节: 指数属性和理性指数 | 10.4 指数属性和理性指数

章节大纲

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

In this section, we discuss how to apply the laws of exponents to rational exponents.
::在本节中,我们讨论如何对理性的推论者适用推论者的法律。

Laws of Exponents
::名人法

When simplifying expressions with rational exponents, all of the laws of exponents are still valid.
::当用理性的推手简化表达方式时,推手的所有法律仍然有效。

Laws of Exponents
::名人法

Product Rule: $\begin{aligned} a^{m} \cdot a^{n} = a^{m + n} \end{aligned}$
::产品规则: AMan=am+n

Quotient Rule : $\begin{aligned} \frac{a^{m}}{a^{n}} = a^{m - n} \end{aligned}$
::引文规则:aman=am-n

Power Rule: $\begin{aligned} (a^{m})^{n} = a^{m n} \end{aligned}$
::权力规则am)n=amn

Other Properties of Note:
::注释的其他属性 :

$\begin{aligned} (a b)^{n} = a^{n} b^{n} \end{aligned}$
:ab)n=anbn

$\begin{aligned} (\frac{a}{b})^{n} = \frac{a^{n}}{b^{n}}, b \neq 0 \end{aligned}$
:ab)n=anbn,b0

$\begin{aligned} a^{- n} = \frac{1}{a^{n}}, a \neq 0 \end{aligned}$
::a-n=1, a=0

$\begin{aligned} \frac{1}{a^{- n}} = a^{n}, a \neq 0 \end{aligned}$
::1个a-n=an, a*0

Notice how the 1st two properties of note relate to properties of radicals that we discussed in a previous section.
::注意我们前一节中讨论过的与激进分子属性有关的第1个注释属性。

$\begin{aligned} \sqrt[n]{a b} & = (a b)^{\frac{1}{n}} = a^{\frac{1}{n}} b^{\frac{1}{n}} = \sqrt[n]{a} \sqrt[n]{b} \\ \sqrt[n]{\frac{a}{b}} & = (\frac{a}{b})^{\frac{1}{n}} = \frac{a^{\frac{1}{n}}}{b^{\frac{1}{n}}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}} \end{aligned}$

::abn=(ab)1n=a1nbn1n=anbnabn=(ab)1n=a1nbn1n=anbn

Example 1
::例1

Simplify.
::简化。

a. $\begin{aligned} x^{\frac{1}{2}} \cdot x^{\frac{3}{4}} \end{aligned}$ .
::a. x12x34。

b. $\begin{aligned} 4 d^{\frac{3}{5}} \cdot 8^{\frac{1}{3}} d^{\frac{2}{5}} \end{aligned}$
::b. 4d35813d25

Solution: a. When two expressions with the same base are multiplied, add the exponents. Here, the exponents do not have the same denominator, so we need to find a common denominator and then add the numerators.
::解决方案 : a. 当两个表达式以同一基数乘以时, 加入引号。这里, 引号没有相同的分母, 所以我们需要找到一个共同的分母, 然后添加数字。

$\begin{aligned} x^{\frac{1}{2}} \cdot x^{\frac{3}{4}} = x^{\frac{2}{4}} \cdot x^{\frac{3}{4}} = x^{\frac{5}{4}} \end{aligned}$

::x12x34=x24x34=x54

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。

b. Change 4 and 8 so they are powers of 2 and then add exponents with the same base.
::b. 更改4和8,因此其权力为2,然后加上同一基数。

$\begin{aligned} 4 d^{\frac{3}{5}} \cdot 8^{\frac{1}{3}} d^{\frac{2}{5}} = 2^{2} d^{\frac{3}{5}} \cdot {(2^{3})}^{\frac{1}{3}} d^{\frac{2}{5}} = 2^{3} d^{\frac{5}{5}} = 8 d \end{aligned}$

::4d35813d25=22d35}(23)13d25=23d55=8d

This video by CK-12 demonstrates how to simplify expressions with rational exponents using the properties of exponents.
::CK-12的这段影片展示了如何使用推手的特性, 以理性推手简化表达方式。

Example 2
::例2

Simplify.
::简化。

a. $\begin{aligned} \frac{w^{\frac{7}{4}}}{w^{\frac{1}{2}}} \end{aligned}$
::a. w74w12

b. $\begin{aligned} \frac{4 x^{\frac{2}{3}} y^{4}}{16 x^{3} y^{\frac{5}{6}}} \end{aligned}$
::b. 4x23y416x3y56

Solution:
::解决方案 :

a. This problem uses the Quotient Rule of Exponents. Subtract the exponents. Change the $\begin{aligned} \frac{1}{2} \end{aligned}$ power to $\begin{aligned} \frac{2}{4} \end{aligned}$ .
::a. 这一问题使用指数的引号规则。减法引号。将12功率改为24。

$\begin{aligned} \frac{w^{\frac{7}{4}}}{w^{\frac{1}{2}}} = w^{\frac{7}{4} - \frac{2}{4}} = w^{\frac{5}{4}} \end{aligned}$

::W74w12=w74-24=w54

b. Subtract the exponents with the same base and reduce $\begin{aligned} \frac{4}{16} \end{aligned}$ .
::b. 以同一基数减去指数并减少416。

$\begin{aligned} \frac{4 x^{\frac{2}{3}} y^{4}}{16 x^{3} y^{\frac{5}{6}}} = \frac{1}{4} x^{(\frac{2}{3}) - 3} y^{4 - (\frac{5}{6})} = \frac{1}{4} x^{(\frac{2}{3}) - (\frac{9}{3})} y^{(\frac{24}{6}) - (\frac{5}{6})} = \frac{1}{4} x^{\frac{- 7}{3}} y^{\frac{19}{6}} \end{aligned}$

::4x23y416x3y56=14x(23)-3y4-(56)=14x(23)-23-(93)y(246)-(56)=14x-73y196

If you are writing your answer in terms of positive exponents, your answer would be $\begin{aligned} \frac{y^{\frac{19}{6}}}{4 x^{\frac{7}{3}}} \end{aligned}$ . Also, notice that when a rational exponent is improper we do not change it to a mixed number.
::如果你用积极的推理来写你的回答, 你的回答是y1964x73。另外, 请注意, 当理性推理不当时, 我们不会把它改成混合数字。

If we were to write the answer using roots, then we would take out the whole numbers. For example, $\begin{aligned} y = \frac{19}{6} \end{aligned}$ can be written as $\begin{aligned} y^{\frac{19}{6}} = y^{3} y^{\frac{1}{6}} = y^{3} \sqrt[6]{y} \end{aligned}$ because 6 goes into 19 three 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 乘以19 3 3 y6 乘以19 3 次, 其余 1 次。

Example 3
::例3

Simplify.
::简化。

a. $\begin{aligned} {(3^{\frac{3}{2}} x^{4} y^{\frac{6}{5}})}^{\frac{4}{3}} \end{aligned}$
::a. (332x4y65)43

b. $\begin{aligned} \frac{{(2 x^{\frac{1}{2}} y^{6})}^{\frac{2}{3}}}{4 x^{\frac{5}{4}} y^{\frac{9}{4}}} \end{aligned}$
::b. (2x12y6)234x54y94

Solution:
::解决方案 :

a. According to the power rule, apply the $\begin{aligned} \frac{4}{3} \end{aligned}$ power to everything inside the " data-term="Parentheses" role="term" tabindex="0"> parentheses and reduce.
::a. 根据权力规则,对括号内的所有东西适用43项权力并减少。

$\begin{aligned} {(3^{\frac{3}{2}} x^{4} y^{\frac{6}{5}})}^{\frac{4}{3}} = 3^{\frac{12}{6}} x^{\frac{16}{3}} y^{\frac{24}{15}} = 3^{2} x^{\frac{16}{3}} y^{\frac{8}{5}} = 9 x^{\frac{16}{3}} y^{\frac{8}{5}} \end{aligned}$

:332x4y65)43=3126x163y2415=32x163y85=9x163y85

b. In the numerator, the entire expression is raised to the $\begin{aligned} \frac{2}{3} \end{aligned}$ power. Distribute this power to everything inside the parentheses. Then, use the Power Rule of Exponents and rewrite 4 as $\begin{aligned} 2^{2} . \end{aligned}$
::b. 在分子中,整个表达式被提升到23强。将这种权力分配到括号内的所有东西。然后,使用指数法则,将4改写为22。

$\begin{aligned} \frac{{(2 x^{\frac{1}{2}} y^{6})}^{\frac{2}{3}}}{4 x^{\frac{5}{4}} y^{\frac{9}{4}}} = \frac{2^{\frac{2}{3}} x^{\frac{1}{3}} y^{4}}{2^{2} x^{\frac{5}{4}} y^{\frac{9}{4}}} \end{aligned}$

:2x12y6) 234x54y94 = 223x1313y422x54y94

Combine terms with the same base by subtracting the exponents.
::减去指数,将条件与同一基数合并。

$\begin{aligned} \frac{2^{\frac{2}{3}} x^{\frac{1}{3}} y^{4}}{2^{2} x^{\frac{5}{4}} y^{\frac{9}{4}}} = (2^{\frac{2}{3} - 2}) (x^{\frac{1}{3} - \frac{5}{4}}) (y^{4 - \frac{9}{4}}) = (2^{\frac{2}{3} - \frac{6}{3}}) (x^{\frac{4}{12} - \frac{15}{12}}) (y^{\frac{16}{4} - \frac{9}{4}}) = 2^{\frac{- 4}{3}} x^{\frac{- 11}{12}} y^{\frac{7}{4}} \end{aligned}$

::223x13y422x54y94=(223-2)(x13-54)(y4-94)=(223-63)(x412-1512)(y164-94)=2-43x1112y74

Finally, rewrite the answer with positive exponents by moving the 2 and $\begin{aligned} x \end{aligned}$ into the denominator. $\begin{aligned} \frac{y^{\frac{7}{4}}}{2^{\frac{4}{3}} x^{\frac{11}{12}}} \end{aligned}$
::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。

by Mathispower4u demonstrates how to simplify expressions with rational exponents.
::由 Mathispower4u 演示如何简化表达式,

Example 4
::例4

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

Solution: Substitute $\begin{aligned} {9.8}^{\frac{8}{3}} \end{aligned}$ for L and solve.
::溶液:L和溶液的替代物9.883。

$\begin{aligned} P = 2 π {(\frac{L}{9.8})}^{\frac{1}{2}} \\ P = 2 π {(\frac{{9.8}^{\frac{8}{3}}}{9.8})}^{\frac{1}{2}} \\ P = 2 π (\frac{{9.8}^{\frac{8}{3}}}{{9.8}^{\frac{3}{3}}})^{\frac{1}{2}} \\ P = 2 π ({9.8}^{\frac{5}{3}})^{\frac{1}{2}} \\ P = 2 π (9.8)^{\frac{5}{6}} \end{aligned}$

::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

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

Feature: Rough Winds
::特点: 狂风

by Denise Huey
::丹妮丝·胡伊(Denise Huey)

Weather forecasts include a variety of information about the weather . A weather forecast on the local news usually includes the temperature, the wind speed , and the humidity in your area. At sea, the Beaufort Scale is usually used to determine how the rough winds are (given in miles per hour).
::天气预报包括关于天气的多种信息。本地新闻上的天气预报通常包括温度、风速和湿度。在海上,波福特天平通常用于确定风速(每小时以英里计 ) 。

The Beaufort Scale is a formula used to determine the intensity of wind for hurricanes and tornadoes. A formula used to measure the intensity of wind on the Beaufort Scale at 10 meters above the sea surface is $\begin{aligned} v = 0.836 B^{(\frac{3}{2})} \end{aligned}$ . "V" represents the wind speed at 10 meters above the sea surface, and "B" represents the Beaufort Scale number. If the news anchor were to mention a "Beaufort 9," should we run for shelter? Let's use this formula to find out.
::Beaufort 比例尺是一种公式,用来确定飓风和龙卷风的风强度。用来测量海面上10米处 Beaufort 比例尺上的风强度的公式是 v=0. 836B(32). “V”表示海面上10米处的风速,“B”表示Beaufort 比例尺号。如果新闻主播提到“Beaufort 9 ” ,我们是否应该运行以避风港?让我们使用这个公式来找出答案。

After doing all the calculations, we find that $\begin{aligned} v = 22.572 \end{aligned}$ meters per second. A wind of this type might blow over trees and signs. Would you run for cover?
::经过所有的计算, 我们发现 v=22.572米每秒。这种风可能会吹过树木和标志。你会跑去寻找掩护吗 ?

What if winds were 30 meters per second? Should we substitute 30 in for $\begin{aligned} B \end{aligned}$ or $\begin{aligned} v \end{aligned}$ ? Since we are given the wind speed, we should substitute this into the equation for " $\begin{aligned} v \end{aligned}$ ."
::如果风每秒30米怎么办?我们是否应该取代B或V中的30米?既然我们得到了风速,我们应该用这个公式来代替V。

$\begin{aligned} 30 = 0.836 B^{(\frac{3}{2})} \end{aligned}$

::30=0.836B(32)

We 1st divide both sides by 0.836, then square both sides, and then take the cube root of both sides to solve for $\begin{aligned} B \end{aligned}$ .
::我们首先将双方除以0.836,然后将双方平方,然后用双方的立方根解决B问题。

$\begin{aligned} \frac{30}{0.836} & = B^{(\frac{3}{2})} \\ 35.885 & = B^{(\frac{3}{2})} \\ {(35.885)}^{2} & = B^{3} \\ 1287.733 & = B^{3} \\ {1287.733}^{(\frac{1}{3})} & = B \\ 10.879 & = B \end{aligned}$

::300.836=B(32)35.885=B(32)(35.885)2=B31287.733=B31287.733(13)=B10.879=B

The number 10 is pretty high on the Beaufort Scale, especially since the numbers range from 0 to 12! At Beaufort 10, you'd expect a really violent storm, with uprooted trees and shingles flying off roofs!
::特别是因为数字在0到12之间! 在Beaufort 10,你会期待一场非常激烈的暴风雨, 连根拔起的树木和闪电从屋顶上飞走!

by Hailey Gilbney explains the Beaufort Scale.
::Hailey Gilbney解释博福特比例尺。

Summary
::摘要

The laws of exponents, particularly the product, quotient, and power rules, apply to rational exponents as well.
::推手的法律,特别是产品、商数和权力规则,也适用于理性推手。

Review
::回顾

Simplify each expression. Reduce all rational exponents and write your answer with positive exponents.
::简化每个表达式。减少所有理性的提示, 并用积极的提示书写您的答复。

1. $\begin{aligned} \frac{1}{5} a^{\frac{4}{5}} 25^{\frac{3}{2}} a^{\frac{3}{5}} \end{aligned}$
::1. 15a452532a35

2. $\begin{aligned} 7 b^{\frac{4}{3}} 49^{\frac{1}{2}} b^{- \frac{2}{3}} \end{aligned}$
::2. 7b434912b-23

3. $\begin{aligned} \frac{m^{\frac{8}{9}}}{m^{\frac{2}{3}}} \end{aligned}$
::3.89m23

4. $\begin{aligned} \frac{x^{\frac{4}{7}} y^{\frac{11}{6}}}{x^{\frac{1}{14}} y^{\frac{5}{3}}} \end{aligned}$
::4. 47y116x114y53

5. $\begin{aligned} \frac{8^{\frac{5}{3}} r^{5} s^{\frac{3}{4}} t^{\frac{1}{3}}}{2^{4} r^{\frac{21}{5}} s^{2} t^{\frac{7}{9}}} \end{aligned}$
::5. 853r5s34t1324r215s279

6. $\begin{aligned} {(a^{\frac{3}{2}} b^{\frac{4}{5}})}^{\frac{10}{3}} \end{aligned}$
::6. (a32b45)103

7. $\begin{aligned} {(5 x^{\frac{5}{7}} y^{4})}^{\frac{3}{2}} \end{aligned}$
::7. (5x57y4)32

8. $\begin{aligned} {(\frac{4 x^{\frac{2}{5}}}{9 y^{\frac{4}{5}}})}^{\frac{5}{2}} \end{aligned}$
::8. (4x259y45)52

9. $\begin{aligned} {(\frac{75 d^{\frac{18}{5}}}{3 d^{\frac{3}{5}}})}^{\frac{5}{2}} \end{aligned}$
:9,75d1853d35)52

10. $\begin{aligned} {(\frac{81^{\frac{3}{2}} a^{3}}{8 a^{\frac{9}{2}}})}^{\frac{1}{3}} \end{aligned}$
::10. (8132a38a92)13

11. $\begin{aligned} 27^{\frac{2}{3}} m^{\frac{4}{5}} n^{- \frac{3}{2}} 4^{\frac{1}{2}} m^{- \frac{2}{3}} n^{\frac{8}{5}} \end{aligned}$
::11. 2723m45n-32412m-23n85

12. $\begin{aligned} {(\frac{3 x^{\frac{3}{8}} y^{\frac{2}{5}}}{5 x^{\frac{1}{4}} y^{- \frac{3}{10}}})}^{2} \end{aligned}$
::12. (3x38y255x14y-310)2

Explore More
::探索更多

1. Matt is making a tile decoration for his kitchen wall. Using square tiles of different sizes, Matt created one decoration that is five tiles across, with sides touching. The 1st tile is 15 in ² , the 2nd tile is 20 in ² , the 3rd is 35 in ² , the 4th is 20 in ² , and the 5th is 15 in ² . What is the length of the decoration? (Round your answer to the nearest tenth.)
::1. 马特正在为厨房墙做瓷砖装饰。马特用不同大小的平方瓷砖制作了一个五张瓷砖,两侧相交。第一瓷砖是15英寸2,第二瓷砖是20英寸2,第三瓷砖是35英寸2,第四瓷砖是20英寸2,第五瓷砖是15英寸2。装饰的长度是多少? (您对最近的十分的答复是10英寸)。

2. In your own words, explain how to rationalize the denominator of a fraction containing the sum or difference of square roots in the denominator. Why does this work?
::2. 用你自己的话说,请解释如何使含有分母中平方根之和或差的分母的分母合理化。

3. In your own words, explain why, in general, $\begin{aligned} (a + b)^{3} \neq a^{3} + b^{3} . \end{aligned}$
::3. 用你自己的话说,请解释为什么一般而言(a+b)3a3+b3。

Answers for Review and Explore More Problems
::回顾和探讨更多问题的答复

Please see the Appendix.
::请参看附录。

References
::参考参考资料

1. "Pendulum," last edited May 30, 2017,
::1. 2017年5月30日上一次编辑的“Pendulum”,

章节大纲

Laws of Exponents ::名人法

Laws of Exponents ::名人法

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Feature: Rough Winds ::特点: 狂风

Summary ::摘要

Review ::回顾

Explore More ::探索更多

Answers for Review and Explore More Problems ::回顾和探讨更多问题的答复

References ::参考参考资料

Laws of Exponents
::名人法

Laws of Exponents
::名人法

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Feature: Rough Winds
::特点: 狂风

Summary
::摘要

Review
::回顾

Explore More
::探索更多

Answers for Review and Explore More Problems
::回顾和探讨更多问题的答复

References
::参考参考资料