INDICES

“Indices” is another word for “Powers”

For example: 85 is read as:

“Eight to the power of 5”

And means: 8 × 8 × 8 × 8 × 8

( which is 32768 )

Multiplying numbers in index form:

( Note, Index is the singular of Indices. )

To simplify 73 × 72

We simply add the indices:

i.e. 73 × 72 = 75

The reason is: 73 × 72 = ( 7 × 7 × 7 ) × ( 7 × 7 ) = 75.

ap × aq = ap + qGenerally:

Dividing numbers in index form:

To simplify: 87 ÷ 83 87

83or,

We simply subtract the indices:

87

83= 84

The reason is: 87

83

8 × 8 × 8 × 8 × 8 × 8 × 8 8 × 8 × 8

= = 84

Generally: ap

aq = ap – q

Example 1: Simplify a) 32 × 35 b) 85 × 82 × 83

a) 32 × 35

b) 85 × 82 × 83

= 37

= 810

Example 2: Simplify a) 59

53b) 72

75

a) 59

53= 56

b) 72

75= 7–3

Negative Indices:

Looking at the last example again:

72

75= 7–3

But 72

75=

7 × 7 7 × 7 × 7 × 7 × 7

173

=

i.e. 7–3 173

=

5–2 152

=So: 4–1 141

= 1 4

=and,

Example 1: Work out the following as a fraction:

a) 7–1 b) 2–3

a) 7–1 171

= 1 7

=

b) 2–3 123

= 1 8

=

a–n 1an

=Generally:

The form (a p) : q

To simplify ( 7 3

)2

We simply multiply the indices:

i.e ( 7 3

)2

= 76

The reason is: = 76 ( 7 3

)2

= 73 73 ×

Generally: ( a p

)q

= a pq

Fractional Indices:

The power 12

means the square root.

The reason is: 5½ × 5½ = 51 (We add the indices)

i.e. 5½ × 5½ = 5 So 5½ must be the square root of 5.

Similarly, the power13

means the cube root.

Example 1: Work out the value of the following:

21

31

21

16 c) 27 b) 49 a)

21

49 a) 49 = 7

27 b) 31 3 27 = 3

2

116 c)

161

21 16

141

2 8

3d)

2 8

3d) =

12

8 3=

1 22

=14

13 8

2 =

Algebraic Examples:

Example 1: Solve 23x – 1 = 32

Example 2: Solve 42x – 1 = 8 x + 1

23x – 1 = 25

3x – 1 = 5

3x = 6

2( ) 2 2x – 1 2( ) 3 x + 1

=

24x – 2 = 23x + 3

4x – 2 = 3x + 3

( Noting that 32 is a power of 2 ).

( Noting that 4 and 8 are both powers of 2 ).

x = 2

x = 5

The form (a p) (b

p) :

We simply multiply 2 by 7:

i.e 2 3

× 7 3

= 143

The reason is: 73 = 23 ×

To simplify 73 23 ×

2 × 2 × 2 × 7 × 7 × 7

= ( 2 × 7 ) × ( 2 × 7 ) × ( 2 × 7 )

= 143

Generally: a p × b

p = ( ab ) p

Example 1: Simplify 2x × 3x

2x × 3x = 6 x

Example 2: Simplify ( 2x3 )4

( 2x3 )4 = 24 ( x3

)4 = 16x12

a p × b

p = ( ab ) p Using:

( ab ) p = a

p × b pUsing the above law in reverse:

Summary of key points:

This PowerPoint produced by R.Collins ; Updated Sep. 2011

The power 1n

means the nth root:

A negative index means the reciprocal: e.g. 5–2 152

=

ap × aq = ap + q

ap

aq = ap – q

( a p

)q

= a pq

The laws of indices are:

xne.g. x1n =

a p × b

p = ( ab ) p