19: laws of indices © christine crisp “teach a level maths” vol. 1: as core modules

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

Laws of Indices

Module C1

Edexcel

OCR

MEI/OCR

Module C2AQA

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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

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(

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(

Laws of IndicesPowers of

Powers24 )3(e.g.

44 33 by rule

(1)83

243

nmnm aa )0( a

)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 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

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(

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

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.

Laws of IndicesSUMMARY

The following are the laws of indices:

nmnm aaa nmnm aaa

nmnm aa

10 a

nn

aa

1

pqaa q

p

Laws of Indices

1.

2.

3.

05 1

21

25 525

7

9

3

3932

Examples

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