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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 \begin{align*}P = 2\pi{(\frac{L}{9.8})}^{\frac{1}{2}}\end{align*}P=2π(L9.8)12. If the period of a pendulum is \begin{align*}10\pi\end{align*}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 \begin{align*}2x^{\frac{3}{2}}-19=35\end{align*}2x3219=35.

Substitute in x and see if the equation holds.

\begin{align*}2(9)^{\frac{3}{2}}-19&=35 \\ 2 \cdot 27 -19 &= 35 \\ 54 - 19 &= 35 \end{align*}2(9)3219227195419=35=35=35

9 is a solution to this equation.

Now, let's solve the following equations for x.

  1. \begin{align*}3x^{\frac{5}{2}}=96\end{align*}3x52=96

First, divide both sides by 3 to isolate \begin{align*}x\end{align*}x.

\begin{align*}3x^{\frac{5}{2}}&=96\\ x^{\frac{5}{2}}&=32 \end{align*}3x52x52=96=32

\begin{align*}x\end{align*}x is raised to the five-halves power. To cancel out this exponent, we need to raise everything to the two-fifths power.

\begin{align*}\left(x^{\frac{5}{2}}\right)^{\frac{2}{5}}&=32^{\frac{2}{5}}\\ x&=32^{\frac{2}{5}}\\ x&=\sqrt[5]{32}^2=2^2=4\end{align*}(x52)25xx=3225=3225=3252=22=4

Check: \begin{align*}3(4)^{\frac{5}{2}}=3 \cdot 2^5=3 \cdot 32=96\end{align*}3(4)52=325=332=96

  1. \begin{align*}-2(x-5)^{\frac{3}{4}}+48=-202\end{align*}2(x5)34+48=202

Isolate \begin{align*}(x-5)^{\frac{3}{4}}\end{align*}(x5)34 by subtracting 48 and dividing by -2.

\begin{align*}-2(x-5)^{\frac{3}{4}}+48&=-202\\ -2(x-5)^{\frac{3}{4}}&=-250\\ (x-5)^{\frac{3}{4}}&=-125\end{align*}2(x5)34+482(x5)34(x5)34=202=250=125

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

\begin{align*}\left[ \left(x-5 \right)^{\frac{3}{4}}\right]^{\frac{4}{3}}&=\left(-125 \right)^{\frac{4}{3}}\\ x-5&=625\\ x&=630\end{align*}[(x5)34]43x5x=(125)43=625=630

Check: \begin{align*}-2(630-5)^{\frac{3}{4}}+48=-2 \cdot 625^{\frac{3}{4}}+48=-2 \cdot 125+48=-250+48=-202\end{align*}2(6305)34+48=262534+48=2125+48=250+48=202

Examples

Example 1

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

We need to plug 156.8 in to the equation \begin{align*}P = 2\pi{(\frac{L}{9.8})}^{\frac{1}{2}}\end{align*}P=2π(L9.8)12 for L and solve. If our answer equals \begin{align*}10\pi\end{align*}10π, then the given length is correct.

\begin{align*}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\end{align*}P=2π(L9.8)122π(156.89.8)122π(16)122π(4)=8π

\begin{align*}8\pi\end{align*}8π does not equal \begin{align*}10\pi\end{align*}10π, so the length cannot be 156.8.

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

Example 2

\begin{align*}8(3x-1)^{\frac{2}{3}}=200\end{align*}8(3x1)23=200

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

\begin{align*}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\end{align*}8(3x1)23[(3x1)23]323x13xx=200=(25)32=125=126=42

Check: \begin{align*}8(3(42)-1)^{\frac{2}{3}}=8(126-1)^{\frac{2}{3}}=8(125)^{\frac{2}{3}}=8 \cdot 25=200\end{align*}8(3(42)1)23=8(1261)23=8(125)23=825=200

Example 3

\begin{align*}6x^{\frac{3}{2}}-141=1917\end{align*}6x32141=19172. 

Here, only the \begin{align*}x\end{align*} 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.

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

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

Review

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

  1. \begin{align*}x=32\end{align*}
  2. \begin{align*}x=-32\end{align*}
  3. \begin{align*}x=8\end{align*}

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

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

Answers for Review Problems

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

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