEST YOURSELF 1 41

CHALLENGE EXERCISE 1 43

Chapter 2: Algebra and Surds 44

INTRODUCTION 45

SIMPLIFYING EXPRESSIONS 45

BINOMIAL PRODUCTS 51

FACTORISATION 55

COMPLETING THE SQUARE 69

ALGEBRAIC FRACTIONS 71 SUBSTITUTION 73 SURDS 76 TEST YOURSELF 2 90 CHALLENGE EXERCISE 2 93 Chapter 3: Equations 94 INTRODUCTION 95 SIMPLE EQUATIONS 95 SUBSTITUTION 100 INEQUATIONS 103

EQUATIONS AND INEQUATIONS INVOLVING ABSOLUTE VALUES 107

EXPONENTIAL EQUATIONS 114 QUADRATIC EQUATIONS 118 FURTHER INEQUATIONS 125 QUADRATIC INEQUATIONS 129 SIMULTANEOUS EQUATIONS 132 TEST YOURSELF 3 138 CHALLENGE EXERCISE 3 139

PREFACE

This book covers the Preliminary syllabus for Mathematics and Extension 1. The extension material is easy to see as it has green headings and there is green shading next to all extension question and answers. The syllabus is available through the NSW Board of Studies website on www.boardofstudies. nsw.edu.au. You can also access resources, study techniques, examination technique, sample and past examination papers through other websites such as www.math.nsw.edu.au and www.csu.edu. au. Searching the Internet generally will pick up many websites supporting the work in this course.

Each chapter has comprehensive fully worked examples and explanations as well as ample sets of graded exercises. The theory follows a logical order, although some topics may be learned in any order. Each chapter contains Test Yourself and Challenge exercises, and there are several practice assessment tasks throughout the book.

If you have trouble doing the Test Yourself exercises at the end of a chapter, you will need to go back into the chapter and revise it before trying them again. Don’t attempt to do the Challenge exercises until you are confi dent that you can do the Test Yourself exercises, as these are more diffi cult and are designed to test the more able students who understand the topic really well.

ACKNOWLEDGEMENTS

Thanks go to my family, especially my husband Geoff, for supporting me in writing this book.

CREDITS

Fairfax Photos: p 327 Istockphoto: p 101, p 171

Margaret Grove: p 37, p 163, p 206, p 246, p 260, p 291, p308 (bottom), p 310, p 311, p 313, p 316,

p 391, p 499, p 543, p 591, p 717, p 719, p 726, p 729, p 730, p 739

Photolibrary: p 205

Shutterstock: p 74, p 164, p 229, p 308 (top), p 580

FEATURES OF THIS BOOK

This second edition retains all the features of previous Maths in Focus books while adding in new improvements.

The main feature of Maths in Focus is in its readability, its plentiful worked examples and straightforward language so that students can understand it and use it in self-paced learning. The logical progression of topics, the comprehensive fully worked examples and graded exercises are still major features.

A wide variety of questions is maintained, with more comprehensive and more diffi cult questions included in each topic. At the end of each chapter is a consolidation set of exercises (Test yourself) in no particular order that will test whether the student has grasped the concepts contained in the chapter. There is also a challenge set for the more able students.

The four practice assessment tasks provide a comprehensive variety of mixed questions from various chapters. These have been extended to contain questions in the form of sample examination questions, including short answer, free response and multiple-choice questions that students may encounter in assessments.

The second edition also features a short summary of general study skills that students will fi nd useful, both in the classroom and when doing assessment tasks and examinations. These study skills are also repeated in the HSC book.

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A syllabus matrix is included to show where each syllabus topic fi ts into the book. Topics are generally arranged in a logical order. For example, arithmetic and algebra are needed in most, if not all other topics, so these are treated at the beginning of the book.

Some teachers like to introduce particular topics before others, e.g. linear functions before more general functions. However, part of the work on gradient requires some knowledge of trigonometry and the topic of angles of any magnitude in trigonometry needs some knowledge of functions. So the order of most chapters in the book have been carefully thought out. Some chapters, however, could be covered in a different order, such as geometry which is covered in Chapter 4, and quadratic functions and locus, which are near the end of the book.

SYLLABUS MATRIX

This matrix shows how the syllabus is organised in the chapters of this book.

Mathematics (2 Unit)

Basic arithmetic and algebra (1.1 – 1.4) Chapter 1: Basic arithmetic

Chapter 2: Algebra and surds Chapter 3: Equations

Real functions (4.1 – 4.4) Chapter 5: Functions and graphs

Trigonometric ratios (5.1 – 5.5) Chapter 6: Trigonometry

Linear functions (6.1 – 6.5, 6.7) Chapter 7: Linear functions

The quadratic polynomial and the parabola (9.1 – 9.5) Chapter 10: The quadratic function

Chapter 11: Locus and the parabola

Plane geometry (2.1 – 2.4) Chapter 4: Geometry 1

Tangent to a curve and derivative of a function (8.1 – 8.9) Chapter 8: Introduction to calculus

Extension 1

Other inequalities (1.4E) Chapter 3: Equations

Circle geometry (2.6 – 2.10E) Chapter 9: Properties of the circle

Further trigonometry (5.6 – 5.9E) Chapter 6: Trigonometry

Angles between two lines (6.6E) Chapter 7: Linear functions

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Internal and external division of lines into given ratios (6.7E) Chapter 7: Linear functions

Parametric representation (9.6E) Chapter 11: Locus and the parabola

Permutations and combinations (18.1E)

Chapter 13: Permutations and combinations

Polynomials (16.1 – 16.3E) Chapter 12: Polynomials 1

STUDY SKILLS

You may have coasted through previous stages without needing to rely on regular study, but in this course many of the topics are new and you will need to systematically revise in order to build up your skills and to remember them.

The Preliminary course introduces the basics of topics such as calculus that are then applied in the HSC course. You will struggle in the HSC if you don’t set yourself up to revise the preliminary topics as you learn new HSC topics.

Your teachers will be able to help you build up and manage good study habits. Here are a few hints to get you started.

There is no right or wrong way to learn. Different styles of learning suit different people. There is also no magical number of hours a week that you should study, as this will be different for every student. But just listening in class and taking notes is not enough, especially when learning material that is totally new.

You wouldn’t go for your driver’s licence after just one trip in the car, or enter a dance competition after learning a dance routine once. These skills take a lot of practice. Studying mathematics is just the same.

If a skill is not practised within the fi rst 24 hours, up to 50% can be forgotten. If it is not practised within 72 hours, up to 85–90% can be forgotten! So it is really important that whatever your study timetable, new work must be looked at soon after it is presented to you.

With a continual succession of new work to learn and retain, this is a challenge. But the good news is that you don’t have to study for hours on end!

In the classroom

In order to remember, fi rst you need to focus on what is being said and done. According to an ancient proverb:

‘I hear and I forget I see and I remember I do and I understand’

If you chat to friends and just take notes without really paying attention, you aren’t giving yourself a chance to remember anything and will have to study harder at home.

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If you have just had a fi ght with a friend, have been chatting about weekend activities or myriad other conversations outside the classroom, it helps if you can check these at the door and don’t keep chatting about them once the lesson starts.

If you are unsure of something that the teacher has said, the chances are that others are also not sure. Asking questions and clarifying things will ultimately help you gain better results, especially in a subject like mathematics where much of the knowledge and skills depends on being able to understand the basics.

Learning is all about knowing what you know and what you don’t know. Many students feel like they don’t know anything, but it’s surprising just how much they know already. Picking up the main concepts in class and not worrying too much about other less important parts can really help. The teacher can guide you on this.

Here are some pointers to get the best out of classroom learning:

Take control and be responsible for your own learning ■

Clear your head of other issues in the classroom ■

Active, not passive, learning is more memorable ■

Ask questions if you don’t understand something ■

Listen for cues from the teacher ■

Look out for what are the main concepts

■

Note taking varies from class to class, but there are some general guidelines that will help when you come to read over your notes later on at home:

Write legibly ■

Use different colours to highlight important points or formulae ■

Make notes in textbooks (using pencil if you don’t own the textbook) ■

Use highlighter pens to point out important points ■

Summarise the main points ■

If notes are scribbled, rewrite them at home

■

At home

You are responsible for your own learning and nobody else can tell you how best to study. Some people need more revision time than others, some study better in the mornings while others do better at night, and some can work at home while others prefer a library.

There are some general guidelines for studying at home:

Revise both new and older topics regularly ■

Have a realistic timetable and be fl exible ■

Summarise the main points ■

Revise when you are fresh and energetic ■

Divide study time into smaller rather than longer chunks ■

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■

Have a balanced life and don’t forget to have fun! ■

If you are given exercises out of a textbook to do for homework, consider asking the teacher if you can leave some of them till later and use these for revision. It is not necessary to do every exercise at one sitting, and you learn better if you can spread these over time.

People use different learning styles to help them study. The more variety the better, and you will fi nd some that help you more than others. Some people (around 35%) learn best visually, some (25%) learn best by hearing and others (40%) learn by doing.

Here are some ideas to give you a variety of ways to study:

Summarise on cue cards or in a small notebook ■

Use colourful posters ■

Use mindmaps and diagrams ■

Discuss work with a group of friends ■

Read notes out aloud ■

Make up songs and rhymes ■

Do exercises regularly ■

Role play teaching someone else

■

Assessment tasks and exams

Many of the assessment tasks for maths are closed book examinations.

You will cope better in exams if you have practised doing sample exams under exam conditions. Regular revision will give you confi dence and if you feel well prepared, this will help get rid of nerves in the exam. You will also cope better if you have had a reasonable night’s sleep before the exam.

One of the biggest problems students have with exams is in timing. Make sure you don’t spend too much time on questions you’re unsure about, but work through and fi nd questions you can do fi rst.

Divide the time up into smaller chunks for each question and allow some extra time to go back to questions you couldn’t do or fi nish. For example, in a 2 hour exam with 6 questions, allow around 15 minutes for each question. This will give an extra half hour at the end to tidy up and fi nish off questions.

Here are some general guidelines for doing exams:

Read through and ensure you know how many questions there are ■

Divide your time between questions with extra time at the end ■

Don’t spend too much time on one question ■

Read each question carefully, underlining key words ■

Show all working out, including diagrams and formulae ■

Cross out mistakes with a single line so it can still be read ■

Write legibly ■

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And fi nally…

Study involves knowing what you don’t know, and putting in a lot of time into concentrating on these areas. This is a positive way to learn. Rather than just saying, ‘I can’t do this’, say instead, ‘I can’t do this yet’, and use your teachers, friends, textbooks and other ways of fi nding out.

With the parts of the course that you do know, make sure you can remember these easily under exam pressure by putting in lots of practice.

Remember to look at new work

today ■ tomorrow ■ in a week ■ in a month ■

Some people hardly ever fi nd time to study while others give up their outside lives to devote their time to study. The ideal situation is to balance study with other aspects of your life, including going out with friends, working and keeping up with sport and other activities that you enjoy.

Good luck with your studies!

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TERMINOLOGY

1

Basic Arithmetic

Absolute value: The distance of a number from zero on

the number line. Hence it is the magnitude or value of a number without the sign

Directed numbers: The set of integers or whole numbers f- - -3, 2, 1, 0, 1, 2, 3,f

Exponent: Power or index of a number. For example 23

has a base number of 2 and an exponent of 3 Index: The power of a base number showing how many times this number is multiplied by itself e.g. 23 2 2 2.

# #

= The index is 3

Indices: More than one index (plural)

Recurring decimal: A repeating decimal that does not terminate e.g. 0.777777 … is a recurring decimal that can be written as a fraction. More than one digit can recur e.g. 0.14141414 ...

Scientiﬁ c notation: Sometimes called standard notation. A standard form to write very large or very small numbers as a product of a number between 1 and 10 and a power of 10 e.g. 765 000 000 is 7.65 108

# in scientifi c notation

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INTRODUCTION

THIS CHAPTER GIVES A review of basic arithmetic skills, including knowing the correct order of operations, rounding off, and working with fractions, decimals and percentages. Work on signifi cant fi gures, scientifi c notation and indices is also included, as are the concepts of absolute values. Basic calculator skills are also covered in this chapter.

Real Numbers

Types of numbers

Irrational numbers Unreal or imaginary numbers Integers Rational numbers Real numbers

Integers are whole numbers that may be positive, negative or zero. e.g. -4 7 0, , ,-11

Rational numbers can be written in the form of a fraction

b a

where a and b are integers, b!0. e.g. 1 , . , . ,

4 3

3 7 0 5• -5

Irrational numbers cannot be written in the form of a fraction

b a

(that is, they are not rational) e.g. 2,r

EXAMPLE

Which of these numbers are rational and which are irrational? 3 1 3, . , , , , . 5 3 9 4 2 65 • r

Solution

3 4

and r are irrational as they cannot be written as fractions (r is irrational).

. , . 1 3 1 3 1 ₉ 1 3 _{2 65} ₂ 20 13 and •

= = - = - so they are all rational.

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Order of operations

1. Brackets: do calculations inside grouping symbols ﬁ rst. (For example, a fraction line, square root sign or absolute value sign can act as a grouping symbol.)

2. Multiply or divide from left to right.

3. Add or subtract from left to right.

EXAMPLE

Evaluate 40-3 5] +4g.

Solution

40 3 (5 4) 40 3 9 40 27 13 # - + = -= -=

PROBLEM

What is wrong with this calculation?

Evaluate 1 2 19 4 +- - + Press 19 - 4 ' 1 + 2 = 19 4'1 2 ₁₇

What is the correct answer?

BRACKETS KEYS

Use ( and ) to open and close brackets. Always use them in pairs. For example, to evaluate 40-3]5+4g

press 40 3 ( 5 4 ) 13 # - + = = To evaluate 1.69 2.77 5.67 3.49 +

- _{correct to 1 decimal place}

press : ( ( 5.67 - 3.49 ) ' ( 1.69 + 2.77 ) ) = 0.7 correct to 1 decimal place

=

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Rounding off

Rounding off is often done in everyday life. A quick look at a newspaper will give plenty of examples. For example in the sports section, a newspaper may report that 50 000 fans attended a football match.

An accurate number is not always necessary. There may have been exactly 49 976 people at the football game, but 50 000 gives an idea of the size of the crowd.

EXAMPLES

1. Round off 24 629 to the nearest thousand.

Solution

This number is between 24 000 and 25 000, but it is closer to 25 000.

24 629 25 000

` = to the nearest thousand

CONTINUED

MEMORY KEYS

Use STO to store a number in memory.

There are several memories that you can use at the same time—any letter from A to F, or X, Y and M on the keypad.

To store the number 50 in, say, A press 50 STO A To recall this number, press ALPHA A ₌ To clear all memories press SHIFT CLR

X

-1

_KEY

Use this key to fi nd the reciprocal of x . For example, to evaluate

7.6 2.1 1 # - 0.063 = -press ( ( ) 7.6- # 2.1 ) x-1 =

(correct to 3 decimal places)

Different calculators use different keys so check the instructions for your calculator.

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2. Write 850 to the nearest hundred.

Solution

This number is exactly halfway between 800 and 900. When a number is halfway, we round it off to the larger number.

850 900

` = to the nearest hundred

In this course you will need to round off decimals, especially when using trigonometry or logarithms.

To round a number off to a certain number of decimal places, look at the next digit to the right. If this digit is 5 or more, add 1 to the digit before it and drop all the other digits after it. If the digit to the right is less than 5, leave the digit before it and drop all the digits to the right.

EXAMPLES

1. Round off 0.6825371 correct to 1 decimal place.

Solution

.

. .

0 6825371

0 6825371 0 7 correct to 1 decimal place

` # =

2. Round off 0.6825371 correct to 2 decimal places.

Solution

.

. .

0 6825371

0 6825371 0 68 correct to 2 decimal places

` # =

3. Evaluate 3 56. '2 1. correct to 2 decimal places.

Solution

. . . 5

.

3 56 2 1 1 69 238095

1 70 correct to 2 decimal places

' =

= #

Drop off the 2 and all digits to the right as 2 is smaller than 5.

Add 1 to the 6 as the 8 is greater than 5.

Check this on your calculator. Add 1 to the 69 as 5 is too large to just drop off.

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While using a ﬁ xed number of decimal places on the display, the calculator still keeps track internally of the full number of decimal places.

EXAMPLE

Calculate 3 25. '1 72. #5 97. +7 32. correct to 2 decimal places.

Solution

. . . . . . . 3 25 1 72 5 97 7 32 1 889534884 5 97 7 32 11 28052326 7 32 18 60052326

18.60 correct to 2 decimal places

' # + = # +

= +

= =

If the FIX key is set to 2 decimal places, then the display will show 2 decimal places at each step.

3.25 1.72 5.97 7.32 1.89 5.97 7.32 . . . 11 28 7 32 18 60 ' # + = # + = + =

If you then set the calculator back to normal, the display will show the full answer of 18.60052326.

Don’t round off at each step of a series of calculations.

The calculator does not round off at each step. If it did, the answer might not be as accurate. This is an important point, since some students round off each step in calculations and then wonder why they do not get the same answer as other students and the textbook.

1.1 Exercises

FIX KEY

Use MODE or SET UP to fi x the number of decimal places (see the

instructions for your calculator). This will cause all answers to have a fi xed number of decimal places until the calculator is turned off or switched back to normal.

1. State which numbers are rational and which are irrational.

(a) 169 0.546 (b) (c) -17 (d) 3 r (e) 0 34. • (f) 218 (g) 2 2 (h) 27 1 17.4% (i) (j) 5 1 ch1.indd 7 5/20/09 3:06:03 PM

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2. Evaluate (a) 20-8'4 (b) 3#7-2#5 (c) 4#]27' '3g 6 (d) 17+3#-2 (e) 1 9. -2#3 1. (f) 1 3 14'7 - + (g) 2 5 3 5 1 3 2 # - (h) 6 5 1 4 3 8 1 (i) 4 1 8 1 8 5 6 5 ' + (j) 1 4 1 2 1 3 5 1 10 7

3. Evaluate correct to 2 decimal places. (a) 2.36+4.2'0.3 (b) ]2 36. +4 2. g'0 3. (c) 12.7#3.95'5.7 (d) 8.2'0.4+4.1#0.54 (e) ]3 2. -6 5. g#]1 3. +2 7. g (f) 4.7 1.3 1 + (g) 4.51 3.28 1 + (h) 5.2 3.6 0.9 1.4 -+ (i) 1.23 3.15 5.33 2.87 -+ (j) 1.72₊8.92_-3.942

4. Round off 1289 to the nearest hundred.

5. Write 947 to the nearest ten.

6. Round off 3200 to the nearest thousand.

7. A crowd of 10 739 spectators attended a tennis match. Write this ﬁ gure to the nearest thousand.

8. A school has 623 students. What is this to the nearest hundred?

9. A bank made loans to the value of $7 635 718 last year. Round this off to the nearest million.

10. A company made a proﬁ t of $34 562 991.39 last year. Write this to the nearest hundred thousand.

11. The distance between two cities is 843.72 km. What is this to the nearest kilometre?

12. Write 0.72548 correct to 2 decimal places.

13. Round off 32.569148 to the nearest unit.

14. Round off 3.24819 to 3 decimal places. 15. Evaluate 2.45#1.72 correct to 2 decimal places. 16. Evaluate 8.7'5 correct to 1 decimal place.

17. If pies are on special at 3 for $2.38, ﬁ nd the cost of each pie.

18. Evaluate 7.48 correct to 2 decimal places. 19. Evaluate 8 6.4+2.3_{correct to} 1 decimal place.

20. Find the length of each piece of material, to 1 decimal place, if 25 m of material is cut into 7 equal pieces.

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DID YOU KNOW?

In building, engineering and other industries where accurate measurements are used, the number of decimal places used indicates how accurate the measurements are.

For example, if a 2.431 m length of timber is cut into 8 equal parts, according to the calculator each part should be 0.303875 m. However, a machine could not cut this accurately. A length of 2.431 m shows that the measurement of the timber is only accurate to the nearest mm (2.431 m is 2431 mm). The cut pieces can also only be accurate to the nearest mm (0.304 m or 304 mm).

The error in measurement is related to rounding off, as the error is half the smallest measurement. In the above example, the measurement error is half a millimetre. The length of timber could be anywhere between 2430.5 mm and 2431.5 mm.

Directed Numbers

Many students use the calculator with work on directed numbers (numbers that can be positive or negative). Directed numbers occur in algebra and other topics, where you will need to remember how to use them. A good understanding of directed numbers will make your algebra skills much better.

-^ h

KEY

Use this key to enter negative numbers. For example,

press ( ) 3- =

21. How much will 7.5 m of tiles cost, at $37.59 per m 2_?

22. Divide 12.9 grams of salt into 7 equal portions, to 1 decimal place.

23. The cost of 9 peaches is $5.72. How much would 5 peaches cost?

24. Evaluate correct to 2 decimal places. (a) 17.3-4.33#2.16 (b) 8 72. #5 68. -4 9. #3 98. (c) 5.6 4.35 3.5 9.8 ++ (d) 7.63 5.12 15.9 6.3 7.8 -+ - (e) 6.87 3.21 1 - 25. Evaluate . . . . 5 39 9 68 5 47 9 91 2 --] g

correct to 1 decimal place.

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Adding and subtracting

To add: move to the right along the number line To subtract: move to the left along the number line

Subtract Add -4 -3 -2 -1 0 1 2 3 4 Same signs Different signs = + + + = + - = = + = + = -- +

EXAMPLES

Evaluate 1. - +4 3

Solution

Start at -4 and move 3 places to the right.

-4 -3 -2 -1 0 1 2 3 4

4 3 1 - + = - 2. - -1 2

Solution

Start at -1 and move 2 places to the left.

-4 -3 -2 -1 0 1 2 3 4

1 2 3 - - = -

Multiplying and dividing

To multiply or divide, follow these rules. This rule also works if there are two signs together without a number in between e.g. 2- -3

You can also do these on a calculator, or you may have a different way of working these out.

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EXAMPLES

Evaluate

1. -2#7

Solution

Different signs ( -2and+7 ) give a negative answer. 2# 7 14

- = -

2. -12'-4

Solution

Same signs ( -12and-4 ) give a positive answer. 12' 4 3

- - =

3. - -1 -3

Solution

The signs together are the same (both negative) so give a positive answer.

1 3 2 = - + = 1 3 - - 1. - +2 3 2. - -7 4 3. 8# -7 4. 7- ]-3g 5. 28' -7 6. -4 9. +3 7. 7. -2 14. -5 37. 8. 4 8. # -7 4. 9. 1 7. - -] 4 87. g 10. 5 3 1 3 2 - - 11. 5-3#4 12. - +2 7#-3 13. 4-3#-2 14. - - -1 2 15. 7+-2 16. 2- -] 1g 17. - +2 15'5 18. -2# #6 -5 19. -28'-7#-5 20. _]_-3_g2

1.2 Exercises

Evaluate

Start at -1 and move 3 places to the right.

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Fractions, Decimals and Percentages

EXAMPLES

1. Write 0.45 as a fraction in its simplest form.

Solution

. 0 45 100 45 5 5 20 9 ' = = 2. Convert 8 3 to a decimal.

Solution

. . . 8 3 000 0 375 8 3 0 375 So =

g

3. Change 35.5% to a fraction.

Solution

. % . 35 5 100 35 5 2 2 200 71 # = = 4. Write 0.436 as a percentage.

Solution

. . % . % 0 436 0 436 100 43 6 # = =

5. Write 20 g as a fraction of 1 kg in its simplest form.

Solution

1kg=1000g 1 20 1000 20 50 1 kg g g g = = Multiply by 100% to change a fraction or decimal to a percentage.

Conversions

You can do all these conversions on your calculator using the ab_c or S+D key.

8 3

means 3'8.

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Sometimes decimals repeat, or recur. Example . 0. 3 1 0 33333333f 3• = =

There are different methods that can be used to change a recurring decimal into a fraction. Here is one way of doing it. Later you will discover another method when studying series. (See HSC Course book, Chapter 8.)

EXAMPLES

1. Write 0 4.• as a rational number.

Solution

. ( ) . ( ) ( ) ( ): n n n n 0 44444 1 10 4 44444 2 2 1 9 4 9 4 Let Then f f = = - = = 2. Change 1 329. • • to a fraction.

Solution

. ( ) . ( ) ( ) ( ): . . n n n n 1 3292929 1 100 132 9292929 2 2 1 99 131 6 99 131 6 10 10 990 1316 1 495 163 Let Then # f f = = - = = = = A rational number is any number that can be written as a fraction.

Check this on your calculator by dividing 4 by 9.

Try multiplying n by 10. Why doesn’t this work?

6. Find the percentage of people who prefer to drink Lemon Fuzzy, if 24

out of every 30 people prefer it.

Solution

% % 30 24 1 100 80 # = CONTINUED ch1.indd 13 7/8/09 10:56:31 AM

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1. Write each decimal as a fraction in its lowest terms.

0.64 (a) 0.051 (b) 5.05 (c) 11.8 (d)

2. Change each fraction into a decimal. (a) 5 2 (b) 1 8 7 (c) 12 5 (d) 11 7

3. Convert each percentage to a fraction in its simplest form.

2% (a) 37.5% (b) 0.1% (c) 109.7% (d)

4. Write each percentage as a decimal. 27% (a) 109% (b) 0.3% (c) 6.23% (d)

5. Write each fraction as a percentage. (a) 20 7 (b) 3 1 (c) 2 15 4 (d) 1000 1

6. Write each decimal as a percentage. 1.24 (a) 0.7 (b) 0.405 (c) 1.2794 (d)

7. Write each percentage as a decimal and as a fraction.

52% (a) 7% (b) 16.8% (c) 109% (d) 43.4% (e) (f) 12 % 4 1

8. Write these fractions as recurring decimals. (a) 6 5 (b) 7 99 (c) 99 13 (d) 6 1 (e) 3 2

1.3 Exercises

Another method

Let . . ( ) . ( ) ( ) ( ): n n n n n 1 3292929 10 13 2929292 1 1000 1329 292929 2 2 1 990 1316 990 1316 1 495 163 Then and f f f = = = - = = = This method avoids decimals

in the fraction at the end.

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Investigation

Explore patterns in recurring decimals by dividing numbers by 3, 6, 9, 11, and so on.

Can you predict what the recurring decimal will be if a fraction has 3 in the denominator? What about 9 in the denominator? What about 11?

Can you predict what fraction certain recurring decimals will be? What denominator would 1 digit recurring give? What denominator would you have for 2 digits recurring?

Operations with fractions, decimals and percentages

You will need to know how to work with fractions without using a calculator, as they occur in other areas such as algebra, trigonometry and surds.

(f) 33 5 (g) 7 1 (h) 1 11 2

9. Express as fractions in lowest terms. (a) 0 8.• (b) 0 2.• (c) 1 5.• (d) 3 7.• (e) 0 67.• • (f) 0 54.• • (g) 0 15. • (h) 0 216. • (i) 0 2 19. • • (j) 1 074. • •

10. Evaluate and express as a decimal. (a) 3 6 5 + (b) 8-3'5 (c) 12 3 4 7 + + (d) 1 99 31 - (e) 7 4 13 6 ++

11. Evaluate and write as a fraction. (a) 7 5. ']4 1. +7 9. g (b) 4.5 1.3 15.7 8.9 -- (c) 12.3 8.9 7.6 6.3 1.7 -+ + (d) . . . 11 5 9 7 4 3 - (e) 8100 64

12. Angel scored 17 out of 23 in a class test. What was her score as a percentage, to the nearest unit?

13. A survey showed that 31 out of 40 people watched the news on Monday night. What percentage of people watched the news?

14. What percentage of 2 kg is 350 g?

15. Write 25 minutes as a percentage of an hour.

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DID YOU KNOW?

Some countries use a comma for the decimal point—for example, 0,45 for 0.45.

This is the reason that our large numbers now have spaces instead of commas between digits—for example, 15 000 rather than 15,000.

EXAMPLES

1. Evaluate 1 . 5 2 4 3 -

Solution

1 5 2 4 3 5 7 4 3 20 28 20 15 20 13 - = -= -= 2. Evaluate 2 2 1 3 ' .

Solution

2 2 1 3 2 5 1 3 2 5 3 1 5 6 ' ' # = = = 3. Evaluate 0 056. #100.

Solution

. . 0 056#100=5 6 Move the decimal point

2 places to the right.

The examples on fractions show how to add, subtract, multiply or divide fractions both with and without the calculator. The decimal examples will help with some simple multiplying and the percentage examples will be useful in Chapter 8 of the HSC Course book when doing compound interest.

Most students use their calculators for decimal calculations. However, it is important for you to know how to operate with decimals. Sometimes the calculator can give a wrong answer if the wrong key is pressed. If you can estimate the size of the answer, you can work out if it makes sense or not. You can also save time by doing simple calculations in your head.

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4. Evaluate 0 02. #0 3. .

Solution

. . . 0 02#0 3=0 006 5. Evaluate 10 8.753 .

Solution

. . 8 753'10=0 8753

6. The price of a $75 tennis racquet increased by 5 %. 2 1

Find the new price.

Solution

_% _$ _. _$ $ . 5 75 0 055 75 4 13 of ` = # = % . % $ . $ $ . 5 2 1 _{0 055} ₁₀₅ 2 1 ₇₅ _{1 055} ₇₅ 79 13 2 1 or of # = = =

So the price increases by $4.13 to $79.13.

7. The price of a book increased by 12%. If it now costs $18.00, what did it cost before the price rise?

Solution

The new price is 112% (old price 100%, plus 12%)

1% $ . 100% $ . $16.07 112 18 00 112 18 00 1 100 ` # = = =

So the old price was $16.07.

1.4 Exercises

1. Write 18 minutes as a fraction of 2 hours in its lowest terms.

2. Write 350 mL as a fraction of 1 litre in its simplest form.

3. Evaluate (a) 5 3 4 1 + (b) 3 5 2 2 10 7 - (c) 4 3 1 5 2 # (d) 7 3 4 ' (e) 1 5 3 2 3 2 '

Multiply the numbers and count the number of decimal places in the question.

Move the decimal point 1 place to the left.

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4. Find 5 3 of $912.60. 5. Find 7 5 of 1 kg, in grams correct to 1 decimal place. 6. Trinh spends 3 1_{of her day} sleeping, 24 7_{at work and} 12 1

eating. What fraction of the day is left?

7. I get $150.00 a week for a casual job. If I spend 10 1 _{on bus fares,} 15 2_{on lunches and} 3 1_{on outings,}

how much money is left over for savings?

8. John grew by 200

17 _{of his height}

this year. If he was 165 cm tall last year, what is his height now, to the nearest cm? 9. Evaluate (a) 8.9+3 (b) 9-3.7 (c) 1 9. #10 (d) 0 032. #100 (e) 0 7. #5 (f) 0 8. #0 3. (g) 0 02. #0 009. (h) 5 72. #1000 (i) 100 8.74 (j) 3 76. #0 1. 10. Find 7% of $750. 11. Find 6.5% of 845 mL. 12. What is 12.5% of 9217 g? 13. Find 3.7% of $289.45.

14. If Kaye makes a profi t of $5 by selling a bike for $85, fi nd the profi t as a percentage of the selling price. 15. Increase 350 g by 15%. 16. Decrease 45 m by 8 %. 2 1

17. The cost of a calculator is now $32. If it has increased by 3.5%, how much was the old cost?

18. A tree now measures 3.5 m, which is 8.3% more than its previous year’s height. How high was the tree then, to 1 decimal place?

19. This month there has been a 4.9% increase in stolen cars. If 546 cars were stolen last month, how many were stolen this month?

20. George’s computer cost $3500. If it has depreciated by 17.2%, what is the computer worth now?

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Powers and Roots

A power (or index ) of a number shows how many times a number is multiplied by itself.

PROBLEM

If both the hour hand and minute hand start at the same position at 12 o’clock, when is the ﬁ rst time, correct to a fraction of a minute, that the two hands will be together again?

EXAMPLES

1. 43₌4_{# #}4 4₌64

2. 25=2# # # #2 2 2 2=32

In 43 the 4 is called the base number and the 3 is called the index or power.

A root of a number is the inverse of the power.

EXAMPLES

1. 36 =6 since 62=36

2. 3 8₌2_since₂3₌₈

3. 6 64 ₌2_since₂6₌₆₄

DID YOU KNOW?

Many formulae use indices (powers and roots).

For example the compound interest formula that you will study in Chapter 8 of the HSC Course book is A=P^1+rhn

Geometry uses formulae involving indices, such as 3 4 V₌ _rr3

. Do you know what this formula is for?

In Chapter 7, the formula for the distance between 2 points on a number plane is d (x₂ x₁)2 (y y)

2 1

2

= - + -

See if you can fi nd other formulae involving indices.

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Proof

( ) ( ) ( ) a a a a a a a a a a m n a a a m n a 1 times times times m n n m m n ' # # # # # # # # #f f f = = = -=

Index laws

There are some general laws that simplify calculations with indices.

am_#an₌am+n

Proof

( ) ( ) a a a a a a a a a a a a m n m n m n m n times times times # # # # # # # # # # # f f f = = = ++ 14444 44442 3 14444 44442 3 14444 44442 3

These laws work for any m and n , including fractions and negative numbers. am_'an₌am-n

(am n) =amn

Proof

( ) ( ) ( ) a a a a a n a n a times times m n m m m m m m m m mn # # # #f = = = f + + + +

POWER AND ROOT KEYS

Use the x2

and x3

keys for squares and cubes. Use the xy or ^ key to fi nd powers of numbers. Use the key for square roots.

Use the 3 _{key for cube roots.}

Use the x _{for other roots.}

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(ab)n₌a bn n

Proof

( ) ( ) ( ) ( ) ab ab ab ab ab n a a a b b b a b times n n n ntimes ntimes # # # # # # # # # # #f f f = = = 14444 44442 3 14444 44442 3 b a b a n n n = c m

Proof

( ) ( ) ( ) b a b a b a b a b a n b b b b a a a a n n b a times times times n n n # # # # # # # # # # # # f f f = = = c m

EXAMPLES

Simplify 1. m9_#m7_'m2

Solution

m m m m m 9 7 2 9 7 2 14 # ' = = + 2. ₍₂_y4₎3

Solution

( y ) ( )y y y 2 2 2 8 4 3 3 4 3 3 4 3 12 = = = # CONTINUED ch1.indd 21 5/20/09 3:06:11 PM

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1. Evaluate without using a calculator. (a) 53_#22 (b) 34₊82 (c) 4 1 3 c m (d) 3 27 (e) 4 16

2. Evaluate correct to 1 decimal place. (a) 3.72 (b) 1.061.5 (c) 2.3-0.2 (d) 3 19 (e) 3 34 8. _-1 2. _#43 1. (f) 0.99 5.61 1 3 ₊ 3. Simplify (a) a6_{# #}a9 a2 (b) y3_#y-8_#y5 (c) a-1_#a-3 (d) _w2_#_w2 1 1 (e) x6_'x (f) p3_'p-7 (g) y y 5 11 (h) ( )x7 3 (i) (2 )x5 2 (j) (3y-2 4) (k) a3_{# '}a5 a7 (l) y x 9 2 5 f p (m) w w w 3 6_# 7 (n) ( ) p p p 9 2_# 3 4 (o) x x x 2 6_' 7 (p) ( ) a b a b 4 9 2 2 6 # # (q) ( ) ( ) x y x y 1 4 2 3 3 2 # # 4. Simplify (a) x5_#x9 (b) a-1_#a-6 (c) m m 3 7 (d) k13_{# '}k6 k9 (e) a-5_{# #}a4 a-7 (f) _x5_#_x5 2 3 (g) m n m n 4 2 5 4 # #

1.5 Exercises

3. ( ) y y y 5 6 3_# -4

Solution

( ) y y y y y y y y y y y ( ) 5 6 3 4 5 18 4 5 18 4 5 14 9 # # = = = = - + ch1.indd 22 5/20/09 3:06:12 PM

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(h) 2 2 p p p 2 # (i) (3x11 2) (j) ( ) x x 3 4 6 5. Simplify (a) ₍_pq3₎5 (b) b a 8 c m (c) 4 b a 4 3 d n (7 (d) a 5_{b )}2 (e) (2 ) m m 4 7 3 (f) xy _xy(xy ) 3_# 2 4 (g) ₃ 4 ( ) ( ) k k 6 2 3 8 (h) 2y y 8 5 7 12 # _ i (i) a a a 11 6 4 3 # -e o (j) x y xy 5 8 3 9 3 # f p

6. Evaluate a 3_b2_when_a₌₂_and

4 3 b= . 7. If 3 2 x= and 9 1 , y= ﬁ nd the value of xy x y 5 3 2 . 8. If 2 1 , 3 1 a= b= and 4 1 , c= evaluate c a b 4 2 3 as a fraction . 9. (a) Simplify a b a b 8 7 11 8 . Hence evaluate (b) a b a b 8 7 11 8 when 5 2 a= and 8 5 b= as a fraction . 10. (a) Simplify p q r p q r 4 6 2 . (b) Hence evaluate p q r p q r 4 6 2 5 8 4 as a fraction when 8 7_, 3 2 p= q= and 4 3 r= . 11. Evaluate ( )a4 3_when 6 . a 3 2 = 1 c m 12. Evaluate b a b 4 3 6 when a 2 1 = and b 3 2 = . 13. Evaluate x y x y 5 5 4 7 when x 3 1 = and y 9 2 = . 14. Evaluate k k 9 5 when k . 3 1 = 15. Evaluate ( ) a b a b 3 2 2 4 6 when a 4 3 = and b 9 1 = . 16. Evaluate a b a b 5 2 6 3 # # _{as a fraction} when a 9 1 = and b 4 3 = . 17. Evaluate a b a b 3 2 7 as a fraction in

index form when a

5 2 4 = c m and b 8 5 3 = c m . 18. Evaluate ( ) ( ) a b c a b c 2 4 3 3 2 4 as a fraction when a , 3 1 = b 7 6 = and c 9 7 = .

ch1.indd 23 5/20/09 3:06:12 PM

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Proof

x x x x x x x x x 1 1 n n n n n n n n 0 0 ' ' ` = = = = =

Negative and zero indices

Class Investigation

Explore zero and negative indices by looking at these questions.

For example simplify x3_'x5_{using (i) index laws and (ii) cancelling.}

(i) x3_'x5₌x-2_{by index laws}

(ii) x x x x x x x x x x x 1 5 3 2 # # # ## # = = x x 1 So 2 2 = -

Now simplify these questions by (i) index laws and (ii) cancelling. (a) x2_'x3 (b) x2_'x4 (c) x2_'x5 (d) x3_'x6 (e) x3_'x3 (f) x2_'x2 (g) x_'x2 (h) x5_'x6 (i) x4_'x7 (j) x_'x3

Use your results to complete:

x x 0 n = = - x0₌1

ch1.indd 24 5/20/09 3:06:13 PM

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1 x x n n = -

Proof

x x x x x x x x x x x 1 1 n n n n n n n n 0 0 0 0 ' ' ` = = = = =

EXAMPLES

1. Simplify . abc ab c 4 5 0 e o

Solution

1 abc ab c 4 5 0 = e o 2. Evaluate 2-3.

Solution

2 2 1 8 1 3 3 = =

3. Write in index form.

(a) 1 x2 (b) 3 x5 (c) 5 1 x (d) x 1 1 +

CONTINUED ch1.indd 25 5/20/09 3:06:13 PM

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1. Evaluate as a fraction or whole number. (a) 3-3 (b) 4-1 (c) 7-3 (d) 10-4 (e) 2-8 6 (f) 0 (g) 2-5 (h) 3-4 (i) 7-1 (j) 9-2 (k) 2-6 (l) 3-2 4 (m) 0 (n) 6-2 (o) 5-3 (p) 10-5 (q) 2-7 (r) 20 (s) 8-2 (t) 4-3 2. Evaluate (a) 20 (b) 2 1 -4 c m (c) 3 2 -1 c m (d) 6 5 -2 c m (e) 3 2 x y x y 0 -+ f p (f) 5 1 -3 c m (g) 4 3 -1 c m (h) 7 1 -2 c m (i) 3 2 -3 c m (j) 2 1 -5 c m (k) 7 3 -1 c m

1.6 Exercises

Solution

(a) 1 x2 x 2 = - (b) x x x 3 3 1 3 5 5 5 # = = (c) _x _x x 5 1 5 1 1 5 1 1 # = = (d) ( ) x _x x 1 1 1 1 1 1 1 + = ₊ =_] + _g

4. Write a −3_{without the negative index.}

Solution

a a 1 3 3 = -

ch1.indd 26 5/20/09 3:06:13 PM

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(l) 9 8 c m (m) 7 6 -2 c m (n) 10 9 -2 c m (o) 11 6 0 c m (p) 4 1 2 - -c m (q) 5 2 3 - -c m (r) 3 7 2 1 - -c m (s) 8 3 0 -c m (t) 1 4 1 2 - -c m

3. Change into index form. (a) 1 m3 (b) 1_x (c) 1 p7 (d) 1 d9 (e) 1 k5 (f) 1 x2 (g) 2 x4 (h) 3 y2 (i) 2 1 z6 (j) 5 3 t8 (k) 7 2 x (l) 2 5 m6 (m) 3 2 y7 (n) (3 4) 1 x₊ 2 (o) ( ) 1 a₊b 8 (p) 2 1 x- (q) (5p 1) 1 3 + (r) (4 9) 2 t_- 5 (s) (x ) 4 1 1 11 + (t) 9 ( 3 ) 5 a₊ b 7

4. Write without negative indices. (a) t-5 (b) x-6 (c) y-3 (d) n-8 (e) w-10 (f) 2x-1 (g) 3m-4 (h) 5x-7 (i) _]2x_g-3 (j) _]4n_g-1 (k) _]x₊1_g-6 (l) _^8y₊z_h-1 (m) _]k_-3_g-2 (n) _^3x₊2y_h-9 (o) 1_x 5 -b l (p) 1_y 10 -c m (q) 2_p 1 -d n (r) 1 a b 2 + -c m (s) _xx _yy 1 -+ -e o (t) 3 2 x y w z 7 + -e o

ch1.indd 27 5/20/09 3:06:14 PM

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Proof

n n a a a a a a by index laws n n n n ` = = = 1 1 ` _^ ^ j _h h

Fractional indices

Class Investigation

Explore fractional indices by looking at these questions. For example simplify (i) _x2 2

1 ` j_{and (ii)}_^ _x_h2_. 2 ( ) x x x i by index laws 2 1 = = 1 ` j _^ _h 2 2 ( ) x x x x x x x ii So 2 2 2 ` = = = = 1 1 ^ ` _^ h j _h

Now simplify these questions.

(a) _x2 2 1 ^ h (b) x2 (c) _x3 3 1 ` j (d) _x3 3 1 ^ h (e) ^3 xh3 (f) 3 x3 (g) x4 4 1 ` j (h) _x4 4 1 ^ h (i) ^4 xh4 (j) 4 x4

Use your results to complete:

n x = 1 n a1₌n a ch1.indd 28 5/20/09 3:06:14 PM

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EXAMPLES

1. Evaluate (a) 492 1 (b) ₂₇3 1

Solution

(a) 492 49 7 = = 1 (b) 273 27 3 3 = = 1

2. Write 3x-2 in index form.

Solution

₃_x_{- =}₂ ₍₃_x_-₂₎21

3. Write ₍_a₊_b₎7 1

without fractional indices.

Solution

7 (a₊b)1 ₌7 a₊b

Proof

n n n n a a a a a m n m m m n = = a = = m m 1 1 ` ^ ^ j h h

Putting the fractional and negative indices together gives this rule.

- n a a 1 n = 1

Here are some further rules.

n ( ) a a a m n n m = = m

ch1.indd 29 5/20/09 3:06:14 PM

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b a a b n n = -c m b l

EXAMPLES

1. Evaluate (a) ₈34 (b) ₁₂₅-3 1 (c) 3 2 -3 c m

Solution

(a) ₈3 ₍ ₈_{) (} ₈ ₎ 2 16 or 3 4 3 4 4 = = = 4 (b) -3 3 125 125 1 125 1 5 1 3 = = = 1 1

Proof

b a b a b a b a a b a b a b 1 1 1 1 n n n n n n n n n n n ' # = = = = = = -c c b m m l ch1.indd 30 5/20/09 3:06:15 PM

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(c) 3 2 2 3 8 27 3 8 3 3 3 = = = -c m c m

2. Write in index form. (a) x5 (b) (4x 1) 1 2 2 3 -

Solution

(a) _x5 ₌_x2 5 (b) -3 3 ( ) ₍ ₎ ( ) x _x x 4 1 1 4 1 1 4 1 2 2 3 ₂ 2 - = -= -2 2

3. Write _r-5 3

without the negative and fractional indices.

Solution

-₅ 5 r r r 1 1 3 5 = = 3 3

DID YOU KNOW?

Nicole Oresme (1323–82) was the fi rst mathematician to use fractional indices.

John Wallis (1616–1703) was the fi rst person to explain the signifi cance of zero, negative

and fractional indices. He also introduced the symbol 3 for infi nity.

Do an Internet search on these mathematicians and fi nd out more about their work and backgrounds. You could use keywords such as indices and infi nity as well as their names to fi nd this information.

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1. Evaluate (a) ₈₁21 (b) ₂₇3 1 (c) ₁₆21 (d) ₈3 1 (e) ₄₉21 (f) ₁₀₀₀3 1 (g) ₁₆41 (h) ₆₄2 1 (i) ₆₄31 (j) ₁7 1 (k) ₈₁41 (l) ₃₂5 1 (m) ₀81 (n) ₁₂₅3 1 (o) ₃₄₃31 (p) ₁₂₈7 1 (q) ₂₅₆41 (r) ₉2 3 (s) ₈-31 (t) ₆₄-3 2

2. Evaluate correct to 2 decimal places. (a) ₂₃4 1 (b) 4 45.8 (c) 7 1.24₊4.32 (d) 12.9 1 5 (e) . . . . 1 5 3 7 3 6 1 4 8 + - (f) . . . . 8 79 1 4 5 9 3 7 4 _# -

3. Write without fractional indices. (a) _y31 (b) _y3 2 (c) _x- 12 (d) ₍₂_x₊₅₎2 1 (e) ₍₃_x_-₁₎-21 (f) ₍₆_q₊_r₎3 1 (g) ₍_x₊₇₎-52

4. Write in index form. (a) t (b) 5 y (c) x3 (d) 3 9_-x (e) 4s+1 (f) 2 3 1 t+ (g) (5 ) 1 x_-y 3 (h) (3x₊1)5 (i) ( 2) 1 x 2 3 - (j) 2 7 1 y+ (k) 4 5 x 3 ₊ (l) y 3 1 2 2_- (m) 5 ( 2) 3 x2 3 4 +

5. Write in index form and simplify. (a) x x (b) _xx (c) x x 3 (d) x x 3 2 (e) x4 x

1.7 Exercises

ch1.indd 32 5/20/09 3:06:15 PM

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index form. (a) ( x₊x)2 (b) (3 a₊3 b) (3 a_-3 b) (c) p 1 p 2 + f p (d) ( x 1 ) x 2 + (e) ( ) x x x 3x 1 3 2_- ₊ negative indices. (a) ₍_a_-₂_b₎-3 1 (b) ₍_y_-₃₎- 23 (c) _{4 6}₍ _a₊₁₎-7 4 (d) -₄ (x y) 3 + 5 (e) -₉ ( x ) 7 6 3 +8 2

Scientiﬁ c notation (standard form)

Very large or very small numbers are usually written in scientiﬁ c notation to make them easier to read. What could be done to make the ﬁ gures in the box below easier to read?

DID YOU KNOW ?

The Bay of Fundy, Canada, has the largest tidal changes in the world. About 100 000 000 000 tons of water are moved with each tide change.

The dinosaurs dwelt on Earth for 185 000 000 years until they died out 65 000 000 years ago. The width of one plant cell is about 0.000 06 m.

In 2005, the total storage capacity of dams in Australia was 83 853 000 000 000 litres and households in Australia used 2 108 000 000 000 litres of water.

A number in scientiﬁ c notation is written as a number between 1 and 10 multiplied by a power of 10.

EXAMPLES

1. Write 320 000 000 in scientiﬁ c notation.

Solution

.

320 000 000₌3 2_#108

2. Write 7 1. _#10-5_{as a decimal number.}

Solution

. . . 7 1 10 7 1 10 0 000 071 5 5 # = ' =

Write the number between 1 and 10 and count the decimal places moved.

Count 5 places to the left.

Maths in Focus Ext 1 Preliminary-4

maths

Margaret Grove

maths

Contents

PREFACE

ACKNOWLEDGEMENTS

CREDITS

FEATURES OF THIS BOOK

SYLLABUS MATRIX

Mathematics (2 Unit)

Extension 1

STUDY SKILLS

In the classroom

At home

Assessment tasks and exams

And fi nally…

1

Basic Arithmetic

INTRODUCTION

Real Numbers

Types of numbers

EXAMPLE

Solution

Order of operations

EXAMPLE

Solution

PROBLEM

BRACKETS KEYS

Rounding off

EXAMPLES

Solution

MEMORY KEYS

X

KEY

Solution

EXAMPLES

Solution

Solution

Solution

EXAMPLE

Solution

1.1

Exercises

FIX KEY

DID YOU KNOW?

Directed Numbers

KEY

Adding and subtracting

EXAMPLES

Solution

Solution

Multiplying and dividing

EXAMPLES

Solution

Solution

Solution

1.2

Exercises

Fractions, Decimals and Percentages

EXAMPLES

Solution

Solution

g

Solution

Solution

Solution

Conversions

EXAMPLES

Solution

Solution

Solution

1.3

Exercises

Investigation

Operations with fractions, decimals and percentages

DID YOU KNOW?

EXAMPLES

Solution

Solution

_KEY