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19: Laws of Indices19: Laws of Indices
© Christine Crisp
““Teach A Level Maths”Teach A Level Maths”
Vol. 1: AS Core Vol. 1: AS Core ModulesModules
Laws of Indices
Module C1
Edexcel
OCR
MEI/OCR
Module C2AQA
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Laws of Indices
Generalizing this, we get:
Multiplying with Indices
e.g.1 43 22 2222222
72432
e.g.2
32 )1()1( )1()1()1()1()1( 5)1(32)1(
nmnm aaa
Laws of Indices
If m and n are not integers, a must be positive
nmnm aaa
e.g.3
23
21
22
23
21
2
22
Multiplying with Indices
nmnm aaa )0( a
)1(
Laws of Indices
33
33333
Generalizing this, we get:
Dividing with Indices
1Cance
l
1
1 1
e.g. 25 33
33253
nmnm aaa )0( a
)2(
Laws of IndicesPowers of
Powers24 )3(e.g.
44 33 by rule
(1)83
243
nmnm aa )0( a
)3(
Laws of IndicesExercise
sWithout using a calculator, use the laws of indices to express each of the following as an integer
1.
2.
3.
73 22
1642
232 6426
5
7
4
4
1024210
Laws of IndicesA Special
Casee.g. Simplify 44 22
Using rule (3)
44 22 442 02
2222
2222
1
Also, 44 22
Laws of Indices
1
02
e.g. Simplify
Also,
44 22 44 22
Using rule (2)
442
2222
2222
44 22
So, 02 1Generalizing this, we
get:
A Special Case
10 a )4(
Laws of Indices
5555555
555
Another Special Case
1
1 1
1 1
1
e.g. Simplify 73 55 Using rule
(3)735 73 55 45
Also, 73 55
45
1
Laws of Indices
73 55
735 73 55
5555555
555
e.g. Simplify
Using rule (3)
Also,1
1 1
1 1
1
73 55
45
45
1
So, 45 45
1
Another Special Case
Laws of Indices
Generalizing this, we get:
e.g. 1 34 34
1
64
1
e.g. 2 32
1 32 8
Another Special Case
nn
aa
1 )5(
Laws of IndicesRational
Numbers
A rational number is one that can be written as
where p and q are integers and
q
p
0q
e.g. an
dare rational
numbers7
43
1
3
are not rational numbers
and
2
Laws of Indices
The definition of a rational index is that
p is the powerq is the roote.g.1 2
1
4 24
e.g.2 32
27 23 27 932
e.g.3 21
16 21
16
1
4
1
16
1
Rational Numbers
pqaa q
p
)6(
Laws of Indices
SUMMARYThe following are the laws of indices:
nmnm aaa nmnm aaa
nmnm aa
10 a
nn
aa
1
pqaa q
p
Laws of IndicesExercise
sWithout using a calculator, use the laws of indices to express each of the following as an integer
1.
2.
3.
05 1
21
25 525
7
9
3
3932
Laws of IndicesExercise
sWithout using a calculator, use the laws of indices to express each of the following as an integer or fraction
4.
5.
6.
34
8
23
23
9
1628 443
9
1
3
12
27
1
3
1
9
1
9
1332
23
Laws of Indices
