Section: 理性指数 | 10.3 理性指数

Section outline

A planet's maximum distance from the sun (in astronomical units) is given by the formula $\begin{aligned} d = p^{\frac{2}{3}}, \end{aligned}$ where p is the period (in years) of the planet's orbit around the sun ¹ . If a planet's orbit around the sun is 27 years, what is its distance from the sun?
::一种行星与太阳的最大距离(天文单位)由公式d=p23给出,p是太阳周围行星轨道的周期(年)1。如果一个行星围绕太阳的轨道为27年,那么它与太阳的距离是多少?

In this section, we cover how to interpret rational exponents like the one in the formula.
::在本节中,我们将论述如何解释像公式中那样的合理推手。

Rational Exponents
::理性指数

Let's look at the square root and see if we can use the properties of exponents to determine what exponent taking a square root is equivalent to.
::让我们看看平方根看看我们能否使用指数的属性来确定何者代表平方根。

Investigation: Writing the Square Root as an Exponent
::调查:将广场根写成指数

1. Evaluate $\begin{aligned} {(\sqrt{x})}^{2} \end{aligned}$ . What happens?
::1. 评价(x)2. 会发生什么?

The $\begin{aligned} \sqrt{} \end{aligned}$ and the $\begin{aligned} ^{2} \end{aligned}$ undo each other, $\begin{aligned} ({\sqrt{x}}^{2}) = x \end{aligned}$ .
::和二人互相残杀,(x2)=x。

2. Recall that according to the power rule, when a power is raised to another power, we multiply the exponents. Therefore , we can rewrite the exponent 2 and the exponent for the root $\begin{aligned} n \end{aligned}$ as an equation , $\begin{aligned} n \cdot 2 = 1 \end{aligned}$ . Solve for $\begin{aligned} n \end{aligned}$ .
::2. 回顾根据权力规则,当权力被提升到另一权力时,我们乘以推手,因此,我们可以重写推手2和根的推手n,作为方程式,n2=1. 解决n。

$\begin{aligned} \frac{n \cdot 2}{2} & = \frac{1}{2} \\ n & = \frac{1}{2} \end{aligned}$

::n22=12n=12

3. From Number 2, we can conclude that $\begin{aligned} \sqrt{x} = x^{\frac{1}{2}} . \end{aligned}$
::3. 从2号,我们可以得出x=x12的结论。

$\begin{aligned} {(\sqrt{x})}^{2} = {(x^{\frac{1}{2}})}^{2} = x^{(\frac{1}{2}) \cdot 2} = x^{1} = x \end{aligned}$

:x)2=(x12)2=x(12)2=x(x)2=x1=x

From this investigation, we see that $\begin{aligned} \sqrt{x} = x^{\frac{1}{2}} \end{aligned}$ . We can extend this idea to the other roots as well: $\begin{aligned} \sqrt[3]{x} = x^{\frac{1}{3}}, \sqrt[4]{x} = x^{\frac{1}{4}}, \dots, \sqrt[n]{x} = x^{\frac{1}{n}} . \end{aligned}$
::从这次调查中,我们可以看到 x=x12。我们可以把这个想法扩展至其它根 : x3=x13, x4=x14,..., xn=x1n。

The Definition of $\begin{aligned} a^{\frac{1}{n}} \end{aligned}$
::A1n 的定义

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

::a1n=an

Notice that the denominator of the fraction becomes the index of the radical .
::注意分数分母成为激进指数

Example 1
::例1

Find $\begin{aligned} 256^{\frac{1}{4}} . \end{aligned}$
::找25614号

Solution: Rewrite this expression in terms of roots. A number to the 1/4th power is the same as the 4th root .
::解析 : 用根重写此表达式。第 1/ 4 位数与第 4 根相同。

$\begin{aligned} 256^{\frac{1}{4}} = \sqrt[4]{256} = \sqrt[4]{4^{4}} = 4 \end{aligned}$

Therefore, $\begin{aligned} 256^{\frac{1}{4}} = 4 \end{aligned}$ .
::因此,25614=4。

Power Rule of Radicals
::激进力量的统治

$\begin{aligned} a^{\frac{m}{n}} \end{aligned}$ = $\begin{aligned} \sqrt[n]{a^{m}} \end{aligned}$
::AM = AMn = AMn

For any real number $\begin{aligned} a, \end{aligned}$ if $\begin{aligned} \frac{m}{n} \end{aligned}$ is in lowest terms an d m and n are int egers, then we can write
::对于任何真实数字a,如果 mn 以最低值计算, m 和 n 是整数,那么我们可以写

$\begin{aligned} a^{\frac{m}{n}} = \sqrt[n]{a^{m}} = (\sqrt[n]{a})^{m} . \end{aligned}$

::AMN=AMN=(an)m。 (一)m。

Note if n is even, then $\begin{aligned} a \geq 0 \end{aligned}$ .
::注意 n 是偶数, 然后 a% 0 。

This property follows from the power rule of exponents: $\begin{aligned} b^{\frac{m}{n}} = (b^{m})^{n} . \end{aligned}$ To evaluate expressions with fractional exponents, we can apply the root and the integer exponent separately. We can see this approach in the following examples:
::此属性源自引言方的权力规则 : bmn= (bm)n。要用分数引言方来评价表达式, 我们可以分别应用 root 和整数引言方。我们可以在以下示例中看到此方法 :

Example 2
::例2

Find $\begin{aligned} 49^{\frac{3}{2}} . \end{aligned}$
::找4932号

Solution: Rewrite the expression using the power rule.
::解答: 使用权力规则重写表达式。

$\begin{aligned} 49^{\frac{3}{2}} = {(49^{\frac{1}{2}})}^{3} = {\sqrt{49}}^{3} = 7^{3} = 343 \end{aligned}$
It is usually easier to compute the root 1st and then raise the result to an integer power.
::4932=(4912)3=493=73=343 通常更容易计算根1st,然后将结果提高到整数功率。

o by Mathispower4u demonstrates how to evaluate an expression with rational exponents.
::o Mathispower4u 演示如何用理性的推理来评价表达式。

Example 3
::例3

Evaluate.
::评估。

a. $\begin{aligned} 125^{\frac{4}{3}} \end{aligned}$
::a. 12543

b. $\begin{aligned} 256^{- \frac{5}{8}} \end{aligned}$
::b. 256-58

c. $\begin{aligned} \sqrt{81^{\frac{1}{2}}} \end{aligned}$
::c. 8112

Solutions:
::解决办法:

a. $\begin{aligned} 125^{\frac{4}{3}} = {(\sqrt[3]{125})}^{4} = 5^{4} = 625 \end{aligned}$
::a. 12543=(1253)4=54=625

b. $\begin{aligned} 256^{- \frac{5}{8}} = {(\sqrt[8]{256})}^{- 5} = 2^{- 5} = \frac{1}{2^{5}} = \frac{1}{32} \end{aligned}$
::b. 256-58=(2568-5=2-5=125=132)

Note the index of a radical cannot be negative, so the factor of -1 needs to be included with the exponent in the numerator of the fraction.
::请注意,激进分子的指数不能是负的,所以 -1 的系数需要与分数的分子数中的指数一起包含。

c. $\begin{aligned} \sqrt{81^{\frac{1}{2}}} = \sqrt{\sqrt{81}} = (81^{\frac{1}{2}})^{\frac{1}{2}} = 81^{\frac{1}{4}} = \sqrt{9} = 3 \end{aligned}$
::c. 8112=81=81=(8112)12=8114=9=3

Example 4
::例4

A planet's maximum distance from the sun (in astronomical units) is given by the formula $\begin{aligned} d = p^{\frac{2}{3}}, \end{aligned}$ where $\begin{aligned} p \end{aligned}$ is the period (in years) of the planet's orbit around the sun. If a planet's orbit around the sun is 27 years, what is its distance from the sun?
::公式 d=p23 给出了行星与太阳的最大距离( 天文单位) 。公式 d=p23 给出了行星绕太阳运行的周期( 年) 。如果行星绕太阳运行的轨道为27年,那么它与太阳的距离是多少?

Solution: Substitute 27 for p and solve.
::解决方案:p和p和p的替代方案27。

$\begin{aligned} d & = 27^{\frac{2}{3}} \\ 27^{\frac{2}{3}} & = {(27^{\frac{1}{3}})}^{2} = {\sqrt[3]{27}}^{2} \\ {(\sqrt[3]{27})}^{2} & = 3^{2} = 9 \end{aligned}$
Therefore, the planet's distance from the sun is 9 astronomical units.
::d=27232723=(2713)2=27322(273)2=32=9,因此,行星与太阳的距离为9个天文单位。

by Mathispower4u demonstrates how to evaluate radical expressions on the TI-83/84.
::Mathispower4u展示了如何评价TI-83/84的激进言论。

Rationalizing the Denominator
::合理解析符号

We often prefer not to leave a radical in the denominator because it can complicate future calculations. The process to get rid of the radical in the denominator of a fraction is called rationalizing the denominator . Let's take a look at some examples.
::我们常常不愿在分母中留下激进的分母,因为这会使未来的计算复杂化。清除分母分母中的激进的过程叫做分母合理化。让我们来看一下一些例子。

Example 5
::例5

Rationalize the denominator of $\begin{aligned} \frac{2}{\sqrt[3]{5}} \end{aligned}$ .
::理顺253的分母

Solution: To eliminate the radical in the denominator, we will multiply the numerator and the denominator of the fraction by enough factors of 1 to create $\begin{aligned} \sqrt[3]{5^{3}} \end{aligned}$ in the denominator.
::解答:为了消除分母中的激进, 我们将将分母的分子和分母乘以 1 的足够系数, 从而在分母中产生 533 。

$\begin{aligned} \frac{2}{\sqrt[3]{5}} \cdot \frac{\sqrt[3]{5}}{\sqrt[3]{5}} \cdot \frac{\sqrt[3]{5}}{\sqrt[3]{5}} = \frac{2 \sqrt[3]{5^{2}}}{\sqrt[3]{5^{3}}} = \frac{2 \sqrt[3]{5^{2}}}{5} \end{aligned}$

There is no longer a radical in the denominator, so this fraction is rationalized.
::分母中不再有一个激进的分母, 所以这个分母被合理化了。

Example 6
::例6

Rationalize the denominator of $\begin{aligned} \frac{4}{3 - \sqrt{2}} \end{aligned}$ .
::理顺43-2分母。

Solution: To rationalize a fraction with two terms in the denominator, multiply by the conjugate of $\begin{aligned} 3 - \sqrt{2} \end{aligned}$ , which is $\begin{aligned} 3 + \sqrt{2} \end{aligned}$ .
::解决办法:使分母分母有两个术语的分母合理化,乘以3-2的组合,即3+2。

$\begin{aligned} \frac{4}{3 - \sqrt{2}} \cdot \frac{3 + \sqrt{2}}{3 + \sqrt{2}} & = \frac{4 (3 + \sqrt{2})}{(3 - \sqrt{2}) (3 + \sqrt{2})} \\ = \frac{12 + 4 \sqrt{2}}{9 - 2} \\ = \frac{12 + 4 \sqrt{2}}{7} \end{aligned}$

by Mathispower4u demonstrates how to rationalize the denominator of a radical expression .
::Mathispower4u 展示了如何使激进表达方式的分母合理化。

Summary
::摘要

Rational exponents of the form $\begin{aligned} a^{\frac{1}{n}} \end{aligned}$ correspond to roots of the form $\begin{aligned} \sqrt[n]{a} \end{aligned}$ .
::窗体 a1n 的理性引言与窗体 a 的根对应。

To evaluate rational exponents, use the power rule of exponents to take the root and then apply the integer exponent.
::为了评估理性指数,使用指数规则根根,然后应用整数指数。

To rationalize a denominator with one term, multiply the numerator and the denominator by the radical the number of times in the index.
::要合理使用一个术语来理顺一个分母,将分子和分母乘以指数中的极端次数。

To rationalize a denominator with two terms, multiply the numerator and the denominator by the conjugate of the denominator.
::要使一个分母合理化有两个术语, 乘以分子和分母乘以分母的组合。

Review
::回顾

Write the expressions below using roots, and then evaluate using a calculator. Answers should be rounded to the nearest hundredth.
::使用根写下面的表达式,然后使用计算器进行评估。答案应该四舍五入到最接近的百位。

1. $\begin{aligned} 72^{\frac{5}{3}} \end{aligned}$

2. $\begin{aligned} 95^{\frac{2}{3}} \end{aligned}$

3. $\begin{aligned} 125^{\frac{3}{4}} \end{aligned}$

Evaluate the following without a calculator:
::在无计算器的情况下评价以下内容:

4. $\begin{aligned} 64^{\frac{2}{3}} \end{aligned}$

5. $\begin{aligned} 27^{\frac{4}{3}} \end{aligned}$

6. $\begin{aligned} 16^{\frac{5}{4}} \end{aligned}$

7. $\begin{aligned} 25^{\frac{3}{2}} \end{aligned}$

8. $\begin{aligned} 9^{\frac{5}{2}} \end{aligned}$

9. $\begin{aligned} 32^{\frac{2}{5}} \end{aligned}$

10. $\begin{aligned} 81^{\frac{3}{4}} \end{aligned}$

Explore More
::探索更多

1. According to Kepler's Third Law, $\begin{aligned} T^{2} = A^{3}, \end{aligned}$ where $\begin{aligned} T \end{aligned}$ is the orbital period of the planet in years, and $\begin{aligned} A \end{aligned}$ is the mean distance that the planet is from the sun in astronomical units (or, the semi-major axis of the ellipse, which is equal to half the sum of smallest and greatest distance from the sun) ¹ .
::1. 根据开普勒第三定律,T2=A3,其中T是行星多年的轨道周期,A是行星与太阳之间天文单位的平均距离(或椭圆半主轴,等于离太阳最小和最大距离之和的一半)。 1

a. The Hohmann orbit is considered the most efficient orbit to reach Mars. To accomplish this orbit, a spaceship must 1st get free of Earth's gravity. Next it must increase its speed so that part of its orbit around the sun just grazes the orbit of Mars. A is 1.262 AU for the Hohmann Transfer Orbit. How long does it take to travel the Hohmann Transfer Orbit?
::a. 霍赫曼轨道被认为是进入火星的最有效轨道。为了实现这一轨道,航天器必须首先摆脱地球的引力。接下来,它必须加快速度,使其围绕太阳的部分轨道只是擦拭火星轨道。A是霍赫曼转移轨道的1.262AU。要穿越霍赫曼转移轨道需要多长时间?

b. To reach the sun directly from Earth, a spacecraft would need to be free of Earth's gravity. The orbit that is most efficient to reach the sun has a mean distance of A = 0.5 AU. How long would it take for a spacecraft from Earth to reach the sun?
::b. 要直接从地球到达太阳,航天器必须不受地球引力的影响,到达太阳最高效的轨道平均距离为A=0.5AU。从地球到太阳的航天器需要多长时间?

The growth and decay formula is $\begin{aligned} y = a b^{\frac{t}{p}} \end{aligned}$ , where $\begin{aligned} a = \end{aligned}$ the initial amount, b = the growth factor (or decay factor, if b < 1), t = the time that has passed, p = the period for the growth or decay factor (the growth or decay interval), and y = the amount after the time that has passed.
::生长和衰变公式是 y=abtp,其中 a= 初始值,b= 生长系数(或衰变系数,如果b < 1),t= 已经通过的时间,p = 生长或衰变系数(生长或衰变间隔)的周期,y = 过了一段时间后的数量。

2. If a gerbil population triples every 4 years, and the population starts with 20 gerbils, how many gerbils will there be in 12 years?
::2. 如果每4年,沙鼠人口增加三倍,人口从20个沙鼠开始,12年内将有多少沙鼠?

3. If a scientist has 40 grams of a element that has a half-life of 6 hours, how much will exist after 18 hours?
::3. 如果科学家拥有40克半衰期为6小时的元素,18小时后将有多少?

4. A certain bacteria triples every 8 hours. If there are currently 100 bacteria, how many bacteria will there be in 18 hours?
::4. 某些细菌每8小时增加3倍,如果目前有100种细菌,18小时内将有多少细菌?

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。

PLIX
::PLIX

Try these interactives that reinforce the concepts explored in this section:
::尝试这些强化本节所探讨概念的交互作用 :

References
::参考参考资料

1. "Kepler's Three Laws of Planetary Motion," last updated March 21, 2005,
::1. 2005年3月21日更新的2005年3月21日《凯勒三部行星运动法则》

Section outline

Rational Exponents ::理性指数

Investigation: Writing the Square Root as an Exponent ::调查:将广场根写成指数

The Definition of a 1 n ::A1n 的定义

Example 1 ::例1

Power Rule of Radicals ::激进力量的统治

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Rationalizing the Denominator ::合理解析符号

Example 5 ::例5

Example 6 ::例6

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

PLIX ::PLIX

References ::参考参考资料

Rational Exponents
::理性指数

Investigation: Writing the Square Root as an Exponent
::调查:将广场根写成指数

The Definition of $\begin{aligned} a^{\frac{1}{n}} \end{aligned}$
::A1n 的定义

Example 1
::例1

Power Rule of Radicals
::激进力量的统治

Example 2
::例2

Example 3
::例3

Example 4
::例4

Rationalizing the Denominator
::合理解析符号

Example 5
::例5

Example 6
::例6

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

PLIX
::PLIX

References
::参考参考资料