Unit 3 Section 2 : Laws of Indices
There are three rules that should be used when working with indices:
When
m and
n are positive integers,
1.  a^{m} × a^{n} = a^{m + n} 
2.  a^{m} ÷ a^{n} = a^{m – n} or 
 = a^{m – n} (m ≥ n) 
3.  (a^{m})^{n} = a^{m × n} 
These three results are logical consequences of the definition of a^{n} , but really need a formal proof. You can 'verify' them with particular examples as below, but this is not a proof:
2^{7} × 2^{3}  =  (2 × 2 × 2 × 2 × 2 × 2 × 2) × (2 × 2 × 2)  
 =  2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2  
 =  2^{10}  (here m = 7, n = 3 and m + n = 10) 
or,
2^{7} ÷ 2^{3}  =  2 × 2 × 2 × 2 × 2 × 2 × 2  2 × 2 × 2 
 
 =  2 × 2 × 2 × 2  
 =  2^{4}  (again m = 7, n = 3 and m – n = 4) 
Also,
(2^{7})^{3}  =  2^{7} × 2^{7} × 2^{7} 
 =  2^{21}  (using rule 1) (again m = 7, n = 3 and m × n = 21) 
The proof of the first rule is given below:
Proof
a^{m} × a^{n} 
= 
a × a × ... × a
m of these 
× 
a × a × ... × a
n of these 

= 
a × a × ... × a × a × a × ... × a
(m+n) of these 

= 
a^{m+n} 
The second and third rules can be shown to be true for all positive integers
m and
n in a similar way.
We can see an important result using rule 2:
  = x^{n – n} = x^{0} 
but   = 1,  so 
x^{0} = 1
This is true for any nonzero value of x, so, for example, 3^{0} = 1, 27^{0} = 1 and 1001^{0} = 1.