Simplify Fraction Exponents

Formula and examples of how to simplify Fraction exponents

Simplify Rational Expressions worksheet
(Free pdf with answer key on this page's topic: how to simplify fractional exponents)

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 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 9½) is three and that 3*3 = 9, right?

Well, what about the next lines? What about 81/3? 81/3 is another way of asking:

"What can you multiply by itself three times to get 8?"

What number is that? The number 2! Remember 2*2*2 = 8. Therefore, 81/3 = 2
By the same logic we can determine that 81 ¼ is the number 3(3*3*3*3 = 81).

8¼ is another way of asking:

"What number can you multiply by itself four times to get 8?"

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 numerators Below is the general formula for a fractional exponent with a numerator of

1/n is another way of asking:

"What number can you multiply by itself n times to get x?"

Practice Problems with fracitonal exponents whose numerator is 1

Simplify each fraction exponent

Problem 1

Simplify 1251/3

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.

Example of fraction exponent
Problem 2

Simplify 1252/3

Problem 3

Simplify 642/3

Problem 4

Simply 81¾