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Laws of Indices or Laws of Indices or PowersPowers
© Christine Crisp
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
nmnm aaa
e.g.3
108 22
1082
182
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 )2(
Laws of IndicesPowers of
Powers24 )3(e.g.
44 33 by rule
(1)83
243
nmnm aa )3(
Laws of IndicesPowers of
Powers34 )4(e.g.
444 444 by rule
(1)124
344
nmnm aa )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 Indices
1 112 2x x x
1
2 so x = x
and
1nn
mmnn
In general x x
x x
Fractional Powers
1 1 11 33 3 3x x x x x
1
3 so x
2 12 233 3x (x ) ( x )
2
3 so x
i.e. the 4
Laws of Indices
The definition of a rational index is that
p is the powerq is the root
e.g.1 21
9 39
e.g.2 32
27 23 27 932
e.g.3 21
16 21
16
1
4
1
16
1
Rational Numbers
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