Powers
If a is any number and n any positive integer (whole
number ) then the product of a with itself n times, , is called a raised to the power n,
and written a^{ n} ; i.e.,
The following important rules apply to powers.
We want these rules to be true for all positive values of a
and all values of m and n:
We shall first look at the simpler cases.
Examples
(a) 10^{ 2} Ã— 10^{ 3} = (10
Ã— 10) Ã— (10 Ã— 10 Ã— 10) = 10^{ 5} = 10^{ 2+3}:
(b) 2^{ 5} Ã· 2^{ 3} = 32 Ã·
8 = 4 = 2^{ 2} = 2^{ 53}
(c) ( 3^{ 2} )^{ 3} = (3 Ã—
3)^{ 3} = (3 Ã— 3) Ã— (3 Ã— 3) Ã— (3 Ã— 3) = 3^{ 6}
= 3^{ 2 Ã— 3}
(d) From rule 2, a^{ n + 1 }Ã· a^{
n} = a^{ ( n + 1)  n }= a^{ 1}. Also,
(e) We have a^{ n} Ã— a^{ 0}
= a^{ n +0} = a n = a^{ n} Ã— 1 : Thus a^{ 0}
= 1.
