Index Notation
5^{2} 
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  
Rules  Examples  
Expanding  a^{n} = a × a × a × ...... × a  3^{5} = 3 × 3 × 3 × 3 × 3 = 243  
Multiplying  a^{x} × a^{y} = a^{(x+y)}  3^{6} × 3^{2} = 3^{(6+2)} = 3^{8} = 6561  
Dividing  a^{x} ÷ a^{y} = a^{(x–y)}  3^{6} ÷ 3^{2} = 3^{(6–2)} = 3^{4} = 81  
Fractional index  a^{(1/x)} = x^{th} root of a  9^{(1/2)} = √9 = 3  
Negative index 


Example One  Expanding
Expand 5^{3}
Answer:
5^{3}
= 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 email 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?
Answer:
2 to the power of ? = 4000
2^{12} = 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?
Answer:
No. of "splits"
= 2 hours ÷ 20 minutes
= 6 "splits"
No. of bacteria after 2 hours
= 2^{6}
= 64 bacteria
Questions
Expand these.
Q1. 5^{5}
Q2. 10^{6}
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 centimetre^{2} of your skin.
 An adult has about 2 metres^{2} (20 000 centimetres^{2}) 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 2^{3} × 2^{6}. Leave the answer in index form.
Answer:
2^{3} × 2^{6}
= 2^{3+6}
= 2^{9}
Example Five  Multiplying Indices
Calculate 2^{3} × 5^{2} × 2^{4}. Leave the answer in index form.
Answer:
2^{3} × 2^{4} × 5^{2}
= 2^{3+4} × 5^{2}
= 2^{7} × 5^{2}
Questions
Multiply these indices. Leave the answer in index form.
Q1. 3^{2} × 3^{5}
Q2. 4^{3} × 4^{6}
Answers
A1. 3^{7}
A2. 4^{9}
Maths Fun  Computer Bytes
Computer bytes do not use decimal numbers (base10 numbers). They use binary numbers (base2 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 2^{7} ÷ 2^{4}. Leave the answer in index form.
Answer:
2^{7} ÷ 2^{4}
=  2 × 2 × 2 × 2 × 2 × 2 × 2 
2 × 2 × 2 × 2 
= 2 × 2 × 2
= 2^{3}
Example Seven  Dividing Indices
Calculate 10^{5} ÷ 10^{3}. Leave the answer in index form.
Answer:
10^{5} ÷ 10^{3}
=  10 × 10 × 10 × 10 × 10 
10 × 10 × 10 
= 10 × 10
= 10^{2}
Questions
Divide these indices. Leave the answer in index form.
Q1. 3^{6} ÷ 3^{2}
Q2. 10^{5} ÷ 10^{2}
Answers
A1. 3^{4}
A2. 10^{3}
Example Eight  Fractional Indices
Calculate 100^{1/2}
Answer:
100^{(1/2)}
= √100
= 10
Example Nine  Fractional Indices
Calculate 27^{1/3}
Answer:
27^{(1/3)}
= ^{3}√27
= 3
Questions
Calculate these.
Q1. 64^{1/2}
Q2. 343^{1/3}
Answers
A1. 8
A2. 7
Example Ten  Negative Indices
Calculate 5 ^{–2}
Answer:
5 ^{–2}
= 1 ÷ 5^{2}
= 1 ÷ 25
= ^{1}⁄_{25}
Example Eleven  Negative Indices
Calculate 10^{2} × 10 ^{–4}
Answer:
10^{2} × 10 ^{–4}
= 10^{2} × (1 ÷ 10^{4})
= 100 ÷ 10000
= ^{1}⁄_{100}
Questions
Calculate these.
Q1. 5 ^{–3}
Q2. 4 ^{–3} × 4 ^{–2}
Answers
A1. ^{1}⁄_{125}
A2. ^{1}⁄_{1024}
Maths Fun  A Nerdy Joke
There are 10 kinds of people  those who can understand BINARY and those who can't!