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) am × an = am+n
(2) ( am )n = amn
(3) a0= 1, a ¹ 0
(4) a-n = 1 / an , a ¹ 0
(5) am / an = am-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.
