Simplify Fraction Exponents

This page explains fractional exponents with a numerator of 1. For instance, we will look at 81^¼. You should first understand this concept before attemtping to understand exponents with fractions whose numerators are not one like 8^¾

If you have ever calculated the square root of a number then you were actually using a fraction exponent! The square root of a number is the same as raising that number (3 in our example) to an exponent of ½

What about other fractions in the exponent?

The cube root of 8 or 81/3

The fourth root of 81 or 81 ¼

To try to understand the idea behind fraction exponents let's examine a property of the fractional exponents. The math below comes directly from the basic laws of exponents.


So we all know that the square root of 9 (or 9^½) is three and that 3*3 = 9, right?

Well, what about the next lines? What about 8^1/3? What number is that? Well, we know, from the laws of exponents, that 8^1/3 can be multiplied with itself three times (8^1/3*8^1/3*8^1/3) to get a product of 8....well, wait a minute. You already know a number that can be multiplied by itself three times to get 8: the number 2! Remember 2*2*2 = 8. Therefore, 8^1/3 = 2 .

By the same logic we can determine that 81 ^¼ is the number 3(3*3*3*3 = 81).

General Formula for fractional exponents

I want to point out that we have only been dealing with fractional exponents with a numerator of 1! Later on we will deal with fractional exponents with other numbrators

Simplify 125^1/3

Answer

Simplify 125^2/3

Answer

Formula when the fraction in the exponent is not one

Below is a specific example illustrating the formula for fraction exponents when the numerator is not one. There are two ways to simplify a fraction exponent such 2/3. You can either apply the numerator first or the denominator. See the example below.

Simplify 64^2/3

Answer

Simply 81¾

Answer