节: 承认并应用引用属性的权力 | 12.9 承认和应用引物权

章节大纲

Professor Smith works in a laboratory and is training a new intern, Rakesh, in a specific task. Professor Smith tells Rakesh that when she herself did the task, she started with a very small sample of cobalt. She had 10 grams of it and took one-third of one-third of one-third of one-third of it. How can Rakesh figure out how many grams of the sample Professor Smith actually ended up using?
::Smith教授在实验室工作,正在培训一名新的实习生,Rakesh,从事一项具体的任务。Smith教授告诉Rakesh,当她本人完成这项任务时,她从一个很小的钴样本开始,她有10克的钴,占三分之一的三分之一。Rakesh如何知道Smith教授到底用了多少克?

In this concept, you will learn to recognize and apply the power of a quotient property .
::在此概念中,您将学会承认和运用商数财产的力量。

Power of a Quotient Property
::引物权

Exponents can be applied to both fractions and quotients. For example, $\begin{aligned} (\frac{1}{2}) (\frac{1}{2}) (\frac{1}{2}) (\frac{1}{2}) = {(\frac{1}{2})}^{4} \end{aligned}$ . To evaluate this multiplication , find the product of the numerators and the product of the denominators.
::指数可以适用于分数和商数。例如, (12) (12) (12) (12) (12) = (12) 4。要评估此乘法, 请找到分子的产物和分母的产物。

$\begin{aligned} \begin{array}{rcl} \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} & = & {(\frac{1}{2})}^{4} \\ = & \frac{1^{4}}{2^{4}} \\ = & \frac{1}{16} \end{array} \end{aligned}$

The Power of a Quotient Property says that for any nonzero numbers $\begin{aligned} a \end{aligned}$ and $\begin{aligned} b \end{aligned}$ and any integer $\begin{aligned} n \end{aligned}$ :
::引号属性的功率表示,对于任何非零数a和b以及任何整数n:

$\begin{aligned} {(\frac{a}{b})}^{n} = \frac{a^{n}}{b^{n}} \end{aligned}$

:ab)n=anbn

Let’s look at an example.
::让我们举个例子。

Simplify: $\begin{aligned} {(\frac{5}{3})}^{4} \end{aligned}$ .
::简化53)4。

First, apply the product of a quotient property .
::首先,应用商数属性的产物。

$\begin{aligned} {(\frac{5}{3})}^{4} = \frac{5^{4}}{3^{4}} \end{aligned}$

Next, expand to simplify.
::下一步,扩展到简化。

$\begin{aligned} \begin{array}{rcl} \frac{5^{4}}{3^{4}} & = & \frac{5 \cdot 5 \cdot 5 \cdot 5}{3 \cdot 3 \cdot 3 \cdot 3} \\ = & \frac{625}{81} \end{array} \end{aligned}$

The answer is $\begin{aligned} \frac{625}{81} \end{aligned}$ .
::答案是62581。

Let’s look at another example.
::让我们再看看另一个例子。

Simplify: $\begin{aligned} {(\frac{3 k}{2 j})}^{4} \end{aligned}$ .
::简化( 3k2j) 4 。

First, apply the product of a quotient property.
::首先,应用商数属性的产物。

$\begin{aligned} {(\frac{3 k}{2 j})}^{4} = \frac{(3 k)^{4}}{(2 j)^{4}} \end{aligned}$

:3k2j)4=(3k)4(2j)4

Next, expand to simplify.
::下一步,扩展到简化。

$\begin{aligned} \begin{array}{rcl} \frac{(3 k)^{4}}{(2 j)^{4}} & = & \frac{3^{4} k^{4}}{2^{2} j^{4}} \\ = & \frac{3 \cdot 3 \cdot 3 \cdot 3 \cdot k \cdot k \cdot k \cdot k}{2 \cdot 2 \cdot 2 \cdot 2 \cdot j \cdot j \cdot j \cdot j} \\ = & \frac{81 k^{4}}{16 j^{4}} \end{array} \end{aligned}$

:3k)4(2j)4=34k422j4=3}3k422j4=3}333333kkk}k}k}k}k}k}2}2222}j}j}j}j}j}j}=81k416j4}

The answer is $\begin{aligned} \frac{81 k^{4}}{16 j^{4}} \end{aligned}$ .
::答案是81k416j4。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Professor Smith and the original 10 grams of cobalt.
::早些时候,你被问及史密斯教授和最初的10克钴的问题。

Rakesh needs to figure out how much of the cobalt Professor Smith actually ended up using after she took one-third of one-third of one-third of one-third of 10 grams.
::拉凯什需要弄清楚史密斯教授在拿了10克中三分之一的三分之一的三分之一后究竟用了多少钴教授

To figure out the number of grams in the sample, he must use monomials and powers.
::为了找出样本中的克数,他必须使用单项和单项权力。

First, set up the problem to model the information given in the story.
::首先,设置问题来模拟故事中所提供的信息。

$\begin{aligned} 10 \times {(\frac{1}{3})}^{4} \end{aligned}$

Next, apply the product of a quotient property.
::其次,应用商数属性的产物。

$\begin{aligned} 10 \times {(\frac{1}{3})}^{4} = 10 \times \frac{1^{4}}{3^{4}} \end{aligned}$

Then, expand to simplify.
::然后,扩展到简化。

$\begin{aligned} \begin{array}{rcl} 10 \times \frac{1^{4}}{3^{4}} & = & 10 \times \frac{1 \cdot 1 \cdot 1 \cdot 1}{3 \cdot 3 \cdot 3 \cdot 3} \\ = & 10 \times \frac{1}{81} \\ = & \frac{10}{81} \end{array} \end{aligned}$

The answer is $\begin{aligned} \frac{10}{81} \end{aligned}$ .
::答案是1081。

Professor Smith’s sample size was $\begin{aligned} \frac{10}{81} \end{aligned}$ grams.
::Smith教授的样本规模为1081克。

Example 2
::例2

Simplify the following quotient.
::简化下列商数。

$\begin{aligned} {(\frac{- 4 x}{3 y})}^{3} \end{aligned}$

:-4x3y)3

First, apply the product of a quotient property.
::首先,应用商数属性的产物。

$\begin{aligned} {(\frac{- 4 x}{3 y})}^{3} = \frac{(- 4 x)^{3}}{(3 y)^{3}} \end{aligned}$

:-4x3y)3=(-4x3y)3(-3y)3

Next, expand to simplify.
::下一步,扩展到简化。

$\begin{aligned} \begin{array}{rcl} \frac{(- 4 x)^{3}}{(3 y)^{3}} & = & \frac{- 4 \cdot - 4 \cdot - 4 \cdot x \cdot x \cdot x}{3 \cdot 3 \cdot 3 \cdot y \cdot y \cdot y} \\ = & \frac{- 64 x^{3}}{27 y^{3}} \end{array} \end{aligned}$

The answer is $\begin{aligned} \frac{- 64 x^{3}}{27 y^{3}} \end{aligned}$ .
::答案是 - 64x327y3。

Example 3
::例3

Simplify the quotient: $\begin{aligned} {(\frac{4}{5})}^{3} \end{aligned}$ .
::简化商数453)

First, apply the product of a quotient property.
::首先,应用商数属性的产物。

$\begin{aligned} {(\frac{4}{5})}^{3} = \frac{4^{3}}{5^{3}} \end{aligned}$

Next, expand to simplify.
::下一步,扩展到简化。

$\begin{aligned} \begin{array}{rcl} \frac{4^{3}}{5^{3}} & = & \frac{4 \cdot 4 \cdot 4}{5 \cdot 5 \cdot 5} \\ = & \frac{64}{125} \end{array} \end{aligned}$

The answer is $\begin{aligned} \frac{64}{125} \end{aligned}$ .
::答案是64125。

Example 4
::例4

Simplify the quotient: $\begin{aligned} {(\frac{2 a}{3 b})}^{2} \end{aligned}$ .
::简化商数 : (2a3b) 2。

First, apply the product of a quotient property.
::首先,应用商数属性的产物。

$\begin{aligned} {(\frac{2 a}{3 b})}^{2} = \frac{2^{2} a^{2}}{3^{2} b^{2}} \end{aligned}$

:2a3b)2=22a232b2

Next, expand to simplify.
::下一步,扩展到简化。

$\begin{aligned} \begin{array}{rcl} \frac{2 a^{2}}{3^{2} b^{2}} & = & \frac{2 \cdot 2 \cdot a \cdot a}{3 \cdot 3 \cdot b \cdot b} \\ = & \frac{4 a^{2}}{9 b^{2}} \end{array} \end{aligned}$

The answer is $\begin{aligned} \frac{4 a^{2}}{9 b^{2}} \end{aligned}$ .
::答案是4a29b2。

Example 5
::例5

Simplify the quotient: $\begin{aligned} {(\frac{a}{5 b})}^{3} \end{aligned}$ .
::简化商数a5b) 3。

First, apply the product of a quotient property.
::首先,应用商数属性的产物。

$\begin{aligned} {(\frac{a}{5 b})}^{3} = \frac{a^{3}}{5^{3} b^{3}} \end{aligned}$

:a5b)3=a353b3

Next, expand to simplify.
::下一步,扩展到简化。

$\begin{aligned} \begin{array}{rcl} \frac{a^{3}}{5^{3} b^{3}} & = & \frac{a \cdot a \cdot a}{5 \cdot 5 \cdot 5 \cdot b \cdot b \cdot b} \\ = & \frac{a^{3}}{125 b^{3}} \end{array} \end{aligned}$

::a353b3=aaaa55_5_5}bb=a3125b3=a3125b3

The answer is $\begin{aligned} \frac{a^{3}}{125 b^{3}} \end{aligned}$ .
::答案是a3125b3。

Review
::回顾

Simplify.
::简化。

$\begin{aligned} {(\frac{2}{3})}^{4} \end{aligned}$

$\begin{aligned} {(\frac{1}{3})}^{3} \end{aligned}$

$\begin{aligned} {(\frac{7}{8})}^{2} \end{aligned}$

$\begin{aligned} {(\frac{2}{5})}^{4} \end{aligned}$

$\begin{aligned} {(\frac{7 k}{- 2 m})}^{3} \end{aligned}$
:7k-2m)3

$\begin{aligned} {(\frac{3 x}{- 2 y})}^{3} \end{aligned}$
:3x-2y)3

$\begin{aligned} {(\frac{4 x}{- 3 y})}^{4} \end{aligned}$
:4x-3y)4

$\begin{aligned} {(\frac{5 y}{- 2 z})}^{5} \end{aligned}$
:5y-2z)5

$\begin{aligned} {(\frac{- 2 y}{4 z})}^{4} \end{aligned}$
:-2y4z)4

$\begin{aligned} {(\frac{4 x y}{- 2 z^{5}})}^{5} \end{aligned}$
:4xy-2z5)5

$\begin{aligned} {(\frac{12 x^{2} y^{4}}{- 6 z^{3}})}^{2} \end{aligned}$
:12x2y4-6z3)2

$\begin{aligned} {(\frac{7 x^{2} y}{- 2 z^{3}})}^{3} \end{aligned}$
:7x2y-2z3)3

$\begin{aligned} {(\frac{2 x^{3} y^{2}}{- 2 z^{3}})}^{3} \end{aligned}$
:2x3y2-2z3)3

$\begin{aligned} {(\frac{x^{11}}{y^{9}})}^{5} \end{aligned}$
:x119)5

$\begin{aligned} {(\frac{- 5 x^{3}}{3 h^{2} j^{8}})}^{5} \end{aligned}$
:-5x33h2j8)5

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。

章节大纲

Power of a Quotient Property ::引物权

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Review ::回顾

Review (Answers) ::回顾(答复)

Power of a Quotient Property
::引物权

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Example 5
::例5

Review
::回顾

Review (Answers)
::回顾(答复)