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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 P=2π(L9.8)12. If the period of a pendulum is 10π 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 2x3219=35.

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.

  1. 3x52=96

First, divide both sides by 3 to isolate x.


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


Check: 3(4)52=325=332=96

  1. 2(x5)34+48=202

Isolate (x5)34 by subtracting 48 and dividing by -2.


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


Check: 2(6305)34+48=262534+48=2125+48=250+48=202


Example 1

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

We need to plug 156.8 in to the equation P=2π(L9.8)12 for L and solve. If our answer equals 10π, then the given length is correct.


8π does not equal 10π, 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: 8(3(42)1)23=8(1261)23=8(125)23=825=200

Example 3


Here, only the x 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: 6(49)32141=6343141=2058141=1917


Determine if the following values of x are solutions to the equation 3x35=24

  1. x=32
  2. x=32
  3. x=8

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

  1. 2x32=54
  2. 3x13+5=17
  3. (7x3)25=4
  4. (4x+5)12=x4
  5. x52=16x12
  6. (5x+7)35=8
  7. 5x23=45
  8. (7x8)23=4(x5)23
  9. 7x37+9=65
  10. 4997=5x323
  11. 2x34=686
  12. x3=(4x3)32

Answers for Review Problems

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

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