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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