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How to Solve Fractional Exponents

Instructor: Ellen Manchester
Fractional exponents, or rational exponents, are a special breed of exponents. These exponents allow us to calculate radicals in a very simplistic way. Once we change radicals to fractional exponents we can use the exponent rules to simplify an expression.

Complicated Radical: What to Do?

How would you solve a problem like this?

Complex Radical

This is a complex radical expression that would be quite difficult to solve as it sits. Fractional exponents allow us to work with radicals while using exponent rules. A fractional exponent is just another way to write a radical. We will get back to this problem later in the lesson.

Fractional Exponents Are Disguised Radicals

Fractional exponents are just another way to write a radical. Remember a radical, or root, is the one number we multiplied together to find a value. For instance, if we had the value 25, what did we multiply together to get 25? Since the numbers have to be the same, the only solution would be 5 x 5. So the root, or number, multiplied is 5. What would be the root of 36? What did we multiply together to get 36? Of course, 6, since 6 x 6 = 36. So a radical is how we find the roots. We use the radical symbol to find the root. The number under the radical sign is called the radicand. A square root does not show the index of 2, but it is implied. The index tells us how many times we use the root. When there is no index, it means we are finding the value that is multiplied twice, or square root.

Radical set up

All radicals are written with an exponent on the radicand. This exponent can be seen or hidden. If there is no exponent on the radicand, then it is a 1. This set up is the basis for writing a radical as a fractional exponent. Another way to say fractional exponent is rational exponent, since a rational number can be written as a fraction.

Using the same terms as above, the radical can be written with a fractional exponent this way:

Radical rewritten

As you can see, the exponent is the numerator and the index is the denominator. The base is the radicand. Writing fractional exponents is just another way to work with radicals.

Now that we know how to set up fractional exponents, let's try a few:

Fractional exponent example 1

Notice the radicand, 4, is the base, the numerator is the exponent, 1, on the radicand, and the denominator is the index, 2.

fractionalexponentex2

Again, the radicand is the base, the numerator is the exponent on the radicand, 2, and the denominator is the index, 3.

Now Let's Calculate

As you work through these problems, remember to re-think how you see a fractional exponent. The denominator tells you which root you are looking for. If the denominator is a 2, you are looking for the square root. If a 3, you are looking for the third root.

Once you find which root you are looking for, then apply the numerator to the root you find.

For instance,

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