NEW
CENTURY
MATHS Advanced
9
For the
australian
curriculum
NSW Stages 5.2/5.3
Sample Chapter
NEW CENTURY MATHS 9 ADVANCED
NSW STAGES 5.2/5.3
ISBN: 978 0 17 019308 5
Klaas Bootsma
David Badger
Sarah Hamper
Robert Yen Series Editor
Customer Service: 1300 790 853
www.nelsonsecondary.com.au
Preface
First published in 1996, New Century Maths Advanced 9 Stages 5.2/5.3 has now been completely
rewritten to reflect the spirit of the new Australian curriculum and the NSW syllabus.
For Years 9 and 10, we publish two levels of mathematics books:
t New Century Maths Stages 5.1/5.2
t New Century Maths Advanced Stages 5.2/5.3
The NSW mathematics syllabus describes Stage 5 (Years 9–10) as a continuum with content classified
into three sub-stages, Stage 5.1, Stage 5.2 and Stage 5.3, with each sub-stage including the content of
the previous sub-stage. This book has been designed for students and classes progressing along Stages
5.2 or 5.3 of the continuum.
We have retained those familiar features that have made New Century Maths a leading series: clear
worked examples, graded exercises, multiple-choice questions, Investigation, Technology, Mental skills,
SkillCheck pre-chapter exercise, Power Plus extension exercise, Language of Maths, Topic overview
with mind map, glossary, video tutorials and worksheets.
New features include:
t
t
t
t
t
t
)PNFXPSLTIFFUT
8PSLFETPMVUJPOTUPTFMFDUFERVFTUJPOT
GeoGebra activities in our Technology sections
$IBQUFSPVUMJOFTTIPXJOHDPWFSBHFPGUIF8PSLJOH.BUIFNBUJDBMMZQSPGJDJFODJFT
NelsonNet student and teacher websites of print and multimedia resources
NelsonNetBook digital version of this book with interactive features
We always aim to write textbooks to provide flexibility for teachers to plan and teach for a variety of
pathways. We thank our families and teaching colleagues for their continued support and patience, and
editors Anna Pang and Alan Stewart for transforming our manuscript into this fine text. Finally, we wish all
teachers and students using this book every success in embracing the new Australian mathematics curriculum.
About the authors
Klaas Bootsma XBTIFBEUFBDIFSPGNBUIFNBUJDTBU"NCBSWBMF)JHI4DIPPMJO$BNQCFMMUPXOBOE
IBTUBVHIUBU-VSOFBBOE(SBOUIBNIJHITDIPPMT)FXBTBTFOJPS)4$FYBNJOFSBOEIBTXPSLFEPO
UIF)4$"EWJDF-JOFBOE4DIPPM$FSUJGJDBUFNBSLJOH,MBBTIBTXJEFFYQFSJFODFUFBDIJOHBMMUZQFTPG
students, and his interests are in curriculum and the use of technology (ICT) in learning.
David Badger is principal of Toongabbie Christian School, was deputy principal at Mt Annan Christian
$PMMFHFBOEIFBEUFBDIFSPGNBUIFNBUJDTBU&BHMF7BMF)JHI4DIPPMJO$BNQCFMMUPXO)FIBTCFFO
JOWPMWFEJO)4$BOE4DIPPM$FSUJGJDBUFNBSLJOHBOEIBTXPSLFEPOUIF)4$"EWJDF-JOF%BWJET
passion is to make mathematics interesting, practical and accessible to all students.
Sarah Hamper teaches at Abbotsleigh School in Wahroonga and has taught at Meriden and Tara
"OHMJDBOTDIPPMT4IFIBTBOJOUFSFTUJOHJGUFEBOEUBMFOUFE ("5 TUVEFOUTBOEHJSMTFEVDBUJPO4BSBIT
expertise is in using modelling, problem solving and ICT for the effective learning of mathematics, and
she has presented workshops for MANSW and nationally.
Series editor Robert Yen has UBVHIUBU)VSMTUPOF"HSJDVMUVSBM &BHMF7BMFBOE"NCBSWBMFIJHITDIPPMT
JOTPVUIXFTU4ZEOFZ)FDPXSPUFNew Century Maths Essentials 9–10 and Mathematics General 2,
writes and presents for MANSW, and has co-edited its journal, Reflections)FOPXXPSLTGPS$FOHBHF
Learning as an associate publisher.
SAMPLE CHAPTER
Contributing author
Megan Boltze wrote and edited many of the NelsonNet print resources (blackline masters) and is head
UFBDIFSPGNBUIFNBUJDTBU"TIDSPGU)JHI4DIPPM
9780170193085
iii
Contents
Preface
About the authors
Curriculum grids
User’s guide
Acknowledgements
iii
iii
viii
xii
xiv
Mental skills 2A: Squaring a
number ending in 5, 1 or 9
2-05 Simple interest
2-06 Ratios and rates
2-07 Converting rates
Mental skills 2B: Estimating
square roots
Power plus
Chapter 2 review
69
69
71
Chapter 3: Products and
factors
74
9NA211
8NA188
9NA208
* ¼ STAGE 5.3
Chapter 1: Pythagoras’
theorem and
surds
9MG222
9MG222
10ANA264
10ANA264
10ANA264
10ANA264
9MG222
9MG222
9MG222
SkillCheck
1-01 Finding the hypotenuse
1-02 Finding a shorter side
1-03 Surds and irrational
numbers*
pffiffiffi
Investigation: Proof that 2
is irrational
1-04 Simplifying surds*
Mental skills 1A: Multiplying
and dividing by a power of 10
Investigation: A formula for
calculating square roots
1-05 Adding and subtracting
surds*
1-06 Multiplying
and dividing surds*
1-07 Pythagoras’ theorem
problems
1-08 Testing for right-angled
triangles
Mental skills 1B: Multiplying
and dividing by a multiple
of 10
1-09 Pythagorean triads
Power plus
Chapter 1 review
2
4
4
10
15
16
18
NSW
8NA187
8NA189
iv
SkillCheck
2-01 Terminating and
recurring decimals
2-02 Converting recurring
decimals to fractions*
2-03 Operations with
percentages
2-04 Percentages and money
8NA192
10NA232
10NA232
20
21
22
9NA213
8NA191
9NA213
23
25
10NA233
31
34
34
37
39
Chapter 2: Working with
numbers
42
8NA184
8NA192
10NA233
10NA233
10ANA269
44
10ANA269
45
SkillCheck
3-01 Adding and subtracting
terms
3-02 Multiplying and dividing
terms
3-03 Adding and subtracting
algebraic fractions
3-04 Multiplying and dividing
algebraic fractions
Mental skills 3A: Multiplying
and dividing by 5, 15, 25 and 50
3-05 Expanding expressions
3-06 Factorising expressions
3-07 Expanding binomial
products
Investigation: Expanding
perfect squares
3-08 Perfect squares*
Investigation: Squaring a
number ending in 5
Mental skills 3B: Multiplying
by 9, 11, 99 and 101
Investigation: Expanding
sums by differences
3-09 Difference of two squares*
3-10 Mixed expansions*
3-11 Factorising special
binomial products*
Investigation: Factorising
quadratic expressions
3-12 Factorising quadratic
expressions*
Investigation: Factorising
quadratic trinomials by
grouping in pairs
3-13 Factorising quadratic
expressions of the form
ax2 þ bx þ c*
SAMPLE CHAPTER
47
48
53
10ANA269
57
59
62
68
76
77
79
82
84
85
87
90
93
97
97
100
100
101
101
103
104
106
107
110
110
9780170193085
N E W C E N T U R Y M AT H S A D V A N C E D
for the A
10ANA269
NSW
3-14 Mixed factorisations*
3-15 Factorising algebraic
fractions*
Power plus
Chapter 3 review
114
116
118
119
9NA209
122
9NA209
Chapter 4: Trigonometry 126
9NA210
9NA210
Mixed revision 1
9MG223
9MG223
9MG223
9MG223
9MG224
9MG224
9MG224
10MG245
10MG245
10MG245
SkillCheck
128
4-01 The sides of a rightangled triangle
129
4-02 The trigonometric ratios 132
4-03 Similar right-angled
triangles
135
Technology: Similar rightangled triangles
138
4-04 Trigonometry on a
calculator
140
Mental skills 4: Estimating
answers
143
4-05 Finding an unknown side 144
Investigation: Calculating the
height of an object
150
4-06 Finding more unknown
sides
151
Investigation: Finding an angle,
given a trigonometric ratio
155
4-07 Finding an unknown angle 156
4-08 Angles of elevation and
depression
161
4-09 Bearings
165
Investigation: Compass walks 169
4-10 Problems involving
bearings
169
Power plus
172
Chapter 4 review
174
Chapter 5: Indices
9NA212
9NA212
9NA212
9NA212
9780170193085
178
SkillCheck
Investigation: Multiplying and
dividing terms with powers
5-01 Multiplying and dividing
terms with the same base
Investigation: Powers of powers
5-02 Power of a power
5-03 Powers of products and
quotients
Investigation: The power of zero
5-04 The zero index
Mental skills 5: Adding or
multiplying in any order
180
10ANA264
10ANA264
NSW
Investigation: Negative powers
Technology: Negative powers
5-05 Negative indices
5-06 Fractional indices n1*
190
191
193
197
5-07 Fractional indices mn *
5-08 Summary of the index laws
5-09 Significant figures
5-10 Scientific notation
5-11 Scientific notation on a
calculator
Investigation: A lifetime of
heartbeats
Power plus
Chapter 5 review
199
202
204
206
218
220
223
Technology: Sketching
parallelograms and rectangles
6-02 Quadrilateral geometry
228
230
166
7MG 165,
166
NSW
NSW
NSW
NSW
Investigation: Angle sum of a
polygon
6-03 Angle sum of a polygon
Investigation: Exterior angle
sum of a convex polygon
6-04 Exterior angle sum
of a convex polygon
Mental skills 6: Dividing
decimals
Power plus
Chapter 6 review
Chapter 7: Equations
8NA194
8NA194,
9NA215
8NA194
181
184
184
10NA240
SkillCheck
7-01 Equations with variables
on both sides
7-02 Equations with
brackets
7-03 Equation problems
7-04 Equations with algebraic
fractions
Mental skills 7A: Fraction of a
quantity
Technology: Solving equations
on a graphics calculator
Investigation: Solving x2 ¼ c
7-05 Simple quadratic
equations ax2 ¼ c
SAMPLE CHAPTER
190
213
214
215
SkillCheck
6-01 Triangle geometry
7MG 165,
9
210
Chapter 6: Geometry
180
186
188
188
ustralian Curriculum
10NA241
238
239
243
244
246
247
248
252
254
254
257
259
263
266
267
267
268
v
Contents
NSW
NSW
10NA234,
235
NSW
Investigation: Solving x3 ¼ c
7-06 Simple cubic equations
ax3 ¼ c*
7-07 Equations and formulas
270
270
272
7-08 Changing the subject of
a formula*
Mental skills 7B: Percentage of
a quantity
Power plus
Chapter 7 review
275
277
278
279
Mixed revision 2
Chapter 8: Earning
money
NSW
NSW
NSW
NSW
NSW
NSW
NSW
NSW
NSW
9SP227
284
288
SkillCheck
8-01 Wages and salaries
8-02 Overtime pay
Technology: Calculating incomes
8-03 Commission, piecework
and leave loading
8-04 Income tax
Technology: Online income
tax calculators
Mental skills 8: Percentage
increase and decrease
8-05 PAYG tax and net pay
Technology: Online PAYG
tax calculator
Power plus
Chapter 8 review
290
290
295
299
7SP171,
8SP207
9SP282,
228
9MG219
NSW
9MG216
8MG196
300
306
9MG216
308
9MG218
309
310
9MG217
8MG198
313
314
315
9SP282
9SP283
8SP284,
9SP228
vi
320
321
10AMG271
330
331
338
340
344
347
356
358
359
366
368
374
376
379
383
384
391
398
399
401
406
407
409
410
413
419
421
Mixed revision 3
426
Chapter 11: Coordinate
geometry
and graphs
430
9NA214
349
356
SkillCheck
10-01 Metric units
10-02 Limits of accuracy of
measuring instruments
10-03 Perimeters and areas of
composite shapes
10-04 Areas of quadrilaterals
Mental skills 10A: Finding
10%, 20% and 5%
10-05 Circumferences and
areas of circular shapes
10-06 Surface area of a prism
Investigation: Surface area
of a cylinder
10-07 Surface area of a cylinder
10-08 Volumes of prisms and
cylinders
Investigation: Volume vs
surface area
Technology: Drawing prisms
and cylinders
Mental skills 10B: Finding
15%, 212%, 25% and 1212%
Technology: Approximating
the volume of a pyramid
10-09 Volumes of pyramids
and cones*
Power plus
Chapter 10 review
SAMPLE CHAPTER
348
352
353
Chapter 10: Surface Area
and Volume 364
Chapter 9: Investigating
data
318
SkillCheck
9-01 The mean, median,
mode and range
Technology: Most valuable
player
9-02 Histograms
and stem-and-leaf plots
Technology: Histograms
9-03 The shape of a
distribution
9-04 Comparing data sets
Technology: Comparing
relative humidities
Mental skills 9: Finding a
percentage of a multiple of 10
9-05 Sampling and types
of data
Investigation: Australian
Bureau of Statistics
9-06 Bias and questionnaires
Investigation: Media reports of
surveys
Investigation: Year 9 student
survey
Power plus
Chapter 9 review
SkillCheck
11-01 The length of an
interval
432
433
9780170193085
N E W C E N T U R Y M AT H S A D V A N C E D
for the A
9NA294
9NA294
9NA215
NSW
NSW
8NA194
9NA208
9NA296
9NA296
Technology: The length of an
interval
Investigation: The midpoint of
an interval
Technology: The midpoint of
an interval
11-02 The midpoint of an
interval
11-03 The gradient of a line
Technology: The gradient of
collinear points
11-04 Graphing linear
equations
Technology: Graphing linear
equations
Mental skills 11: Divisibility
tests
Investigation: Comparing
gradients and y-intercepts
11-05 The gradient-intercept
formula y ¼ mx þ b
11-06 Finding the equation
of a line
11-07 Solving linear equations
graphically
11-08 Direct proportion
Investigation: Graphing
y ¼ x2 and y ¼ ax2
Technology: Graphing
y ¼ ax2
Investigation: Graphing
y ¼ ax2 þ c
11-09 Graphing quadratic
equations
11-10 Graphing circles
Power plus
Chapter 11 review
Chapter 12: Probability
8SP204
9SP226
9SP226
9SP226
SkillCheck
12-01 Probability
12-02 Relative frequency
Technology: Throwing a coin
12-03 Venn diagrams
12-04 Two-way tables
Investigation: Using two-way
tables
Mental skills 12: The unitary
method with percentages
Investigation: Are the Lotto
numbers equally likely?
9SP225
437
ustralian Curriculum
12-05 Two-step experiments
Power plus
Chapter 12 review
511
517
518
438
439
440
444
Chapter 13: Congruent
and similar
figures
449
8MG201
450
10MG243
456
10MG243
457
458
459
9MG220
462
464
465
9MG220,
221
468
469
9MG221
470
471
475
479
480
484
486
487
492
497
500
506
509
10MG244
9MG220
10MG244
SkillCheck
Investigation: Congruent
triangles
13-01 Tests for congruent
triangles
13-02 Congruent triangle
proofs
13-03 Using congruence to
prove geometrical
properties
Investigation: Same shape,
different size
13-04 Similar figures
Mental skills 13: Simplifying
fractions and ratios
Investigation: Properties of
similar figures
13-05 Properties of
similar figures
Investigation: Are all shapes
similar?
13-06 Scale diagrams
Investigation: Floor plans
Investigation: Areas of similar
figures
Technology: Areas of similar
figures
13-07 Areas of similar
figures*
Investigation: Similar triangles
13-08 Tests for similar
triangles
13-09 Similar triangle proofs*
Power plus
Chapter 13 review
509
511
522
524
525
526
531
536
539
540
547
548
549
554
554
559
560
561
563
566
567
571
576
577
Mixed revision 4
582
General revision
Glossary
Answers
Index
586
592
606
00
SAMPLE CHAPTER
9780170193085
9
vii
Curriculum grid: Australian curriculum
Strand and substrand
New Century Maths
Advanced 9
Stages 5.2/5.3
chapter
New Century Maths
Advanced 10
Stages 5.2/5.3
chapter
NUMBER AND ALGEBRA
Real numbers
1
2
5
11
Pythagoras’ theorem and surds
Working with numbers
Indices
Coordinate geometry and
graphs
1 Surds
7 Graphs
8 Equations and logarithms
Money and financial mathematics
2 Working with numbers
2 Interest and depreciation
Patterns and algebra
3 Products and factors
5 Indices
7 Equations
5 Products and factors
8 Equations and logarithms
14 Polynomials
Linear and non-linear relationships
7 Equations
11 Coordinate geometry and
graphs
3
5
7
8
10
11
Coordinate geometry
Products and factors
Graphs
Equations and logarithms
Simultaneous equations
Quadratic equations and the
parabola
14 Polynomials
MEASUREMENT AND GEOMETRY
Using units of measurement
10 Surface area and volume
Geometric reasoning
6 Geometry
13 Congruent and similar figures
Pythagoras and trigonometry
1 Pythagoras’ theorem and surds
4 Trigonometry
4 Surface area and volume
13 Geometry
15 Circle geometry
9 Trigonometry
STATISTICS AND PROBABILITY
Chance
12 Probability
Data representation and interpretation
9 Investigating data
12 Probability
6 Investigating data
SAMPLE CHAPTER
viii
9780170193085
Curriculum grid: NSW syllabus
Strand and substrand
New Century Maths
Advanced 9
Stages 5.2/5.3
chapter
New Century Maths
Advanced 10
Stages 5.2/5.3
chapter
NUMBER AND ALGEBRA
Financial mathematics
(Stage 5.2)
2 Working with numbers
8 Earning money
2 Interest and depreciation
Indices (Stage 5.2)
5 Indices
5 Products and factors
Linear relationships
(Stages 5.2, 5.3)
7 Equations
11 Coordinate geometry and
graphs
Non-linear relationships
(Stages 5.2, 5.3)
7 Equations
11 Coordinate geometry and
graphs
Ratios and rates
(Stages 5.2, 5.3)
2 Working with numbers
11 Coordinate geometry and graphs
Algebraic techniques (Stages 5.2, 5.3)
3 Products and factors
Equations
(Stages 5.2, 5.3)
7 Equations
Surds and indices
(Stage 5.3)
1 Pythagoras’ theorem and surds
5 Indices
Polynomials (Stage 5.3)
3 Coordinate geometry
3 Coordinate geometry
7 Graphs
11 Quadratic equations and the
parabola
7 Graphs
5 Products and factors
8 Equations and logarithms
10 Simultaneous equations
11 Quadratic equations and the
parabola
5 Products and factors
14 Polynomials
Logarithms (Stage 5.3)
8 Equations and logarithms
Functions and other graphs (Stage 5.3)
16 Functions
MEASUREMENT AND GEOMETRY
Area and surface area
(Stages 5.2, 5.3)
10 Surface area and volume
Numbers of any magnitude
(Stage 5.1)
5 Indices
10 Surface area and volume
Right-angled triangles (Trigonometry)
(Stages 5.2, 5.3)
1 Pythagoras’ theorem and surds
4 Trigonometry
Properties of geometrical figures
(Stages 5.2, 5.3)
6 Geometry
13 Congruent and similar figures
Volume (Stages 5.2, 5.3)
10 Surface area and volume
Trigonometry and Pythagoras’
theorem (Stage 5.3)
4 Surface area and volume
9 Trigonometry
4 Surface area and volume
13 Geometry
4 Surface area and volume
9 Trigonometry
Circle geometry (Stage 5.3)
15 Circle geometry
STATISTICS AND PROBABILITY
Single variable data analysis
(Stages 5.2, 5.3)
9 Investigating data
6 Investigating data
SAMPLE CHAPTER
Probability (Stages 5.1, 5.2)
Double variable data analysis
(Stages 5.2, 5.3)
9780170193085
12 Probability
12 Probability
6 Investigating data
ix
Curriculum grid: Year 9 content descriptions
This is an extract from the Australian Curriculum.
Content description
New Century Maths
Advanced 9 Stages 5.2/5.3
chapter
NUMBER AND ALGEBRA
Real numbers
ACMNA208: Solve problems involving direct proportion. Explore the
relationship between graphs and equations corresponding to simple rate
problems
11 Coordinate geometry and
graphs
ACMNA209: Apply index laws to numerical expressions with integer indices
5 Indices
ACMNA210: Express numbers in scientific notation
5 Indices
YEAR 10A ACMNA264: Define rational and irrational numbers and perform
operations with surds and fractional indices
1 Pythagoras’ theorem and
surds
5 Indices
Money and financial mathematics
ACMNA211: Solve problems involving simple interest
2 Working with numbers
NSW STAGE 5.1: Solve problems involving earning money
8 Earning money
Patterns and algebra
ACMNA212: Extend and apply the index laws to variables, using positive
integer indices and the zero index
5 Indices
ACMNA213: Apply the distributive law to the expansion of algebraic
expressions, including binomials, and collect like terms where appropriate
3 Products and factors
YEAR 10 ACMNA233: Expand binomial products and factorise monic
quadratic expressions using a variety of strategies
3 Products and factors
YEAR 10 ACMNA234: Substitute values into formulas to determine an
unknown;
7 Equations
YEAR 10A ACMNA269: Factorise monic and non-monic quadratic
expressions and solve a wide range of quadratic equations derived from a
variety of contexts
3 Products and factors
Linear and non-linear relationships
ACMNA214: Find the distance between two points located on a Cartesian
plane using a range of strategies, including graphing software
11 Coordinate geometry and
graphs
ACMNA215: Sketch linear graphs using the coordinates of two points and
solve linear equations
7 Equations
11 Coordinate geometry and
graphs
YEAR 10 ACMNA235: Solve problems involving linear equations, including
those derived from formulas
7 Equations
ACMNA294: Find the midpoint and gradient of a line segment (interval) on
the Cartesian plane using a range of strategies, including graphing software
11 Coordinate geometry and
graphs
NSW STAGE 5.2: Interpret and graph linear relationships using the
gradient-intercept form of the equation of a straight line
11 Coordinate geometry and
graphs
SAMPLE CHAPTER
ACMNA296: Graph simple non-linear relations with and without the use of
digital technologies and solve simple related equations
x
11 Coordinate geometry and
graphs
9780170193085
9
NEW CENTURY MATHS ADVANCED
for the
Content description
A u s t ra l i a n C u r r i cu l u m
New Century Maths
Advanced 9 Stages 5.2/5.3
chapter
MEASUREMENT AND GEOMETRY
Using units of measurement
ACMMG216: Calculate the areas of composite shapes
10 Surface area and volume
ACMMG217: Calculate the surface area and volume of cylinders and solve
related problems
10 Surface area and volume
ACMMG218: Solve problems involving the surface area and volume of right prisms
10 Surface area and volume
ACMMG219: Investigate very small and very large time scales and intervals
10 Surface area and volume
NSW STAGE 5.1: Round numbers to a specified number of significant
figures, describe the limits of accuracy of measuring instruments
10 Surface area and volume
Geometric reasoning
ACMMG220: Use the enlargement transformation to explain similarity and
develop the conditions for triangles to be similar
13 Congruent and similar figures
ACMMG221: Solve problems using ratio and scale factors in similar figures
13 Congruent and similar figures
YEAR 10 ACMMG243: Formulate proofs involving congruent triangles and angle
properties; ACMMG244: Apply logical reasoning, including the use of congruence
and similarity, to proofs and numerical exercises involving plane shapes
13 Congruent and similar figures
NSW STAGE 5.2: Apply interior and exterior angle sum results for polygons
to find the sizes of unknown angles
6 Geometry
Pythagoras and trigonometry
ACMMG222: Investigate Pythagoras’ theorem and its application to solving
simple problems involving right-angled triangles
1 Pythagoras’ theorem and
surds
ACMMG223: Use similarity to investigate the constancy of the sine, cosine
and tangent ratios for a given angle in right-angled triangles
4 Trigonometry
ACMMG224: Apply trigonometry to solve right-angled triangle problems
4 Trigonometry
YEAR 10 ACMMG245: Solve right-angled triangle problems including those
involving direction and angles of elevation and depression
4 Trigonometry
STATISTICS AND PROBABILITY
Chance
ACMSP225: List all outcomes for two-step chance experiments, both with
and without replacement using tree diagrams or arrays. Assign probabilities
to outcomes and determine probabilities for events
12 Probability
ACMSP226: Calculate relative frequencies from given or collected data to
estimate probabilities of events involving ‘and’ or ‘or’
12 Probability
ACMSP227: Investigate reports of surveys in digital media and elsewhere for
information on how data were obtained to estimate population means and medians
12 Probability
Data representation and interpretation
ACMSP228: Identify everyday questions and issues involving at least one
numerical and at least one categorical variable, and collect data directly
from secondary sources
9 Investigating data
ACMSP282: Construct back-to-back stem-and-leaf plots and histograms and
describe data, using terms including ‘skewed’, ‘symmetric’ and ‘bi-modal’
9 Investigating data
ACMSP283: Compare data displays using mean, median and range to describe
and interpret numerical data sets in terms of location (centre) and spread
9 Investigating data
SAMPLE CHAPTER
9780170193085
xi
New Century Maths User’s guide
Coverage of the Australian curriculum and NSW syllabus
t N
ew Century Maths Advanced 9Stages 5.2/5.3 DPWFSTCPUIUIF"VTUSBMJBODVSSJDVMVNBOE
UIF/48TZMMBCVT BTTIPXOCZUIFUBCMFPGDPOUFOUTBOEDVSSJDVMVNHSJETPOUIFQSFWJPVT
QBHFT5IFQSFWJPVTUXPQBHFTMJTUBMMPGUIF:FBS"VTUSBMJBO$VSSJDVMVNDPOUFOUEFTDSJQUJPOT
FYQMJDJUMZ
t 5
IJTCPPLDPOUBJOT4UBHFT BOEDPOUFOU n
JODMVEJOHUIPTF/48TZMMBCVTDPOUFOUUIBUJTOPU
10ANA264 5 06 Fract
199
ces mn *
indi
nal
ctio
Fra
5-07
DPWFSFECZUIF"VTUSBMJBODVSSJDVMVN/48POMZ
A264
10AN
202
laws
x
5-08 Summary of the inde
DPOUFOUJTIJHIMJHIUFEJOorange XIJMF4UBHF
9NA209
204
res
figu
nt
ifica
5-09 Sign
NSW
DPOUFOUJTøNBSLFEXJUIBOBTUFSJTL 206
tion
nota
c
ntifi
Scie
5-10
9NA210
t 5IFSFBSFUISFFDPOUFOUTUSBOETJO.BUIFNBUJDT
a
on
tion
nota
tific
5 11 S i
NA/VNCFSBOE"MHFCSB
MG.FBTVSFNFOUBOE(FPNFUSZ
SP4UBUJTUJDTBOE1SPCBCJMJUZ
t &
BDIDIBQUFSCFHJOTXJUIBchapter outline
UIBUJODMVEFTUIF8PSLJOH.BUIFNBUJDBMMZ
QSPGJDJFODJFTDPWFSFEJOFBDITFDUJPO
U6OEFSTUBOEJOH
F 'MVFODZ
PS 1SPCMFNTPMWJOH
R3FBTPOJOH
C$PNNVOJDBUJOH
n Chapter outline
Proficiency strands
U F
e
1-01 Finding the hypotenus
U F
1-02 Finding a shorter side
R C
U F
ers*
1-03 Surds and irrational numb
R
U F
1-04 Simplifying surds*
R
g surds* U F
1-05 Adding and subtractin
ing
divid
and
g
iplyin
Mult
1-06
R
U F
surds*
1-07 Pythagoras’ theorem
C
PS
F
problems
1-08 Testing for right-angled
C
U F
triangles
C
U F
1-09 Pythagorean triads
'VSUIFSSFGFSFODFTUPUIF8PSLJOH.BUIFNBUJDBMMZQSPGJDJFODJFTDBOCFGPVOEJOUIFUFBDIJOH
QSPHSBN
Working mathematically explained
8IJMFUIFUISFFcontent strandsBSFUIFAOPVOTPGUIFNBUIFNBUJDTDVSSJDVMVN UIFGJWFWorking
Mathematically proficienciesBSFUIFAWFSCTUIFEPJOHBOEUIJOLJOHQSPDFTTFTUIBUHPIBOEJOIBOE
XJUIUIFDPOUFOUCFJOHUBVHIU
t Understanding JTALOPXJOHBOESFMBUJOHNBUIT*UJTNPSFUIBOKVTUMFBSOJOHGBDUT*UTEFFQ
VOEFSTUBOEJOH TFFJOHIPXNBUIFNBUJDBMDPOUFOUJTJOUFSDPOOFDUFE LOPXJOHAXIZBTXFMMBTAIPX
t F
luency JTABQQMZJOHNBUIT *UJTCFJOHBCMFUPVTFNBUIFNBUJDTDPNQFUFOUMZBOEFGGFDUJWFMZ8IFO
ZPVBSFGMVFOUJOBMBOHVBHF ZPVIBWFNBTUFSFEJUTPUIBUZPVDBOJNQSPWJTFBOEDPOGJEFOUMZVTF
UIFDPSSFDUXPSEPSQISBTF'MVFODZJONBUITJTDIPPTJOHBOBQQSPQSJBUFTLJMM NFUIPEPSGPSNVMB
UPVTFBUUIFSJHIUQMBDFBOEUJNF
t Problem solving JTANPEFMMJOHBOEJOWFTUJHBUJOHXJUINBUIT*UJOWPMWFTJOUFSQSFUJOHBSJDI FMBCPSBUFQSPCMFN TFMFDUJOHBOBQQSPQSJBUFTUSBUFHZPSNPEFM TPMWJOHUIFQSPCMFN UIFO
FWBMVBUJOH DPNNVOJDBUJOHBOEKVTUJGZJOHUIFTPMVUJPO
t Reasoning JTAHFOFSBMJTJOHBOEQSPWJOHXJUINBUIT VTJOHIJHIFSPSEFSUIJOLJOHUPDPOOFDUTQFDJGJD
GBDUTUPHFOFSBMQSJODJQMFT VTJOHBMHFCSB MPHJD QSPPGBOEKVTUJGJDBUJPO
t Communicating JTAEFTDSJCJOHBOEFYQMBJOJOHNBUIT SFQSFTFOUJOHNBUIFNBUJDBMUIFPSZBOE
TPMVUJPOTJOXPSET BMHFCSBJDTZNCPMT TQFDJBMOPUBUJPOT EJBHSBNT HSBQITBOEUBCMFT
SAMPLE CHAPTER
xii
9780170193085
9
NEW CENTURY MATHS ADVANCED
for the
In each chapter
A u s t ra l i a n C u r r i cu l u m
n In this chapter you will:
t
t
t
t
t
t
t
t
t
t W
ordbankJTBTBNQMFHMPTTBSZPGo
UFSNTGSPNUIFDIBQUFS
t 5IFSFJTBMTPBGVMMHMPTTBSZBUUIFCBDL
PGUIFCPPLJOUIJTCPPL BMMUFSNT
QSJOUFEJOredBQQFBSJOUIFHMPTTBSZ
t "VTUSBMJBO$VSSJDVMVNDPOUFOU
EFTDSJQUJPOT QSJOUFEJOCMVFBOE
NBSLFE"$ BSFMJTUFEBUUIF
CFHJOOJOHPGFBDIDIBQUFS
t SkillCheckBOEStartUp assignment
SFWJFXQSFSFRVJTJUFTLJMMTBOE
LOPXMFEHFGPSUIFDIBQUFS
BQQMZ JOEFY MBXT UP OVNFSJDBM
FYQSFTTJPOT XJUI JOUFHFS JOEJDF
T
TJNQMJGZ BMHFCSBJD QSPEVDUT BOE
RVPUJFOUT VTJOH JOEFY MBXT
FYQSFTT OVNCFST JO TDJFOUJmD
OPUBUJPO
JOUFSQSFU BOE VTF [FSP BOE OFHBUJ
WF JOEJDFT
45"(& JOUFSQSFU BOE VTF
GSBDUJPOBM JOEJDFT
SPVOE OVNCFST UP TJHOJmDBOU
mHVSFT
JOUFSQSFU XSJUF BOE PSEFS OVNC
FST JO TDJFOUJmD OPUBUJPO
JOUFSQSFU BOE VTF TDJFOUJmD OPUBUJP
O PO B DBMDVMBUPS
TPMWF QSPCMFNT JOWPMWJOH TDJFOUJ
mD OPUBUJPO
SkillCheck
Worksheet
StartUp assignment 5
MAT09NAWK10050
Worksheet
Powers review
MAT09NAWK10051
Skillsheet
Indices
MAT09NASS10020
1
'PS FBDI UFSN
i TUBUF UIF CBTF
ii TUBUF UIF JOEFY
iii XSJUF UIF FYQSFTTJPO JO XPSET
a 84
b 48
c h
2 &YQSFTT FBDI SFQFBUFE NVMUJQ
d h
MJDBUJPO JO JOEFY OPUBUJPO
a 232323232
b 33337373
c 333333
7
838
d 10 3 x 3 x 3 x 3 x 3 x
e 636363k3k
f x3y3x3y3x
g a3b3b3b3a
h 3n33n3n
i q3p3q3p3q3q
Summary
t * NQPSUBOUGBDUTBOEGPSNVMBTBSF
IJHIMJHIUFEJOBSummary CPY
Stage 5.3
See Example 10
Exercise 1-04
Generally, any number raised to the power of 1
n is the nth root of that number:
pffiffiffi
1
an ¼ n a
Simplifying surds
1
Simplify each expression.
pffiffiffi2
pffiffiffi2
a
pffiffiffi
2
b
5
c 3 3 2
pffiffiffiffiffiffiffiffiffi2
pffiffiffi
e
pffiffiffi2
0:09
f 2 7 2
g
3 5
See Example 11 2 Simpli
fy each surd.
pffiffiffi
pffiffiffiffiffi
a
pffiffiffiffiffi
8
b
27
pffiffiffiffiffiffiffi
ffi
c
24
pffiffiffiffiffi
e
pffiffiffiffiffi
243
f
45
pffiffiffiffiffi
g
pffiffiffiffiffi
48
i
pffiffiffiffiffiffiffi
96
ffi
j
63
pffiffiffiffiffi
k
288
pffiffiffiffiffiffiffi
ffi
m 75
pffiffiffiffiffi
n
147
pffiffiffiffiffiffiffiffi
o
32
pffiffiffiffiffiffiffi
ffi
q
p
162
ffiffiffiffiffiffiffi
ffi
r
245
s
125
3 Simplify each expression.
pffiffiffiffiffi
pffiffiffi
a 5 50
pffiffiffiffiffi
b
3
8
pffiffiffiffiffi
c 4 27
pffiffiffiffiffiffiffiffi
pffiffiffiffiffi
40
243
e
f
28
2
g
9
pffiffiffiffiffiffiffiffiffiffi
6
pffiffiffiffiffi
3125
i 9 68
pffiffiffiffiffi
j
1
k
10
2 72
pffiffiffiffiffiffiffiffi
pffiffiffiffiffi
m 10 160
pffiffiffiffiffi
n 3 75
o 7 68
4 Decide whether each statem
ent is true (T) or false (F).
pffiffiffi pffiffiffiffiffi
p
ffiffiffiffiffi
a 3 7 ¼ 21
pffiffiffiffiffiffiffi2
b
12 ¼ 6
pffiffiffiffiffi
c
9:4 ¼ 9:4
pffiffiffi
pffiffiffi
d
75 ¼ 5 3
e
3
17
t 1
BHFTDPOUBJOJOHStage 5.3 contentBSF
NBSLFECZBTIBEFENBSHJO
t (SBEFEFYFSDJTFTBSFMJOLFEUPXPSLFE
FYBNQMFTBOEJODMVEFNVMUJQMFDIPJDF
RVFTUJPOT FYBNTUZMFQSPCMFNTBOE
SFBMJTUJDBQQMJDBUJPOT
t 8PSLFETPMVUJPOTGPSTFMFDUFEFYFSDJTF
RVFTUJPOTBSFQSPWJEFE
pffiffiffiffiffi2
5 10
pffiffiffi
h 5 2 2
d
d
h
l
p
t
pffiffiffiffiffi
54
pffiffiffiffiffiffiffi
ffi
200
pffiffiffiffiffiffiffi
ffi
108
pffiffiffiffiffiffiffi
ffi
242
pffiffiffiffiffiffiffi
ffi
512
pffiffiffiffiffi
d 8 98
pffiffiffiffiffi
h 3 24
pffiffiffiffiffi
l 3 48
4
pffiffiffiffiffi
52
p
6
Investigation: A lifetime of heartbeats
t I nvestigationsFYQMPSFUIF
TZMMBCVTJONPSFEFUBJM UISPVHI
HSPVQXPSL EJTDPWFSZBOE
NPEFMMJOHBDUJWJUJFT
How many times does your heart beat in an average lifetime of 80 years?
1 Work in pairs and copy this table.
Name
Trial 1
Trial 2
Average beats per minute
2 Use two fingers to measure your pulse. Have your partner time you for a minute. Do this
twice, record your results in the table and find the average.
Just for the record
Hairy numbers
t J ust for the record DPOUBJOTJOUFSFTUJOHGBDUT
BOEBQQMJDBUJPOTPGUIFNBUIFNBUJDTMFBSOU
JOUIFDIBQUFS
SAMPLE CHAPTER
Straight hair
+ round follicle
Wavy hair
oval follicle
Curly hair
flat follicle
There are about 110 000 hairs on your head. Each hair grows at the rate of about 1.3 3 103 cm
per hour. A single hair lasts about six years. Every day you lose between 30 and 60 hairs. Each
hair grows from a small depression in the skin called a follicle (a gland) After the hair falls out
9780170193085
xiii
New Century Maths User’s guide
t T
echnology promotes ICT in the
classroom, using spreadsheets,
GeoGebra and the Internet
Mental skills 4
Technology Similar right-ang
led triangles
In this activity you will use GeoG
ebra to measure and calculate
trigonometric ratios.
1 a Before you start, set angle
s to measure in degrees. Click
Options, Rounding and
1 Decimal Place.
Maths without calculators
Estimating answers
t .
FOUBMTLJMMTSFJOGPSDFNFOUBM
calculation strategies (‘maths
XJUIPVUDBMDVMBUPST
A quick way of estimating an answer is to round each number in the calculation.
1
Study each example.
a 631 þ 280 þ 51 þ 43 þ 96 600 þ 300 þ 50 þ 40 þ 100
¼ ð600 þ 300 þ 100Þ þ ð50 þ 40Þ
¼ 1000 þ 90
¼ 1090 ðActual answer ¼ 1101Þ
NelsonNet resources
.BSHJOJDPOTMJOLUPQSJOU 1%' BOENVMUJNFEJBSFTPVSDFTGPVOEPOUIF/FMTPO/FUXFCTJUF www.nelsonnet.com.au. These include:
Worksheet
Venn diagrams
Puzzle sheet
Geometry crossword
Skillsheet
Indices
Homework sheet
Pythagoras’
theorem 1
Video tutorial
Worksheets
Puzzle sheets of matching
activities and crosswords
Skillsheets of examples and
exercises of prerequisite skills
and knowledge
Homework sheets for weekly
practice and revision, including
mental calculation, numeracy and
literacy questions
Negative indices
Worked solutions
Exercise 8-10
Technology worksheet
Video tutorials of worked
examples
Worked solutions of selected
exercise questions
Technology worksheets:
Excel
Graphing data
additional technology activities
Quiz
ExamView quizzes: interactive
Equations
and self-marking
&BDISFTPVSDFIBTBVOJRVFJEFOUJåFSDPEF'PSFYBNQMF UIFWJEFPUVUPSJBMA1ZUIBHPSBTUIFPSFN
has the code MAT09MGVT10001, which stands for Mathematics, Year 9, Measurement and
Geometry strand, Video Tutorial 10001.
SAMPLE CHAPTER
xiv
9780170193085
9
NEW CENTURY MATHS ADVANCED
for the
A u s t ra l i a n C u r r i cu l u m
At the end of each chapter
t Power plus is an extension/challenge exercise
t Language of maths has a chapter word list
and literacy questions
t Topic overview has reflection questions and an
incomplete mind map
t Chapter revision is a review exercise with links to
each exercise set of the chapter
t Mixed revision is a review exercise after every 3–4
chapters
n Language of maths
descending
fractional power
estimate
indices
quotient
negative power
reciprocal
power
index notation
product
scientific notation
significant figures
term
zero power
ascending
expanded form
base
index laws
exponent
index
1
1 What does a power of 2 mean?
‘power’?
2 Which two words from the list mean
key on a calculator used for?
or
3 What is the
with the same base?
Wh i h i d law for dividing terms
At the end of the book
t General revision exercise
t Instructional and Mathematical glossaries (in this book, words printed in red also appear in the
glossary)
t Answers and Index
NelsonNetBook
t N
elsonNetBook is the interactive digital version of this
textbook found on NelsonNet, containing margin icons that
link directly to NelsonNet resources
t 5PFBDIQBHFPG/FMTPO/FU#PPLZPVDBOBEEOPUFT WPJDF
and sound bites, highlighting, weblinks and bookmarks
t Zoom and Search functions
t $IBQUFSTDBOCFDVTUPNJTFEGPSEJGGFSFOUHSPVQTPGTUVEFOUT
SAMPLE CHAPTER
9780170193085
xv
Path: K:/CLA-NCM_09A-13-0303/Application/CLA-NCM_09A-13-0303005.3d
Date: 6th May 2013
Time: 16:32
User ID: elangok 1BlackLining
Disabled
Number and Algebra
5
Indices
The speed of light is about 300 000 000 metres per second.
In one year, light travels approximately 9 460 000 000 000 km.
Light from the stars travels for many years before it is seen on
Earth. Even light from the Sun takes eight minutes to reach
the Earth. Powers or indices provide a way to work easily with
very large and very small numbers.
SAMPLE CHAPTER
Path: K:/CLA-NCM_09A-13-0303/Application/CLA-NCM_09A-13-0303005.3d
Date: 6th May 2013
Time: 16:33
User ID: elangok 1BlackLining
Disabled
N E W C E N T U R Y M AT H S A D V A N C E D
for the A
n Chapter outline
n Wordbank
Proficiency strands
5-01 Multiplying and dividing
terms with the same base
5-02 Power of a power
5-03 Powers of products and
quotients
5-04 The zero index
5-05 Negative indices
1
5-06 Fractional indices *
n
m
5-07 Fractional indices *
n
5-08 Summary of the index laws
5-09 Significant figures
5-10 Scientific notation
5-11 Scientific notation on a
calculator
*STAGE 5.3
9780170193085
ustralian Curriculum
U F
U F
R C
R C
U F
U F
U F
R C
R C
R C
U F
R C
U F
R C
U F
U F
U F
R C
R C
R C
U F
PS R C
base A number that is raised to a power, meaning it is multiplied
by itself repeatedly, for example, in 25, the base is 2.
index laws Rules for simplifying algebraic expressions involving
powers of the same base, for example, am 4 an ¼ amn.
index notation A way of writing repeated multiplication using
indices (powers), in the form an, for example 2 3 2 3 2 3 2 3 2
in index notation is 25.
negative power A power that is a negative number, as in the
term 32.
power (or index or exponent) The number of times a base
appears in a repeated multiplication, for example, in 25, the
power is 5.
scientific notation A shorter way of writing very large or
very small numbers using powers of 10. For example,
9 460 000 000 000 in scientific notation is 9.46 3 1012.
SAMPLE CHAPTER
significant figures Meaningful digits in a numeral that tell ‘how
many’. For example, 28 000 000 has two significant figures:
2 and 8.
9
Path: K:/CLA-NCM_09A-13-0303/Application/CLA-NCM_09A-13-0303005.3d
Date: 6th May 2013
Time: 16:33
User ID: elangok 1BlackLining
Disabled
Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13
Indices
n In this chapter you will:
•
•
•
•
•
•
•
•
•
apply index laws to numerical expressions with integer indices
simplify algebraic products and quotients using index laws
express numbers in scientific notation
interpret and use zero and negative indices
(STAGE 5.3) interpret and use fractional indices
round numbers to significant figures
interpret, write and order numbers in scientific notation
interpret and use scientific notation on a calculator
solve problems involving scientific notation
SkillCheck
Worksheet
StartUp assignment 5
1
i state the base
ii state the index
iii write the expression in words.
MAT09NAWK10050
Worksheet
Powers review
MAT09NAWK10051
For each term:
2
b 48
c h5
a 84
Express each repeated multiplication in index notation.
a
c
e
g
i
Skillsheet
Indices
MAT09NASS10020
3
232323232
535353535353838
636363k3k
a3b3b3b3a
q3p3q3p3q3q
b 72
c d5
d k2
c ð33 Þ2
g 24 4 2
d 60
h ð8Þ2
c 216 ¼ 6?
g 64 ¼ 2?
d 144 ¼ 12?
h 625 ¼ 5?
Evaluate each expression.
a 42 3 43
e 91
5
3333333373737
10 3 x 3 x 3 x 3 x 3 x
x3y3x3y3x
53n353n3n
Write each term in expanded form.
a 93
4
b
d
f
h
d 5h
b 106 4 10 2
f 55 3 5
For each equation, find the missing power.
a 8 ¼ 2?
e 4096 ¼ 2?
b 81 ¼ 3?
f 2401 ¼ 7?
Investigation: Multiplying and dividing terms with powers
1 Write each expression in expanded form, then evaluate it.
a i 22 3 23
ii 25
b i 34 3 33
3
3
6
c i 4 34
ii 4
d i 55 3 53
2 What do you notice about each pair of answers in question 1?
3 Is it true that 24 3 26 ¼ 210? Give a reason for your answer.
SAMPLE CHAPTER
180
ii 37
ii 58
9780170193085
Path: K:/CLA-NCM_09A-13-0303/Application/CLA-NCM_09A-13-0303005.3d
Date: 6th May 2013
Time: 16:33
User ID: elangok 1BlackLining
Disabled
N E W C E N T U R Y M AT H S A D V A N C E D
for the A
ustralian Curriculum
9
4 Determine whether each equation is true (T) or false (F). Justify your answer.
b 63 3 67 ¼ 621
a 25 3 25 ¼ 210
3
9
27
c 4 34 ¼4
d 35 3 310 ¼ 315
5 Write in words and as a formula the rule for multiplying am and an, two terms with the
same base.
6 Use the rule to copy and complete each equation.
a 5 4 3 5 2 ¼ 5…
b 45 3 43 ¼ 4…
c 105 3 107 ¼ …
e n 3 3 n8 ¼ …
f p 3 3 p7 ¼ …
d 93 3 92 ¼ …
7 Evaluate each expression.
a i 36 4 33
ii 33
b i 28 4 26
ii 22
8
3
5
8
4
c i 5 45
ii 5
d i 10 4 10
ii 104
8 What do you notice about each pair of answers in question 7?
9 Is it true that 48 4 46 ¼ 42? Give a reason for your answer.
10 Determine whether each equation is true (T) or false (F). Justify your answer.
a 310 4 36 ¼ 34
b 48 4 42 ¼ 44
c 212 4 23 ¼ 24
d 610 4 65 ¼ 65
11 Write in words and as a formula the rule for dividing am and an, two terms with the same
base.
12 Use the rule to copy and complete each equation.
a 26 4 23 ¼ 2…
b 108 4 106 ¼ 10…
c 37 4 32 ¼
11
6
8
5
d 4 44 ¼…
e x 4x ¼…
f g12 4 g10 ¼ …
Multiplying and dividing terms with
5-01 the same base
Video tutorial
Simplifying with the
index laws
MAT09NAVT00002
Consider 5 4 3 5 3 ¼ ð5 3 5 3 5 3 5Þ 3 ð5 3 5 3 5Þ
¼ 5353535353535
¼ 57
) 5 4 3 5 3 ¼ 5 4þ3
¼ 57
Summary
SAMPLE CHAPTER
When multiplying terms with the same base, add the powers:
am 3 an ¼ amþn
9780170193085
181
Path: K:/CLA-NCM_09A-13-0303/Application/CLA-NCM_09A-13-0303005.3d
Date: 6th May 2013
Time: 16:33
User ID: elangok 1BlackLining
Disabled
Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13
Indices
The rule above is called an index law. Index is another name for power. The plural of index is
indices (pronounced ‘in-de-sees’).
Proof:
3a3
3ffla} 3 a|fflfflfflfflfflfflfflfflfflfflffl
3a3
3ffla}
a m 3 a n ¼ a|fflfflfflfflfflfflfflfflfflfflffl
ffl{zfflfflfflfflfflfflfflfflfflfflffl
ffl{zfflfflfflfflfflfflfflfflfflfflffl
m factors
n factors
¼ a|fflfflfflfflfflfflfflfflfflfflffl
3a3
3ffla}
ffl{zfflfflfflfflfflfflfflfflfflfflffl
ðm þ nÞ factors
¼ a mþn
Example
1
Simplify each expression, writing the answer in index notation.
a 84 3 85
b 10 3 103
c d3 3 d5
d 4m 2 3 3m6
e 3r 2t 3 6r4t 3
Solution
a 8 4 3 8 5 ¼ 8 4þ5
b 10 3 10 3 ¼ 101 3 10 3
¼ 89
d 3 3 d 5 ¼ d 3þ5
¼ 101þ3
d 4m 2 3 3m6 ¼ ð4 3 3Þ 3 ðm 2 3 m 6 Þ
Consider
c
¼ d8
¼ 104
e 3r 2 t 3 6rt 3 ¼ ð3 3 6Þ 3 ðr 2 3 r 1 Þ 3 ðt 1 3 t 3 Þ
¼ 12m 2þ6
¼ 18r 2þ1 t 1þ3
¼ 12m8
¼ 18r 3 t 4
6
56 4 54 ¼ 54
5
653653653653535
¼
65365365365
¼ 535
¼ 52
6
4
) 5 4 5 ¼ 5 64
¼ 52
Summary
When dividing terms with the same base, subtract the powers:
m
a m 4 a n ¼ aa n ¼ a mn
This is another index law.
m
a m 4 a n ¼ aa n
Proof:
6a36a3a3a3a3 3a
¼
6a36a3a3 3a
ðm factors)
ðn factors)
½ðm nÞ factors]
SAMPLE CHAPTER
¼ a3a3 3a
¼ a mn
182
9780170193085
Path: K:/CLA-NCM_09A-13-0303/Application/CLA-NCM_09A-13-0303005.3d
Date: 6th May 2013
Time: 16:33
User ID: elangok 1BlackLining
Disabled
N E W C E N T U R Y M AT H S A D V A N C E D
for the A
Example
ustralian Curriculum
9
2
Simplify each expression, writing the answer in index notation.
8
a 8 5 4 83
b 10
10
8x 3 y7
d 20w 10 4 5w 2
e
24x 2 y
c d 20 4 d 4
Solution
a 8 5 4 83 ¼ 853
b
¼ 82
20w 10
2
16 5w
¼ 4w 102
d 20w 10 4 5w 2 ¼
4
e
10 8 ¼ 1081
10
¼ 107
d 20 4 d 4 ¼ d 204
c
¼ d 16
1
xy 6
8x 3 y 7
6 8x 32 y 71
¼
¼
3
24x 2 y1
3 24
¼ 4w 8
Exercise 5-01
Multiplying and dividing terms
with the same base
1
Which expression is equal to 512 3 53? Select the correct answer A, B, C or D.
B 515
C 2515
D 2536
A 59
2
Simplify each expression, writing the answer in index notation.
a
d
g
j
m
p
103 3 102
74 3 7
6 3 62 3 63 3 64
x 3 x4
b3 3 b10
y 3 y3 3 y2
b
e
h
k
n
q
2 3 24
8 3 83 3 8 4
44 3 44 3 44
g4 3 g4
p10 3 p10
m 3 3 m 3 m4
c
f
i
l
o
r
32 3 35
54 3 5 3 54
34 3 30 3 37
w7 3 w
r3r
n8 3 n2
3
Which expression is equal to 104 3 10? Select the correct answer A, B, C or D.
B 100 4
C 10 4
D 10 5
A 100 5
4
Simplify each expression.
a 3p2 3 2p5
d h3 3 5h8
g 5n8t 3 6n8t 4
j 8p4m5 3 4p3m5
5
4y10 3 3y 2
3q 3 8q8
2ab3 3 15ab
16qr8 3 3q7
c
f
i
l
6m 3 3m8
2a 2 3 5a 5
3e4g 3 3 e6g 2
9u 3v 3 6uv 2w8
Which expression is equal to 512 4 5 3 ? Select the correct answer A, B, C or D.
4
A 5
6
b
e
h
k
9
4
B 5
C 1
See Example 1
See Example 2
9
D 1
Simplify each expression, writing the answer in index notation.
a 107 4 10 5
8
d 52
5
g 74 4 73
9780170193085
b 85 4 8
12
e 93
9
20
h 2
2
c 2015 4 20 5
27
f 23
2
SAMPLE CHAPTER
i 114 4 114
183
Path: K:/CLA-NCM_09A-13-0303/Application/CLA-NCM_09A-13-0303005.3d
Date: 6th May 2013
Time: 16:33
User ID: elangok 1BlackLining
Disabled
Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13
Indices
j p15 4 p10
k n7 4 n
l
20
m h4
h
n
y8
y2
25
q ww
12
o a4
a
p b16 4 b15
w 24 4 w6
r m16 4 m16
7
Which expression is equal to 10 4 4 10? Select the correct answer A, B, C or D.
B 14
C 13
D 10 3
A 10 4
8
Simplify each expression.
a 10y15 4 5y 3
b 20w 9 4 4w 3
c 24r 8 4 3r
4
d 30x3
x
10
e 10m
2m
f
g 14d 4h10 4 7hd 2
h 15x6y8 4 15xy 4
i 6e 25d 40 4 18e 5d 4
10 8
k 45a 5b
5a
l
j
12q 5 t 4
16q 4 t 3
4g12
8g 6
36pq3 r5
24qr
Investigation: Powers of powers
1 Write each expression in expanded form, then evaluate it.
a i (23)2
ii 26
b i (34)3
ii 312
2 3
6
5 4
c i (5 )
ii 5
d i (2 )
ii 220
2 What do you notice about each pair of answers in question 1?
3 Is it true that: (27)3 ¼ 221? Give a reason for your answer.
4 Determine whether each equation is true (T) or false (F). Justify your answer.
a (35)3 ¼ 315
b (23)2 ¼ 25
c (210)4 ¼ 214
2 5
10
3 6
18
d (4 ) ¼ 4
e (3 ) ¼ 3
f (52)4 ¼ 56
5 Write in words and as a formula the rule for raising am to a power of n, that is, (am)n.
6 Use the rule to copy and complete each equation.
a (37)2 ¼ 3…
b (52)6 ¼ 5…
c (45)2 ¼ 4…
3 4
3 7
d (a ) ¼ a…
e (8 ) ¼ …
f (k4)6 ¼ …
Puzzle sheet
Indices puzzle
5-02 Power of a power
MAT09NAPS10053
Consider
Video tutorial
Simplifying with the
index laws
MAT09NAVT00002
ð5 3 Þ4 ¼ 5 3 3 5 3 3 5 3 3 5 3
¼ ð5 3 5 3 5Þ 3 ð5 3 5 3 5Þ 3 ð5 3 5 3 5Þ 3 ð5 3 5 3 5Þ
¼ 53535353535353535353535
¼ 512
) ð53 Þ4 ¼ 53 3 4
SAMPLE CHAPTER
¼ 512
184
9780170193085
Path: K:/CLA-NCM_09A-13-0303/Application/CLA-NCM_09A-13-0303005.3d
Date: 6th May 2013
Time: 16:33
User ID: elangok 1BlackLining
Disabled
N E W C E N T U R Y M AT H S A D V A N C E D
for the A
ustralian Curriculum
9
Summary
When raising a term with a power to another power, multiply the powers:
ða m Þn ¼ a m 3 n
Proof:
ða m Þn ¼ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl
a m 3 a m ffl3
3 afflm}
{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl
n factors
¼ a|fflfflfflfflfflfflfflfflfflfflffl
3a3
3ffla} 3 a|fflfflfflfflfflfflfflfflfflfflffl
3a3
3ffla} 3 |fflfflfflfflfflfflfflfflfflfflfflfflfflffl
3 a 3 affl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl
3 3ffla}
ffl{zfflfflfflfflfflfflfflfflfflfflffl
ffl{zfflfflfflfflfflfflfflfflfflfflffl
m factors
m factors
m factors
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
n lots of m factors
¼ am3n
Example
3
Simplify each expression, writing the answer in index notation.
a (8 5)2
d (5v4)3
b (d 3)5
e (n)6
c (2g)4
f (3t 4)3
a ð8 5 Þ2 ¼ 25 3 2
¼ 210
b ðd 3 Þ5 ¼ d 3 3 5
¼ d 15
c
ð2gÞ4 ¼ 24 3 g 4
¼ 16g 4
d ð5v 4 Þ3 ¼ 5 3 3 ðv 4 Þ3
¼ 125 3 v 4 3 3
¼ 125v12
e ðnÞ6 ¼ ð1Þ6 3 n6
¼ 1 3 n6
¼ n6
f
ð3t 4 Þ3 ¼ ð3Þ3 3 ðt 4 Þ3
¼ 27 3 t 4 3 3
¼ 27t 12
Solution
Exercise 5-02
1
Which expression is equal to (10 3)3? Select the correct answer A, B, C or D.
A 30 3
2
3
B 100
C 109
See Example 3
D 106
Simplify each expression, writing the answer in index notation.
a (43)2
g (100)2
b (52)8
h (64)5
c (33)4
i (53)5
d (27)4
j (e 2)4
e (21)2
k (t 5)5
f (9)3
l (y 3)7
m (c1)5
s (2x)10
n (m7)5
t (5n3)8
o (y4)4
u (4d 3)3
p (h0)6
v (k 5)9
q (q6)3
w (d 3)4
r (w4)1
x (2a8)8
Which expression is equal to (3)5? Select the correct answer A, B, C or D.
A 36
4
Power of a power
B 35
C 35
D 15
Simplify each expression.
a (2d 3)4
g (10d 5)4
m (2r)4
s (4w5)4
9780170193085
b (5m 3)2
h (3e)3
c (4y5)2
i (2b4)1
d (3x2)4
j (6d 6)2
e (5u6)5
k (3f 4)5
f (2w5)3
l (2c3)10
n (5t)3
o (3m 3)2
p (y 3)12
q (x)3
r (m 3)10
t (3f )5
u (3p2)3
v (3h5)4
w (10k)2
x (8y 3)1
SAMPLE CHAPTER
185
Path: K:/CLA-NCM_09A-13-0303/Application/CLA-NCM_09A-13-0303005.3d
Date: 6th May 2013
Time: 16:33
User ID: elangok 1BlackLining
Disabled
Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13
Indices
Video tutorial
Simplifying with the
index laws
MAT09NAVT0002
5-03 Powers of products and quotients
Consider
ð2 3 5Þ3 ¼ ð2 3 5Þ 3 ð2 3 5Þ 3 ð2 3 5Þ
¼ 23232353535
Homework sheet
¼ 23 3 5 3
Indices 1
) ð2 3 5Þ3 ¼ 23 3 5 3
MAT09NAHS10005
Summary
When raising a product of terms to a power, raise each term to that power:
ðabÞn ¼ a n bn
Proof:
ðabÞn ¼ ab
3 ab 3 3 ab
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
n factors
¼ a|fflfflfflfflfflfflfflfflfflfflffl
3a3
3ffla} 3 b|fflfflfflfflfflfflfflfflfflfflffl
3b3
3fflb}
ffl{zfflfflfflfflfflfflfflfflfflfflffl
ffl{zfflfflfflfflfflfflfflfflfflfflffl
n factors
¼ an bn
Example
n factors
4
Simplify each expression.
a (2gh2)5
b (p3q4)2
Solution
a ð2ghÞ5 ¼ ð2Þ5 3 g 5 3 ðh2 Þ5
Consider
b ð p 3 q 4 Þ2 ¼ ð p 3 Þ2 3 ðq 4 Þ2
¼ 32 3 g 5 3 h 2 3 5
¼ p 3 3 2 3 q4 3 2
¼ 32g 5 h10
¼ p6 q8
6
5 ¼53535353535
8
8
8
8
8
8
8
5
3
5
3
5
3
5
3
5
3
5
¼
83838383838
6
5
¼ 6
8
6
5
56
)
¼ 6
8
8
SAMPLE CHAPTER
186
9780170193085
Path: K:/CLA-NCM_09A-13-0303/Application/CLA-NCM_09A-13-0303005.3d
Date: 6th May 2013
Time: 16:33
User ID: elangok 1BlackLining
Disabled
N E W C E N T U R Y M AT H S A D V A N C E D
for the A
ustralian Curriculum
9
Summary
When raising a quotient of terms to a power, raise each term to that power:
n
a ¼ an
b
bn
n
a
a
a
a
¼ 3 3 3
b
b|fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}
b
b
n factors
a 3 a 3 3 a ðn factorsÞ
¼
b 3 b 3 3 b ðn factorsÞ
n
a
¼ n
b
Proof:
Example
5
Simplify each expression.
2
7c
a
d
b
4k 2
5
3
Solution
a
7c
d
2
ð7cÞ2
¼ 2
d
72 c 2
¼ 2
d
2
49c
¼ 2
d
b
4
¼
ð4k 2 Þ3
53
43 ðk 2 Þ3
125
6
¼ 64k
125
Powers of products and quotients
B 16 3 25
Simplify each expression.
b (x 2y)5
a (ab)3
g (ek 3)3
3
3
Which expression is equal to ð4 3 5Þ2 ? Select the correct answer A, B, C or D.
A 16 3 25
2
4k 2
5
¼
Exercise 5-03
1
c (l 3m 5)6
C 8 3 10
d (6dp2)4
See Example 4
D 8 3 10
e (8k 4y 5)2
f (3m 2n)5
h (w 3x4)7
i (8d 3y 5)2
j (4b2c 3)4
k (3a 3d)3
l (2p2q3)4
3
Which expression is equal to 3 ? Select the correct answer A, B, C or D.
4
9
9
B
C 27
D 27
A 12
12
64
64
Simplify each expression.
5
4
2
3
4
5 8
2
6
m
5
2n
w
m2
a
b
c
d
e
f
3
p
x
7
2
4n
t
4
2
4
3
7k 4
3r 2
a2 b
2 5
g 2
h 5h
i
j
k
l
3
10
6
t2
d5
3c 2
See Example 5
SAMPLE CHAPTER
9780170193085
187
Path: K:/CLA-NCM_09A-13-0303/Application/CLA-NCM_09A-13-0303005.3d
Date: 6th May 2013
Time: 16:33
User ID: elangok 1BlackLining
Disabled
Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13
Indices
Worked solutions
Exercise 5-03
MAT09NAWS10503
5
Simplify each expression.
a (2x10y15)3 3 5x 2y 3
c 18q5r 8 4 (3qr 2)2
d (18q5r 8 4 3qr 2)2
b (2x10y15 3 5x 2y 3)3
3
3a 5 x6
e
ax
g (4p3h10)2 3 2p2h9
h (4p3h10)2 4 2p2h9
i (4p3h10 3 2p2h9)2
f
3a 5 x6
ðaxÞ4
Investigation: The power of zero
What is the value of a number raised to a power of 0, for example, 20?
1 Copy and complete each table of decreasing powers. Notice the pattern in your answers.
a
b
Power of 2 Number
Power of 3 Number
25
35
32
243
4
2
34
16
33
23
2
2
32
1
2
31
20
30
2 Simplify each expression in index notation.
a 34 3 30
b 52 3 50
c 20 3 27
d 70 3 73
e 45 3 40
f 50 3 57
5
0
5
0
2
0
3
0
6
0
g 2 42
h 3 43
i 4 44
j 9 49
k 5 45
l 84 4 80
3 Any number will remain unchanged when multiplied by what?
4 Any number will remain unchanged when divided by what?
5 What is the answer when any number is raised to the power of 0, that is, a0? Justify your
answer.
5-04 The zero index
3
5
Consider 5 3 4 5 3 ¼ 3
5
¼ 1 Any number divided by itself equals 1.
But also
5 3 4 5 3 ¼ 533
¼ 50
) 50 ¼ 1
Summary
Any number raised to the power of zero is equal to 1.
SAMPLE CHAPTER
a0 ¼ 1
188
9780170193085
Path: K:/CLA-NCM_09A-13-0303/Application/CLA-NCM_09A-13-0303005.3d
Date: 6th May 2013
Time: 16:33
User ID: elangok 1BlackLining
Disabled
N E W C E N T U R Y M AT H S A D V A N C E D
for the A
am 4 am ¼ 1
Proof:
m
m
a 4a ¼ a
But also
ustralian Curriculum
9
Any number divided by itself equals 1.
mm
¼ a0
) a0 ¼ 1:
Example
6
Simplify each expression.
a 110
d (3r)0
b (8)0
e 3r 0
c g0
f 80
a 110 ¼ 1
b (8)0 ¼ 1
c g0 ¼ 1
d (3r)0 ¼ 1
e 3r 0 ¼ 3 3 r 0
f
Solution
80 ¼ 1 3 80
¼ 331
¼3
Exercise 5-04
1
2
¼ 1 3 1
¼ 1
The zero index
Simplify each expression.
See Example 6
a 20
b (2)0
c 20
0
2
g
3
0
p
k
3
d (m)0
e m0
f ð4aÞ0
i 10000
j ðp þ 3Þ0
m (9k)0
n (x 2y)0
o (xyw)0
p (ab)0
q (6r)0
r (6r)0
s 6r 0
t 6(r)0
u (cd)0
v (7x 2)0
w 3(a 2b3)0
x (5v 5w4)0
c 2m0 þ (2m)0
d 2m0 (2m)0
Exercise 5-04
0
0
MAT09NAWS10504
h 7x0
l
2b0
Simplify each expression.
a 70 þ 20
0
Worked solutions
b 70 20
0
0
0
e (6a) þ 6a
f (6a) 6x
i 30 3 50
j 32 3 50
m 2w0 3 3p0
q
12p0
ð2pÞ0
0
g (5y) 4
0
1
1
k
þ y0
2
2
h (5y) 4
0 0
1
1
y
l
þ
2
2
n 12u0 4 3
o (5d 0)3
p 8b0 (3b0)2
r 6n3 4 2n3
s
12q 5
36q 5
t ð3x 3 Þ3 4 x9
SAMPLE CHAPTER
9780170193085
189
Path: K:/CLA-NCM_09A-13-0303/Application/CLA-NCM_09A-13-0303005.3d
Date: 6th May 2013
Time: 16:33
User ID: elangok 1BlackLining
Disabled
Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13
Indices
Mental skills 4
Maths without calculators
Adding or multiplying in any order
Numbers can be added or multiplied in any order.
We can use this property to make our calculations simpler.
1
Study each example.
a 19 þ 5 þ 5 þ 1 ¼ ð19 þ 1Þ þ ð5 þ 5Þ
¼ 20 þ 10
¼ 30
b 13 þ 8 þ 20 þ 27 þ 80 ¼ ð13 þ 27Þ þ ð20 þ 80Þ þ 8
¼ 40 þ 100 þ 8
¼ 148
c
2 3 36 3 5 ¼ ð2 3 5Þ 3 36
¼ 10 3 36
¼ 360
d 25 3 11 3 4 3 7 ¼ ð25 3 4Þ 3 ð11 3 7Þ
¼ 100 3 77
¼ 7700
2
3
Now evaluate each sum.
a 45 þ 16 þ 45 þ 4 þ 7
c 18 þ 91 þ 9 þ 20
e 24 þ 16 þ 80 þ 44 þ 10
g 100 þ 36 þ 200 þ 10 þ 90
i 70 þ 50 þ 30 þ 25 þ 25
Now evaluate each product.
a 83435
d 5 3 11 3 40
g 3 3 20 3 7 3 5
b
d
f
h
j
38 þ 600 þ 50 þ 12 þ 40
75 þ 33 þ 7 þ 25
56 þ 5 þ 20 þ 15 þ 4
54 þ 27 þ 9 þ 16 þ 3
32 þ 120 þ 40 þ 80 þ 40
b 50 3 7 3 2
e 12 3 2 3 3
h 6383532
c 33536
f 2 3 4 3 25 3 8
i 2 3 3 3 2 3 11
Investigation: Negative powers
What is the value of a number raised to a negative power, for example, 21 or 22?
1 Copy and complete each table showing decreasing powers. Notice the pattern in your answers.
a
Power of 2
23
22
21
20
21
22
23
Number
8
4
b
Power of 10
103
102
101
100
101
102
103
Number
1000
SAMPLE CHAPTER
190
9780170193085
Path: K:/CLA-NCM_09A-13-0303/Application/CLA-NCM_09A-13-0303005.3d
Date: 6th May 2013
Time: 16:33
User ID: elangok 1BlackLining
Disabled
N E W C E N T U R Y M AT H S A D V A N C E D
for the A
ustralian Curriculum
9
2 Copy and complete this table showing decreasing powers in expanded form. Notice the
pattern in your answers.
a
Power of 5
33
32
31
30
31
32
33
Expanded form
33333
333
3
1
1
3
1 ¼ 1
3 3 3 32
1
¼ 1
3 3 3 3 3 33
b
Power of 5
53
52
51
50
51
52
53
54
55
Expanded form
53535
34
35
3 If 32 ¼ 12 and 53 ¼ 13 , then write each negative power in a similar way.
3
5
1
a 4
b 74
c 26
4 Simplify each expression in index notation.
a 104 4 107
5 Consider
b 23 4 28
c 34 4 35
d 52 4 58
e a4 4 a6
f a 4 a4
10 3 10 3 10 3 10
104 ¼
107 10 3 10 3 10 3 10 3 10 3 10 3 10
1
¼
10 3 10 3 10
¼ 13
10
104
47
But also 107 ¼ 10
¼ 103
) 103 ¼ 1 3
10
Use the method above to show that:
3
4
2
4
c 58 ¼ 52 ¼ 16
a 28 ¼ 25 ¼ 15
b 35 ¼ 31 ¼ 1
d a6 ¼ a2 ¼ 12
3
2
a
2
3
5
5
a
6 Write in words and as a formula the rule for raising a to a negative power n, that is, an.
Technology Negative powers
In this activity we will discover the pattern for negative powers. We will consider base values
from 2 to 10 shown in column A and indices (powers) 1, 2 and 3 shown in row 1. On a
spreadsheet, the symbol for power is ^ (called a carat, press SHIFT 6). For example, 31 is
entered as 3^ 1.
SAMPLE CHAPTER
9780170193085
191
Path: K:/CLA-NCM_09A-13-0303/Application/CLA-NCM_09A-13-0303005.3d
Date: 6th May 2013
Time: 16:33
User ID: elangok 1BlackLining
Disabled
Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13
Indices
1 Create a spreadsheet as shown below.
2 We will first examine the power of 1. In cell B4, enter 5A4^ $B$1 to calculate 21.
$B$1 is an absolute cell reference, which ensures that the cell does not change when
a formula is copied. This means that in column B, the power will always refer to cell
B1 (1) only. Fill Down from cell B4 to B12.
3 Use Format cells to set column B decimals to Fraction and Up to three digits.
4 Compare your answers in column B with the original values in column A. Can you describe
the pattern when a base is raised to a power of 1?
5 Now consider powers of 2. Adapt steps from 1 to 3 for column C. Use Fill Down from
cell C4 to C12.
6 Compare your answers in column C with the original values in column A. Can you describe
the pattern when a base is raised to a power of 2?
SAMPLE CHAPTER
7 Now consider powers of 3. Adapt steps for column D. In cell D4, enter the formula
5A4^ $D$1.
Note: D12’s fraction is missing as it has 4 digits in the denominator, which the spreadsheet
doesn’t allow for. Can you figure out what the fraction should be?
192
9780170193085
Path: K:/CLA-NCM_09A-13-0303/Application/CLA-NCM_09A-13-0303005.3d
Date: 6th May 2013
Time: 16:33
User ID: elangok 1BlackLining
Disabled
N E W C E N T U R Y M AT H S A D V A N C E D
for the A
ustralian Curriculum
9
8 Compare your answers in column D with the original values in column A. Can you describe
the pattern when a base is raised to a power of 3?
9 Write a rule for negative powers, given the answers you have found in this activity. Discuss
with other students in your class.
Worksheet
5-05 Negative indices
Power calculations
MAT09NAWK10056
0
2
Consider 20 4 23 ¼ 3
2
¼ 13
2
Video tutorial
Negative indices
MAT09NAVT10010
But also 20 4 23 ¼ 203
¼ 23
¼ 13
2
) 23
Summary
A number raised to a negative power gives a fraction (with a numerator of 1):
an ¼ a1n
Proof:
But also
0
a0 4 a n ¼ aa n
¼ a1n
a0 4 a n ¼ a0n
) an
Example
¼ an
¼ a1n
7
Simplify each expression using a positive index (power).
a 53
b 3n2
c ð3nÞ2
b 3n2 ¼ 3 3 n2
¼ 3 3 12
1
n
3
¼ 2
n
c
d p2q3
Solution
a 53 ¼ 13
5
1
ð3nÞ2
¼ 12
9n
ð3nÞ2 ¼
1
2 3
3
d p q ¼ p2 3 q
q3
¼ 2
p
SAMPLE CHAPTER
9780170193085
193
Path: K:/CLA-NCM_09A-13-0303/Application/CLA-NCM_09A-13-0303005.3d
Date: 6th May 2013
Time: 16:33
User ID: elangok 1BlackLining
Disabled
Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13
Indices
The reciprocal as a power
Consider 91 ¼
1
9
1 is the reciprocal of 9.
9
1
2
¼ 1 Consider 3
2
3
¼ 14 2
3
¼ 13 3
2
3
¼
2
¼ 11
2
Summary
A number raised to a power of 1 gives its reciprocal.
a1 ¼ 1a
1
a
¼ ba
b
Example
8
Simplify each expression.
1
4
a
3
b
1
y
5
b
1
y
5
¼y
5
Solution
a
Stage 5.3
1
4
3
¼
3
4
Negative powers of quotients
2
4
¼ 12
Consider 5
4
5
1
¼
16
25
¼ 1 4 16
25
SAMPLE CHAPTER
194
9780170193085
Path: K:/CLA-NCM_09A-13-0303/Application/CLA-NCM_09A-13-0303005.3d
Date: 6th May 2013
Time: 16:33
User ID: elangok 1BlackLining
Disabled
N E W C E N T U R Y M AT H S A D V A N C E D
for the A
ustralian Curriculum
¼ 1 3 25
16
25
¼
16
2
¼ 52
4
2
¼ 5
4
9
Stage 5.3
Summary
A number raised to a power of –n gives its reciprocal raised to the power of n.
n n
n
a
¼ ba ¼ ba n
b
Proof:
n
a
¼ 1n
a
b
b
1
¼ n
a
bn
bn
¼ an
n
¼ ba
Example
9
Simplify each expression.
3
4
a
3
b
Solution
a
3 3
4
¼ 3
3
4
27
¼
64
b
21
2
2
2 2
21
¼ 5
2
2
2
2
¼
5
4
¼
25
c
c
3a 2
b4
2 4 2
3a
¼ b
3a
b4
8
¼ b2
9a
SAMPLE CHAPTER
9780170193085
195
Path: K:/CLA-NCM_09A-13-0303/Application/CLA-NCM_09A-13-0303005.3d
Date: 6th May 2013
Time: 16:33
User ID: elangok 1BlackLining
Disabled
Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13
Indices
Exercise 5-05
See Example 7
1
Simplify each expression using a positive index.
a 62
e g5
i a4
2
3
Worked solutions
4
Exercise 5-05
MAT09NAWS10505
See Example 8
Stage 5.3
See Example 9
5
6
Negative indices
b 57
f z1
j 53
c 31
g n3
k yd
d 102
h t2
l rm
Evaluate each expression, giving your answers in fraction form.
b 54
c 61
a 32
1
7
e 25
f 2
g 43
10
3
i 2
j 3
k 62
d 72
h 106
l 94
Write each expression using a negative index.
a 12
b 1n
c 13
n
8
1
2
1
e
f
g
3
105
a4
i 6a
k 25
j 42
t
w
d 1
8
h 1
b
l 53
d
Simplify each expression using positive indices.
b 2b5
c 3e3
a 5h1
d 4n2
e pb2
f r 2s4
g w2y
h d 3y 3
i (2m)1
j
k (4h)2
l
m 3m 3p2
n 15k1w4
o 12x2y3
p 12x2y 3
q (3h)2
r (4k)3
s (2c)4
t (8y)1
u 4pq3
v 4p1q3
w vm2
x v1m2
(xy)1
Simplify each expression.
1
1
b 8
a 2
7
5
1
1
5
e 3
f
4
2
1
1
m
5r
i
j
2
4
Simplify each expression.
2
2
b 2
a 1
4
3
5
4
4
5
e
f
3
4
2
3
k
3
i
j
x
3
2
2 5
2d
h
m
n
5t
m3
9 1
10
1
g x
3
1
2
k
3z
c
1 6
10
2
1
g 2
4
4
a2
k
4
3 2
5d
o
3p4
c
(5k)3
1
3
2
1
h 5a
1
1
l
v
d
3
5
2
3
2
h 1
5
2
4
l
3g3
3 3
3c
p
4a 2
d
SAMPLE CHAPTER
196
9780170193085
Path: K:/CLA-NCM_09A-13-0303/Application/CLA-NCM_09A-13-0303005.3d
Date: 6th May 2013
Time: 16:33
User ID: elangok 1BlackLining
Disabled
N E W C E N T U R Y M AT H S A D V A N C E D
for the A
ustralian Curriculum
1
5-06 Fractional indices n
9
Stage 5.3
We know now the meaning of zero and negative indices, that is, a0, a1 and an.
1
1
What is the meaning of fractional indices, that is, a 2 and a n ?
1 2
1
Consider 25 2 ¼ 25 2 3 2
Power of a power
1
¼ 25
¼ 25
pffiffiffiffiffi 2
but
25 ¼ 25
pffiffiffiffiffi
1
) 25 2 ¼ 25 ¼ 5
Summary
Any number raised to the power of 1 is the square root of that number:
2
pffiffiffi
1
a2 ¼ a
1 2
1
a2 ¼ a232
¼ a1
pffiffiffi 2 ¼ a
But ð aÞ ¼ a
pffiffiffi
1
) a2 ¼ a
1 3
1
Now consider 27 3 ¼ 273 3 3
¼ 271
ffiffiffiffiffi 3 ¼ 27
p
3
but
27 ¼ 27
pffiffiffiffiffi
1
) 27 3 ¼ 3 27 ¼ 9
Proof:
Power of a power
Summary
Any number raised to the power of 1 is the cube root of that number:
3
pffiffiffi
1
a3 ¼ 3 a
1 3
1
a3 ¼ a333
¼ a1
pffiffiffi 3 ¼ a
But ð 3 aÞ ¼ a
1
pffiffiffi
) a3 ¼ 3 a
1 5
1
Now consider 32 5 ¼ 32 5 3 5
¼ 321
¼ 32
Proof:
Power of a power
SAMPLE CHAPTER
9780170193085
197
Path: K:/CLA-NCM_09A-13-0303/Application/CLA-NCM_09A-13-0303005.3d
Date: 6th May 2013
Time: 16:33
User ID: elangok 1BlackLining
Disabled
Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13
Indices
Stage 5.3
1 5
pffiffiffiffiffi
1
If 32 5 ¼ 32, then 32 5 is called the 5th root of 32, written 5 32.
25 ¼ 32
pffiffiffiffiffi
1
) 32 5 ¼ 5 32 ¼ 2
Summary
Generally, any number raised to the power of 1n is the nth root of that number:
pffiffiffi
1
an ¼ n a
Proof:
1 n
1
an ¼ an 3 n
¼ a1
¼a
p
ffiffi
ffi
n
But ð n aÞ ¼ a
1
pffiffiffi
) an ¼ n a
Example
10
Evaluate each expression.
1
1
a 900 2
Solution
1
a 900 2 ¼
pffiffiffiffiffiffiffiffi
900
1024
1
10
c 1024 10
1
b 125 3 ¼
¼ 30
c
1
b 125 3
ffiffiffiffiffiffiffiffi
p
3
125
¼5
pffiffiffiffiffiffiffiffiffiffi
¼ 10 1024
Enter on calculator: 10
¼2
10
because 2
3
1024
=
¼ 1024
Summary
On a calculator, the nth root key is 3
or
before pressing
or yx respectively.
Example
c
ffiffiffi
p
4
n
d
SAMPLE CHAPTER
Solution
198
SHIFT
pffiffiffi
1
8 ¼ 82
or
2ndF
key
11
Write each expression using a fractional index.
pffiffiffiffiffi
pffiffiffi
a
8
b 3 36
a
, found by pressing the
b
p
ffiffiffiffiffi
1
3
36 ¼ 36 3
c
1
ffiffiffi
p
4
n ¼ n4
d
ffiffiffiffiffi
p
7
ab
ffiffiffiffiffi
p
1
7
ab ¼ ðabÞ7
9780170193085
Path: K:/CLA-NCM_09A-13-0303/Application/CLA-NCM_09A-13-0303005.3d
Date: 6th May 2013
Time: 16:33
User ID: elangok 1BlackLining
Disabled
N E W C E N T U R Y M AT H S A D V A N C E D
for the A
Exercise 5-06
1
1
a 25
1
2
e 32
1
5
See Example 10
b 343
1
3
c 625
f ð0:027Þ
1
1
3
1
2
d 1000
1
2
1
g ð0:04Þ
1
h 64 3
1
j ð729Þ3
1
3
k 256 8
l
1
3125 5
Write each expression using a radical (root) sign.
1
1
1
b 12 3
c g2 1
a 10 2
1
1
e ð8rÞ2
f ð6hÞ6
g ð5j 8 Þ5
d m4
1
h 90ab 9
3
Write each expression using a fractional index.
p
pffiffiffi
ffiffiffiffiffi
3
c
a p5ffiffiffiffiffi
b p
49
ffiffiffiffiffi
e 6 66
f 4 64
g
pffiffiffi
pffiffiffi
k
i
a
j 3q
pffi
pffiffiffiffiffi
m 5t
n
xy
o
pffiffiffiffiffi
20
p
ffiffiffiffiffiffiffiffi
8
144
ffiffiffi
p
7
h
ffiffiffiffiffiffiffiffiffiffi
p
4
100f
pffiffiffiffiffiffiffiffi
d 5p400
ffiffiffiffiffiffiffiffiffiffi
h 10 1000
p
ffiffiffi
ffi
6
l p
w
ffiffiffiffiffiffiffiffi
ffi
p 3 2mn
4
Evaluate each expression correct to 2 decimal places.
p
ffiffiffiffiffiffiffiffi
1
1
3
2
a 20 3
b 215
c p
144
p
ffiffiffiffiffiffiffiffiffiffi
ffiffiffiffiffiffiffiffiffiffiffi
1
g 5 754
f 4 1111
e ð666Þ3
pffiffiffiffiffiffiffiffiffiffi
d p2001
ffiffiffiffiffiffiffiffiffiffiffi
h 6 0:008
5
Simplify each expression.
1
1
1
1
1
b e3 3 e 33 e 3
a b2 3 b2
1
3
2
3
1
9 3
e 2t 3 5t
i 8v 6 w
f
9
Stage 5.3
Fractional indices n
Evaluate each expression.
i ð8Þ3
2
ustralian Curriculum
6a
j 40a
1
10
3
2
1
3
c y3y5
2
4 8a
1
g ðn
1
10
12
1
1
3
l
Worked solutions
2
d m5 3 m5
h 16a 2 b6
m 4 Þ4
k 35x 4 5x
See Example 11
1
2
Exercise 5-06
MAT09NAWS10506
3
4
36y 4 4y
m
5-07 Fractional indices n
2
3
What is the meaning of fractional indices such as a 3 and a 2 ?
1 3
3
Power of a power
Consider 32 5 ¼ 32 5
pffiffiffiffiffi3
¼ 5 32
¼ 23
¼8
1
3
or consider 32 5 ¼ 323 5
ffiffiffiffiffiffiffi
p
5
¼ 323
pffiffiffiffiffiffiffiffiffiffiffiffiffi
¼ 5 32 768
¼8
Power of a power
Summary
SAMPLE CHAPTER
m
an ¼
9780170193085
p
ffiffiffi
n
a
m
or
ffiffiffiffiffiffi
p
n m
a
199
Path: K:/CLA-NCM_09A-13-0303/Application/CLA-NCM_09A-13-0303005.3d
Date: 6th May 2013
Time: 16:33
User ID: elangok 1BlackLining
Disabled
Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13
Indices
Stage 5.3
1 m
1
m
or ða m Þn
an ¼ an
pffiffiffiffiffiffi
pffiffiffi m
¼ n a or n a m
Proof:
Note: Taking the root first often makes the calculation simpler.
Example
12
Evaluate each expression.
2
4
1
Solution
2
a 83 ¼
pffiffiffi2
3
8
4
b 27 3 ¼
¼ 22
¼4
Example
3
c 643
b 27 3
a 83
pffiffiffiffiffi4
3
27
c
¼ 34
¼ 81
d 164
1
643 ¼ 1 1
64 3
1ffiffiffiffiffi
¼p
3
64
1
¼
4
3
d 164 ¼ 1 3
16 4
1
¼ pffiffiffiffiffi
4
16
¼ 13
2
1
¼
8
3
13
3
Evaluate 300 5 correct to two decimal places.
Solution
3
Enter on calculator: 300
300 5 ¼ 30:63887063 . . .
3
5
=
30:64
Example
14
Write each expression using a fractional index.
pffiffiffiffiffi
pffiffiffiffiffi
a 4 p3
b
b7
Solution
a
p
ffiffiffiffiffi 4
p3 ¼ p3
3
¼ p4
1
4
b
pffiffiffiffiffi b7 ¼ b7
7
¼ b2
1
2
1ffiffiffiffiffi
c p
3
q4
c
1ffiffiffiffiffi ¼ 1
p
1
3
q 4 ðq 4 Þ 3
4
¼ 14 or q3
q3
SAMPLE CHAPTER
200
9780170193085
Path: K:/CLA-NCM_09A-13-0303/Application/CLA-NCM_09A-13-0303005.3d
Date: 6th May 2013
Time: 16:33
User ID: elangok 1BlackLining
Disabled
N E W C E N T U R Y M AT H S A D V A N C E D
for the A
Example
ustralian Curriculum
9
Stage 5.3
15
Simplify each expression.
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
4
a
ð16r 2 Þ3
b
ffiffiffiffiffiffiffiffi
p
3
27k
b
pffiffiffiffiffiffiffiffi2
2
3
27k ¼ ð27k Þ3
2
c ð32a 5 Þ
35
Solution
a
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4
ð16r 2 Þ3 ¼ 16r 2
3
4
3
3
4 234
3
2
¼ 9k
3
5
3
3
3
¼ 32 5 a 5 3 5
¼ 8a3
m
n
See Example 12
5
3
f 64
2
c 128
4
3
g 32
2
i 1000 3
3
3
4
d 27
3
5
3
1
l 814
4
3
p 31255
4
t 102410
o 2564
3
7
s 1287
r 4002
Evaluate each expression correct to two decimal places.
7
3
5
b 85
c 50 4
a 15 4
e 100
34
f 16
3
5
e
p
ffiffiffiffiffiffi
3
m5
f
p
ffiffiffiffiffiffi
5
m3
2
3
a ð16n Þ
b ð8wÞ
pffiffiffiffiffiffiffiffiffiffi5
4
e
81r 4
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
4
f
ð81h12 Þ3
i
125b6
23
d 6
2
g 12
Simplify each expression.
3
4 4
See Example 13
23
32
Write each expression using a fractional index.
pffiffiffiffiffi
pffiffiffiffiffi
a 7 g2
b
e5
j ð625t 2 Þ
34
2
3
h 81 4
1
n 362
q 10245
5
7
k 83
j 125 3
m 252
4
1
¼
ð32a 5 Þ5
2
3
Fractional indices
b 8
e 1024
3
35
Evaluate each expression.
a 4
2
32a 5
¼ 27 k
3
2
Exercise 5-07
1
2 2
3 3
¼ 16 r
¼ 8r
c
h 179 5
See Example 14
pffiffiffiffiffiffi
c 6 x18
1ffiffiffiffiffiffi
d p
4 16
y
1ffiffiffiffiffi
g p
4 3
n
1ffiffiffiffiffi
h p
3 4
n
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
5
ð32d 10 Þ3
c
pffiffiffiffiffiffiffiffiffiffiffi3
d
64m8
1
g pffiffiffiffiffiffi
ffi 4
3
8s6
h
3
k ð49p4 q10 Þ2
l
1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
p
5
1024x15
1000x 3 y 6
See Example 15
2
23
SAMPLE CHAPTER
9780170193085
201
Path: K:/CLA-NCM_09A-13-0303/Application/CLA-NCM_09A-13-0303005.3d
Date: 6th May 2013
Time: 16:33
User ID: elangok 1BlackLining
Disabled
Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13
Indices
Worksheet
Index laws review
5-08 Summary of the index laws
MAT09NAWK10054
Puzzle sheet
Indices squaresaw
MAT09NAPS10055
Homework sheet
Indices 2
MAT09NAHS10006
Summary
am 3 an ¼ amþn
m
a m 4 a n ¼ aa n ¼ a mn
a0 ¼ 1
ða m Þn ¼ a m 3 n
an ¼ a1n
1
a
¼ ba
b
n n
n
a
¼ ba ¼ ba n
b
pffiffiffiffiffiffi
m
pffiffiffi m
a n ¼ ð n aÞ or n a m
a1 ¼ 1a
ðabÞn ¼ a n b n
n
a ¼ an
b
bn
pffiffiffi 1 pffiffiffi 1 pffiffiffi
1
a 2 ¼ a, a 3 ¼ 3 a, a n ¼ n a
Exercise 5-08
Summary of the index laws
1 Simplify each expression.
a a4 3 a 3
b t8 3 t
f (g 3)6
e (w 2)4
12
i 30c 8
j (5b4)4
5c
2 Evaluate each expression.
a 40
e (2)3
b (4)0
f (3)2
i (72)0
j
45 4 42
c n8 4 n2
d p3 4 p
g 2b 2 3 3b 5
h 4d 7 3 5d 6
k 24m6 4 8m4
l
c 7 3 20
g (5 2)2
d (7 3 2)0
h 2 4 3 23
k 42 4 45
0
m 5 2 4 50
n 102 4 10 2
o 1
2
3 Evaluate each expression, giving your answers in fraction form.
b 25
c 201
a 52
l
(3a)2
10 3 4 10 3
p 102 3 10 2
SAMPLE CHAPTER
202
d 103
9780170193085
Path: K:/CLA-NCM_09A-13-0303/Application/CLA-NCM_09A-13-0303005.3d
Date: 6th May 2013
Time: 16:33
User ID: elangok 1BlackLining
Disabled
N E W C E N T U R Y M AT H S A D V A N C E D
for the A
ustralian Curriculum
4 Evaluate each expression.
1
1
b 27 3 þ 40
a 16 2
1
1
e ð82 Þ3
f ð 9 3 Þ2
1
5
c 25 2
3
3
g 81 4
h ð32Þ5
b 8a 2w 2 3 5a 3w 7
c (4a 2b 5)4
d
f 6c 2d 0
g
48u 5 v 4
16uv
4
7x 2 y 6
2
k
3
35x 5 y 3
3
1
m (2p 3q2)5
n
o 2n0
7n
6 Simplify each expression using a positive index.
i (4n2t)3
j
a 87
e (5b)2
i 11t3
b 35
f 5b2
j (11t)3
c y1
g (ab)1
k p 3q5
m 8u3v4
n 2r6y5
o 10e1f 3
7 Simplify each expression.
1
1
7
a
b 5
4
2
1
1
r
1
e
f
8
10p
8 Write each expression using a negative index.
1
1
a 3
b
2
4
1
9
f
e
k
k4
9 Simplify each expression.
b d 3 3 d 7
a q 5 3 q2
e 5g 3 3 6g1
f 8a2 3 3a 3
5t 3
10t1
10 Write each expression using a fractional index.
pffiffiffi
pffiffiffi
a
5
b 3d
qffiffiffiffiffiffiffiffiffiffi
pffiffiffi 2
3 p
e
f
ðxyÞ5
i 48q 4 3q2
j
11 Simplify each expression.
3
2
4
b ð8c 3 Þ3
a
5a
2
3
49 2
a3 b9 3
e
f
d2
c6
2
p
ffiffiffiffiffiffiffiffiffiffiffiffi
1
5
10
j
i
32m
3g 2
Stage 5.3
1
d 83 þ 42
5 Simplify each expression.
a (3mn3)2
3
4
e
5
9
1
2
3
1
6y
g
z
c
1
10 4
1
g 7
x
20p 3 q 8
5p 2 q 6
2
3x
h
10
5 2
p
l
9y
p
ða 2 bÞ4 3 a3
b5
d x3
h ab1
i mw3
p 1 k 4 n7
2
1
1
7
1
2
h
5a
d
1
92
5
h 3
p
c
d
c m6 4 m 5
d t 4 t1
g 7x2 3 4x
h
k 2(b1)4
l (3h)2
pffiffiffiffiffi
3y
qffiffiffiffiffiffiffiffiffiffiffi
4
g
ð5aÞ3
ffiffiffiffiffi
p
4
10
qffiffiffi5
y
h
6
c
10 2
c
7m
ffiffiffiffiffiffiffiffiffiffiffiffiffi
p
g 4 625m6
5
64p1
16p 2
Stage 5.3
d
Worked solutions
Exercise 5-08
MAT09NAWS10508
d
5
ð25w 5 Þ2
h
64
y3
2
2a3
c2
3
4
SAMPLE CHAPTER
9780170193085
k
ð16x 8 Þ4
l
203
Path: K:/CLA-NCM_09A-13-0303/Application/CLA-NCM_09A-13-0303005.3d
Date: 6th May 2013
Time: 16:33
User ID: elangok 1BlackLining
Disabled
Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13
Indices
5-09 Significant figures
NSW
Worksheet
Significant figures
MAT09NAWK10058
A way of rounding a number is to give the most relevant or important digits of the number. For
example, a crowd of 47 321 people can be written as 47 000, which is rounded to the nearest
thousand, or to two significant figures.
The first significant figure in a number is the
first non-zero digit. For example, the significant
figures are shown in bold in this table:
First
significant
digit
4
4
1
2
Number
47 321
47 000
0.000 159 2
0.000 2
•
•
•
•
FPO
Number of
significant
digits
5
2
4
1
When rounding to significant figures, start counting from the first digit that is not 0.
If it is a large number, you may need to insert 0s at the end as placeholders.
Zeros at the end of a whole number or at the beginning of a decimal are not significant: they
are necessary placeholders.
Zeros between significant figures or at the end of a decimal are significant. For example, the
significant figures are shown in bold in this table.
Number
809 000
0.020 70
First significant digit
8
2
Example
Number of significant digits
3
4
16
State the number of significant figures in each number.
a 63.70
b 0.003 05
c 7600
Solution
a The zero after 7 is significant.
[ 63.70 has four significant figures.
b The first significant figure is 3, and the zero between 3 and 5 is significant.
[ 0.003 05 has three significant figures.
c The zeros after 6 are not significant.
[ 7600 has two significant figures.
SAMPLE CHAPTER
204
9780170193085
Path: K:/CLA-NCM_09A-13-0303/Application/CLA-NCM_09A-13-0303005.3d
Date: 6th May 2013
Time: 16:34
User ID: elangok 1BlackLining
Disabled
N E W C E N T U R Y M AT H S A D V A N C E D
for the A
Example
ustralian Curriculum
9
17
Round each number to three significant figures.
a 56.357
b 9.249
c 548 307
Solution
a 56.357 56.4 b 9.249 9.25 c 548 307 548 000
Example
The zeros here are not
significant, but they are
placeholders that are
necessary for showing the
place values of the 5, 4 and 8.
18
Write each number correct to one significant figure.
a 0.007 39
b 0.025
c 0.963
Solution
a 0.007 39 0.007
b 0.025 0.03
Exercise 5-09
c 0.963 1
The zeros at the beginning of
a decimal are not significant:
they are placeholders.
Significant figures
1 State the number of significant figures in each number.
a 457
b 0.23
c 15 000
d 4.0004
g 0.002 07
h 89 072
i 0.040
j 76 000 000
2 Round each number to three significant figures.
a 37.609
b 9435
d 2.813
e 15.99
g 1 769 000
h 385 764
See Example 16
e 0.0005
k 0.000 328
f 5000
l 169.320
See Example 17
c 168.39
f 60 522
i 10.2717
3 Write each number correct to two significant figures.
a 0.0637
d 0.000 158
g 0.2795
b 0.903
e 0.007 625
h 0.018 944
See Example 18
c 0.084 55
f 0.038 71
i 0.3145
4 What is 45 067 853 rounded to 3 significant figures? Select the correct answer A, B, C or D.
A 45 167 853
B 45 100 000
C 45 067 900
D 45 070 000
5 What is 0.005 605 0 rounded to 2 significant figures? Select the correct answer A, B, C or D.
A 0.01
B 0.010 000 0
C 0.0056
D 0.005 600 0
6 Round each number to one significant figure.
a 9.478
d 0.007 66
g 1856.78
b 57.12
e 0.5067
h 0.000 28
c 0.0367
f 10 675
i 56 239 400
7 A company makes a profit of $35 754 125.
a Round the profit to the nearest million and state the number of significant figures in the
answer.
SAMPLE CHAPTER
b Round the profit to the nearest ten million and state the number of significant figures in
the answer.
9780170193085
205
Path: K:/CLA-NCM_09A-13-0303/Application/CLA-NCM_09A-13-0303005.3d
Date: 6th May 2013
Time: 16:34
User ID: elangok 1BlackLining
Disabled
Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13
Indices
8 Australia’s population in 2010 was 21 387 000. To how many significant figures has this
number been written?
9 A total of 21 558 people attended a local football match. Express this number to three
significant figures.
10 Evaluate each expression, correct to the number of significant figures shown in the brackets.
a 45.6 3 8.7 2.75 3 78.32
c (63.73 27.89) 4 5.82
(2)
(3)
e 9:732 þ 2:765 (1)
12:27 3 15:8
1 þ 253
g
(3)
0:941 0:0076
Just for the record
b 15.5 9.87 4 0.24 þ 8.43 3 2.4
d 63:25 þ 76:03 (4)
55:89 89:24
f 78.91 4 (23.6 þ 94.7)
h
pffiffiffiffiffiffiffiffiffi
84:3 3 0:0715
(1)
(2)
(4)
Big numbers
The table below lists the names of some big numbers and their meanings.
Name
million
billion
trillion
quadrillion
quintillion
sextillion
septillion
octillion
nonillion
decillion
Numeral
106 ¼ 1 000 000
109 ¼ 1 000 000 000
1012
1015
1018
10 21
10 24
10 27
10 30
10 33
According to the Guinness Book of Records, the largest number for which there is an
accepted name is the centillion, first recorded in 1852. It is equal to 10 303.
What special name for the number 10100?
Puzzle sheet
Scientific notation
puzzle
MAT09NAPS10059
5-10 Scientific notation
Scientific notation is a short way of writing very large or
very small numbers using powers of 10. It was invented
in the early twentieth century when scientists needed to
describe very large values, such as astronomical distances
and very small values such as the masses of atoms.
SAMPLE CHAPTER
206
9780170193085
Path: K:/CLA-NCM_09A-13-0303/Application/CLA-NCM_09A-13-0303005.3d
Date: 6th May 2013
Time: 16:34
User ID: elangok 1BlackLining
Disabled
N E W C E N T U R Y M AT H S A D V A N C E D
for the A
ustralian Curriculum
9
Summary
Numbers written in scientific notation are expressed in the form
m 3 10 n
where m is a number between 1 and 10 and n is an integer.
Example
19
Video tutorial
Express each number in scientific notation.
a 764 000 000 000
b 6000
Scientific notation
c 0.0008
d 0.000 000 472
MAT09NAVT10011
Solution
a Use the significant figures in the number to write a value between 1 and 10: 7.64
Count how many places the decimal point moves to the right to make 764 000 000 000.
11 places
or count the number of
764 000 000 000
places after the first significant
figure, 7
[ 764 000 000 000 ¼ 7.64 3 1011
b Use the significant figures in the number to write a value between 1 and 10: 6
Count how many places the decimal point moves to the right to make 6000.
3 places
6000
or count the number of
places after the first significant
figure, 6
[ 6000 ¼ 6 3 10 3
c Use the significant figures in the number to write a value between 1 and 10: 8
Count how many places the decimal point moves to the left to make 0.0008.
4 places
0.0008
or count the number of
decimal places to the first
significant figure, 8
[ 0.0008 ¼ 8 3 104
Note that small numbers are written with negative powers of 10.
d Use the significant figures in the number to write a value between 1 and 10: 4.72
Count the number of places the decimal point moves to the left to make 0.000 000 472.
7 places
or count the number of
decimal places to the first
significant figure, 4
SAMPLE CHAPTER
0.000 000 472
0.000 000 472 ¼ 4.72 3 107
9780170193085
207
Path: K:/CLA-NCM_09A-13-0303/Application/CLA-NCM_09A-13-0303005.3d
Date: 6th May 2013
Time: 16:34
User ID: elangok 1BlackLining
Disabled
Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13
Indices
Example
20
Express each number in decimal form.
a 2.7 3 10 4
b 3.56 3 102
Solution
a 2.7 × 10 4 = 2.7000
Move the decimal point 4 places to the right.
b 3.56 × 10 –2 = 0.0356
Move the decimal point 2 places to the left.
= 27000
= 0.0356
Example
21
a Which number is the larger: 3.65 3 1012 or 8.1 3 1012?
b Write these numbers in ascending order: 4.3 3 10 6, 2.8 3 107, 1.9 3 107
Solution
To compare numbers in scientific notation, first compare the powers of ten.
If the powers of ten are the same, then compare the decimal parts.
a The powers of ten are the same. Compare the decimal parts: 8.1 > 3.65.
[ The larger number is 8.1 3 1012
b Compare the powers of ten: 10 6 < 107.
Then compare the two numbers with 107: 1.9 < 2.8.
[ The numbers in ascending order are 4.3 3 10 6, 1.9 3 107, 2.8 3 107.
Exercise 5-10
See Example 19
1
Express each number in scientific notation.
a
e
i
m
q
u
2
Scientific notation
2400
7.8
3 000 000 000
0.035
0.000 003
0.000 000 1
b
f
j
n
r
v
786 000
348 000 000
80
0.000 076
0.913
0.000 89
c
g
k
o
s
w
55 000 000
59 670
763
0.8
0.000 007 146
0.000 000 078
d
h
l
p
t
x
95
15
10
0.0713
0.009
0.1
Express each measurement in scientific notation.
a The world’s largest mammal is the blue whale,
which can weigh up to 130 000 kg.
b The diameter of an oxygen molecule
is 0.000 000 29 cm.
SAMPLE CHAPTER
c The thickness of a human hair is 0.000 08 m.
d Light travels at a speed of 300 000 000 m/s.
208
9780170193085
Path: K:/CLA-NCM_09A-13-0303/Application/CLA-NCM_09A-13-0303005.3d
Date: 6th May 2013
Time: 16:34
User ID: elangok 1BlackLining
Disabled
N E W C E N T U R Y M AT H S A D V A N C E D
for the A
ustralian Curriculum
9
e The nearest star to Earth, excluding the Sun, is Alpha Centauri, which
is 40 000 000 000 000 km away.
f The thickness of a typical piece of paper is 0.000 12 m.
g The small intestine of an adult is approximately 610 cm long.
h The diameter of a hydrogen atom is 0.000 000 0001 m.
i The diameter of our galaxy, the Milky Way, is 770 000 000 000 000 000 000 m.
j A microsecond means 0.000 001 s.
k The Andromeda Galaxy is the most remote body visible to the naked eye, at a distance of
2 200 000 light years away.
3
4
Express each number in decimal form.
b 7.1 3 10 3
a 6 3 10 5
0
d 3.14 3 10
e 6 3 105
8
g 3.02 3 10
h 5.9 3 1010
4
j 4 3 10
k 5 3 10 3
m 8.03 3 101
n 6.32 3 10 4
7
p 2.2 3 10
q 9.0 3 106
See Example 20
c
f
i
l
o
r
8
3.02 3 10
7.1 3 103
1.1 3 1012
4.76 3 104
1.6 3 102
1.11 3 101
For each pair of numbers, write the larger one.
a
c
e
g
i
5
5
3 3 10 or 4 3 10
8.4 3 100 or 1.3 3 107
9.3 3 109 or 7.6 3 109
3.04 3 100 or 3.04 3 104
2 3 1015 or 2 3 1017
See Example 21
b
d
f
h
j
5
6
8.4 3 10 or 2.7 3 10
3.6 3 107 or 6.3 3 107
3.5 3 106 or 9.3 3 10 2
4.5 3 105 or 3.7 3 107
6.23 3 105 or 9.7 3 105
SAMPLE CHAPTER
9780170193085
209
Path: K:/CLA-NCM_09A-13-0303/Application/CLA-NCM_09A-13-0303005.3d
Date: 6th May 2013
Time: 16:35
User ID: elangok 1BlackLining
Disabled
Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13
Indices
5
Write each set of numbers in ascending order.
a 3.8 3 109, 7.3 3 109, 5.5 3 109
b 2.2 3 104, 5.8 3 106, 7 3 104
c 3.5 3 100, 5.3 3 10 2, 4.9 3 10 2
6
Write each set of numbers in descending order.
a 6 3 10 5, 2.9 3 10 2, 1 3 10 2
b 1.2 3 109, 6.3 3 10 2, 8.1 3 104
c 4.1 3 101, 9.5 3 101, 6.4 3 103
Worksheet
Scientific notation
problems
MAT09NAWK10060
Homework sheet
Indices 3
MAT09NAHS10007
Homework sheet
5-11 Scientific notation on a calculator
To enter a number in scientific notation on a calculator, use the
Example
or
key.
22
Evaluate each expression using scientific notation.
a (4.25 3 107) 3 (8.2 3 106)
b (1.08 3 1015) 4 (3 3 1011)
c (4.9 3 107)2
Indices revision
MAT09NAHS10008
Solution
a Enter 4.25
×
7
8.2
7
6
=
6
(4.25 3 10 ) 3 (8.2 3 10 ) ¼ 3.485 3 1014
−
b Enter 1.08
15
÷
3
11
(1.08 3 1015) 4 (3 3 1011) ¼ 3.6 3 1027
c Enter 4.9
7
=
=
Note that with scientific
notation on a calculator, there
is no need to enter brackets
) around the
(
numbers.
7 2
(4.9 3 10 ) ¼ 2.401 3 1015
Example
23
Estimate the value of each expression in scientific notation, then evaluate it correct to three
significant figures.
9
a 9:2 3 10 5
b ð8:5 3 10 4 Þ 3 ð6:3 3 107 Þ
c ð6:08 3 10 3 Þ2
2:7 3 10
Solution
Estimate
9:2 3 10 9 9 3 10 9
a
2:7 3 10 5 3 3 10 5
9
¼ 9 3 10 5
3
10
¼ 3 3 10 4
Calculated answer
9:2 3 10 9 ¼ 34 074:074 07
2:7 3 10 5
34 000
SAMPLE CHAPTER
210
¼ 3:4 3 10 4
9780170193085
Path: K:/CLA-NCM_09A-13-0303/Application/CLA-NCM_09A-13-0303005.3d
Date: 6th May 2013
Time: 16:35
User ID: elangok 1BlackLining
Disabled
N E W C E N T U R Y M AT H S A D V A N C E D
for the A
Estimate
b ð8:5 3 10 4 Þ 3 ð6:3 3 107 Þ
ustralian Curriculum
9
Calculated answer
ð8:5 3 10 4 Þ 3 ð6:3 3 107 Þ ¼ 5:355 3 1012
5:36 3 1012
ð9 3 10 4 Þ 3 ð6 3 107 Þ
¼ ð9 3 6Þ 3 ð10 4 3 107 Þ
¼ 54 3 1011
¼ 5:4 3 10 3 1011
¼ 5:4 3 1012
ð6:08 3 10 5 Þ3 ¼ 2:24755 . . . 3 1017
c ð6:08 3 10 5 Þ 3 ð6 3 10 5 Þ3
2:25 3 1017
¼ 6 3 3 ð10 5 Þ3
¼ 216 3 1015
¼ 2:16 3 10 2 3 1015
¼ 2:16 3 1017
Exercise 5-11
Scientific notation on a calculator
1 Evaluate each expression using scientific notation.
b (8 3 107) 4 (4 3 10 2)
a (2 3 10 3) 3 (3 3 10 5)
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
d
9 3 1012
c (2 3 10 5)3
7
8
f (1 3 108) 4 (2 3 10 3)
e (4 3 10 ) 3 (6 3 10 )
3 5
g (4 3 10 )
h 24.08 4 (8 3 106)
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
j (2 3 10 5)2
i
3:969 3 1019
p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
9
l 7:62 3 10
k 3 8 3 109
2 3 104
2 Estimate the value of each expression in scientific notation, then evaluate correct to three
significant figures.
a
c
e
g
(5.7 3 10 3) 3 (2.3 3 10 5)
(9.1 3 10 20) 4 (3.2 3 10 5)
(7.13 3 1010) 3 (9.8 3 108)
(5.85 3 10 4)3
b
d
f
h
See Example 22
See Example 23
(8 3 10 5) 3 (3.7 3 107)
(1.2 3 108)2
(1.9 3 1011) 4 (2.1 3 107)
(6 3 1012) 4 (2.8 3 10 3)
3 The human body consists of approximately 6 3 109 cells, and each cell consists of 6.3 3 109
atoms. Roughly how many atoms are there in a human body?
Worked solutions
Exercise 5-11
MAT09NAWS10511
SAMPLE CHAPTER
9780170193085
211
Path: K:/CLA-NCM_09A-13-0303/Application/CLA-NCM_09A-13-0303005.3d
Date: 6th May 2013
Time: 16:35
User ID: elangok 1BlackLining
Disabled
Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13
Indices
4 A telephone book is 4.5 cm thick. There are 2000 pages in it. Find the thickness, in
millimetres, of one page in scientific notation.
5 Evaluate each expression in scientific notation, correct to two significant figures.
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a (7.4 3 10 30) (3.59 3 10 29)
b (1.076 3 1017) þ (2.3 3 1016)
c
6:6 3 1027
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
d (7.5 3 10 23) 4 (3.3 3 1013)
e (8.17 3 1016)3
f
2:69 3 1045
ð5:6 3 10 4 Þ 3 ð3:9 3 105 Þ
ð2:3 3 107 Þ
g (7.05 3 10 3) 4 (3.9 3 107)
h
j 5 20
m 99
k 8011
n (0.7)5
i 1595 3 1959
l 310
Express the answers for questions 6 to 10 in scientific notation correct to two significant figures if
necessary.
Worked solutions
Exercise 5-11
MAT09NAWS10511
6 The Earth is 1.50 3 108 km from the Sun and the speed of light is 3 3 10 5 km/s. How long
does it take for light to travel from the Sun to Earth? Express your answer in:
a seconds
b minutes.
7 The Sun burns 6 million tonnes of hydrogen a second. Calculate how many tonnes of
hydrogen it burns in a year (that is, 365.25 days).
8 Sound travels at approximately 330 metres per second. If Mach 1 is the speed of sound, how
fast is Mach 5? Convert your answer to kilometres per second.
9 The distance light travels in one year is called a light year. If the speed of light is approximately
3 3 10 5 km per second, how far does light travel in a leap year?
10 A thunderstorm is occurring 30 km from where you are standing. Use the speed of light
(3 3 10 5 km per second) and the speed of sound (330 metres per second) to calculate
in seconds:
a how long the light from the lightning takes to reach you
b how long the sound from the thunder takes to reach you.
11 a What is the largest number that can be displayed on your calculator?
b What is the smallest number that can be displayed?
SAMPLE CHAPTER
212
9780170193085
Path: K:/CLA-NCM_09A-13-0303/Application/CLA-NCM_09A-13-0303005.3d
Date: 6th May 2013
Time: 16:36
User ID: elangok 1BlackLining
Disabled
N E W C E N T U R Y M AT H S A D V A N C E D
for the A
ustralian Curriculum
9
Investigation: A lifetime of heartbeats
How many times does your heart beat in an average lifetime of 80 years?
1 Work in pairs and copy this table.
Name
Trial 1
Trial 2
Average beats per minute
2 Use two fingers to measure your pulse. Have your partner time you for a minute. Do this
twice, record your results in the table and find the average.
3 Repeat Step 2 for your partner.
4 Calculate how many times your heart (and your partner’s heart) beats in the following
periods. Write your answers in scientific notation correct to two significant figures.
a an hour
b a day
c a week
d a year (use 365.25 days)
e an average lifetime of 80 years
Just for the record
Straight hair
• round follicle
Hairy numbers
Wavy hair
oval follicle
Curly hair
flat follicle
There are about 110 000 hairs on your head. Each hair grows at the rate of about 1.3 3 103 cm
per hour. A single hair lasts about six years. Every day you lose between 30 and 60 hairs. Each
hair grows from a small depression in the skin called a follicle (a gland). After the hair falls out,
the follicle rests for about three to four months before the next hair starts growing. Hair follicles
are either oval, flat or round in shape. How straight, wavy or curly your hair is depends on the
shape of your hair follicles.
How many hairs are on all the heads in China if its population is approximately 1.435 3 109?
Answer in both scientific notation and decimal notation.
SAMPLE CHAPTER
9780170193085
213
Path: K:/CLA-NCM_09A-13-0303/Application/CLA-NCM_09A-13-0303005.3d
Date: 6th May 2013
Time: 16:36
User ID: elangok 1BlackLining
Disabled
Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13
Indices
Power plus
Worksheet
1
Write each number in scientific notation.
a 438.2 3 109
d 2013 3 103
g 6.7 millionths
Binary number system
MAT09NAWK10057
2
3
c 0.0004 3 1012
f 57.8 thousandths
i 3.2 billionths
b 0.52 3 107
e 57.8 thousand
h 3.2 billion
Evaluate each expression.
pp
ffiffiffiffiffiffiffiffiffiffiffi
pp
ffiffiffiffiffiffiffiffiffi
ffiffiffiffiffiffiffiffi
ffiffiffiffiffi
81
a
b
625
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
p
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffiffiffiffiffiffiffiffiffiffiffi
ffi
pffiffiffiffiffiffiffiffiffiffi
4 pffiffiffiffiffiffiffi
e
6561
f
256
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pp
ffiffiffiffiffiffiffiffiffiffiffi
ffiffiffiffiffiffiffiffi
256
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p
3
1 000 000
g
c
Find values of a, m and n so that each equation is true.
m
m
m
b an ¼ 3
c a n ¼ 64
a an ¼ 2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p
ffiffiffiffiffiffiffiffiffiffi
5
1024
p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 000 000
h
d
m
d a n ¼ 125
4
For how many values of a and b does ab ¼ ba?
5
The terms in the pattern 3, 5, 17, 257, 65 537,… can all be generated by a simple
method, using only the numbers 1 and 2.
a What is this method?
b What is the next number in the sequence?
SAMPLE CHAPTER
214
9780170193085
Path: K:/CLA-NCM_09A-13-0303/Application/CLA-NCM_09A-13-0303005.3d
Date: 6th May 2013
Time: 16:36
User ID: elangok 1BlackLining
Disabled
Chapter 5 review
n Language of maths
Puzzle sheet
ascending
base
descending
estimate
expanded form
exponent
fractional power
index
index laws
index notation
indices
negative power
power
product
quotient
reciprocal
scientific notation
significant figures
term
zero power
Indices crossword
MAT09NAPS10061
1 What does a power of 12 mean?
2 Which two words from the list mean ‘power’?
3 What is the
or
key on a calculator used for?
4 What is the index law for dividing terms with the same base?
5 Which digits in 0.006 701 are significant figures?
6 What power is associated with the reciprocal of a term or number?
7 What type of numbers when written in scientific notation have negative powers of 10?
n Topic overview
•
•
•
•
•
•
What was this topic about? What was the main theme?
What content was new and what was revision?
What are the index laws?
Write 10 questions (with solutions) that could be used in a test for this chapter.
Include some questions that you have found difficult to answer.
List the sections of work in this chapter that you did not understand. Follow up this work with
a friend or your teacher.
Worksheet
Mind map: Indices
MAT09NAWK10063
Copy and complete this mind map of the topic, adding detail to its branches and using pictures,
symbols and colour where needed. Ask your teacher to check your work.
Multiplying and
dividing terms
with the same base
Scientific
notation
Powers of
products
and quotients
Power of
a power
INDICES
Index or
power
Base
Zero and
negative
indices
SAMPLE CHAPTER
Significant
figures
9780170193085
Fractional
indices
215
Path: K:/CLA-NCM_09A-13-0303/Application/CLA-NCM_09A-13-0303005.3d
Date: 6th May 2013
Time: 16:36
User ID: elangok 1BlackLining
Disabled
Chapter 5 revision
See Exercise 5-01
See Exercise 5-02
1 Simplify each expression, writing the answer in index notation.
b 420 4 44
c
a 10 3 3 107
8
2
d h 3h
e 3n3 3 4n
f
g 20m9 4 4m
h 3v4w 2 3 2v 3w 5
i
6 10
8 8
p
q
j 24t4 h2
k 2 2
l
3t h
p q
2 Simplify each expression, writing the answer in index notation.
a (22)3
d (2y 3)10
g ð2Þ5
See Exercise 5-03
3 Simplify each expression.
a (ab 2)4
d (4h2g)3
5
g 3m
2
See Exercise 5-04
b (k 5)5
e (5t 2)2
h ð2k Þ5
c (x)4
f (10g)3
i ð5m3 Þ2
b (5x 3y 2)2
4
e a
7
c (4t 2)3
f (2pqr)5
4
2a7
i
b
h (3np2)4
8 6 3
b y
k
8b 2 y
j (4t 4u 5)3 3 8t 2u
4 Simplify each expression.
a 70
l 45c6d 8 4 (3cd 2)2
b (7)0
c e0
d (e)0
See Exercise 5-05
e e0
0
2p
h
g (gh)0
3
5 Simplify each expression using a positive index.
b 192
c x1
a 83
e (4m)1
See Exercise 5-05
Stage 5.3
See Exercise 5-05
See Exercise 5-06
See Exercise 5-07
f (4m)2
g
1
3
i 2x4
j
k
5a
6 Write each expression using a negative index.
a 13
b 15
c
10
r
2
using a positive index.
7 Simplify 8
3x
8 Write each expression using a radical (root) sign.
1
1
b u2
c
a q3
3
2
216
i
2p0
3
d p5
(5b)1
h 5b1
100 1
l
9
c4d 2
1
r
d 3
b
1
1
ð2qÞ3
d ðarÞ2
3
c 362
b ð32Þ5
10 Simplify each expression.
4
a ð125d 15 Þ3
See Exercise 5-09
f g0h
9 Evaluate each expression.
a 64 3
See Exercise 5-07
a12 4 a 2
10d 15 4 5d 3
5x 5y 2 3 3xy
100a 2 b4
5ab 2
1
b ð16y 20 Þ4
2
c ð32x 8 Þ5
d
p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3
64n12
11 Round each value correct to the number of significant figures shown in the brackets.
a 8.5678 (2)
b 15 712 (3)
c 476 (1)
d 0.007 126 6 (4)
e 0.9041 (3)
f 301 378 (2)
g 4805.28 (3)
h 0.000 87 (1)
i 67 000 000 (1)
SAMPLE CHAPTER
9780170193085
Path: K:/CLA-NCM_09A-13-0303/Application/CLA-NCM_09A-13-0303005.3d
Date: 6th May 2013
Time: 16:36
User ID: elangok 1BlackLining
Disabled
Chapter 5 revision
12 Express each number in scientific notation.
a 37 000
b 0.61
d 0.000 49
e 13
c 250 000
f 0.000 000 000 08
13 Express each number in decimal form.
b 6 3 107
a 8.1 3 10 3
d 8.1 3 103
e 6 3 107
c 3.075 3 100
f 3.075 3 102
See Exercise 5-10
See Exercise 5-10
14 Write these numbers in ascending order: 3 3 10 3, 9.1 3 108, 2.4 3 10 3.
See Exercise 5-10
15 Evaluate each expression using scientific notation.
22
8
a (3.65 3 10 ) 3 (7.4 3 10 )
c (5 3 10 5)3
See Exercise 5-11
10
4
b p
(1.44
3 10 )ffi 4 (3.6 3 10 )
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
d
6:25 3 108
16 Estimate the value of each expression in scientific notation, then evaluate correct to two
significant figures.
a (8.9 3 109) 3 (1.1 3 107)
b (9.3 3 1015) 3 (4 3 10 2)
See Exercise 5-09
c (3.1 3 10 4)2
SAMPLE CHAPTER
9780170193085
217