节: 负指数 | 8.5负指数

章节大纲

Negative Exponents
::负指数

The product and quotient rules for exponents lead to many interesting concepts. For example, so far we’ve mostly just considered positive, whole numbers as exponents, but you might be wondering what happens when the exponent isn’t a positive whole number. What does it mean to raise something to the power of zero, or -1, or $\begin{aligned} \frac{1}{2} \end{aligned}$ ? In this lesson, we’ll find out.
::引言者的产品和商数规则引出了许多有趣的概念。比如,到目前为止,我们大部分只是将正数、整数视为推理,但你也许在想,当推论者不是正数时会发生什么情况。提高零或-1或12的功率意味着什么? 在这个教训中,我们会发现。

Simplify Expressions With Negative Exponents
::简化带有负指数的表达式

When we learned the quotient rule for exponents $\begin{aligned} (\frac{x^{n}}{x^{m}} = x^{(n - m)}) \end{aligned}$ , we saw that it applies even when the exponent in the denominator is bigger than the one in the numerator. Canceling out the factors in the numerator and denominator leaves the leftover factors in the denominator, and subtracting the exponents leaves a negative number. So negative exponents simply represent fractions with exponents in the denominator. This can be summarized in a rule:
::当我们学习了指数者(xnxm=x(n-m))的商数规则(xnxm=x(n-m))时,我们看到,即使分母中的指数比分子中的指数大,该规则也适用。取消分子和分母中的因素留下分母中的剩余系数,减去指数后留下负数。因此负指数只是分数中的分数,分母中带有指数。这可以用一条规则来概括:

Negative Power Rule for Exponents : $\begin{aligned} x^{- n} = \frac{1}{x^{n}} \end{aligned}$ , where $\begin{aligned} x \neq 0 \end{aligned}$
::对指数的负功率规则:x-n=1xn, 其中 x0

Negative exponents can be applied to products and quotients also. Here’s an example of a negative exponent being applied to a product:
::负指数也可以适用于产品和商数。以下是对产品应用负指数的例子:

$\begin{aligned} (x^{3} y)^{- 2} = x^{- 6} y^{- 2} & using the power rule \\ x^{- 6} y^{- 2} = \frac{1}{x^{6}} \cdot \frac{1}{y^{2}} = \frac{1}{x^{6} y^{2}} & using the negative power rule separately on each variable \end{aligned}$

:x3y)-2=x-6-6y-2使用功率规则-6y-2=1x6}1x6}1y2=1x6y2 分别使用对每个变量的负功率规则

And here’s one applied to a quotient:
::以下是一个商数:

$\begin{aligned} {(\frac{a}{b})}^{- 3} = \frac{a^{- 3}}{b^{- 3}} & using the power rule for quotients \\ \frac{a^{- 3}}{b^{- 3}} = \frac{a^{- 3}}{1} \cdot \frac{1}{b^{- 3}} = \frac{1}{a^{3}} \cdot \frac{b^{3}}{1} & using the negative power rule on each variable separately \\ \frac{1}{a^{3}} \cdot \frac{b^{3}}{1} = \frac{b^{3}}{a^{3}} & simplifying the division of fractions \\ \frac{b^{3}}{a^{3}} = {(\frac{b}{a})}^{3} & using the power rule for quotients in reverse. \end{aligned}$

:ab)-3=a-3=a-3-b-3=3使用系数规则的功率规则;a-3=3b-3=3=a-31b-3=1a3b-31,对每个变量分别使用负功率规则;a3=b3a3=3a3简化分数的分数b3a3=(ba)3使用功率规则对反向位数使用功率规则。

That last step wasn’t really necessary, but putting the answer in that form shows us something useful: $\begin{aligned} {(\frac{a}{b})}^{- 3} \end{aligned}$ is equal to $\begin{aligned} {(\frac{b}{a})}^{3} \end{aligned}$ . This is an example of a rule we can apply more generally:
::这最后一步并非真正必要, 但以这种形式提出答案向我们展示了一些有用的东西ab) -3等于(ba)3。

Negative Power Rule for Fractions: $\begin{aligned} {(\frac{x}{y})}^{- n} = {(\frac{y}{x})}^{n}, \end{aligned}$ where $\begin{aligned} x \neq 0, y \neq 0 \end{aligned}$
::分数的负功率规则 : (xy)-n=(yx)n, 其中 x0, y0

This rule can be useful when you want to write out an expression without using fractions.
::当您想要在不使用分数的情况下写入表达式时, 此规则是有用的。

Writing Expressions Without Fractions
::无分数的书写表达式

Write the following expressions without fractions.
::撰写以下表达式,不包含分数。

a) $\begin{aligned} \frac{1}{x} \end{aligned}$
::a) 1x 1x

$\begin{aligned} \frac{1}{x} = x^{- 1} \end{aligned}$
::1x=x-1

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

$\begin{aligned} \frac{2}{x^{2}} = 2 x^{- 2} \end{aligned}$
::2x2=2x-2

c) $\begin{aligned} \frac{x^{2}}{y^{3}} \end{aligned}$
::c) x2y3

$\begin{aligned} \frac{x^{2}}{y^{3}} = x^{2} y^{- 3} \end{aligned}$
::x2y3=x2y-3

d) $\begin{aligned} \frac{3}{x y} \end{aligned}$
:d) 3xy

$\begin{aligned} \frac{3}{x y} = 3 x^{- 1} y^{- 1} \end{aligned}$
::3xy=3x-1y-1

Simplifying Expressions
::简化表达式

Simplify the following expressions and write them without fractions.
::简化以下表达式, 并写入这些表达式, 不带分数。

a) $\begin{aligned} \frac{4 a^{2} b^{3}}{2 a^{5} b} \end{aligned}$
::a) 4a2b32a5b

Reduce the numbers and apply the quotient rule to each variable separately:
::分别对每个变量减少数字并应用商数规则:

$\begin{aligned} \frac{4 a^{2} b^{3}}{2 a^{5} b} = 2 \cdot a^{2 - 5} \cdot b^{3 - 1} = 2 a^{- 3} b^{2} \end{aligned}$

::4a2b32a5b=2a2-5b3-1=2a-3b2

b) $\begin{aligned} {(\frac{x}{3 y^{2}})}^{3} \cdot \frac{x^{2} y}{4} \end{aligned}$
:x3y2)3x2y4

Apply the power rule for quotients first:
::先对商数应用权力规则 :

$\begin{aligned} {(\frac{2 x}{y^{2}})}^{3} \cdot \frac{x^{2} y}{4} = \frac{8 x^{3}}{y^{6}} \cdot \frac{x^{2} y}{4} \end{aligned}$

:2xy2)3x2x2y4=8x3y6x2y4

Then simplify the numbers, and use the product rule on the $\begin{aligned} x \end{aligned}$ ’s and the quotient rule on the $\begin{aligned} y \end{aligned}$ ’s:
::然后简化数字, 并使用产品规则对 x 和 y 的商数规则 :

$\begin{aligned} \frac{8 x^{3}}{y^{6}} \cdot \frac{x^{2} y}{4} = 2 \cdot x^{3 + 2} \cdot y^{1 - 6} = 2 x^{5} y^{- 5} \end{aligned}$

::8x3y6x2y4=2x3+2y1-6=2x5y-5

You can also use the negative power rule the other way around if you want to write an expression without negative exponents.
::如果您想要写一个表达式而没有负引言,您也可以使用负权力规则绕过另一边。

Writing Expressions Without Negative Exponents
::无负指数的写法表达式

Write the following expressions without negative exponents.
::撰写以下表达式,无负引号。

a) $\begin{aligned} 3 x^{- 3} \end{aligned}$
:a) 3x-3

$\begin{aligned} 3 x^{- 3} = \frac{3}{x^{3}} \end{aligned}$
::3x-3=3x3 3x3 3x3 3x3 3x3 3x3 3x3 3x3 3x3 3x3 3x3 3x3 3x3 3x3 3x3 3x3 3x3 3x3 3x3 3x3 3x3 3x3 3x3

b) $\begin{aligned} a^{2} b^{- 3} c^{- 1} \end{aligned}$
:b) a2b-3c-1

$\begin{aligned} a^{2} b^{- 3} c^{- 1} = \frac{a^{2}}{b^{3} c} \end{aligned}$
::a2b-3c-1=a2b3c (千兆赫)

c) $\begin{aligned} 4 x^{- 1} y^{3} \end{aligned}$
:c) 4x-1y3

$\begin{aligned} 4 x^{- 1} y^{3} = \frac{4 y^{3}}{x} \end{aligned}$
::4 - 1y3=4y3x

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

$\begin{aligned} \frac{2 x^{- 2}}{y^{- 3}} = \frac{2 y^{3}}{x^{2}} \end{aligned}$
::2 - 2y- 3= 2y3x2

Examples
::实例

Simplify the following expressions and write the answers without negative powers.
::简化以下表达式,并在无负功率的情况下写出答案。

Example 1
::例1

$\begin{aligned} {(\frac{a b^{- 2}}{b^{3}})}^{2} \end{aligned}$
:b2b3)2

Apply the quotient rule inside the " data-term="Parentheses" role="term" tabindex="0"> parentheses : $\begin{aligned} {(\frac{a b^{- 2}}{b^{3}})}^{2} = (a b^{- 5})^{2} \end{aligned}$
::括号内应用商数规则ab-2b3)2=(ab-5)2

Then apply the power rule: $\begin{aligned} (a b^{- 5})^{2} = a^{2} b^{- 10} = \frac{a^{2}}{b^{10}} \end{aligned}$
::然后应用权力规则ab-5)2=a2b-10=a2b10

Example 2
::例2

$\begin{aligned} \frac{x^{- 3} y^{2}}{x^{2} y^{- 2}} \end{aligned}$
::x-3y2x2y-2

Apply the quotient rule to each variable separately: $\begin{aligned} \frac{x^{- 3} y^{2}}{x^{2} y^{- 2}} = x^{- 3 - 2} y^{2 - (- 2)} = x^{- 5} y^{4} = \frac{y^{4}}{x^{5}} \end{aligned}$
::对每个变量分别应用商数规则: x- 3y2x2x2y-2=x- 3--2y2- (-2)=x- 5y4=y4x5

Review
::回顾

Simplify the following expressions in such a way that there aren't any negative exponents in the answer.
::简化以下表达式, 使答案中没有任何负面的缩写。

$\begin{aligned} x^{- 1} y^{2} \end{aligned}$
::x- 1y2 级

$\begin{aligned} x^{- 4} \end{aligned}$
::x-4 4个

$\begin{aligned} \frac{x^{- 3}}{x^{- 7}} \end{aligned}$
::x-3x-7 级

$\begin{aligned} \frac{x^{- 3} y^{- 5}}{z^{- 7}} \end{aligned}$
::x-3y-5z-7 级

$\begin{aligned} {(\frac{a}{b})}^{- 2} \end{aligned}$
:a)-2

$\begin{aligned} (3 a^{- 2} b^{2} c^{3})^{3} \end{aligned}$
:3a-2b2c3)3

Simplify the following expressions in such a way that there aren't any fractions in the answer.
::简化以下表达式, 使答案中没有任何分数。

$\begin{aligned} \frac{a^{- 3} (a^{5})}{a^{- 6}} \end{aligned}$
::a-3(a5)a-6

$\begin{aligned} \frac{5 x^{6} y^{2}}{x^{8} y} \end{aligned}$
::5x6y2x8y

$\begin{aligned} \frac{(4 a b^{6})^{3}}{(a b)^{5}} \end{aligned}$
:4ab6)3(ab)5

$\begin{aligned} \frac{(3 x^{3}) (4 x^{4})}{(2 y)^{2}} \end{aligned}$
:3x3)(4x4)(2y)2

$\begin{aligned} \frac{a^{- 2} b^{- 3}}{c^{- 1}} \end{aligned}$
::a-2b-3c-1

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

Negative Exponents ::负指数

Simplify Expressions With Negative Exponents ::简化带有负指数的表达式

Writing Expressions Without Fractions ::无分数的书写表达式

Simplifying Expressions ::简化表达式

Writing Expressions Without Negative Exponents ::无负指数的写法表达式

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Review ::回顾

Review (Answers) ::回顾(答复)

Negative Exponents
::负指数

Simplify Expressions With Negative Exponents
::简化带有负指数的表达式

Writing Expressions Without Fractions
::无分数的书写表达式

Simplifying Expressions
::简化表达式

Writing Expressions Without Negative Exponents
::无负指数的写法表达式

Examples
::实例

Example 1
::例1

Example 2
::例2

Review
::回顾

Review (Answers)
::回顾(答复)