Section: 解决理性指数方程式 | 7.9 解决理性指数方

Section outline

The period (in seconds) of a pendulum with a length of L (in meters) is given by the formula $\begin{align*}P = 2\pi{(\frac{L}{9.8})}^{\frac{1}{2}}\end{align*}$ . If the period of a pendulum is $\begin{align*}10\pi\end{align*}$ is the length of the pendulum 156.8?
::P=2(L9.8)12的公式给出长于L(米)的钟表周期(秒)。如果钟表周期为10,则时长为156.8?

Solving Rational Exponent Equations
::解决理性指数方程式

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.
::当解决一个理性的速率方程式时,你首先孤立变量。然后,为了消除指数,你需要将一切提升到对等权力。

Let's determine if x = 9 is a solution to $\begin{align*}2x^{\frac{3}{2}}-19=35\end{align*}$ .
::让我们确定 x = 9 是 2x32-19 = 35 的解决方案。

Substitute in x and see if the equation holds.
::以 x 代替, 看看方程式是否有效。

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

9 is a solution to this equation.
::9是这个等式的解决方案。

Now, let's solve the following equations for x.
::现在,让我们解决x的以下方程式。

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

First, divide both sides by 3 to isolate $\begin{align*}x\end{align*}$ .
::首先,将双方除以3,分离x。

$\begin{aligned} 3 x^{\frac{5}{2}} & = 96 \\ x^{\frac{5}{2}} & = 32 \end{aligned}$
$\begin{align*}3x^{\frac{5}{2}}&=96\\ x^{\frac{5}{2}}&=32 \end{align*}$
::3x52=96x52=32

$\begin{align*}x\end{align*}$ is raised to the five-halves power. To cancel out this exponent, we need to raise everything to the two-fifths power.
::X 被提升到五分之一的电源。要取消这个提示, 我们需要将一切提升到五分之二的电源。

$\begin{aligned} {(x^{\frac{5}{2}})}^{\frac{2}{5}} & = 32^{\frac{2}{5}} \\ x & = 32^{\frac{2}{5}} \\ x & = {\sqrt[5]{32}}^{2} = 2^{2} = 4 \end{aligned}$
$\begin{align*}\left(x^{\frac{5}{2}}\right)^{\frac{2}{5}}&=32^{\frac{2}{5}}\\ x&=32^{\frac{2}{5}}\\ x&=\sqrt[5]{32}^2=2^2=4\end{align*}$
:x5225=3225x=3225x=3252=22=4)

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

$\begin{align*}-2(x-5)^{\frac{3}{4}}+48=-202\end{align*}$
:2x-5-5)34+48202

Isolate $\begin{align*}(x-5)^{\frac{3}{4}}\end{align*}$ by subtracting 48 and dividing by -2.
::孤立(x-5-5)34,减去48,除以-2。

$\begin{aligned} - 2 (x - 5)^{\frac{3}{4}} + 48 & = - 202 \\ - 2 (x - 5)^{\frac{3}{4}} & = - 250 \\ (x - 5)^{\frac{3}{4}} & = - 125 \end{aligned}$
$\begin{align*}-2(x-5)^{\frac{3}{4}}+48&=-202\\ -2(x-5)^{\frac{3}{4}}&=-250\\ (x-5)^{\frac{3}{4}}&=-125\end{align*}$
::- (x-5)34+48202-2(x-5)34250(x-5)34125

To undo the three-fourths power, raise everything to the four-thirds power.
::为了推翻四分之三的权力, 将一切提升到四分之三的权力。

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

Check: $\begin{align*}-2(630-5)^{\frac{3}{4}}+48=-2 \cdot 625^{\frac{3}{4}}+48=-2 \cdot 125+48=-250+48=-202\end{align*}$
::检查:- (2) 630- 5) 34+48 @2} @262534+48}2125+48}250+48202

Examples
::实例

Example 1
::例1

Earlier, you were asked to verify the length of the pendulum.
::早些时候,有人要求你核查钟摆的长度。

We need to plug 156.8 in to the equation $\begin{align*}P = 2\pi{(\frac{L}{9.8})}^{\frac{1}{2}}\end{align*}$ for L and solve. If our answer equals $\begin{align*}10\pi\end{align*}$ , then the given length is correct.
::我们需要将156.8插入L的 P=2Q(L9.8)12 等式并解析。如果我们的答案等于 10Q, 那么给定的长度是正确的。

$\begin{aligned} P = 2 π {(\frac{L}{9.8})}^{\frac{1}{2}} \\ 2 π {(\frac{156.8}{9.8})}^{\frac{1}{2}} \\ 2 π (16)^{\frac{1}{2}} \\ 2 π (4) = 8 π \end{aligned}$
$\begin{align*}P = 2\pi{(\frac{L}{9.8})}^{\frac{1}{2}}\\ 2\pi{(\frac{156.8}{9.8})}^{\frac{1}{2}}\\ 2\pi (16)^{\frac{1}{2}}\\ 2\pi (4) = 8 \pi\end{align*}$
::P=2(L9.8) 122(156.89.8) 122(16) 122(4)=8

$\begin{align*}8\pi\end{align*}$ does not equal $\begin{align*}10\pi\end{align*}$ , so the length cannot be 156.8.
::8不等于10,所以长度不能为156.8。

Solve the following rational exponent equations and check for extraneous solutions.
::解决以下合理的推理方程式,并检查不相干的解决办法。

Example 2
::例2

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

Divide both sides by 8 and raise everything to the three-halves power.
::将两边除以8 把每样东西都加到三座的威力上

$\begin{aligned} 8 (3 x - 1)^{\frac{2}{3}} & = 200 \\ {[{(3 x - 1)}^{\frac{2}{3}}]}^{\frac{3}{2}} & = (25)^{\frac{3}{2}} \\ 3 x - 1 & = 125 \\ 3 x & = 126 \\ x & = 42 \end{aligned}$
$\begin{align*}8(3x-1)^{\frac{2}{3}}&=200\\ \left[ \left(3x-1 \right)^{\frac{2}{3}}\right]^{\frac{3}{2}}&=(25)^{\frac{3}{2}}\\ 3x-1&=125\\ 3x&=126\\ x&=42\end{align*}$
::8( 3x- 1) 23=200[ (3x- 1) 23] 32=( 25) 323x- 1=1=1253x=126x=42

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

Example 3
::例3

$\begin{align*}6x^{\frac{3}{2}}-141=1917\end{align*}$ 2.
::6x32-141=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.
::在这里, 只有 x 才能升到三座。从两侧减141 , 然后除以 6 。然后, 通过将双方提升到三分之二的权力来消除引力。

$\begin{aligned} 6 x^{\frac{3}{2}} - 141 & = 1917 \\ 6 x^{\frac{3}{2}} & = 2058 \\ x^{\frac{3}{2}} & = 343 \\ x & = 343^{\frac{2}{3}} = 7^{2} = 49 \end{aligned}$
$\begin{align*}6x^{\frac{3}{2}}-141&=1917 \\ 6x^{\frac{3}{2}}&=2058 \\ x^{\frac{3}{2}}&=343 \\ x&=343^{\frac{2}{3}}=7^2=49\end{align*}$
::6x32-141=19176x32=2058x32=343x=343x=34323=72=49

Check: $\begin{align*}6(49)^{\frac{3}{2}}-141=6 \cdot 343-141=2058-141=1917\end{align*}$
::查询: 6( 4932) -141=6}343 -141=2058 - 141=1917

Review
::回顾

Determine if the following values of x are solutions to the equation $\begin{align*}3x^{\frac{3}{5}}=-24\end{align*}$
::确定以下 x 值是否是方程 3x35 的解决方案 24

$\begin{align*}x=32\end{align*}$
::x=32x=32

$\begin{align*}x=-32\end{align*}$
::x32

$\begin{align*}x=8\end{align*}$
::x=8x=8

Solve the following equations. Round any decimal answers to 2 decimal places.
::解决以下方程式。将小数点后的任何答案四舍五入到小数点后两位。

$\begin{align*}2x^{\frac{3}{2}}=54\end{align*}$
::2x32=54

$\begin{align*}3x^{\frac{1}{3}}+5=17\end{align*}$
::3x13+5=17

$\begin{align*}(7x-3)^{\frac{2}{5}}=4\end{align*}$
:7x-3)25=4

$\begin{align*}(4x+5)^{\frac{1}{2}}=x-4\end{align*}$
:4x+5)12=x-4

$\begin{align*}x^{\frac{5}{2}}=16x^{\frac{1}{2}}\end{align*}$
::x52=16x12

$\begin{align*}(5x+7)^{\frac{3}{5}}=8\end{align*}$
:5x+7)35=8

$\begin{align*}5x^{\frac{2}{3}}=45\end{align*}$
::5x23=45

$\begin{align*}(7x-8)^{\frac{2}{3}}=4(x-5)^{\frac{2}{3}}\end{align*}$
:7x-8)23=4(x-5)23

$\begin{align*}7x^{\frac{3}{7}}+9=65\end{align*}$
::7x37+9=65

$\begin{align*}4997=5x^{\frac{3}{2}}-3\end{align*}$
::4997=5x32-3

$\begin{align*}2x^{\frac{3}{4}}=686\end{align*}$
::2x34=686

$\begin{align*}x^3=(4x-3)^{\frac{3}{2}}\end{align*}$
::x3=( 4x-3) 32

Review (Answers)
::回顾(答复)

Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键。

Section outline

Solving Rational Exponent Equations ::解决理性指数方程式

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Review ::回顾

Review (Answers) ::回顾(答复)

Solving Rational Exponent Equations
::解决理性指数方程式

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Review
::回顾

Review (Answers)
::回顾(答复)