Fractional Exponents at a Glance

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)} = x¹ = 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 x³. 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 (x³)^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 .