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ACT Prep: Help and Review44 chapters | 435 lessons | 26 flashcard sets

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.

How would you solve a problem like this?

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 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.

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:

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:

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

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

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,

- 9^1/2 says to find the square root of 9, which is 3, then 3^1 = 3. So 9^1/2 is the square root of 9 which is 3.
- 25^1/2 says to find the square root of 25, which is 5. So 25^1/2 is the square root of 25, which is 5.
- 8^1/3 says to find the cube root of 8, which is 2 since 2 x 2 x 2 = 8. So 8^1/3 is the cube root of 8.
- 8^2/3 says to first find the cube root of 8, which is 2, then apply the numerator, 2, to your root. So 2^2 = 4.
- 25^3/2 says, find the square root of 25, which is 5, then 5^3 is 5 x 5 x 5 = 125.

Taking each one, one step at a time allows us to first find the root, using the denominator, then use the numerator as an exponent on the root.

Let's get back to this problem:

Since this is a complex radical, let's break it down from the inside out. The innermost radical is 6^1/5. Then on top of that is the radical 1/2, then the outermost radical is 1/4. So using the Power Rule of Exponents, we will multiply whenever we have an exponent on an exponent. (Review the Exponent Rules lesson for help).

Fractional exponents are just another way to work with radicals. It simplifies the process of working with radicals by changing the radical into a fractional exponent.

The index tells us the number of roots we are working with, and it is the denominator of the fractional exponent.

The numerator of the fractional exponent is the exponent on the radicand.

The radicand is the base of the exponent.

Once you change the radical into a fractional exponent, you can use the Exponent Rules to simplify the expression.

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ACT Prep: Help and Review44 chapters | 435 lessons | 26 flashcard sets

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