<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
Skip Navigation
You are viewing an older version of this Concept. Go to the latest version.

Solving Equations with Fractional Exponents

Raise fractional exponents to their inverse power to cancel

Atoms Practice
Practice Solving Equations with Fractional Exponents
Practice Now
Solving Rational Exponent Equations

The period (in seconds) of a pendulum with a length of L (in meters) is given by the formual P = 2\pi{(\frac{L}{9.8})}^{\frac{1}{2}} . If the period of a pendulum is 10\pi 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 2x^{\frac{3}{2}}-19=35 .

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

2(9)^{\frac{3}{2}}-19&=35 \\2 \cdot 27 -19 &= 35 \\54 - 19 &= 35

9 is a solution to this equation.

Example B

Solve 3x^{\frac{5}{2}}=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)^{\frac{5}{2}}=3 \cdot 2^5=3 \cdot 32=96

Example C

Solve -2(x-5)^{\frac{3}{4}}+48=-202 .

Solution: Isolate (x-5)^{\frac{3}{4}} by subtracting 48 and dividing by -2.


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

\left[ \left(x-5 \right)^{\frac{3}{4}}\right]^{\frac{4}{3}}&=\left(-125 \right)^{\frac{4}{3}}\\x-5&=625\\x&=630

Check: -2(630-5)^{\frac{3}{4}}+48=-2 \cdot 625^{\frac{3}{4}}+48=-2 \cdot 125+48=-250+48=-202

Intro Problem Revisit We need to plug 156.8 in to the equation P = 2\pi{(\frac{L}{9.8})}^{\frac{1}{2}} for L and solve. If our answer equals 10\pi , then the given length is correct.

P = 2\pi{(\frac{L}{9.8})}^{\frac{1}{2}}\\2\pi{(\frac{156.8}{9.8})}^{\frac{1}{2}}\\2\pi (16)^{\frac{1}{2}}\\2\pi (4) = 8 \pi

8\pi does not equal 10\pi , so the length cannot be 156.8.

Guided Practice

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

1. 8(3x-1)^{\frac{2}{3}}=200

2. 6x^{\frac{3}{2}}-141=1917


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

8(3x-1)^{\frac{2}{3}}&=200\\\left[ \left(3x-1 \right)^{\frac{2}{3}}\right]^{\frac{3}{2}}&=(25)^{\frac{3}{2}}\\3x-1&=125\\3x&=126\\x&=42

Check: 8(3(42)-1)^{\frac{2}{3}}=8(126-1)^{\frac{2}{3}}=8(125)^{\frac{2}{3}}=8 \cdot 25=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.

6x^{\frac{3}{2}}-141&=1917 \\6x^{\frac{3}{2}}&=2058 \\x^{\frac{3}{2}}&=343 \\x&=343^{\frac{2}{3}}=7^2=49

Check: 6(49)^{\frac{3}{2}}-141=6 \cdot 343-141=2058-141=1917


Determine if the following values of x are solutions to the equation 3x^{\frac{3}{5}}=-24

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

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

  1. 2x^{\frac{3}{2}}=54
  2. 3x^{\frac{1}{3}}+5=17
  3. (7x-3)^{\frac{2}{5}}=4
  4. (4x+5)^{\frac{1}{2}}=x-4
  5. x^{\frac{5}{2}}=16x^{\frac{1}{2}}
  6. (5x+7)^{\frac{3}{5}}=8
  7. 5x^{\frac{2}{3}}=45
  8. (7x-8)^{\frac{2}{3}}=4(x-5)^{\frac{2}{3}}
  9. 7x^{\frac{3}{7}}+9=65
  10. 4997=5x^{\frac{3}{2}}-3
  11. 2x^{\frac{3}{4}}=686
  12. x^3=(4x-3)^{\frac{3}{2}}

Image Attributions


Please wait...
Please wait...

Original text