Laws of indices

1. 19: Laws of Indices19: Laws of Indices
© Christine Crisp
““Teach A Level Maths”Teach A Level Maths”
Vol. 1: AS Core ModulesVol. 1: AS Core Modules

2. Laws of Indices
Module C1
Edexcel
OCR
MEI/OCR
Module C2
AQA
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3. Laws of Indices
Generalizing this, we get:
Multiplying with Indices
e.g.1 =× 43
22 2222222 ××××××
7
2=
43
2 +
=
e.g.2 =−×− 32
)1()1( )1()1()1()1()1( −×−×−×−×−
5
)1(−=
32
)1( +
−=
nmnm
aaa +
=×

4. Laws of Indices
If m and n are not integers, a must be positive
nmnm
aaa +
=×
e.g.3 2
3
2
1
22 ×
2
3
2
1
2
+
=
2
2=
Multiplying with Indices
nmnm
aaa +
=×
)0( >a
)1(−−−−−−

5. Laws of Indices
33
33333
×
××××
Generalizing this, we get:
Dividing with Indices
1
Cancel
1
1 1
e.g. =÷ 25
33
3
3=
25
3 −
=
nmnm
aaa −
=÷
)0( >a
)2(−−−−−−

6. Laws of Indices
Powers of Powers
24
)3(e.g.
44
33 ×=
by rule (1)
8
3=
24
3 ×
=
( ) nmnm
aa ×
=
)0( >a
)3(−−−−−−

7. Laws of Indices
Exercises
Without using a calculator, use the laws of indices to
express each of the following as an integer
1.
2.
3.
73
22 ×
1642
==
( )23
2 6426
==
5
7
4
4
1024210
==

8. Laws of Indices
A Special Case
e.g. Simplify 44
22 ÷
Using rule (3) 44
22 ÷ 44
2 −
=
0
2=
2222
2222
×××
×××
=
1=
Also, 44
22 ÷

9. Laws of Indices
1=
0
2=
e.g. Simplify
Also,
44
22 ÷
44
22 ÷Using rule (2) 44
2 −
=
2222
2222
×××
×××
=44
22 ÷
So, 0
2 1=
Generalizing this, we get:
A Special Case
10
=a )4(−−−−−−

10. Laws of Indices
5555555
555
××××××
××
Another Special Case
1
1 1
1 1
1
e.g. Simplify 73
55 ÷
Using rule (3) 73
5 −
=73
55 ÷
4
5−
=
Also, =÷ 73
55
4
5
1
=

11. Laws of Indices
=÷ 73
55
73
5 −
=73
55 ÷
5555555
555
××××××
××
e.g. Simplify
Using rule (3)
Also,
1
1 1
1 1
1
73
55 ÷
4
5−
=
4
5
1
=
So, 4
5−
4
5
1
=
Another Special Case

12. Laws of Indices
Generalizing this, we get:
e.g. 1 =−3
4 =
3
4
1
64
1
e.g. 2 =
−3
2
1
=3
2 8
Another Special Case
n
n
a
a
1
=−
)5(−−−−−−

13. Laws of Indices
Rational Numbers
A rational number is one that can be written as
where p and q are integers and
q
p
0≠q
e.g. and are rational numbers
7
4
3− 




 −
=
1
3
are not rational numbersand2 π

14. Laws of Indices
The definition of a rational index is that
p is the power
q is the roote.g.1 =2
1
4 24 =
e.g.2 =3
2
27 =
23
27 932
=
e.g.3 =
−
2
1
16 =
2
1
16
1
4
1
16
1
=
Rational Numbers
pq
aa q
p
= )6(−−−−−−

15. Laws of Indices
SUMMARY
The following are the laws of indices:
nmnm
aaa +
=× nmnm
aaa −
=÷
( ) nmnm
aa ×
=
10
=a
n
n
a
a
1
=−
pq
aa q
p
=

16. Laws of Indices
Exercises
Without using a calculator, use the laws of indices to
express each of the following as an integer
1.
2.
3.
0
5 1=
2
1
25 525 ==
7
9
3
3
932
==

17. 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.
5.
6.
3
4
8
2
3−
2
3
9
−
1628 4
43
===
9
1
3
1
2
==
27
1
3
1
9
1
9
1
332
2
3
====

18. Laws of Indices