TOPIC 3
Indices
By the end of this topic, you should be able to:
ü Write a repeated multiplication as a power
ü Write an index in expanded form
ü Use indices to express a number as a product of its factors and of
its prime factors
ü Derive the laws of indices by investigation
ü Apply the laws of integral indices to solve problems that involve
indices
ü Express a number in standard form
ü Solve problems that involve the practical use of standard form.
Oral activity
Discuss indices
1 Discuss what you remember about indices.
2 Write an expression that involves an index.
3 Compare your expression with those of the other members
of your group.
4 Discuss how you would add, subtract, multiply and divide
expressions that involve indices.
GROUP
In mathematics, we can write expressions that involve indices in index notation or in
standard form.
24
TOPIC 3 Indices
Index notation
Key concept
43
In index notation, we write the expression 4 4 4 as
and
3
we read 4 as “four cubed” or as “4 to the power of 3”. In this
expression, 4 is the base and 3 is the index. The index shows the
number of times you multiply the base by itself. So 43 means that you
multiply 4 by itself twice. In expanded form, we write this expression
as 4 4 4.
Read, write and
interpret numbers in
index notation.
Note
In index notation,
the index shows the
number of times we
multiply the base by
itself.
Example 1
1 Find the base of 85.
2 Find the index of 85.
3 Find the value of 85.
Solutions
1 8
2 5
3 32 768
Example 2
1 Write 95 in expanded form.
2 Find the value of 95.
Solutions
1 95 = 9 9 9 9 9
2 95 = 59 049
Exercise 1
1
2
3
4
5
6
7
Work with index notation and
expanded form
a) Find the base of 43.
b) Find the index of 43.
a) Find the fourth power of 2.
b) Find the third power of 8.
Write 10 10 10 in index notation.
Write 75 in expanded form.
a) Find the value of 73.
b) Find the value of 37.
Use a calculator to find the value of each expression.
a) 38
b) 25
c) 95
d) 174
a) Write 5 5 5 5 5 5 5 5 in index notation.
b) Find the value of this expression.
SINGLE
Emerging issue
In science and
technology, we write
very large and very
small numbers in
index notation.
SO 2.1.10.1.1
25
TOPIC 3 Indices
Key concept
Factors and prime factors
Express numbers
as products of their
factors and of their
prime factors.
A factor of a given number divides exactly into that number
without leaving a remainder. For example, 3 is a factor of 12,
because 12 3 = 4. Other factors of 12 are 1, 2, 4, 6 and 12.
Numbers that have more than two factors are composite numbers.
Examples of composite numbers are 6, 8 and 12.
Numbers that have only two different factors are prime numbers.
Examples of prime numbers are 2, 3, 5, 7 and 11. The number one, 1,
is not a prime number because it does not have two different factors.
To express a number as the product of its prime factors, write
12 = 2 2 3. To express a number in index notation, write
12 = 3 22. To find the prime factors of a number, keep dividing the
number by prime numbers until you break down all the composite
factors into prime factors.
Example 3
Express 24 as the product of its prime factors. Then express 24 in
index notation.
Solution
2 24
2 12
2 6
3 3
1
Divide 24 by 2 and write 12 on the next line.
Divide 12 by 2 and write 6 on the next line.
Divide 6 by 2 and write 3 on the next line.
Divide 3 by 3 and write 1 on the next line.
Expressed as the product of its prime factors, 24 = 2 2 2 3.
In index notation, 24 = 23 3.
Exercise 2
1
2
3
4
5
26
SO 2.1.10.1.1; 2.1.10.1.2
Find factors and prime factors
a) Explain the difference between a composite number
and a prime number.
b) Provide one example of each.
Find all the factors of each number.
a) 5
b) 6
c) 8
d) 20
a) Find all the factors of 30.
b) Find the prime factors of 30.
a) Find the prime factors of 36.
b) Write 36 as the product of its prime factors.
a) Find the prime factors of 54.
b) Use index notation to write 54 as the product of its
prime factors.
SINGLE
TOPIC 3 Indices
Laws of indices
Key concept
There are rules that we need to follow when we work with indices.
These rules are the laws of indices. The laws of indices help us
simplify calculations when we need to multiply or divide a power
by another power that has the same base, for example 23 25
or 38 33. They also help us simplify calculations when a power
is raised to another power, for example (42)3.
Identify rules for
working with indices.
Activity 1
Investigate the product of two powers
with the same base
PAIR
1 Copy the table into your exercise book.
Expression Factors in expanded Answer Result
form
in index
notation
31 32
32 33
33 34
34 35
35 36
3m 3n
(3) (3 3)
(3 3) (3 3 3)
…
…
…
(3 3 … m factors)
(3 3 …
n factors)
33
3…
……
……
……
……
Note
Expanded form
shows all the factors
of an expression
that is given in index
notation.
31 32 = 31 2 = 33
32 33 = 32 3 = …
…
…
…
3m 3n = 3… …
2 Complete the table by filling in the missing factors, powers,
indices and results.
3 Look at the last column. Describe how to find the result
when you multiply two powers that have the same base.
4 Fill in the missing word: To multiply two powers that have
the same base, … the indices to get the answer.
5 Copy and fill in the missing numbers: 23 25 = 2… … = 2…
Apply the first law of indices
In multiplication, we can break down numbers into prime factors and
apply the first law of indices to simplify the expression.
Exercise 3
Apply the first law of indices
1 Write each expression as a single power.
a) 53 55
b) 74 73 75
c) a4 a2 a3
2 Write each number as the product of its prime factors.
a) 16
b) 72
c) 8 575
3 Write each multiplication as the product of its prime factors.
a) 81 128
b) 375 98
c) 968 972
4 Write each multiplication as the product of its prime factors.
a) 8 27
b) 48 576
c) 375 2 025
SINGLE
SO 2.1.10.1.2; 2.1.10.1.3
27
TOPIC 3 Indices
Activity 2
Investigate the quotient of two powers
with the same base
PAIR
1 Copy the table into your exercise book.
Cancel factors Answer Result
to simplify
in index
notation
Expression Factors in
expanded
form
4
3
___
32
7
3
___
33
3m
___
3n
3333 3333
33
33
3…
…
…
333…
m factors
333…
n factors
……
…
…
333…
m factors
333…
n factors
34 32
34 2
3…
…
3m n
3m 3n
3… …
2 Complete the table by filling in the missing factors, powers,
indices and results.
3 Look at the last column. Describe how to find the result
when you divide two powers that have the same base.
4 Fill in the missing word: To divide two powers that have
the same base, … the indices to get the answer.
Activity 3
Investigate a power raised to a power
PAIR
1 Copy the table into your exercise book.
Expression Factors in expanded
form
(32)2
32 32
(32)3
32 32 32
…
(32)5
(3m)n
Answer Result
in index
notation
333
34
3
3 3 3 3…
333
…
……
3m 3m 3 3 …
… n factors (m n)
factors
3m n
(32)2 = 32 2
= 34
(32)3 = 3… …
= 3…
(32)5 = 3… …
= 3…
m
n
(3 ) = 3… …
= 3…
2 Complete the table by filling in the missing factors, powers,
indices and results.
3 Look at the last column. Describe how to find the result
when you raise a power to another power.
4 Fill in the missing word: To raise a power to another power,
… the two indices to get the answer.
28
SO 2.1.10.1.3
TOPIC 3 Indices
Summary of the laws of indices
To multiply two powers that have the same base, add the indices to
get the answer, for example 32 34 = 32 4 = 36 = 729.
To divide two powers that have the same base, subtract the indices to
get the answer, for example 56 53 = 56 3 = 53 = 125.
To raise a power to another power, multiply the two indices to get the
answer, for example (42)3 = 42 3 = 46 = 4 096.
Example 4
1 Calculate 23 22.
a) Write the answer in index notation.
b) Calculate the value of 23 22.
2 Calculate 47 44.
a) Write the answer in index notation.
b) Calculate the value of 47 44.
3 Calculate (33)2.
a) Write the answer in index notation.
b) Calculate the value of (33)2.
Solutions
1 a) 23 22 = 23 2
2 a) 47 44 = 47 4 = 43
3 a) (33)2 = 33 2 = 36
Exercise 4
b) 32
b) 64
b) 729
Apply the laws of indices
1 Simplify each expression and leave the answers in index
notation.
a) 22 25
b) 43 42
c) 34 3
d) 52 54
e) 7 75
f) 62 62
8
9
12
2
g) 8 8
h) 2 2
i) 111 15
2 Simplify each expression and write the answers as a single
power.
a) 35 32
b) 27 24
c) 64 62
4
8
5
d) 4 4
e) 5 5
f) 19 14
g) 82 81
h) 77 76
i) 86 84
3 Simplify each expression and leave the answers in index
notation.
a) (55)2
b) (32)4
c) (23)2
d) (72)4
e) (43)5
f) (61)2
5
6
2
5
g) (1 )
h) (8 )
i) (93)3
4 Calculate the value of each expression.
a) 85 82
b) 27 22
c) (43)2
d) 39 35
e) (52)3
f) 44 42
3
2
3
3
g) 1 1
h) (3 )
i) 33 32
SINGLE
SO 2.1.10.1.4
29
TOPIC 3 Indices
The meaning of a0 and a1
Activity 4
Investigate zero indices and negative indices
PAIR
1 Copy the table into your exercise book.
Cancel
factors to
simplify
Answer Result
in index
notation
3333
__________
3333
__________
3333
__________
3333
__________
…
__
…
…
__
…
3 3 3 34 31 27
= 34 1
33
= 33
33
34 32 9
…
3
34 2
3…
…
…
…
…
__
…
…
__
…
Expression Factors in
expanded
form
4
3
___
3
31
4
3
___
33
32
4
3
___
33
4
3
___
34
4
3
33
…
…
3333
3 3 3 3 __
1 31
_____________
_____________
33333 33333 3
3
___
35
Value
…
34 35 __13
34 5
3…
2 Complete the table by filling in the missing factors, powers,
indices, results and values.
3 Fill in the missing numbers: 34 34 = 34 4 = 3…
34 = __________
3 3 3 3 = ___
81 = …
__
4
3333
…
…
Therefore,
4 Fill in the missing numbers: The value of a number raised
to the power of 0 equals …, for example 5… = …
3
30 =
5
2 = 2… … = 2… = …
5 Fill in the missing numbers: __
5
2
6 Fill in the missing numbers: 34 36 = 34 6
=
3…
34 = ________________
3333
1
__
= ___
6
3
333333
Therefore,
…
…
7 Fill in the missing numbers: The value of 53 = ( __15 )… = …
Note
The multiplicative
inverse of x is __1x . The
product of a number
and its multiplicative
inverse is always equal
to 1.
8 Fill in the missing word: When you replace the base of a
power with its reciprocal, the … of the index changes.
9 Fill in the missing numbers in each equation.
a)
c)
e)
g)
30
3… =
SO 2.1.10.1.3
( __14 ) 2 = 4…
( __15 )4 = 5…
( __12 ) 4 = 2…
3 4=… 4
( __
6)
b) 73 = (…)3
d) 32 = __13 …
f)
h)
( )
8… = ( __18 )2
1 …
37 = __
3
TOPIC 3 Indices
Example 5
Fill in the missing numbers in each equation.
1 34 34 = 3…
2 52 … = 1
3 7… = 1
Solutions
1 34 34 = 30
Exercise 5
2 52 52 = 1
3 70 = 1
Use zero indices and negative indices
1 Fill in the missing numbers in each equation.
a) 53 53 = 5…
b) 22 … = 1
26 = 2… = …
d) __
c) 8… = 1
26
2 Find the reciprocal of each number.
a) 2
b) __13
c) __23
d) 5
3 Write each expression with positive indices.
a) 52
b) ( __17 ) 3
d) 101 102
c) 34 36
4 Write each expression as a single power.
a) 32 33
b) (42)2
4
5
c) 2 2
d) 53 53
5 Fill in the missing numbers in each equation.
b) 74 7… = 72
a) 23 = ( __12 )…
c)
( )
1 …
33 = __
3
SINGLE
d) 10 10 = 1… = …
Standard form
Key concept
We write very large numbers and very small numbers in standard
form so that it is easier to work with them. To write a number
in standard form, convert the number to a product of a number
between 1 and 10 and a power of 10, for example 3 170 = 3.17 103.
Use standard form to
read and write large
and small numbers.
Example 6
1 Write 0.00184 in standard form.
2 Convert 3.71 103 to a decimal.
Solutions
1.84 = ____
1.84 = 1.84 103
1 0.00184 = _____
3
1 000
2 3.71
3
10
10
3.71 = _____
3.71 =
= ____
1 000
103
0.00371
We can add, subtract, multiply and divide numbers in standard form.
Example 7 and Exercise 6 will help you understand how to perform
these operations.
SO 2.1.10.1.4; 2.1.10.1.5
31
TOPIC 3 Indices
Note
Example 7
The decimal part of a
number in standard
form is rounded to
two decimal places.
1 Calculate 3.27 105 8.53 103.
2 Calculate 4.85 102 2.57 104.
3 Calculate 5.02 103 8.27 107.
Solutions
1 3.27 105 8.53 103 = 327 000 8 530
= 335 530
≈ 3.36 105
2
4
2 4.85 10 2.57 10 = 12.4645 102 4
= 12.4645 106
≈ 1.25 107
3 5.02 103 8.27 107 = 0.6070 103 7
= 0.6070 104
≈ 6.07 105
Emerging issue
Exercise 6
In environmental
management, we
use standard form to
write large quantities
and volumes. For
example, we estimate
the annual inflow of
the Okavango Delta
to be 1.01 1010 m3.
1 Write each number in standard form.
a) 3 290 000
b) 5 140
c) 904 000 000
d) 21 500
e) 8 000 000
f) 700
2 Write each decimal fraction in standard form.
a) 0.0052
b) 0.088
c) 0.0000152
d) 0.000014
e) 0.0000000076 f) 0.6
3 Write each expression as a whole number.
a) 2.08 104
b) 3.72 103
c) 7.88 106
5
8
d) 1 10
e) 3.8 10
f) 6.504 109
4 Convert each expression to a decimal.
a) 1.84 102
b) 2.12 104
c) 8.01 103
d) 6 105
e) 7.15 107
f) 5.001 102
5 Calculate the value of each expression. Write the answers
in standard form.
a) 8.09 104 3.88 103
b) 5.24 103 7.44 102
c) 3.18 104 6.54 106 d) 9.21 103 4.33 105
6 The electrical currents in two wires are 8.2 1011 amperes
and 7.28 1010 amperes respectively. Calculate the total
current in the two wires.
7 The distance from Earth to a satellite is approximately
2.37 1010 m. A person on Earth sends a radio message,
which travels at a speed of 3 108 m/s, to the satellite.
Calculate how long it will take the message to reach
the satellite.
32
SO 2.1.10.1.5; 2.1.10.1.6
Work with numbers in standard form
SINGLE
TOPIC 3 Indices
Summary
ü Use indices to write a repeated
multiplication as a power. For example,
3 3 3 3 3 = 35.
ü Use indices to express a number as the
product of its prime factors. For example,
108 = 22 33.
ü The laws of indices are as follows.
ü am an = amn
ü am an = am n
ü (am)n = amn
ü You can write very large and very small
numbers in standard form to make
them easier to read and write, and to
simplify calculations. For example,
56 190 000 000 = 5.62 1010 and
0.0007944 = 7.94 104.
Revision
1 Write 4 4 4 4 4 4 in index
notation.
2 a) Write 24 in index notation.
b) Write 18 as the product of its prime
factors.
3 Write each expression as a single power.
a) 35 33
b) 53 57
c) (23)4
d) 22 24
4 Calculate the value of each expression.
a) 53
b) 102
c) 41
d) 70
5 a) Write 4 090 000 000 in standard
form.
b) Convert 3.72 103 to a decimal.
6 Calculate the value of each expression.
Write the answers in standard form.
a) 1.73 106 2.57 106
b) 2.15 103 9.74 104
c) 2.72 103 8.05 104
7 Calculate the value of each expression.
Write the answers in standard form.
a) 3.56 105 9.18 105
b) 6.27 104 5.07 103
c) 2.76 107 6.98 106
8 Calculate the value of each expression.
Write the answers in standard form.
a) 8.74 108 3.22 108
b) 3.86 104 5.72 104
c) 6.90 108 2.15 103
9 We measure the pressure that a standing
person exerts on a floor in N/m2, where N
is the weight of the person. The weight of
a person is equal to their mass 10. For
example, if your mass is 56 kg then your
weight is 560 N. Calculate the pressure
exerted on the floor by a 62-kg person if
the total area of their shoes is 280 cm2.
10 The formula λ = __fc represents the
relationship between the wavelength, λ,
the speed of light, c, and the frequency,
f, of light waves. We measure λ in metres,
c in m/s and f in hertz, Hz. Calculate the
wavelength of light waves that have a
frequency of 4.30 1014 Hz if the speed
of light is 3 108 m/s.
11 One sodium atom has 11 protons,
12 neutrons and 11 electrons.
The mass of 1 electron is about
0.00000000000000000000000000000091094 kg.
The mass of 1 proton is about
0.000000000000000000000000001673 kg.
The mass of 1 neutron is the same as the
mass of 1 proton.
a) Write the total mass of i) electrons,
ii) protons and iii) neutrons in
1 sodium atom in standard form.
b) Calculate the mass of 1 sodium atom.
Write the answer in standard form.
12 In astronomy, we use the astronomical
unit to measure distance. The symbol
for astronomical unit is au.
1 au = 149 598 000 km, which is the
average distance between Earth and the
Sun. On a certain day, Saturn is 8.83 au
from Earth. Calculate the distance between
Earth and Saturn in kilometres. Write the
answer in standard form.
33