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Solving Equations with Fractional Exponents

Raise fractional exponents to their inverse power to cancel

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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.

Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 7.9. 

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