(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 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 Formulafor 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