# Index Notation

 52

In the example shown in the box, the 5 is called the base. The 2 is called the index or power or exponent. 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

53
= 5 × 5 × 5
= 125

## Example Two - Pillow Fight Flash Mob

The world's largest pillow fight flash mob occurred simultaneously on March 22nd, 2008 in 25 cities around the world. In New York City alone, it is estimated that 4000 people enjoyed the experience. Online communication via Facebook and e-mail was used to organize the event.

If each person told exactly 2 people, and each of those 2 people told 2 more people and so forth, what is the approximate number of communications that could contact 4000 people?

2 to the power of ? = 4000
212 = 4096
(or 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 4096)
There would be approximately 12 communications to create this flash mob.

## Example Three - Bacterial Growth

About every 20 minutes, bacteria reproduce by binary fission, similar to splitting in half. If there was 1 bacterium in a petri dish at the start of an experiment, how many would there be after 2 hours?

No. of "splits"
= 2 hours ÷ 20 minutes
= 6 "splits"

No. of bacteria after 2 hours
= 26
= 64 bacteria

Expand these.
Q1. 55
Q2. 106

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 Four - Multiplying Indices

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

23 × 26
= 23+6
= 29

## Example Five - Multiplying Indices

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

23 × 24 × 52
= 23+4 × 52
= 27 × 52

## Questions

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

A1. 37
A2. 49

## Maths Fun - Computer Bytes

Computer bytes do not use decimal numbers (base-10 numbers). They use binary numbers (base-2 numbers). Because computers use binary numbers, a "kilobyte" is not really a thousand bytes, but is 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 or 1,024 bytes. Also, a "megabyte" is not a million bytes, but is 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 or 1,048,576 bytes. Write a kilobyte and a megabyte using indices of base 2.

## Example Six - Dividing Indices

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

27 ÷ 24

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

= 2 × 2 × 2

= 23

## Example Seven - Dividing Indices

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

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

A1. 34
A2. 103

Calculate 1001/2

100(1/2)
= √100
= 10

Calculate 271/3

27(1/3)
= 327
= 3

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

A1. 8
A2. 7

Calculate 5 –2

5 –2
= 1 ÷ 52
= 1 ÷ 25
= 125

## Example Eleven - Negative Indices

Calculate 102 × 10 –4

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

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