# Fractional Exponents

## Relate fractional exponents to nth roots

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Fractional Exponents

Suppose you want to find the cube root of a number, but your calculator only allows you to find square roots. However, it does allow you to find a number raised to any power, including non-integer powers. What would you do? Is there a way that you could rewrite the cube root of a number with a fractional exponent so that you could use your calculator to find the answer?

### Fractional Exponents

The next objective is to be able to use fractions as exponents in an expression.

You can write roots as fractional exponents using the rule .

1.
2.
3.

1.

#### Evaluating Expressions with Exponents

It is important when evaluating expressions that you remember the Order of Operations. Evaluate what is inside the parentheses, then evaluate the exponents, then perform multiplication/division from left to right, and then perform addition/subtraction from left to right.

### Examples

#### Example 1

Earlier, you were asked how you can write the cube root so that you can use your calculator to find cube root of a number.

As shown in this concept, you can write a root as a fractional exponent where the denominator is the power of the root and the numerator is the power that is inside the root.

A cube root can be written as the number to the  power.

#### Example 2

Simplify .

We start by simplifying using the fact that .

Next we rearrange knowing that 6 and 2 have a common factor, 2.

### Review

Simplify the following expressions. Be sure the final answer includes only positive exponents.

Simplify the following expressions without any fractions in the answer.

Evaluate the following expressions to a single number.

Mixed Review

1. A quiz has ten questions: 7 true/false and 3 multiple choice. The multiple choice questions each have four options. How many ways can the test be answered?
2. Simplify .
3. Simplify .
4. Simplify .
5. Solve for .

To view the Review answers, open this PDF file and look for section 8.4.

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