Simplify Fraction Exponents

Formula and examples of how to simplify Fraction 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 8 1/3?

8 1/3 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, 8 1/3 = 2

By the same logic we can determine that 81 1/4 is the number 3 ( 3 * 3 * 3 * 3 = 81)

8 1/4 is another way of asking: " What number can you multiply by itself four times to get 8?"

General Formula

for fractional exponents

With fracitonal exponents whose numerator is 1

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? "

Formula when the fraction in the exponent is not 1

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 Example of fraction exponent

Practice Problems

Problem 1

Simplify 125 1/3

Problem 2

Simplify 125 2/3

Problem 3

Simplify 64 2/3

Problem 4

Simply 81 3/4