Rational Exponents

Quick Review

Now, remember, a rational number is any number that can be written as a fraction. If that's true, what is a rational exponent? A rational exponent is an exponent that is written as a fraction. Here are a couple of examples: x^2/3, y^1/2.

You'll notice that the exponent is a fraction. So what is the fraction telling us? The fraction refers to a radical. Let's look at a radical expression.

Rational to Radical

First, we have the radical symbol. The b is called the index or root number. Inside the radical is the radicand. For our example, x^a is the radicand.

How does that convert to a fractional exponent? Here's what the radical and rational exponent look like together. As you can, see the b in the denominator is the same as the index for the radical. The a in the numerator is the same as the exponent in the radicand.

There are two ways to rewrite a rational exponent: x^a/b = bth root of (x^a) = (bth root of x)^a. These all mean the same!

A Few Examples

Let me show you how this works. So, we have y^2/3 is the same thing as saying the cube root (it's the cube root because we have a 3 here) of y². Well, that's the same thing as saying (cube root of y)².

Let's look at another one. Here we have the fifth root of x³. The fifth root is rewritten as the denominator in the rational exponent fraction. The exponent 3 is written as the numerator in the rational exponent fraction. So, the fifth root of x³ is rewritten as x^3/5.

Here's the next example: the seventh root of the quantity (2x)⁵. How would that look as a rational exponent? The quantity (2x)^5/7. Remember, the denominator of the rational fraction is the radicand exponent and the numerator is the index number.