Integral Exponentiation

As multiplication is related to the concept of "repeated addition", exponentiation involving integers is akin to "repeated multiplication". The use of positive integers as exponents is illustrated as follows:

bⁿ = b × b × .... × b where the number of b's to be multiplied together is n = x

where b is a real number and n is a positive integer. In the above equation, b is the base, n is the exponent, and x is the real number b raised to the nth power, or the nth power of b.

The following formulas are useful in algebraic manipulations involving exponents:

(1) a^m × aⁿ = a^m+n
(2) ( a^m )ⁿ = a^mn
(3) a⁰= 1, a ¹ 0
(4) a^-n = 1 / an , a ¹ 0
(5) a^m / aⁿ = a^m-n , a ¹ 0

where a is a real number, m and n are positive integers, and the above constraints are satisfied.

Formulas (1) and (2) can be derived from the definition of exponents using positive integers. Formula (3) defines exponentiation with zero; any real number raised to the zeroth power is 1. Formula (4) defines exponentiation using negative integers; a real number raised to the nth power, where n < 0, is the reciprocal of the same real number raised to the mth power, where m = -n > 0. Formula (5) is derived from Formulas (1) and (4).

The above formulas also apply when the exponents are positive real numbers. This will be shown in later sections.