What Is An Index Number.You should know that:

8 x 8 x 8 x 8 x 8 x 8 = 8 6 We say“eight to the power of 6”.

The power of 6 is an index number.

The plural (more than one) of index numbers is

indices.Hence indices are index numbers which are powers.

The number eight is the base number.

Multiplication Of Indices.

We know that : 7 x 7x 7 x 7 x 7 x 7 x 7 x 7 = 7 8

But we can also simplify expressions such as :

6 3 x 6 4To simplify:

(1) Expand the expression.= (6 x 6 x 6) x (6 x 6 x 6 x 6)

(2) How many 6’s do you

now have?

7

(3) Now write the expression

as a single power of 6.

= 6 7

Key Result.

6 3 x 6 4 = 6 7

Using the previous example try to simplify the following

expressions:

(1) 3 7 x 3 4

= 3 11

(2) 8 5 x 8 9

= 8 14

(3) 4 11 x 4 7 x 4 8

= 4 26

We can now write down our first rule of index numbers:

Rule 1 : Multiplication of Indices.

a n x a m = a n + m

NB: This rule only applies to indices with a common

base number. We cannot simplify 3 11 x 4 7 as 3 and 4

are different base numbers.

What Goes In The Box ? 1

Simplify the expressions below :

(1) 6 4 x 6 3

(2) 9 7 x 9 2

(3) 11 6 x 11

(4) 14 9 x 14 12

(5) 27 25 x 27 30

(6) 2 2 x 2 3 x 2 5

(7) 8 7 x 8 10 x 8

(8) 5 20 x 5 30 x 5 50

= 6 7

= 9 9

= 11 7

= 14 21

= 27 55

= 2 10

= 8 18

= 5 100

Division Of Indices.Consider the expression:

4788 The expression can be

written as a quotient:

4

7

8

8 Now expand the numerator

and denominator.

8888

8888888

How many eights will

cancel from the top and the

bottom ?

4

Cancel and simplify.

888

=8 3Result:

8 7 8 4= 8 3

Using the previous result simplify the expressions below:

(1) 3 9 3 2

= 3 7

(2) 8 11 8 6

= 8 5

(3) 4 24 4 13

= 4 11

Rule 2 : Division of Indices.

a n a m = a n - m

We can now write down our second rule of index numbers:

What Goes In The Box ? 2Simplify the expressions below :

(1) 5 9 5 2

(2) 7 12 7 5

(3) 19 6 19

(4) 36 15 36 10

(5) 18 40 18 20

(6) 2 32 2 27

(7) 8 70 8 39

(8) 5 200 5 180

=5 7

= 7 7

= 19 5

= 36 5

= 18 20

= 2 5

= 8 31

= 5 20

Negative Index Numbers.

Simplify the expression below:

5 3 5 7 = 5 - 4 To understand this result fully

consider the following:

Write the original expression

again as a quotient:

Expand the numerator and the

denominator:5555555

555

7

3

5

5

Cancel out as many fives as

possible:

5555

1

Write as a power of five:

Now compare the two results:45

1

The result on the previous slide allows us to see the following

results:

Turn the following powers into fractions:

(1) 32

32

1

8

1

(2) 4

3

43

1

81

1

(3)6

10

610

1

1000000

1

We can now write down our third rule of index numbers:

Rule 3 : For negative indices:.

a - mm

a

1

More On Negative Indices.Simplify the expressions below leaving your answer as a

positive index number each time:

(1)5

96

3

33

)5(963

596

3

83

(2)

28

34

77

77

)2(8

34

7

7

6

1

7

7

617

7

7

77

1

What Goes In The Box ? 3

Change the expressions below to fractions:

Simplify the expressions below leaving your answer with a

positive index number at all times:

(1)5

2 (2)

33

3

2

2

4

(3) (4)3

2

3

6

3

65

4

44

(5)

1110

67

77

77

(6) (7)

246

342

333

333

32

1

27

1

2

1

4

3

44 2

7 33

1

Powers Of Indices.Consider the expression below:

( 2 3 ) 2

To appreciate this expression

fully do the following:

Expand the term inside the

bracket.

= ( 2 x 2 x 2 ) 2Square the contents of the bracket.

= ( 2 x 2 x 2 ) x (2 x 2 x 2 ) Now write the

expression as a power

of 2.= 2 6

Result: ( 2 3 ) 2 = 2 6

Use the result on the previous slide to simplify the

following expressions:

(1) ( 4 2 ) 4 (2) ( 7 5 ) 4(3) ( 8 7 ) 6

= 4 8 = 7 20 = 8 42

We can now write down our fourth rule of index numbers:

Rule 4 : For Powers Of Index Numbers.

( a m ) n = a m n

(4) (3 2) -3

= 3 -6

63

1

41)5(

(5)443

)55(

x

45

What Goes In The Box ? 4Simplify the expressions below leaving your answer as a

positive index number.

(1) 54)7(

63)5(

(2) (3)37

)10(

(4)342

)88( (5)523

)77( (6)

1056)1111(

207 18

5

1

2110

188

57

11011

The Rules Of Indices.

Rule 4 : For

Powers Of Index

Numbers.

( a m ) n = a m n

Rule 1 : Multiplication of Indices.

a n x a m = a n + m

Rule 2 : Division of Indices.

a n a m = a n - m

Rule 3 : For negative indices:.

a - m

ma

1

Applying The Rules With Fractions.

We are now going to look at the rules of indices again but

use them with fractions that are obtained from the roots of

numbers.

MultiplicationExample 1.

Simplify:

3 aa

Solution.

3 aa •Change the roots to powers.

3

1

2

1

aa • Select the appropriate rule of indices.

Rule 1 : Multiplication of Indices.

a n x a m = a n + m

•Add the fractions.

6

5

6

23

3

1

2

1

6

5

a56 )( a

Example 2.

Simplify:

3543 )()( aa

Solution.

•Change the roots to powers.3543 )()( aa

5

3

3

4

aa • Select the appropriate rule of indices.

Rule 1 : Multiplication of Indices.

a n x a m = a n + m

•Add the fractions.

15

29

15

920

5

3

3

4

15

29

a

2915)( a

Division.Example 1.

Simplify:

3 gg

Solution.

•Change the roots to powers.3 gg

3

1

2

1

gg

• Select the appropriate rule of indices.

Rule 2 : Division of Indices.

a n a m = a n - m

•Subtract the fractions.

6

1

6

23

3

1

2

1

6

1

g

6 g

Example 2.

Simplify:2534 )()( dd

Solution.

2534 )()( dd •Change the roots to powers.

5

2

4

3

dd • Select the appropriate rule of indices.

Rule 2 : Division of Indices.

a n a m = a n - m

•Subtract the fractions.

20

7

20

815

5

2

4

3

20

7

d

720)( d

Multiplication & DivisionExample 1.

Simplify:

23 )( a

aa

Solution.

23 )( a

aa

•Change the roots to powers.

3

2

2

1

a

aa

• Select the appropriate rule of indices.

Rule 1 : Multiplication of Indices.

a n x a m = a n + m

Rule 2 : Division of Indices.

a n a m = a n - m

•Calculate the fractions.

2

3

2

11

6

5

6

49

3

2

2

3

6

5

a56 )( a

Example 2.

Simplify:

)()(

)()(43

4325

rr

rr

Solution.

)()(

)()(43

4325

rr

rr

•Change the roots to powers.

4

1

2

3

3

4

5

2

rr

rr

• Select the appropriate rule of indices.

Rule 1 : Multiplication of Indices.

a n x a m = a n + m

Rule 2 : Division of Indices.

a n a m = a n - m

•Calculate the fractions.

15

26

15

206

3

4

5

2

4

7

4

1

4

6

4

1

2

3

60

1

60

105104

4

7

15

26

60

1

rRule 3 : For negative indices:.

a - mm

a

160

60

1

11

rr

Example 3.

Simplify:

34

3

)(12

42

d

dd

Solution.

34

3

)(12

42

d

dd

4

3

3

1

2

1

12

42

d

dd

4

3

3

1

2

1

12

8

d

dd

4

3

6

5

12

8

d

d

3

2 4

3

6

5

d

3

2 12

1

d

3

)(212

d

Example 4.

Simplify:

kk

kk

3)(4

2)(53

334

Solution.

kk

kk

3)(4

2)(53

334

2

1

3

1

34

3

12

10

k

k

6

5

4

15

6

5

k

k

6

5 6

5

4

15

k

6

5 12

1045

k

6

5 12

35

k

6

)(53512

k

Power To The Power.Example 1.Simplify:

)4( 3

2

a

Solution.

)4( 3

2

a

•Change the roots to powers.

2

1

3

2

)4( a

• Select the appropriate rule of indices.

Rule 4 : For Powers Of Index Numbers.

( a m ) n = a m n

2

1

3

2

2

1

)()4( a

•Multiply the fractions.

3

1

6

2

23

12

2

1

3

2

3

1

2a

32 a

Example 1.Simplify:

23 23 )))(27(( w

Solution.

23 23 )))(27(( w

•Change the roots to powers.

3

2

3

2

)27( w

• Select the appropriate rule of indices.

Rule 4 : For Powers Of Index Numbers.

( a m ) n = a m n

)()27( 3

2

3

2

3

2

w

•Multiply the fractions.

9

4

3

2

3

2

9

4

9w

49 )(9 w

What Goes In The Box 5?Simplify the expressions below :

34 43 aa (1) (2) )(5)(10 33 aa

(3)42

23

62

)(43

aa

aa

712)(12 a

76)(2 a

12 13

1

a

(4) 33 4 )(27 a

43 a

Summary Of The Rules Of Indices.

Rule 1 : Multiplication of Indices.

a n x a m = a n + m

Rule 2 : Division of Indices.

a n a m = a n - m

Rule 3 : For negative indices:.

a - m

ma

1

Rule 4 : For Powers Of Index Numbers.

( a m ) n = a m n

Rule 5 : For indices which are fractions.

nn aa

1

(The nth root of “a” )

Rule 6 : For indices which are fractions.

(The nth root of “a” to

the power of m)mnn

m

aa )(

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