Overview
Just as positive exponents direct how many times that a base number can be multiplied by itself, zero and negative exponents direct a similar relationship between the base number and its exponent.
Review of Positive Exponents
A nonzero expression that contains an exponent of any value has 2 parts, the base and the exponent. For example, an expression such as 23 has a base of 2 and an exponent of 3, so that 2 is a factor 3 times as in 2·2·2, or 8. If the expression contains a variable, such as x5, the base is a factor the number of times the exponent directs, such as x·x·x·x·x. If the nonzero number or the variable has an exponent of 1, it is the base number, such as 31 =3, 2561 = 256, or y1 = y.
Zero Exponents
Zero exponents follow a convention such that any nonzero base raised to the power of zero has a value of 1, such that 90 equals 1, z0 equals 1, as long as z is not equal to zero, and 92457800 equals 1. Suppose the values of an exponent are expressed such that 24 = 16, because 2·2·2·2 is 16. Dividing 24 by 2 equals 23 or 8, dividing 23 by 2 equals 22 or 4, dividing 22 by 2 equals 21 or 2, and dividing 21 by 2 equals 20, or 1.
Negative Integer Exponents
Negative exponents are the reciprocal of positive exponents, such that if the base x is not equal to 0 and the exponent n is an integer, x-n equals 1/xn. If 2-1 equals 1/2, then continuing the pattern dividing by 2, dividing 20 by 2 equals 2-1 or ½. The number 1 divided by 2 equals ½. Dividing 2-1 by 2 equals 2-2, or ¼, because 22 equals 4. Dividing 2-2 by 2 equals 2-3 or 1/8.
Simplifying Expressions
Expressions with exponents are simplified when there are no negative or zero exponents in the final expression. A negative exponent in the numerator of a fraction can always be moved to the denominator, and a zero exponent can always be simplified to equal 1. For example, an expression 3y-2 can be simplified as 3/y2.
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