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7.9: Solving Rational Exponent Equations

Difficulty Level: At Grade Created by: CK-12
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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?


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 2x3219=35.

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


9 is a solution to this equation.

Example B

Solve 3x52=96.

Solution: 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

Example C

Solve 2(x5)34+48=202.

Solution: 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

Intro Problem Revisit 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.

Guided Practice

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

1. 8(3x1)23=200

2. 6x32141=1917


1. 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

2. 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

Explore More

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 Explore More Problems

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

Image Attributions


Difficulty Level:

At Grade


Date Created:

Mar 12, 2013

Last Modified:

Jun 04, 2015
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