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# Solving Equations with Fractional Exponents

## Raise fractional exponents to their inverse power to cancel

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Practice Solving Equations with Fractional Exponents
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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\begin{align*}P = 2\pi{(\frac{L}{9.8})}^{\frac{1}{2}}\end{align*}. If the period of a pendulum is 10π\begin{align*}10\pi\end{align*} 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\begin{align*}2x^{\frac{3}{2}}-19=35\end{align*}.

Substitute in x and see if the equation holds.

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

9 is a solution to this equation.

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

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

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

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

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

(x52)25xx=3225=3225=3252=22=4\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*}

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

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

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

2(x5)34+482(x5)34(x5)34=202=250=125\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*}

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

[(x5)34]43x5x=(125)43=625=630\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*}

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

### Examples

#### 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\begin{align*}P = 2\pi{(\frac{L}{9.8})}^{\frac{1}{2}}\end{align*} for L and solve. If our answer equals 10π\begin{align*}10\pi\end{align*}, then the given length is correct.

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

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

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

#### Example 2

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

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

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

Check: 8(3(42)1)23=8(1261)23=8(125)23=825=200\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*}

#### Example 3

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

Here, only the x\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.

6x321416x32x32x=1917=2058=343=34323=72=49\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: 6(49)32141=6343141=2058141=1917\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 3x35=24\begin{align*}3x^{\frac{3}{5}}=-24\end{align*}

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

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

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

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

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