Course: 8.3 涉及数字的指数属性

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涉及引号的指数属性
Exponential Properties Involving Quotients
::涉及引号的指数属性

The rules for simplifying quotients of exponents are a lot like the rules for simplifying products.
::简化出价商数的规则与简化产品的规则非常相似。

Let’s look at what happens when we divide $\begin{aligned} x^{7} \end{aligned}$ by $\begin{aligned} x^{4} \end{aligned}$ :
::让我们看看当我们将 x7 除以 x4 时会发生什么:

$\begin{aligned} \frac{x^{7}}{x^{4}} = \frac{x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x}{x \cdot x \cdot x \cdot x} = \frac{x \cdot x \cdot x}{1} = x^{3} \end{aligned}$

::x7x4=xxxxxxxxxxxxxxxxxxxxxxxxxxxxxx=x3

You can see that when we divide two powers of $\begin{aligned} x \end{aligned}$ , the number of $\begin{aligned} x \end{aligned}$ ’s in the solution is the number of $\begin{aligned} x \end{aligned}$ ’s in the top of the fraction minus the number of $\begin{aligned} x \end{aligned}$ ’s in the bottom. In other words, when dividing expressions with the same base, we keep the same base and simply subtract the exponent in the denominator from the exponent in the numerator.
::你可以看到,当我们分割 x 的两个功率时,解决方案中的 x 数是分数顶部的 x 数减去底部的 x 数。换句话说,当用同一基数分隔表达式时,我们保持同一基数,简单地从分子中标注的分母中减去指数。

Quotient Rule for Exponents: $\begin{aligned} \frac{x^{n}}{x^{m}} = x^{(n - m)} \end{aligned}$
::指数的引号规则:xnxm=x(n-m)

When we have expressions with more than one base, we apply the quotient rule separately for each base:
::当我们有一个以上基数的表达式时,我们对每个基数分别适用商数规则:

Now let’s see what happens if the exponent in the denominator is bigger than the exponent in the numerator. For example, what happens when we apply the quotient rule to $\begin{aligned} \frac{x^{4}}{x^{7}} \end{aligned}$ ?
::现在让我们来看看,如果分母中的指数大于分子中的指数,会发生什么情况。比如,如果我们对 x4x7 应用商数规则,会发生什么情况?

The quotient rule tells us to subtract the exponents. 4 minus 7 is -3, so our answer is $\begin{aligned} x^{- 3} \end{aligned}$ . A negative exponent! What does that mean?
::商数规则要求我们减去指数。 4 减 7 是 - 3, 所以我们的答案是 x - 3 。负指数! 这是什么意思 ?

$\begin{aligned} \frac{x^{5} y^{3}}{x^{3} y^{2}} = \frac{x \cdot x \cdot x \cdot x \cdot x}{x \cdot x \cdot x} \cdot \frac{y \cdot y \cdot y}{y \cdot y} = \frac{x \cdot x}{1} \cdot \frac{y}{1} = x^{2} y \end{aligned}$

::x5y3x3y2 =xxxxxxxxxxxxxxxxxxxxxxxxxxx}xx}}{yy}=xxx1}=x2yyy=xxxxx1}=x2yyyyy=xxxxxxx1}=xxxxxxx_y1=yyyy=y

OR
::或

$\begin{aligned} \frac{x^{5} y^{3}}{x^{3} y^{2}} = x^{5 - 3} \cdot y^{3 - 2} = x^{2} y \end{aligned}$

::x5y3x3y2=x5-33-2=x2yy

Well, let’s look at what we get when we do the division longhand by writing each term in factored form :
::那么,让我们看看当我们做这个分界线时,我们得到什么, 以因素化的形式写下每个词:

$\begin{aligned} \frac{x^{4}}{x^{7}} = \frac{x \cdot x \cdot x \cdot x}{x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x} = \frac{1}{x \cdot x \cdot x} = \frac{1}{x^{3}} \end{aligned}$

::x4x7 =xxxxxxxxxxxxxxxxxxxxxxxxx=1xxxxxxxxx=1x3

Even when the exponent in the denominator is bigger than the exponent in the numerator, we can still subtract the powers. The $\begin{aligned} x \end{aligned}$ ’s that are left over after the others have been canceled out just end up in the denominator instead of the numerator. Just as $\begin{aligned} \frac{x^{7}}{x^{4}} \end{aligned}$ would be equal to $\begin{aligned} \frac{x^{3}}{1} \end{aligned}$ (or simply $\begin{aligned} x^{3} \end{aligned}$ ), $\begin{aligned} \frac{x^{4}}{x^{7}} \end{aligned}$ is equal to $\begin{aligned} \frac{1}{x^{3}} \end{aligned}$ . And you can also see that $\begin{aligned} \frac{1}{x^{3}} \end{aligned}$ is equal to $\begin{aligned} x^{- 3} \end{aligned}$ . We’ll learn more about negative exponents shortly.
::即使分母中的指数大于分子中的指数,我们仍然可以减减功率。在其它的取消之后剩下的X,最后被取消,结果在分母中,而不是在分子中。就像x7x4等于x31(或简单的x3),x4x7等于1x3一样,你也可以看到1x3等于x3。我们将很快了解更多关于负指数的情况。

Simplifying Expressions
::简化表达式

Simplify the following expressions, leaving all exponents positive.
::简化以下表达式,使所有引言都为正数。

a) $\begin{aligned} \frac{x^{2}}{x^{6}} \end{aligned}$
::a) x2x6

Subtract the exponent in the numerator from the exponent in the denominator and leave the $\begin{aligned} x \end{aligned}$ ’s in the denominator: $\begin{aligned} \frac{x^{2}}{x^{6}} = \frac{1}{x^{6 - 2}} = \frac{1}{x^{4}} \end{aligned}$
::将分子中的指数从分母中的指数中减去,并将X留在分母中: x2x6=1x6-2=1x4

b) $\begin{aligned} \frac{a^{2} b^{6}}{a^{5} b} \end{aligned}$
:b) a2b6a5b

Apply the rule to each variable separately: $\begin{aligned} \frac{a^{2} b^{6}}{a^{5} b} = \frac{1}{a^{5 - 2}} \cdot \frac{b^{6 - 1}}{1} = \frac{b^{5}}{a^{3}} \end{aligned}$
::对每个变量分别适用此规则: a2b6a5b=1a5-2b6-11=b5a3

Examples
::实例

Simplify each of the following expressions using the quotient rule.
::使用商数规则简化以下每个表达式。

Example 1
::例1

$\begin{aligned} \frac{x^{10}}{x^{5}} \end{aligned}$
::x10x5

$\begin{aligned} \frac{x^{10}}{x^{5}} = x^{10 - 5} = x^{5} \end{aligned}$
::x10x5=x10-5=x5

Example 2
::例2

$\begin{aligned} \frac{a^{6}}{a} \end{aligned}$
::a6a a6a

$\begin{aligned} \frac{a^{6}}{a} = a^{6 - 1} = a^{5} \end{aligned}$
::a6a=a6-6-1=a5

Example 3
::例3

$\begin{aligned} \frac{a^{5} b^{4}}{a^{3} b^{2}} \end{aligned}$
::a5b4a3b2 千兆赫

c) $\begin{aligned} \frac{a^{5} b^{4}}{a^{3} b^{2}} = a^{5 - 3} \cdot b^{4 - 2} = a^{2} b^{2} \end{aligned}$
:c) a5b4a3b2=a5-3b4-2=a2b2

Review
::回顾

Evaluate the following expressions.
::评估以下表达式。

$\begin{aligned} \frac{5^{6}}{5^{2}} \end{aligned}$

$\begin{aligned} \frac{6^{7}}{6^{3}} \end{aligned}$

$\begin{aligned} \frac{3^{4}}{3^{10}} \end{aligned}$

$\begin{aligned} \frac{2^{2} \cdot 3^{2}}{5^{2}} \end{aligned}$

$\begin{aligned} \frac{3^{3} \cdot 5^{2}}{3^{7}} \end{aligned}$

Simplify the following expressions.
::简化下列表达式。

$\begin{aligned} \frac{a^{3}}{a^{2}} \end{aligned}$
::a3a2 个

$\begin{aligned} \frac{x^{5}}{x^{9}} \end{aligned}$
::x5x9

$\begin{aligned} \frac{x^{6} y^{2}}{x^{2} y^{5}} \end{aligned}$
::x6y2x2y5

$\begin{aligned} \frac{6 a^{3}}{2 a^{2}} \end{aligned}$
::6a32a2

$\begin{aligned} \frac{15 x^{5}}{5 x} \end{aligned}$
::15x55x 15x55x

$\begin{aligned} \frac{25 y x^{6}}{20 y^{5} x^{2}} \end{aligned}$
::25yx620y5x2

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

涉及引号的指数属性

Exponential Properties Involving Quotients ::涉及引号的指数属性

Simplifying Expressions ::简化表达式

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Review ::回顾

Review (Answers) ::回顾(答复)

Exponential Properties Involving Quotients
::涉及引号的指数属性

Simplifying Expressions
::简化表达式

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Review
::回顾

Review (Answers)
::回顾(答复)