Determine the exponents on each element in the set. Any quantity that has no exponent, according to mathematical rules, is assumed to have an exponent of "1." In this article, exponents are indicated by a "^" followed by the numeric value of the exponent.
Reverse the sign on the exponents. For example, if the set was all of the integers between three and six, i.e., the numbers four and five, they would be represented as 4^1 and 5^1. Because the exponents on these integers are positive, reversing the exponents yields the numbers 4^-1 and 5^-1.
Simplify the new elements algebraically. In mathematics, numbers with negative exponents are not considered in "simplest form" and must be reduced. This is accomplished by moving the term containing the negative exponent to the opposite side of the fraction that represents it -- sort of like "flipping the fraction" -- and then changing the sign on the exponent. Since all nonrational numbers are really fractions with a denominator of 1, the number 4^-1 is the same as (4^-1)/1. Applying the rules, first 4^-1 is moved, or "flipped," giving 1/(4^-1), and the sign of the exponent is reversed, giving 1/(4^1), which further reduces to 1/4. Thus, the inverse of 4 is 1/4.
Check your work for accuracy. In mathematics, when a number is multiplied by its inverse, the resulting product equals one. Checking the example from above, 4 x (1/4) = 1. This result shows the process was carried out successfully.