节: 乘法表达式中的指数属性 | 9.1 乘法表达式中的指数属性-interactive

章节大纲

Exponents Revisited
::重审

When a number is being multiplied by itself repeatedly, an exponent can be used to express this.
::当一个数字本身被多次乘以时,可以用一个指数来表示这一点。

The exponent is the number which indicates how many times to multiply the base. The base is the number being multiplied. This expression can be written as a product of factors:
::引号是表示基准乘以多少倍的数值。基数是乘以的数值。此表达式可以由下列因素生成:

\begin{aligned} 7^{3} = 7 \cdot 7 \cdot 7 \end{aligned}

Writing exponents in expanded form will help understand the properties for exponent operations .
::扩展格式的书写列表将有助于理解指数操作的属性。

Product of Powers Property
::财产财产财产

In algebraic equations, there will be many instances where you will need to multiply terms with exponents. For example, you might need to multiply the term $\begin{aligned} {2 x}^{4} \end{aligned}$ by $\begin{aligned} {6 x}^{2} . \end{aligned}$ To multiply these expressions, first look at a simpler example.
::在代数方程式中,有很多例子需要用引言来乘法。例如,可能需要将术语 2x4 乘以 6x2 来乘以这些表达式,首先请看一个简单的示例。

Use the interactive below to explore how to multiply exponents.
::使用下面的交互方式来探索如何乘法引言。

INTERACTIVE

Product of Powers

Drag the sliders to observe the multiplication of exponents.
::拖动滑动器以观察指数的乘法。

Th e rule for multiplying exponents with the same base is called the Product of a Power Property . The Product of Powers Property states that when multiplying two exponents with the same base, you can add the exponents and keep the base.
::使用相同基数的乘数指数的规则被称为“电力财产产品 ” 。权力财产产品表示,在用同一基数乘以两个指数时,您可以添加指数并保留基数。

To simplify the product of $\begin{aligned} {2 x}^{4} \end{aligned}$ and $\begin{aligned} {6 x}^{2} \end{aligned}$ , you will keep the base and add the exponent. However, in the expression $\begin{aligned} {2 x}^{4} \end{aligned}$ what is the exponent and what is the base? The exponent is 4, and the base is x. The number 2 is the coefficient . To include the 2 in the base, you will need to write it in parenthesis: $\begin{aligned} {(2 x)}^{4} \end{aligned}$ . Since the coefficients are not part of the base or the exponent you will multiply them normally. This concept is easier to observe when written expanded form:
::要简化 2x4 和 6x2 的产物, 您将保留基数并添加引号。但是, 在表达式 2x4 中, 前面的出处和底数是什么 ? 前面的出处是 4 , 后面的基数是 x 。数字 2 是系数。要将 2 包含在基数中, 您需要将其写成括号 : ( 2x) 4 。由于系数不是基数的一部分, 或者前面的数通常会乘以它们。当书面扩展格式时, 这个概念比较容易观察 :

\begin{aligned} {2 x}^{4} \cdot {6 x}^{2} = (2 \cdot x \cdot x \cdot x \cdot x) \cdot (6 \cdot x \cdot x) \end{aligned}

::2x4}6x2 = (2}xxxxxxxxxx) = (6xxxxxx)

Now you can rearrange this using the commutative property as follows:
::现在您可以使用通量属性重新排列如下:

\begin{aligned} 2 \cdot 6 \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x = {12 x}^{6} \end{aligned}

::26xxxxxxxxx=12x6

Power of a Product Property
::产品产权的功率

A term can be a product of coefficients and variables. What would happen if you took a term to a power? Use the interactive below to help understand this concept.
::一个术语可以是系数和变数的产物。如果您将一个术语带给一个强国, 那么会发生什么? 使用下面的交互式词来帮助理解这个概念。

INTERACTIVE

Power of a Product

Drag the sliders to observe the power of a term.
::拖动滑动器以观察术语的力量。

Th e rule for raising a term to a power is called the Power of a Product Property . The Power of a Product Property states that If a term is being raised to an exponent, you can distribute the exponent to each factor .
::向权力机关提出术语的规则称为 " 产品产权的权力 " , " 产品产权的权力 " 规定,如果一个术语被提给推手,您可以向每个因素分配该术语。

Power of a Power Property
::权力财产权

The final property has to do with raising a power to a power. This property will pop up in many places. You may find yourself having to simplify an expression like $\begin{aligned} {({3 x}^{2})}^{3} \end{aligned}$ . To do this use the interactive below to explore the property.
::最后的属性与将权力提升到权力有关。此属性会在许多地方出现。您可能发现自己需要简化表达式( 3x2) 3 。要做到这一点, 请使用下面的交互方式来探索该属性。

INTERACTIVE

Power of a Power

Drag the sliders to observe what happens when you raise a power to a power.
::拖动滑动器以观察当您向电源增强电源时会发生什么。

Discussion Questions
::讨论问题讨论问题

What is the pattern for raising a power to a power?
::将权力提升到权力的格局是什么?

Why does this work?
::为什么这个工作?

The rule for raising a power to a power is called the Power of a Power Property . The Power of a Power Property states that if an exponent is being raised to another exponent, you can multiply the exponents. You can use this property to solve a problem like $\begin{aligned} {({3 x}^{2})}^{3} \end{aligned}$ . Before applying the property, first write the expression in expanded form.
::将权力赋予权力的规则被称为权力财产的权力。电力财产的权力规定,如果向另一个权力强者提出一个控股人,您可以乘以该控股人。您可以使用该财产来解决类似(3x2)3 的问题。在应用该财产之前,先以扩展的形式写出表达方式。

\begin{aligned} {({3 x}^{2})}^{3} = (3 \cdot x \cdot x) \cdot (3 \cdot x \cdot x) \cdot (3 \cdot x \cdot x) = 3 \cdot 3 \cdot 3 \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x = 3^{3} x^{6} \end{aligned}

3x2)3 = (3xxxx) = (3xxx) = (3xxx) = (3xxx) = (3xxx) = (3xxx) = (3xxx) = (3xxx) = (3xxx) = (3xxx) = (3xx6)

To solve a problem like this, you will need to use two of three properties: you will need to distribute the exponent to the term which will involve taking a power to a power. You must first distribute the outer exponent to each factor in the term:
::要解决这样的一个问题, 您需要使用三个属性中的两个: 您需要分配要包含将权力带给一个权力的术语的引号。您必须首先将外部引号分配给该术语中的每个要素 :

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

::33(x2)3

After distributing the exponent, you will need to simplify the power to a power to obtain the answer.

\begin{aligned} 3^{3} {(x^{2})}^{3} = 3^{3} x^{6} = 27 x^{6} \end{aligned}

::分配前台后, 您需要简化权限, 才能获取答案。 33( x2) 3= 33x6= 27x6

Summary

Exponents represent repeated multiplication. For example: $\begin{aligned} 7^{3} = 7 \cdot 7 \cdot 7 \cdot \end{aligned}$
::指数表示重复乘法。例如: 73= 7= 7+7+7

Use exponent properties 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
- Power of a Product Property: $\begin{aligned} a^{m} \times a^{n} = a^{m + n} \end{aligned}$
  ::产品属性的功率: AMxan=am+n
- Power of a Power Property: $\begin{aligned} (a^{m})^{n} = a^{m n} \end{aligned}$
  ::权力财产权am)n=amn
::用于简化指数表达式的引号属性: Power 产物财产: AMxan=am+n 产品产权的动力: AMxan=am+n Power power power of a power power power: (am)n=amn

章节大纲

Exponents Revisited ::重审

Product of Powers Property ::财产 财产 财产

Power of a Product Property ::产品产权的功率

Power of a Power Property ::权力财产权

Exponents Revisited
::重审

Product of Powers Property
::财产财产财产

Power of a Product Property
::产品产权的功率

Power of a Power Property
::权力财产权