Exponential Functions: Negative and Fractional Exponents

Much of the material in this section is a review of the material covered in the Pre-Algebra SparkNote on Powers, Exponents, and Roots.

Negative Exponents

Taking a quantity to a negative exponent is equivalent to taking the reciprocal of the quantity to the positive opposite of the exponent:

x^-a =

Examples:

4^-3 = ()³ = = .
()^-4 = 5⁴ = 625.
()^-2 = ()² = = .
()^-5 = ()⁵ = = .
(- 2)^-2 = ()² = = .
()^-3 = ()³ = = = - .

Note that a number to a negative exponent is not necessarily a negative number.

Fractional Exponents

In a fractional exponent, the numerator is the power to which the number should be taken and the denominator is the root which should be taken. For example, 125 means "take 125 to the fourth power and take the cube root of the result" or "take the cube root of 125 and then take the result to the fourth power." Order does not matter when evaluating exponents--it is usually easier to take the root first, and then take the power.

Since we cannot take the even root of a negative number, we cannot take a negative number to a fractional power if the denominator of the exponent is even.

A negative fractional exponent works just like an ordinary negative exponent. First, we switch the numerator and the denominator of the base number, and then we apply the positive exponent.

Examples:

49 = 7³ = 343.
81 = 3⁵ = 243.
() = ()⁴ = .
216 = (6)³ = 6³()³ = 216(6) = 1296.
50 = (5)⁵ = 5⁵()⁵ = 3125(4) = 12500.
125^- = () = ()² = .
(- 6) cannot be computed.
()^- = () = ()³ = .