How to Calculate Fractional Exponents

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?