# 7.9: Solving Rational Exponent Equations

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?

### Guidance

This concept is very similar to the previous two. When solving a rational exponent equation, isolate the variable. Then, to eliminate the exponent, you will need to raise everything to the reciprocal power.

#### Example A

Determine if
*
x
*
= 9 is a solution to
.

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

9 is a solution to this equation.

#### Example B

Solve .

**
Solution:
**
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:

#### Example C

Solve .

**
Solution:
**
Isolate
by subtracting 48 and dividing by -2.

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

Check:

**
Intro Problem Revisit
**
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.

### Guided Practice

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

1.

2.

#### Answers

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

Check:

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:

### Explore More

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

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

### Image Attributions

## Description

## Learning Objectives

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

## Difficulty Level:

At Grade## Categories:

## Concept Nodes:

## Date Created:

Mar 12, 2013## Last Modified:

Sep 19, 2014## Vocabulary

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