Laws of Indices
© Christine Crisp
Laws of Indices
Multiplying with Indices
e.g.1
23  24  2 2 2 
 27
2 2 2 2
 2 3 4
e.g.2
(2) ´ (2) = (2) ´ (2) ´ (2) ´ (2) ´ (2)
5
= (2)
= (2)2+3
2
3
Generalizing this, we get:
a a  a
m
n
mn
Laws of Indices
Multiplying with Indices
a a  a
m
n
mn
If m and n are not integers, a must be positive
1
22
e.g.3


1  3
22 2
2
2
a a  a
m
3
22
n
(a  0)
mn
      (1)
Laws of Indices
Dividing with Indices
1
e.g.
35  32 
1
3 3 3 3 3
3 3
 33
3
1
Cancel
1
52
Generalizing this, we get:
a
m
a  a
n
(a  0)
m n
      ( 2)
Laws of Indices
Powers of Powers
e.g.
4 2
(3 )
 34  34
by rule (1)
 38
 3 4 2
a  (a 0a)
m n
mn
      ( 3)
Laws of Indices
Exercises
Without using a calculator, use the laws of indices to
express each of the following as an integer
1.
2.
2 2
3
47
4
3.
5
2 
3 2
7
 210  1024
 4 2  16
 2 6  64
Laws of Indices
A Special Case
e.g. Simplify
Using rule (3)
Also,
24  24
24  24  24  4
24  24
 20
2  2  2 2

2  2  2 2
1
Laws of Indices
A Special Case
e.g. Simplify
24  24
Using rule (2)
Also,
24  24  24  4
 20
2  2  2 2
4
4

2 2
2  2  2 2
1
So,
20  1
Generalizing this, we get:
a 0  1       ( 4)
Laws of Indices
Another Special Case
e.g. Simplify
5 5
3
7
53  57  5 3  7
 54
Using rule (3)
Also, 5 3  5 7 
1
1
555
5555555
1

1
1
54
1
1
Laws of Indices
Another Special Case
e.g. Simplify
5 5
3
7
53  57  5 3  7
 54
Using rule (3)
1
1
1
555
Also, 5  5 
5555555
3
7
1

So,
1
1
1
54
5
4

1
54
Laws of Indices
Another Special Case
Generalizing this, we get:
a
3
e.g. 1
4
e.g. 2
1
2
n
3


1
an
      ( 5)
1
1

3
64
4
3
2
 8

Laws of Indices
Rational Numbers
A rational number is one that can be written as
p
q
where p and q are integers and q  0
e.g.
4
7
and 3
2 and

 3

 are rational numbers
1 

are not rational numbers
Laws of Indices
Rational Numbers
The definition of a rational index is that
a
p
q
e.g.1
1
42

e.g.2
2
27 3

e.g.3
16
1
2


q
p
a
      ( 6)
p is the power
42
3
2
q is the root
27 
3 9
1
1
1
16 2

2
1

16 4
Laws of Indices
SUMMARY
The following are the laws of indices:
a a  a
m
n
mn
a
a 
m n
m
a  a
n
 a mn
a 1
1
n
a 
n
a
p
0
aq 
q
a
p
mn
Laws of Indices
Exercises
Without using a calculator, use the laws of indices to
express each of the following as an integer
1.
5 1
2.
1
25 2
3.
0
39
3
7
 25  5
 32  9
Laws of Indices
Exercises
Without using a calculator, use the laws of indices to
express each of the following as an integer or fraction
4
4
3
4
4.
83
 8  2  16
5.
6.
3
9
2
3
2
1
1


2
9
3

1
3
92
1
1
1



2
3
3
27
3
9
Laws of Indices
Laws of Indices
The following slides contain repeats of
information on earlier slides, shown without
colour, so that they can be printed and
photocopied.
For most purposes the slides can be printed
as “Handouts” with up to 6 slides per sheet.
Laws of Indices
SUMMARY
The following are the laws of indices:
a a  a
m
n
mn
am  an  amn
a 
m n
a
mn
a0  1
1
n
a 
n
a
p
a
q

q
a
p
Laws of Indices
Examples
1.
50  1
2.
1
25 2
3.
3
9
37
 25  5
3 9
2
Laws of Indices
4.
5.
6.
4
83
3
9
3
 8
2
3
2
4
 2  16
4
1
1


2
9
3

1
3
92
1
1
1



2
3
3
27
3
9