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Quick Study Guide: Fractional Exponents

By Keren Perles

You’ve already learned about positive and negative exponents. Did you know that fractions can be exponents too? Fractional exponents can seem complicated, but this study guide will explain how to calculate fractional exponents easily.

The Basics

Fractional exponents can look intimidating, but they’re much simpler than they seem. Remember that ½ is really the reciprocal – or the “opposite” of 2. That’s why multiplying 3 times 2 gives you 6, so if you want to get from 6 back to 3, you need to multiply by the reciprocal of 2: ½. So 6 X ½ = 3.

The same thing applies to fractional exponents. Let’s take a simple example. You know that 3^2 = 9. Why? Because 3 X 3 (3 multiplied two times) is 9. So how can you get from 9 back to 3? You know one simple way – take the square root of 9, and you’ll get 3. Guess what? You could also use a fractional exponent of ½, since it’s the reciprocal of 2: 9^(1/2) is also 3.

In other words, this the basic rule of fractional exponents: To raise a base to a fractional power, following the following steps:

1.Find the reciprocal of the power.

2.Take the resulting root of the base.

A Few Examples

Because this basic rule can be tough to understand, here are a few examples to make it clearer.

1. 16^(1/2) – To find the answer to 16^(1/2), first take the reciprocal of the power. The reciprocal of ½ is 2. Then, take the 2nd root of the base (or the square root). The square root of 16 is 4, so 16^(1/2) = 4.

2. 27^(1/3) – To find the answer to 27^(1/3), first take the reciprocal of the power. The reciprocal of 1/3 is 3. Then, take the 3rd root of the base (or the cubic root). The cubic root of 27 is 3, so 27^(1/3) = 3.

3. 16^(1/4) – To find the answer to 16^(1/4), first take the reciprocal of the power. The reciprocal of 1/4 is 4. Then, take the 4th root of the base. The 4th root of 16 is 2 (2 X 2 X 2 X 2 = 16), so 16^(1/4) = 2.

More Complex Roots

The rule above works if the numerator of the fractional power is 1, but what do you do if the fractional power is more than 1? Easy – treat it as a power.

For example, take the problem 4^(3/2). That’s the same thing as (4^(1/2))^3. So first you’d figure out that 4^(1/2) is 2. Then you’d raise the answer (2) to the third power. 2^3 = 8, so 4^(3/2) = 8 too.

Next time you're not sure how to calculate fractional exponents, follow the simple steps you just learned. There you go, that wasn't so hard, was it?