Fractional exponents are abbreviations for taking the roots of a number. For example, x½ means , the square root of x. This makes sense because, by the rules of addition for exponents,
(x½)(x½) = x(½+½) = x1 = x
In the same way, x⅓ is the cube root of x. We need to multiply x⅓ by itself three times to get back to x. x¼ is the fourth root of x, x⅕ is the fifth root of x, and so on. What's that? You want one more? All right, all right. x⅙ 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 x¾ mean? Well, if we multiply x¾ by itself 4 times, we get x3. Therefore, x¾ is still a root; it's the fourth root of x cubed. By the rules of multiplying exponents, we can also write x¾ as (x3)¼, or as (x¼)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?
8⅔ can also be written as (81/3)2 . Since the cube root of 8 is 2, this simplifies to (2)2=4.
can be written as which simplifies to .