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Cambridge Lower Secondary
Mathematics
LEARNER’S BOOK 9
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Lynn Byrd, Greg Byrd & Chris Pearce
Second edition
Digital Access
Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication.
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Cambridge Lower Secondary
Mathematics
LEARNER’S BOOK 9
SA
M
Greg Byrd, Lynn Byrd and Chris Pearce
Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication.
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Projects and their accompanying teacher guidance have been written by the NRICH
Team. NRICH is an innovative collaboration between the Faculties of Mathematics
and Education at the University of Cambridge, which focuses on problem solving and
on creating opportunities for students to learn mathematics through exploration and
discussion https://nrich.maths.org.
Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication.
Introduction
Introduction
Welcome to Cambridge Lower Secondary Mathematics Stage 9
The Cambridge Lower Secondary Mathematics course covers the Cambridge Lower
Secondary Mathematics curriculum framework and is divided into three stages: 7, 8 and 9.
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During your course, you will learn a lot of facts, information and techniques. You will start
to think like a mathematician. This book covers all you need to know for Stage 9.
The curriculum is presented in four content areas:
•
Number
•
Algebra
•
Geometry and measures
•
Statistics and probability.
This book has 15 units, each related to one of the four content areas. However, there
are no clear dividing lines between these areas of mathematics; skills learned in
one unit are often used in other units. The book encourages you to understand the
concepts that you need to learn, and gives opportunity for you to practise the
necessary skills.
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Many of the questions and activities are marked with an icon that indicates that
they are designed to develop certain thinking and working mathematically skills.
There are eight characteristics that you will develop and apply
throughout the course:
Specialising – testing ideas against specific criteria;
•
Generalising – recognising wider patterns;
•
Conjecturing – forming questions or ideas about mathematics;
•
Convincing – presenting evidence to justify or challenge a mathematical idea;
•
Characterising – identifying and describing properties of mathematical objects;
•
Classifying – organising mathematical objects into groups;
•
Critiquing – comparing and evaluating ideas for solutions;
•
Improving – Refining your mathematical ideas to reach more effective
approaches or solutions.
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•
Your teacher can help you develop these skills, and you will also develop your
ability to apply these different strategies.
We hope you will find your learning interesting and enjoyable.
Greg Byrd, Lynn Byrd and Chris Pearce
3
Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication.
Contents
Unit
6
How to use this book
9–20
1 Number and calculation
1.1 Irrational numbers
1.2 Standard form
1.3 Indices
Number
21–54
2 Expressions and formulae
2.1 Substituting into expressions
2.2 Constructing expressions
2.3 Expressions and indices
2.4 Expanding the product of two linear expressions
2.5 Simplifying algebraic fractions
2.6 Deriving and using formulae
Algebra
55–81
3 Decimals, percentages and rounding
3.1 Multiplying and dividing by powers of 10
3.2 Multiplying and dividing decimals
3.3 Understanding compound percentages
3.4 Understanding upper and lower bounds
Number
82
Project 1 Cutting tablecloths
83–102
4 Equations and inequalities
4.1 Constructing and solving equations
4.2 Simultaneous equations
4.3 Inequalities
Algebra
103–126
5 Angles
5.1 Calculating angles
5.2 Interior angles of polygons
5.3 Exterior angles of polygons
5.4 Constructions
5.5 Pythagoras’ theorem
Geometry and measure
128–137
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127
Strand
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Page
Project 2 Angle tangle
6 Statistical investigations
6.1 Data collection and sampling
6.2 Bias
Statistics and probability
7 Shapes and measurements
7.1 Circumference and area of a circle
7.2 Areas of compound shapes
7.3 Large and small units
Geometry and measure
161–189
8 Fractions
8.1 Fractions and recurring decimals
8.2 Fractions and the correct order of operations
8.3 Multiplying fractions
8.4 Dividing fractions
8.5 Making calculations easier
Number
190
Project 3 Selling apples
191–211
9 Sequences and functions
9.1 Generating sequences
9.2 Using the nth term
9.3 Representing functions
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138–160
Algebra
4
Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication.
Contents
Unit
Strand
212–233
10 Graphs
10.1 Functions
10.2 Plotting graphs
10.3 Gradient and intercept
10.4 Interpreting graphs
Algebra
234
Project 4 Cinema membership
235–249
11 Ratio and proportion
11.1 Using ratios
11.2 Direct and inverse proportion
Number
250–269
12 Probability
12.1 Mutually exclusive events
12.2 Independent events
12.3 Combined events
12.4 Chance experiments
Statistics and probability
270–299
13 Position and transformation
13.1 Bearings and scale drawings
13.2 Points on a line segment
13.3 Transformations
13.4 Enlarging shapes
Geometry and measure
300
Project 5 Triangle transformations
301–316
14 Volume, surface area and symmetry
14.1 Calculating the volume of prisms
14.2 Calculating the surface area of triangular prisms, pyramids and
cylinders
14.3 Symmetry in three-dimensional shapes
Geometry and measure
317–347
15 Interpreting and discussing results
15.1 Interpreting and drawing frequency polygons
15.2 Scatter graphs
15.3 Back-to-back stem-and-leaf diagrams
15.4 Calculating statistics for grouped data
15.5 Representing data
Statistics and probability
Project 6 Cycle training
Glossary
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349–351
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348
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Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication.
How to use this book
How to use this book
In this book you will find lots of different features to help your learning.
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Questions to find out what you
know already.
What you will learn in the unit.
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Important words to learn.
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Step-by-step examples showing
how to solve a problem.
These questions help you to
develop your skills of thinking
and working mathematically.
6
Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication.
How to use this book
Questions to help you think
about how you learn.
This is what you have
learned in the unit.
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An investigation to carry out with a
partner or in groups.
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Questions that cover what you
have learned in the unit. If you
can answer these, you are ready to
move on to the next unit.
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At the end of several units,
there is a project for you to
carry out, using what you
have learned. You might make
something or solve a problem.
7
Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication.
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Acknowledgements
Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication.
Getting started
2
3
4
3
64
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5
Write as a number:
a
122
b
81
c
53
d
8
2 = 256
Use this fact to work out the value of
a
29
b 27
Here is a multiplication: 155 × 152
a
Write the correct answer from this list: 157 1510 307 3010
b Write the answer to 155 ÷ 152 in index form.
Look at these numbers: 4 −4.5 3000 17 3 225
20
a
Which of these numbers are integers?
b Which of these numbers are rational numbers?
Write one million as a power of 10.
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1
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1
Number and
calculation
9
Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication.
1 Number and calculation
1, 4, 9 and 16 are the first four square numbers. They have integer
square roots.
22 = 4 and 4 = 2
12 = 1 and 1 = 1
32 = 9 and 9 = 3
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42 = 16 and 16 = 4
What about 2 ? Is there a rational number n for which n2 = 2?
Remember that you can write a rational number as a fraction.
( )
11
2
2
= 1 1 × 1 1 = 2 1 so 2 must be a little less than 1 1 .
2
2
A closer answer
4
is 1 5
12
because
( )
15
12
2
2
=
2 1 .
144
( 408 )
An even closer answer is 1 169 because 1 169
408
2
=2
1 .
166464
Do you think you can find a fraction which gives an answer of exactly 2
when you square it?
A calculator gives the answer 2 = 1.414213562. This is a rational
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number because you can write it as a fraction: 1 414213562 .
1000000000
Is 1.414213562 × 1.414213562 exactly 2?
In this unit, you will look at numbers such as 2 .
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1.1 Irrational numbers
In this section you will …
Key words
•
learn about the difference between rational numbers and
irrational numbers
irrational number
•
use your knowledge of square numbers to estimate
square roots
surd
•
use your knowledge of cube numbers to estimate
cube roots.
rational number
10
Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication.
1.1 Irrational numbers
Integers are whole numbers. For example, 13, −26 and 100 004 are integers.
You can write rational numbers as fractions. For example, 9 3 , −3 4 and
4
15
18 5 are rational numbers.
11
You can write any fraction as a decimal.
Tip
The set of rational
numbers includes
integers.
9 3 = 9.75 −3 4 = −3.26666666... 18 5 = 18.4545454...
4
15
11
Tip
Square roots of
negative numbers
do not belong to
the set of rational
or irrational
numbers. You
will learn more
about these
numbers if you
continue to study
mathematics to a
higher level.
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The fraction either terminates (for example, 9.75) or it has recurring
digits (for example, 3.266666666666… and 18.45454545454…).
There are many square roots and cube roots that you cannot write
as fractions. When you write these fractions as decimals, they do not
terminate and there is no recurring pattern. For example, a calculator
gives the answer 7 = 2.645751... The calculator answer is not exact. The
decimal does not terminate and there is no recurring pattern. Therefore,
7 is not a rational number.
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Numbers that are not rational are called irrational numbers. 7 , 23,
3
10 and 3 45 are irrational numbers. Irrational numbers that are square
roots or cube roots are called surds.
There are also numbers that are irrational but are not square roots or
cube roots. One of these irrational numbers is called pi, which is the
Greek letter π. Your calculator will tell you that π = 3.14159… You will
meet π later in the course.
Worked example 1.1
Do not use a calculator for this question.
a
Show that 90 is between 9 and 10.
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b N is an integer and 3 90 is between N and N + 1. Find the value of N.
Answer
a
92 = 81 and 102 = 100
81 < 90 < 100
This means 90 is between 81 and 100.
So 81 < 90 < 100
And so 9 < 90 < 10
b 43 = 64 and 53 = 125
64 < 90 < 125 and so
3
64 < 3 90 < 3 125
So 4 < 3 90 < 5 and N = 4
11
Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication.
1 Number and calculation
Exercise 1.1
1
Write whether each of these numbers is an integer or an irrational
number. Explain how you know.
a
9
b
19
c
39
d
49
e
a
Write the rational numbers in this list.
1 7 5 −38 160 − 2.25 − 35
12
b Write the irrational numbers in this list.
0.3333… −16 200 1.21 23 3 343
8
Write whether each of these numbers is an integer or a surd.
Explain how you know.
3
100
100
1000
a
b
c
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2
99
3
3
3 10 000
d
1000
e
f
10 000
Is each of these numbers rational or irrational? Give a reason for each
answer.
a
2+ 2
b
2+2
4
5
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3
c
4+ 3 4
d
4+4
Find
a
two irrational numbers that add up to 0
b two irrational numbers that add up to 2.
Think like a mathematician
6
a
Use a calculator to find
3 × 12
20 × 5
2 × 18
i
ii
iii
iv
8× 2
What do you notice about your answers?
Find another multiplication similar to the multiplications in part a.
Find similar multiplications using cube roots instead of square roots.
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b
c
d
7
8
9
Without using a calculator, show that
a
7 < 55 < 8
b 4 < 3 100 < 5
Without using a calculator, find an irrational number between
a
4 and 5
b 12 and 13.
Without using a calculator, estimate
190 to the nearest integer
a
b
3
190 to the nearest integer.
12
Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication.
1.1 Irrational numbers
10 a
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Use a calculator to find
i ( 2 + 1) × ( 2 −1)
ii ( 3 + 1) × ( 3 − 1)
iii ( 4 + 1) × ( 4 − 1)
b Continue the pattern of the multiplications in part a.
c
Generalise the results to find ( N + 1) × ( N − 1) where N is a positive integer.
d Check your generalisation with further examples.
11 Here is a decimal: 5.020 020 002 000 020 000 020 000 002…
Arun says:
There is a regular pattern:
one zero, then two zeros,
then three zeros, and so on.
This is a rational number.
a
b
Is Arun correct? Give a reason for your answer.
Compare your answer with a partner’s. Do you agree? If not, who is correct?
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In this exercise, you have looked at the properties of rational and
irrational numbers.
Are the following statements true or false?
a
i
The sum of two integers is always an integer.
iiThe sum of two rational numbers is always a rational
number.
iiiThe sum of two irrational numbers is always an irrational
number.
b Here is a calculator answer: 3.646 153 846
The answer is rounded to 9 decimal places.
Can you decide whether the number is rational or irrational?
Summary checklist
I can use square numbers and cube numbers to estimate square roots and
cube roots.
I can say whether a square root or the cube root of a positive integer
is rational or irrational.
13
Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication.
1 Number and calculation
1.2 Standard form
In this section you will …
Key words
•
scientific notation
learn to write large and small numbers in standard form.
standard form
Look at these numbers
4.67 ×10 = 46.7
4.67 ×103 = 4670
4.67 ×106 = 4 670 000
Tip
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4.67 ×10 2 = 467
You can use powers of 10 in this way to write large numbers. For
example, the average distance to the Sun is 149 600 000 km. You can
write this as 1.496 × 108 km. This is called standard form. You write a
number in standard form as a × 10n where 1 ⩽ a < 10 and n is an integer.
You can write small numbers in a similar way, using negative integer
powers of 10. For example:
4.67 ×10 −1 = 0.467
4.67 ×10 −2 = 0.0467
−3
Tip
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4.67 ×10 = 0.004 67
4.67 × 102 is the
same as
4.67 × 100 or
4.67 × 10 × 10
Think of 4.67 × 10−1
as 4.67 ÷ 10
4.67 ×10 −7 = 0.000 004 67
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Small numbers occur often in science. For example, the time for light to
travel 5 metres is 0.000 000 017 seconds. In standard form, you can write
this as 1.7 ×10 −8 seconds.
Worked example 1.2
Write these numbers in standard form.
a
1 million = 1 000 000 or 10
So 256 million = 256 000 000 = 2.56 × 108
6
b 1 billion = 1 000 000 000 or 109
So 25.6 billion = 25 600 000 000 = 2.56 × 1010
c
Standard form is
also sometimes
called scientific
notation.
256 million b 25.6 billion c 0.000 025 6
Answer
a
Tip
0.000 025 6 = 2.56 × 10−5
Tip
Notice that in
every case the
decimal point
is placed after
the 2, the first
non-zero digit.
14
Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication.
1.2 Standard form
Exercise 1.2
2
3
4
Write these numbers in standard form.
a
300 000
b 320 000
c
328 000
d 328 710
Write these numbers in standard form.
a
63 000 000
b 488 000 000
c
3 040 000
d 520 000 000 000
These numbers are in standard form. Write each number in full.
a
5.4 × 103
b 1.41 × 106
c
2.337 × 1010
d 8.725 × 107
Here are the distances of some planets from the Sun.
Write each distance in standard form.
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1
Planet
Mercury
Distance (km) 57 900 000
5
Uranus
2 870 000 000
Here are the areas of four countries.
Country
Area (km2)
a
b
c
China
Indonesia Russia
Kazakhstan
6
6
7
9.6 × 10 1.9 × 10 1.7 × 10 2.7 × 106
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Which country has the largest area?
Which country has the smallest area?
Copy and complete this sentence with a whole number:
The largest country is approximately … times larger than the
smallest country.
Write these numbers in standard form.
a
0.000 007
b 0.000 812
c
0.000 066 91
d 0.000 000 205
These numbers are in standard form. Write each number in full.
a
1.5 × 10−3
b 1.234 × 10−5
c
7.9 × 10−8
d 9.003 × 10−4
The mass of an electron is 9.11 × 10−31 kg.
This is 0.000…911 kg.
a
How many zeros are there between the decimal point and the 9?
b Work out the mass of 1 million electrons.
Give the answer in kilograms in standard form.
SA
6
Mars
227 900 000
7
8
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Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication.
1 Number and calculation
9
Here are four numbers:
w = 9.81 × 10−5
11
12
13
z = 4 × 10−4
a
Which number is the largest?
b Which number is the smallest?
a
Explain why the number 65 × 104 is not in standard form.
b Write 65 × 104 in standard form.
c
Write 48.3 × 106 in standard form.
Write these numbers in standard form.
a
15 × 10−3
b 27.3 × 10−4
c
50 × 10−9
Do these additions. Write the answers in standard form.
a
2.5 × 106 + 3.6 × 106
b 4.6 × 105 + 1.57 × 105
c
9.2 × 104 + 8.3 × 104
Do these additions. Write the answers in standard form.
a
4.5 × 10−6 + 3.1 × 10−6
b 5.12 × 10−5 + 2.9 × 10−5
c
9 × 10−8 + 7 × 10−8
aMultiply these numbers by 10. Give each answer in standard
form.
i 7 × 105
ii
3.4 × 106
iii 4.1 × 10−5
iv 1.37 × 10−4
b Generalise your results from part a.
c
Describe how to multiply or divide a number in standard
form by 1000.
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y = 9.091 × 10−5
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10
x = 2.8 × 10−4
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What are the advantages of writing numbers in standard form?
Summary checklist
I can write large and small numbers in standard form.
16
Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication.
1.3 Indices
1.3 Indices
In this section you will …
use positive, negative and zero indices
•
use index laws for multiplication and division.
This table shows powers of 3.
32
9
33
27
34
81
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•
35
243
36
729
Tip
When you move one column to the right, the index increases by 1 and
the number multiplies by 3.
9 × 3 = 27
27 × 3 = 81 81 × 3 = 243, and so on.
When you move one column to the left, the index decreases by 1 and the
number divides by 3. You can use this fact to extend the table to the left:
3−3
3−2
3−1
30
31
32
33
34
35
36
1
81
1
27
1
9
1
3
1
3
9
27
81
243
729
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3−4
9 ÷ 3 = 3 3 ÷ 3 = 1 1 ÷ 3 = 1 1 ÷ 3 = 1 1 ÷ 3 = 1 , and so on.
3
3
9
9
27
You can see from the table that 3 = 3 and 3 = 1.
1
0
3−2 = 12 3−3 = 13 , and so on.
Also: 3−1 = 1 3
3
3
SA
In general, if n is a positive integer then 3− n = 1n . These results are not
The index is the
small red number.
Tip
30 = 1 seems
strange but it fits
the pattern.
3
only true for powers of 3. They apply to any positive integer.
8−3 = 13 = 1 60 = 1
For example: 5−2 = 12 = 1 5
25
8
512
In general, if a and n are positive integers then a0 = 1 and a − n = 1n .
a
Exercise 1.3
1
2
Write each number as a fraction.
a
4−1
b 2−3
c
−3
−4
d 6
e
10
f
Here are five numbers: 2−4 3−3 4−2 5−1 60
List the numbers in order of size, smallest first.
9−2
2−5
17
Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication.
1 Number and calculation
3
4
Write these numbers as powers of 2.
1
1
a
b
c
64
d
f
8−1
c
f
1
0.000 001
2
1
64
4
e
1
Write each number as a power of 10.
a
100
b 1000
d 0.1
e
0.001
Write 1
64
a
as a power of 64
b as a power of 8
c
as a power of 4
d as a power of 2.
1
6
a
Write as a power of a positive integer.
81
b How many different ways can you write the answer to part a?
7
When x = 6, find the value of
a
x2
b x−2
c
x0
d x−3
8
Write m−2 as a fraction when
a
m=9
b m = 15
c
m=1
d m = 20
2
−2
9
y = x + x and x is a positive number.
a
Write y as a mixed number when
i x=1
ii
x=2
iii x = 3
b Find the value of x when
i y = 25.04
ii
y = 100.01
10 a
Write the answer to each multiplication as a power of 3.
i 32 × 33
ii
34 × 35
iii 36 × 34
iv 3 × 35
b In part a you used the rule 3a × 3b = 3a + b when the indices are
positive integers.
In the following multiplications, a or b can be negative
integers.
Show that the rule still gives the correct answers.
i 32 × 3−1
ii
3−2 × 3
iii 33 × 3−1
iv 3−1 × 3−1
v 3−2 × 3−1
c
Write two examples of your own to show that the rule works.
d Give your work to a partner to check.
11 Write the answer to each multiplication as a power of 5.
a
5 4 × 52
b 54 × 5−2
c
5−4 × 52
d 5−4 × 5−2
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5
Tip
Write out the
numbers and
multiply.
18
Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication.
1.3 Indices
12 Write the answer to each multiplication as a single power.
a
6 −3 × 62
b 75 × 7 −2
c
11−4 ×11−6
d 4 −6 × 42
13 Find the value of x in each case.
a
25 × 2 x = 2 9
b 3x × 3−2 = 34
c
4 x × 4 −3 = 4 −5
d 12 −3 ×12 x = 12 2
14 a
b
c
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Think like a mathematician
Write as a single power
i
25 ÷ 23
ii
45 ÷ 42
iii 56 ÷ 55
iv 210 ÷ 27
The rule for part a is that na ÷ nb = na −b when the indices a and b are positive
integers.
Write some examples to show that this rule also works for indices that are
negative integers.
Give your examples to a partner to check.
SA
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15 Write the answer to each division as a single power.
a
6 2 ÷ 65
b 93 ÷ 94
c
152 ÷156
d 103 ÷108
16 Write the answer to each division as a single power.
a
22 ÷ 2 −3
b 85 ÷ 8−2
5−4 ÷ 52
c
d 12 −3 ÷12 −5
17 Write down
a
83 as a power of 2
b 8−3 as a power of 2
c
272 as a power of 3
d 27−2 as a power of 3
e
272 as a power of 9
f
27−2 as a power of 9.
Summary checklist
I can understand positive, negative and zero indices.
I can use the addition rule for indices to multiply powers of the same number.
I can use the subtraction rule for indices to divide powers of the same number.
19
Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication.
1 Number and calculation
Check your progress
2
3
4
5
6
7
c
6.25
62.5
625
d
e
Write whether each number is rational or irrational. Give a reason for each answer.
32 + 42
9+ 7
a
b
Without using a calculator, find an integer n such that n < 3 50 < n + 1.
Write each number in standard form.
a
86 000 000 000
b 0.000 006 45
Write these numbers in order of size, smallest first.
A = 9 × 10−4
B = 6 × 10−3 C = 8 × 10−5 D = 7.5 × 10−4
Write each number as a fraction.
a
7−2
b 3−4
c
2−7
Write each number as a power of 5.
a
125
b 1
c
0.04
Write the answer to each calculation as the power of a single number.
a
68 × 6 −3
b 12 −2 ×12 −3
c
4 2 ÷ 48
d 15−4 ÷15−6
SA
M
8
Write whether each number is rational or irrational.
5
a
4
b
PL
E
1
20
Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication.