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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 × 10 3 = (10 × 10) × (10 × 10 × 10) = 10 5 = 10 2+3:

(b) 2 5 ÷ 2 3 = 32 ÷ 8 = 4 = 2 2 = 2 5-3

(c) ( 3 2 ) 3 = (3 × 3) 3 = (3 × 3) × (3 × 3) × (3 × 3) = 3 6 = 3 2 × 3

(d) From rule 2, a n + 1 ÷ a n = a ( n + 1) - n = a 1. Also,

(e) We have a n × a 0 = a n +0 = a n = a n × 1 : Thus a 0 = 1.

 
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