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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
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Laws of Indices
Module C1
Edexcel
OCR
MEI/OCR
Module C2AQA
"Certain images and/or photos on this presentation are the copyrighted property of JupiterImages and are being used with permission under license. These images and/or photos may not be copied or downloaded without permission from JupiterImages"
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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
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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(
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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(
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Laws of IndicesPowers of
Powers24 )3(e.g.
44 33 by rule
(1)83
243
nmnm aa )0( a
)3(
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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
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Laws of IndicesA Special
Casee.g. Simplify 44 22
Using rule (3)
44 22 442 02
2222
2222
1
Also, 44 22
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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(
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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
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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
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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(
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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
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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(
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Laws of Indices
SUMMARYThe following are the laws of indices:
nmnm aaa nmnm aaa
nmnm aa
10 a
nn
aa
1
pqaa q
p
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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
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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
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Laws of Indices
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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.
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Laws of IndicesSUMMARY
The following are the laws of indices:
nmnm aaa nmnm aaa
nmnm aa
10 a
nn
aa
1
pqaa q
p
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Laws of Indices
1.
2.
3.
05 1
21
25 525
7
9
3
3932
Examples
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Laws of Indices
4.
5.
6.
34
8
23
23
9
1628 443
9
1
3
12
27
1
3
1
9
1
9
1332
23