# 7.9: Solving Rational Exponent Equations

**At Grade**Created by: CK-12

**Practice**Solving Equations with Exponents

The period (in seconds) of a pendulum with a length of *L* (in meters) is given by the formula . If the period of a pendulum is is the length of the pendulum 156.8?

### Solving Rational Exponent Equations

When solving a rational exponent equation, you first isolate the variable. Then, to eliminate the exponent, you will need to raise everything to the reciprocal power.

Let's determine if *x* = 9 is a solution to .

Substitute in *x* and see if the equation holds.

9 is a solution to this equation.

Now, let's solve the following equations for x.

First, divide both sides by 3 to isolate .

is raised to the five-halves power. To cancel out this exponent, we need to raise everything to the two-fifths power.

Check:

Isolate by subtracting 48 and dividing by -2.

To undo the three-fourths power, raise everything to the four-thirds power.

Check:

### Examples

#### Example 1

Earlier, you were asked to verify the length of the pendulum.

We need to plug 156.8 in to the equation for *L* and solve. If our answer equals , then the given length is correct.

does not equal , so the length cannot be 156.8.

**Solve the following rational exponent equations and check for extraneous solutions.**

#### Example 2

Divide both sides by 8 and raise everything to the three-halves power.

Check:

#### Example 3

2.

Here, only the is raised to the three-halves power. Subtract 141 from both sides and divide by 6. Then, eliminate the exponent by raising both sides to the two-thirds power.

Check:

### Review

Determine if the following values of *x* are solutions to the equation

Solve the following equations. Round any decimal answers to 2 decimal places.

### Answers for Review Problems

To see the Review answers, open this PDF file and look for section 7.9.

### Notes/Highlights Having trouble? Report an issue.

Color | Highlighted Text | Notes | |
---|---|---|---|

Show More |

### Image Attributions

Here you'll learn how to solve equations where the variable has a rational exponent.

## Concept Nodes:

**Save or share your relevant files like activites, homework and worksheet.**

To add resources, you must be the owner of the Modality. Click Customize to make your own copy.