Course: 12.10 承认和适用电力产权的权力

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承认和适用电力产权的权力
Laura had to solve the following problem and then explain it to the class.
::劳拉必须解决以下的问题然后向班级解释

$\begin{aligned} (x^{2} y^{3} z^{3})^{3} \end{aligned}$

:x2y3z3)3

This is a monomial expression that is being raised to the third power. Do you know how to simplify this expression? Can you help Laura so she knows she has the right answer before explaining it to the class?
::这是一个单一的表达式, 正在被提升到第三权力。您知道如何简化这个表达式吗 ? 您能帮助劳拉, 让她知道在向课堂解释之前她有正确的答案吗 ?

In this concept, you will learn to recognize and apply the power of a power property .
::在此概念中,您将学会承认和运用权力财产的权力。

Power of a Power Property
::权力财产权

Exponents are a useful tool in simplifying expressions. When the rules of exponents are followed, the patterns become clear.
::指数是简化表达方式的有用工具。当遵循指数规则时,模式就会变得清晰。

Let’s look at an example where you have to solve powers raised to a power.
::让我们来看看一个例子,其中你必须解决权力被赋予的权力。

Simplify:
::简化 :

$\begin{aligned} {(\frac{x^{7}}{y^{9}})}^{4} \end{aligned}$

:x7y9)4

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

$\begin{aligned} {(\frac{x^{7}}{y^{9}})}^{4} = \frac{(x^{7})^{4}}{(y^{9})^{4}} \end{aligned}$

:x7y9)4=(x7)4(y9)4

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

$\begin{aligned} \begin{array}{rcl} \frac{(x^{7})^{4}}{(y^{9})^{4}} & = & \frac{x^{7} \cdot x^{7} \cdot x^{7} \cdot x^{7}}{y^{9} \cdot y^{9} \cdot y^{9} \cdot y^{9}} \\ = & \frac{x^{7 + 7 + 7 + 7}}{y^{9 + 9 + 9 + 9}} \\ = & \frac{x^{28}}{y^{36}} \end{array} \end{aligned}$

The answer is $\begin{aligned} \frac{x^{28}}{y^{36}} \end{aligned}$ .
::答案是 x28y36。

This is an example of the Power of a Power Property which says for any nonzero numbers $\begin{aligned} a \end{aligned}$ and $\begin{aligned} b \end{aligned}$ and any integer $\begin{aligned} n \end{aligned}$ :
::这是电力财产的权力的例子,它规定任何非零数a和b以及任何整数n:

$\begin{aligned} (a^{m})^{n} = a^{m \times n} \end{aligned}$

:am)n=amxn

Let’s look at an example of the power of a power property.
::让我们看看一个权力财产权力的例子。

Simplify:
::简化 :

$\begin{aligned} (x^{5})^{3} \end{aligned}$

:十五)3

$\begin{aligned} \begin{array}{rcl} (x^{5})^{3} & = & x^{5 \times 3} \\ = & x^{15} \end{array} \end{aligned}$

Let’s look at another example of the power of a power property.
::让我们看看另一个权力财产权力的例子。

Simplify:
::简化 :

$\begin{aligned} (x^{6} y^{3})^{7} \end{aligned}$

:x6y3)7

$\begin{aligned} \begin{array}{rcl} (x^{6} y^{3})^{7} & = & x^{6 \times 7} y^{3 \times 7} \\ = & x^{42} y^{21} \end{array} \end{aligned}$

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Laura and her problem. Laura has to solve the following problem and then explain it to the class. She isn’t quite sure how to solve the problem. Can you help her out?
::早些时候,劳拉和她的麻烦问题给了你一个问题。劳拉必须解决以下的问题,然后向课堂解释。她不太确定如何解决这个问题。你能帮她吗?

She has to apply the power of a power property.
::她必须运用权力财产的权力。

Simplify:
::简化 :

$\begin{aligned} (x^{2} y^{3} z^{3})^{3} \end{aligned}$

:x2y3z3)3

First, apply the power of a power property.
::首先,适用权力财产的权力。

$\begin{aligned} (x^{2} y^{3} z^{3})^{3} = x^{2 \times 3} y^{3 \times 3} z^{3 \times 3} \end{aligned}$

:x2y3z3)3=x2x3y3x3x3x3x3x3x3x3x3x3x3x3x3x3x3x3x3

Next, simplify.
::下一个,简化。

$\begin{aligned} x^{2 \times 3} y^{3 \times 3} z^{3 \times 3} = x^{6} y^{9} z^{9} \end{aligned}$

::x2x3y3x3z3x3=x6y9z9

The answer is $\begin{aligned} x^{6} y^{9} z^{9} \end{aligned}$ .
::答案是 x6y9z9。

Example 2
::例2

Simplify:
::简化 :

$\begin{aligned} (x^{2} y^{4} z^{3})^{4} \end{aligned}$

:x2y4z3)4

First, apply the power of a power property.
::首先,适用权力财产的权力。

$\begin{aligned} (x^{2} y^{4} z^{3})^{4} = x^{2 \times 4} y^{4 \times 4} z^{3 \times 4} \end{aligned}$

:x2y4z3)4=x2x4y4x4x4z3x4

Next, simplify.
::下一个,简化。

$\begin{aligned} x^{2 \times 4} y^{4 \times 4} z^{3 \times 4} = x^{8} y^{16} z^{12} \end{aligned}$

::x2x4y4x4z3x4=x8y16z12

The answer is $\begin{aligned} x^{8} y^{16} z^{12} \end{aligned}$ .
::答案是 x8y16z12。

Example 3
::例3

Apply the Power of a Power Property to:
::1. 将权力财产的权力适用于:

$\begin{aligned} (x^{7})^{3} \end{aligned}$

:十七)3

First, apply the power of a power property.
::首先,适用权力财产的权力。

$\begin{aligned} (x^{7})^{3} = x^{7 \times 3} \end{aligned}$

:x7)3=x7×3

Next, simplify.
::下一个,简化。

$\begin{aligned} x^{7 \times 3} = x^{21} \end{aligned}$

::x7x3=x21

The answer is $\begin{aligned} x^{21} \end{aligned}$ .
::答案是 x21 。

Example 4
::例4

Apply the Power of a Power Property to:
::1. 将权力财产的权力适用于:

$\begin{aligned} (x^{3} y^{6})^{3} \end{aligned}$

:x3y6)3

First, apply the power of a power property.
::首先,适用权力财产的权力。

$\begin{aligned} (x^{3} y^{6})^{3} = x^{3 \times 3} y^{6 \times 3} \end{aligned}$

:x3y6)3=x3x3x3x3y6x3

Next, simplify.
::下一个,简化。

$\begin{aligned} x^{3 \times 3} y^{6 \times 3} = x^{9} y^{18} \end{aligned}$

::x3x33y6x3=x9y18

The answer is $\begin{aligned} x^{9} y^{18} \end{aligned}$ .
::答案是 x9y18 。

Example 5
::例5

Apply the Power of a Power Property to:
::1. 将权力财产的权力适用于:

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

:a7)8

First, apply the power of a power property.
::首先,适用权力财产的权力。

$\begin{aligned} (a^{7})^{8} = a^{7 \times 8} \end{aligned}$

:a7)8=a7x8

Next, simplify.
::下一个,简化。

$\begin{aligned} a^{7 \times 8} = a^{56} \end{aligned}$

::a7x8=a56

The answer is $\begin{aligned} a^{56} \end{aligned}$ .
::答案是56号

Review
::回顾

Simplify each monomial expression by applying the Power of a Power Property.
::通过适用权力财产的权力,简化每一种单一表达方式。

$\begin{aligned} (x^{2})^{2} \end{aligned}$
:x2)2

$\begin{aligned} (y^{4})^{3} \end{aligned}$
:y4)3

$\begin{aligned} (x^{2} y^{4})^{3} \end{aligned}$
:x2y4)3

$\begin{aligned} (x^{3} y^{3})^{4} \end{aligned}$
:x3y3)4

$\begin{aligned} (y^{6} z^{2})^{6} \end{aligned}$
:y6z2)6

$\begin{aligned} (x^{3} y^{4})^{5} \end{aligned}$
:x3y4)5

$\begin{aligned} (a^{5} b^{3})^{3} \end{aligned}$
:a5b3)3

$\begin{aligned} (a^{4} b^{4})^{5} \end{aligned}$
:a4b4)5

$\begin{aligned} (a^{3} b^{6} c^{7})^{3} \end{aligned}$
:a3b6c7)3

$\begin{aligned} (x^{12})^{3} \end{aligned}$
:x12)3

$\begin{aligned} (y^{9})^{6} \end{aligned}$
:y9)6

$\begin{aligned} (a^{2} b^{8} c^{9})^{4} \end{aligned}$
:a2b8c9)4

$\begin{aligned} (x^{4} y^{3} z^{3})^{3} \end{aligned}$
:x4y3z3)3

$\begin{aligned} (a^{4} b^{3} c^{7} d^{8})^{6} \end{aligned}$
:a4b3c7d8)6

$\begin{aligned} (a^{3} b^{11})^{5} \end{aligned}$
:a3b11)5

$\begin{aligned} (x^{6} y^{10} z^{12})^{5} \end{aligned}$
:x6y10z12)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。

Section outline

General

承认和适用电力产权的权力

Power of a Power Property ::权力财产权

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 Power Property
::权力财产权

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)