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