Post on 07-Apr-2018
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Laws of IndicesLaws of Indices
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Laws of Indices
Generalizing this, we get:
Multiplying with Indices
e.g.1 !v 43 22 2222222 vvvvvv7
2!43
2
!
e.g.2 !v 32 )1()1( )1()1()1()1()1( vvvv5)1(!
32)1(
!
nmnmaaa
!v
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Laws of Indices
If m and n are not integers, a must be positive
nmnmaaa
!v
e.g.3 23
2
1
22 v
2
3
2
1
2
!
2
2!
Multiplying with Indices
nmnmaaa
!v
)0( "a
)1(
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Laws of Indices
33
33333
v
vvvv
Generalizing this, we get:
Dividing with Indices
1
Cancel1
1 1
e.g. !z 25
!
25!
nmnm !z
)0( "a)2(
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Laws of Indices
Powers of Powers
24)3(e.g.
4433 v!
by rule (1)8
3!
243!
nmnm
aav
!
)0( "a)3(
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Laws of Indices
Exercises
Without using a calculator, use the laws of indices toexpress each of the following as an integer
1.
2.
3.
7322 v
1642
!!
232 6426 !!
5
7
4
4
1024210 !!
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Laws of Indices
A Special Case
e.g. Simplify 44 22 z
Using rule (3) 44 22 z 442 !
02!
2222
2222
vvv
vvv!
1!
Also, 44 22 z
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Laws of Indices
1!
0!
e.g. Simplify
Also,
44 22 z
4422 zUsing rule (2) 442 !
2222
2222
vvv
vvv!44 22 z
So, 0 1!
Generalizing this, we get:
A Special Case
10!a )4(
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Laws of Indices
5555555
555
vvvvvv
vv
Another Special Case
1
1 1
1 1
1
e.g. Simplify 73 55 z
Using rule (3) 735 73
55 z
45
Also, !73
4!
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Laws of Indices
!73
7373z
vvvvvv
vv
e.g. Simplify
Using rule (3)
Also,1
1 1
1 1
1
73 55 z
4
4
1
So, 45
4
1!
Another Special Case
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Laws of Indices
Generalizing this, we get:
e.g. 1 !3
!
3
1
6
1
e.g. 2 !3
2
1!32 8
Another Special Case
n
n
a
a
1!
)5(
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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 numbers7
43
1
3
are not rational numbersand2 T
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Laws of Indices
The definition of a rational index is that
p is the power
q is the roote.g.1 !21
4 24 !
e.g.2 !37 !3 7 93 !
e.g.3 !
2
1
16 !
2
1
16
1
4
1
16
1!
Rational Numbers
p
aa
p
!)6(
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Laws of Indices
SUMMARY
The following are the laws of indices:
nn v nmnm aaa z
nm
n
m aa v
10!a
n
n
a
a
1!
p
aa
p
!
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Laws of Indices
Exercises
Without using a calculator, use the laws of indices toexpress each of the following as an integer
1.
2.
3.
05 !
2
1
25 525 !!
7
9
3
393
2!!
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Laws of Indices
Exercises
Without using a calculator, use the laws of indices toexpress each of the following as an integer or fraction
4.
5.
6.
3
4
8
23
23
9
16284
43
9
1
3
1
2
27
1
3
1
9
1
9
1
332
2
3