# Fractional Exponents

Fractional exponents are abbreviations for taking the **roots** of a number. For example, *x*^{½} means , the **square root** of *x*. This makes sense because, by the rules of addition for exponents:

(*x*^{½})(*x*^{½}) = *x*^{(½ + ½)} = *x*^{1} = *x*

In the same way, *x*^{⅓} is the **cube root** of *x*. We need to multiply *x*^{⅓} by itself three times to get back to *x*. Moving along, *x*^{¼} is the **fourth root** of *x*, *x*^{⅕} is the **fifth root** of *x*, and so on. What's that? You want one more? All right, all right: *x*^{⅙} is the **sixth root** of *x*, but that's all you're getting from us.

What about fractions that have a number other than 1 in the numerator? Can we ignore them and move on to easier problems? Unfortunately, no. That big gaping hole on your answer sheet will be glaring. What would *x*^{¾} mean? Well, if we multiply *x*^{¾} by itself 4 times, we get *x*^{3}. Therefore, *x*^{¾} is still a root; it's the fourth root of *x* cubed. By the rules of multiplying exponents, we can also write *x*^{¾} as (*x*^{3})^{¼}, or as (*x*^{¼})^{3}.

We could write it many other ways as well, but those other ways would be wrong, and if simplifying an exponent is wrong, we don't want to be right. Wait, what?

### Sample Problems

The expression 8^{⅔} can also be written as
**(8**^{⅓}**) ^{2}** . Since the cube root of 8 is 2, this simplifies to (2)

^{2}= 4.

The expression can be written as which simplifies to .