5^{2} 
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  
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 


Expand 5^{3}
Answer:
5^{3}
= 5 × 5 × 5
= 125
Expand these.
Q1. 5^{5}
Q2. 10^{6}
Answers
A1. 3125
A2. 1 000 000
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Calculate 2^{3} × 2^{6}. Leave the answer in index form.
Answer:
2^{3} × 2^{6}
= 2^{3+6}
= 2^{9}
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}
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}
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}
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}
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}
Calculate 100^{1/2}
Answer:
100^{(1/2)}
= √100
= 10
Calculate 27^{1/3}
Answer:
27^{(1/3)}
= ^{3}√27
= 3
Calculate these.
Q1. 64^{1/2}
Q2. 343^{1/3}
Answers
A1. 8
A2. 7
Calculate 5 ^{–2}
Answer:
5 ^{–2}
= 1 ÷ 5^{2}
= 1 ÷ 25
= ^{1}⁄_{25}
Calculate 10^{2} × 10 ^{–4}
Answer:
10^{2} × 10 ^{–4}
= 10^{2} × (1 ÷ 10^{4})
= 100 ÷ 10000
= ^{1}⁄_{100}
Calculate these.
Q1. 5 ^{–3}
Q2. 4 ^{–3} × 4 ^{–2}
Answers
A1. ^{1}⁄_{125}
A2. ^{1}⁄_{1024}