节: 分区表达式中的指数属性 | 9.2 分部表达式中指数的属性-interactive

章节大纲

Astronomical Numbers
::天文数字

Performing calculations is an important responsibility for astronomers. Astronomers need to calculate distance , volumes, surface areas, speeds and much more for the largest objects in the universe. For example, the distance from the Earth to the International Space Station (ISS) is approximately $\begin{aligned} 3^{5} \end{aligned}$ miles, and the distance from the Earth to Alpha Centauri is approximately $\begin{aligned} 3^{28} \end{aligned}$ miles. How many times further is Alpha Centauri 's distance than the ISS 's distance? How do you find out how many times further something is? What operation must you apply to solve this question?
::执行计算是天文学家的一项重要责任。天文学家需要计算宇宙中最大天体的距离、体积、地表面积、速度等等。例如, 从地球到国际空间站的距离约为35英里, 从地球到阿尔法半人马座的距离约为328英里。阿尔法半人马座的距离比国际空间站的距离还要远多少倍? 您如何找到更多东西? 您必须应用什么操作来解决这个问题 ?

Since you are trying to find out how many times further away Alpha Centauri is, you should divide the numbers . The distance to Alpha Centauri compared to the distance to the ISS would look like this:
::既然你试图找出阿尔法半人马座距离阿尔法半人马座距离有多远, 您应该将数字分开。与阿尔法半人马座距离相比, 与国际空间站距离的距离会是这样的 :

\begin{aligned} \frac{Distance to Alpha Centauri}{Distance to the ISS} \approx \frac{22, 876, 792, 454, 961}{243} \end{aligned}

::距离国际空间站22,876,792,454,961243

However, this number is so big that some calculators would not even be able to display it.
::然而,这个数字太大,有些计算器甚至无法显示。

How could you find a solution without using a calculator?
::不使用计算器,你怎么能找到解决办法?

Dividing With Exponents
::与生物体分离

As done in the lesson , you will begin by writing the problem in factored form . Using a factored form will help you see the problem more clearly. Additionally, using the factored form is a great strategy for solving problems using exponent properties if you ever forget the properties.
::正如在课程中所做的那样,您将首先以计数形式写出问题。使用计数形式将帮助您更清楚地看到问题。此外,使用计数形式是使用计数形式解决问题的好策略,如果您忘记了属性,则使用计数形式解决问题。

Example
::示例示例示例示例

Simplify the expression $\begin{aligned} \frac{4^{6}}{4^{2}} \end{aligned}$ .
::简化表达式4642。

When written in factored form you get the following:
::以计数格式写时,可取得以下文件:

\begin{aligned} \frac{4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4}{4 \cdot 4} \end{aligned}

U se the same property to cancel factors in the numerator and denominator that you used to simplify fractions:
::使用相同的属性来取消您用来简化分数的分子和分母中的系数:

INTERACTIVE

Dividing with Exponents by Factoring

Drag the sliders to observe what happens when you divide terms of the same base by factoring.
::拖动滑动符以观察乘数分隔同一基数时发生的情况。

Discussion Questions
::讨论问题讨论问题

Why does this work?
::为什么这个工作?

Why wouldn’t this rule work if the bases were different from each other?
::如果基础彼此不同, 为什么这项规则不能奏效?

Quotient of Powers Property
::强权引号

The property which was derived in the previous activity is called the quotient of powers property . The Quotient of Powers Property states that when dividing two exponents with the same base, you can subtract the exponents and keep the base. This property will always work as long as the base is not zero because you cannot divide by zero. Use the interactive below to explore this property further.
::先前活动中产生的财产称为权力财产的商数。“权力财产引号”指出,在用同一基数分割两个指数时,可以减去指数并保留基数。只要基数不是零,则该财产始终有效,因为不能除以零。使用下面的交互方式进一步探索该属性。

INTERACTIVE

Quotient of Powers

Drag the sliders to observe what happens when you divide terms with the same base.
::拖动滑块以观察用相同基数分隔条件时发生的情况。

Discussion Questions
::讨论问题讨论问题

Returning to the original question of $\begin{aligned} \frac{22^{7}}{22^{4}}, \end{aligned}$ how could you use the Quotient of Powers Property to solve the problem?
::回到227224的最初问题,你如何利用权力财产的报价来解决这个问题?

Astronomers typically use exponent rules to solve problems with large numbers. Why do you think this is?
::天文学家通常使用速记规则解决大量问题。你认为这为什么?

Dividing With Terms
::与任期

To simplify a quotient of two terms , you will need to use the Quotient of Powers Property to simplify individual parts.
::为了简化两个术语的商数,你将需要使用权力财产的报价来简化各部分。

Use the interactive below to explore this idea.
::使用下面的交互方式来探讨这个想法。

INTERACTIVE

Quotient of Powers with Terms

Drag the sliders to observe what happens when you divide terms.
::拖动滑动器以观察分隔条件时发生的情况。

When terms have coefficients, simplify the coefficient like a regular fraction .
::当条件有系数时,要像正常的分数一样简化系数。

Example
::示例示例示例示例

Simplify the expression $\begin{aligned} \frac{4 r^{3} s^{5}}{6 r s^{5}} \end{aligned}$ .
::简化表达式 4r3s56rs5。

To solve this, you can either write the expression in expanded form or use the Quotient of Powers Property. Here is a look at both.
::要解决这个问题, 您可以以扩展的形式写出表达式, 或者使用“ 权力财产的引号 ” 。下面请看两者。

\begin{aligned} \begin{array}{ll} Expanded Form & Quotient of Powers \\ \frac{4 r^{3} s^{5}}{6 r s^{5}} & \frac{4 r^{3} s^{5}}{6 r s^{5}} \\ = \frac{4 \cdot r \cdot r \cdot r \cdot s \cdot s \cdot s \cdot s \cdot s}{2 \cdot 3 \cdot r \cdot s \cdot s \cdot s \cdot s \cdot s} & \cdot \frac{4}{6} = \frac{2}{3} \\ = \frac{2 \cdot r \cdot r}{3} & \cdot \frac{r^{3}}{r^{1}} = r^{3 - 1} = r^{2} \\ = \frac{2}{3} r^{2} & \cdot \frac{s^{5}}{s^{5}} = 1 \\ = \frac{2}{3} r^{2} \end{array} \end{aligned}

::4r3s56rs54r3s56rs5=4 r r r r r r r r r r s si si si si si si si 2 3 r r si si si si si si si si si si si si si si si si 46= si si si si si si si si 46= = 23=2 r r r r r r3 r3=1 r2= 23r5=1 =2

Summary

Use exponent property to simplify exponential expressions:
- Product of Powers Property: $\begin{aligned} a^{m} \times a^{n} = a^{m + n} \end{aligned}$
  ::功率产值属性: AMxan=am+n
- Quotient of Powers Property: $\begin{aligned} \frac{a^{m}}{a^{n}} = a^{m - n} \end{aligned}$
  ::强权引号财产:aman=am-n
::用于简化指数表达式的引号属性: 权力产财产: AMxan=am+n 权力产财产: man=am-n

The Quotient of Powers Property can be used to simplify fractions with variables by canceling pairs of variables in the numerator and denominator. For example: $\begin{aligned} \frac{x^{3} y^{1}}{x^{2} y^{7}} = \frac{x}{y^{6}} \end{aligned}$
::Power 属性的引号可用于通过取消分子和分母中的对变量来简化变量的分数。例如: x3y1x2y7=xy6

章节大纲

Astronomical Numbers ::天文数字

Dividing With Exponents ::与生物体分离

Quotient of Powers Property ::强权引号

Dividing With Terms ::与任期

Astronomical Numbers
::天文数字

Dividing With Exponents
::与生物体分离

Quotient of Powers Property
::强权引号

Dividing With Terms
::与任期