将指数法适用于理性指数

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The period (in seconds) of a pendulum with a length of L (in meters) is given by the formula $\begin{array}{r}P=2\pi {(\frac{L}{9.8})}^{\frac{1}{2}}\end{array}$ . If the length of a pendulum is $\begin{array}{r}{9.8}^{\frac{8}{3}}\end{array}$ , what is its period?
::P=2(L9.8)12的公式给出了长度为L(米)的钟摆的时段(秒)。如果钟摆的时段长度为9.883,其时段是多少?

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Applying the Laws of Exponents to Rational Exponents
::将指数法适用于理性指数

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When simplifying expressions with rational exponents, all the laws of exponents that were learned previously are still valid. On top of that, the rules of fractions apply as well.
::当用理性的推手简化表达方式时,所有先前学到的推手法则依然有效。 除此之外,分数规则也适用。

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Simplify: $\begin{array}{r}{x}^{\frac{1}{2}}\cdot x^{\frac{3}{4}}\end{array}$
::简化: x12x34

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Recall from the Product Property of Exponents, that when two numbers with the same base are multiplied we add the exponents. Here, the exponents do not have the same base, so we need to find a common denominator and then add the numerators.
::回顾指数产品属性时,当两个以同一基数为单位的数字乘以相同基数时,我们增加引数。这里,引数没有相同的基数,所以我们需要找到一个共同的分母,然后添加点数。

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$\begin{array}{r}{x}^{\frac{1}{2}}\cdot x^{\frac{3}{4}}=x^{\frac{2}{4}}\cdot x^{\frac{3}{4}}=x^{\frac{5}{4}}\end{array}$
::x12x34=x24x34=x54

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This rational exponent does not reduce, so we are done.
::这个理性的推手不会减少,所以我们做到了。

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Simplify: $\begin{array}{r}\frac{4x^{\frac{2}{3}}{y}^4}{16x^3{y}^{\frac{5}{6}}}\end{array}$
::简化: 4x2323y416x3y56

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This problem utilizes the Quotient Property of Exponents. Subtract the exponents with the same base and reduce $\begin{array}{r}\frac{4}{16}\end{array}$ .
::这个问题使用了指数的引号属性。 以相同的基数将指数减为416, 并减少416。

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$\begin{array}{r}\frac{4x^{\frac{2}{3}}{y}^4}{16x^3{y}^{\frac{5}{6}}}=\frac{1}{4}{x}^{\left(\frac{2}{3}\right)-3}{y}^{\frac{4-5}{6}}=\frac{1}{4}{x}^{\frac{-7}{3}}{y}^{\frac{19}{6}}\end{array}$
::4x23y416x3y56=14x(23)-3y4-56=14x-73y196

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If you are writing your answer in terms of positive exponents, your answer would be $\begin{array}{r}\frac{y^{\frac{19}{6}}}{4x^{\frac{7}{3}}}\end{array}$ . Notice, that when a rational exponent is improper we do not change it to a mixed number.
::请注意,当一个理性的推论不恰当时,我们不会把它改成混合数字。

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If we were to write the answer using roots, then we would take out the whole numbers. For example, $\begin{array}{r}y=\frac{19}{6}\end{array}$ can be written as $\begin{array}{r}{y}^{\frac{19}{6}}=y^3{y}^{\frac{1}{6}}=y^3\sqrt[6]{y}\end{array}$ because 6 goes into 19, 3 times with a remainder of 1.
::如果我们用根写出答案, 那么我们就会取出整个数字。 例如, y= 196 可以写为y196=y3y16=y3y6, 因为 6 到 19, 3 次, 其余 1 次 。

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Simplify: $\begin{array}{r}\frac{{\left(2x^{\frac{1}{2}}{y}^6\right)}^{\frac{2}{3}}}{4x^{\frac{5}{4}}{y}^{\frac{9}{4}}}\end{array}$
::简化 2x12y6) 234x54y94

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On the numerator, the entire expression is raised to the $\begin{array}{r}\frac{2}{3}\end{array}$ power. Distribute this power to everything inside the parenthesis. Then, use the Powers Property of Exponents and rewrite 4 as $\begin{array}{r}{2}^2\end{array}$ .
::在分子上, 整个表达式被提升到 23 种权力 。 将此项权力分配到括号内的所有事物 。 然后, 使用 指数 的 Power 属性, 重写 4 作为 22 。

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$\begin{array}{r}\frac{{\left(2x^{\frac{1}{2}}{y}^6\right)}^{\frac{2}{3}}}{4x^{\frac{5}{4}}{y}^{\frac{9}{4}}}=\frac{2^{\frac{2}{3}}{x}^{\frac{1}{3}}{y}^4}{2^2{x}^{\frac{5}{4}}{y}^{\frac{9}{4}}}\end{array}$
:2x12y6) 234x54y94 = 223x1313y422x54y94

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Combine like terms by subtracting the exponents.
::将类似术语合并, 减去指数 。

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$\begin{array}{r}\frac{2^{\frac{2}{3}}{x}^{\frac{1}{3}}{y}^4}{2^2{x}^{\frac{5}{4}}{y}^{\frac{9}{4}}}=2^{\left(\frac{2}{3}\right)-2}{x}^{\left(\frac{1}{3}\right)-\left(\frac{5}{4}\right)}{y}^{4-\left(\frac{9}{4}\right)}=2^{\frac{-4}{3}}{x}^{\frac{-11}{12}}{y}^{\frac{7}{4}}\end{array}$
::223x13y422x54y94=2(23)-2x(13)-(54y4-(94)=2-43x-1112y74)

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Finally, rewrite the answer with positive exponents by moving the 2 and $\begin{array}{r}x\end{array}$ into the denominator. $\begin{array}{r}\frac{y^{\frac{7}{4}}}{2^{\frac{4}{3}}{x}^{\frac{11}{12}}}\end{array}$
::最后,将 2 和 x 移动到分母 y74243x1112 中, 重写答案, 以正面的表情重写答案 。

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Examples
::实例

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Example 1
::例1

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Earlier, you were asked to find  the period of the pendulum. 
::早些时候,有人要求你找到钟摆的时间

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Substitute $\begin{array}{r}{9.8}^{\frac{8}{3}}\end{array}$ for L and solve.
::替代9.883的L和溶解。

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::P=2(L9.8)12P=2(9.8839.8)12P=2(9.88.39.8)12P=2(9.88.39.833)12P=2(9.853)12P=2(9.8)56

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Therefore , the period of the pendulum is $\begin{array}{r}P=2\pi (9.8{)}^{\frac{5}{6}}\end{array}$ .
::因此,工作周期为P=2(9.8)56。

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Example 2
::例2

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Simplify and write final answers using positive exponents: $\begin{array}{r}4d^{\frac{3}{5}}\cdot 8^{\frac{1}{3}}{d}^{\frac{2}{5}}\end{array}$
::使用正表征简化和写入最后答案:4d35813d25

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Change 4 and 8 so that they are powers of 2 and then add exponents with the same base.
::更改4和8, 使它们具有2的权责, 然后用同样的基数加入推手。

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$\begin{array}{r}4d^{\frac{3}{5}}\cdot 8^{\frac{1}{3}}{d}^{\frac{2}{5}}=2^2{d}^{\frac{3}{5}}\cdot {\left(2^3\right)}^{\frac{1}{3}}{d}^{\frac{2}{5}}=2^3{d}^{\frac{5}{5}}=8d\end{array}$
::4d35813d25=22d35}(23)13d25=23d55=8d

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Example 3
::例3

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Simplify and write final answers using positive exponents: $\begin{array}{r}\frac{w^{\frac{7}{4}}}{w^{\frac{1}{2}}}\end{array}$
::使用正表征简化并写入最终答案: w74w12

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Subtract the exponents. Change the $\begin{array}{r}\frac{1}{2}\end{array}$ power to $\begin{array}{r}\frac{2}{4}\end{array}$ .
::减速 。 将 12 功率 改为 24 功率 。

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$\begin{array}{r}\frac{w^{\frac{7}{4}}}{w^{\frac{1}{2}}}=\frac{w^{\frac{7}{4}}}{w^{\frac{2}{4}}}=w^{\frac{5}{4}}\end{array}$
::W74w12=W74w24=w54

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Example 4
::例4

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Simplify and write final answers using positive exponents: $\begin{array}{r}{\left(3^{\frac{3}{2}}{x}^4{y}^{\frac{6}{5}}\right)}^{\frac{4}{3}}\end{array}$
::使用正表征简化和写入最后答案332x4y65)43

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Distribute the $\begin{array}{r}\frac{4}{3}\end{array}$ power to everything inside the parenthesis and reduce.
::将43项权力分配到括号内的一切 并减少

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$\begin{array}{r}{\left(3^{\frac{3}{2}}{x}^4{y}^{\frac{6}{5}}\right)}^{\frac{4}{3}}=3^{\frac{12}{6}}{x}^{\frac{16}{3}}{y}^{\frac{24}{15}}=3^2{x}^{\frac{16}{3}}{y}^{\frac{8}{5}}=9x^{\frac{16}{3}}{y}^{\frac{8}{5}}\end{array}$
:332x4y65)43=3126x163y2415=32x163y85=9x163y85

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Review
::回顾

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Simplify each expression. Reduce all rational exponents and write final answer using positive exponents.
::简化每个表达式。 减少所有合理的提示, 并使用正提示写入最后答案 。

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1. $\begin{array}{r}\frac{1}{5}{a}^{\frac{4}{5}}{25}^{\frac{3}{2}}{a}^{\frac{3}{5}}\end{array}$
   ::15a452532a35
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2. $\begin{array}{r}7b^{\frac{4}{3}}{49}^{\frac{1}{2}}{b}^{-\frac{2}{3}}\end{array}$
   ::7b4344912b-23
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3. $\begin{array}{r}\frac{m^{\frac{8}{9}}}{m^{\frac{2}{3}}}\end{array}$
   ::m89m23
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4. $\begin{array}{r}\frac{x^{\frac{4}{7}}{y}^{\frac{11}{6}}}{x^{\frac{1}{14}}{y}^{\frac{5}{3}}}\end{array}$
   ::x47y116x114y53
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5. $\begin{array}{r}\frac{8^{\frac{5}{3}}{r}^5{s}^{\frac{3}{4}}{t}^{\frac{1}{3}}}{2^4{r}^{\frac{21}{5}}{s}^2{t}^{\frac{7}{9}}}\end{array}$
   ::853r5s34t1324r215s279
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6. $\begin{array}{r}{\left(a^{\frac{3}{2}}{b}^{\frac{4}{5}}\right)}^{\frac{10}{3}}\end{array}$
   :a32b45)103
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7. $\begin{array}{r}{\left(5x^{\frac{5}{7}}{y}^4\right)}^{\frac{3}{2}}\end{array}$
   :5x57y4)32
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8. $\begin{array}{r}{\left(\frac{4x^{\frac{2}{5}}}{9y^{\frac{4}{5}}}\right)}^{\frac{5}{2}}\end{array}$
   :4x259y45)52
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9. $\begin{array}{r}{\left(\frac{75d^{\frac{18}{5}}}{3d^{\frac{3}{5}}}\right)}^{\frac{5}{2}}\end{array}$
   :75d1853d35)52
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10. $\begin{array}{r}{\left(\frac{{81}^{\frac{3}{2}}{a}^3}{8a^{\frac{9}{2}}}\right)}^{\frac{1}{3}}\end{array}$
    :8132a38a92)13
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11. $\begin{array}{r}{27}^{\frac{2}{3}}{m}^{\frac{4}{5}}{n}^{-\frac{3}{2}}{4}^{\frac{1}{2}}{m}^{-\frac{2}{3}}{n}^{\frac{8}{5}}\end{array}$
    ::2723m45n-32412m-323n85
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12. $\begin{array}{r}{\left(\frac{3x^{\frac{3}{8}}{y}^{\frac{2}{5}}}{5x^{\frac{1}{4}}{y}^{-\frac{3}{10}}}\right)}^2\end{array}$
    :3x38y255x14y-310)2
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13. Rewrite your answer from Problem #1 using radicals.
    ::使用激进剂重写您在问题1中的答案。
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14. Rewrite your answer from Problem #4 using radicals.
    ::使用激进剂重写问题4的答复。
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15. Rewrite your answer from Problem #4 using one radical.
    ::重写您在4号问题中的答案 使用一个激进。

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Review (Answers)
::回顾(答复)

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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 。