# Exponent Rules

 52

In the example shown in the box, the 5 is called the base. The 2 is called the exponent or index or power. 5 to the power of 2 = 5 × 5 = 25

INDEX RULES
RulesExamples
Expandingan = a × a × a × ...... × a35 = 3 × 3 × 3 × 3 × 3 = 243
Multiplyingax × ay = a(x+y)36 × 32 = 3(6+2) = 38 = 6561
Dividingax ÷ ay = a(x–y)36 ÷ 32 = 3(6–2) = 34 = 81
Fractional indexa(1/x) = xth root of a9(1/2) = √9 = 3
Negative index
 a –1 = 1 ax
 3 –2 = 1 = 1 32 9

Expand 53

Answer:
53
= 5 × 5 × 5
= 125

Expand these.
Q1. 55
Q2. 106

Answers
A1. 3125
A2. 1 000 000

## Maths Fun - Body Bacteria

How many bacteria are on your body NOW? Here are some facts to help you work it out...

• There about 5 million bacteria on every centimetre2 of your skin.
• An adult has about 2 metres2 (20 000 centimetres2) of skin.

That probably took a minute for you to work out. How many bacteria are on your skin NOW (a minute later)? Bacteria reproduce by binary fission (splitting in half) every 20 minutes. Therefore, all those bacteria have reproduced (doubled in number) 3 times in the last minute!

## Example Two - Multiplying Indices

Calculate 23 × 26. Leave the answer in index form.

Answer:
23 × 26
= 23+6
= 29

## Example Three - Multiplying Indices

Calculate 23 × 52 × 24. Leave the answer in index form.

Answer:
23 × 24 × 52
= 23+4 × 52
= 27 × 52

## Questions

Multiply these indices. Leave the answer in index form.
Q1. 32 × 35
Q2. 43 × 46

Answers
A1. 37
A2. 49

## Example Four - Dividing Indices

Calculate 27 ÷ 24. Leave the answer in index form.

Answer:
27 ÷ 24

 = 2 × 2 × 2 × 2 × 2 × 2 × 2 2 × 2 × 2 × 2

= 2 × 2 × 2

= 23

## Example Five - Dividing Indices

Calculate 105 ÷ 103. Leave the answer in index form.

Answer:
105 ÷ 103

 = 10 × 10 × 10 × 10 × 10 10 × 10 × 10

= 10 × 10

= 102

## Questions

Divide these indices. Leave the answer in index form.
Q1. 36 ÷ 32
Q2. 105 ÷ 102

Answers
A1. 34
A2. 103

Calculate 1001/2

Answer:
100(1/2)
= √100
= 10

Calculate 271/3

Answer:
27(1/3)
= 327
= 3

Calculate these.
Q1. 641/2
Q2. 3431/3

Answers
A1. 8
A2. 7

Calculate 5 –2

Answer:
5 –2
= 1 ÷ 52
= 1 ÷ 25
= 125

## Example Nine - Negative Indices

Calculate 102 × 10 –4

Answer:
102 × 10 –4
= 102 × (1 ÷ 104)
= 100 ÷ 10000
= 1100

Calculate these.
Q1. 5 –3
Q2. 4 –3 × 4 –2

Answers
A1. 1125
A2. 11024