(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 8^{1/3}

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}? **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 ^{¼} is the number 3(3*3*3*3 = 81).

**8**

^{¼}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

### 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.