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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?

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. 

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