小数指数

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Fractional Exponents 
::小数指数

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In a previous section we looked at the quotient rule for exponents: $\begin{array}{r}\left(\frac{x^n}{x^m}=x^{(n-m)}\right)\end{array}$ . Consider what happens when $\begin{array}{r}n=m\end{array}$ .
::在前一节中,我们研究了引言人的商数规则xnxm=x(n-m)),考虑n=m时会发生什么。

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For example, what happens when we divide $\begin{array}{r}{x}^4\end{array}$ by $\begin{array}{r}{x}^4\end{array}$ ? Applying the quotient rule tells us that $\begin{array}{r}\frac{x^4}{x^4}=x^{(4-4)}=x^0\end{array}$ —so what does that zero mean?
::例如, 当我们将 x4 除以 x4 时会发生什么? 应用商数规则告诉我们 x4x4=x( 4- 4) =x0 - 那么零意味着什么 ?

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Well, we first discovered the quotient rule by considering how the factors of $\begin{array}{r}x\end{array}$ cancel in such a fraction . Let’s do that again with our example of $\begin{array}{r}{x}^4\end{array}$ divided by $\begin{array}{r}{x}^4\end{array}$ :
::那么,我们首先通过考虑x的因子如何在如此小的分数中取消来发现商数规则。 让我们用我们的x4除以x4的例子来再次这样做:

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::x4x4=xxxxxxxxxxxxxxx=1

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So $\begin{array}{r}{x}^0=1!\end{array}$ You can see that this works for any value of the exponent , not just 4:
::x0=1! 您可以看到, 这对指数值的任何值有效, 而不仅仅是 4 :

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::xnxn=x(n-n)=x0

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Since there is the same number of $\begin{array}{r}x\end{array}$ ’s in the numerator as in the denominator, they cancel each other out and we get $\begin{array}{r}{x}^0=1\end{array}$ . This rule applies for all expressions:
::由于分子中的x数与分母中的x数相同,它们相互取消,我们得到x0=1。 这一规则适用于所有表达式:

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Zero Rule for Exponents: $\begin{array}{r}{x}^0=1\end{array}$ , where $\begin{array}{r}x\ne 0\end{array}$
::指数为x0=1的零规则

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Simplify Expressions With Fractional Exponents
::以小数指数简化表达式

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So far we’ve only looked at expressions where the exponents are positive and negative integers . The rules we’ve learned work exactly the same if the powers are fractions or irrational numbers—but what does a fractional exponent even mean? Let’s see if we can figure that out by using the rules we already know.
::到目前为止,我们只研究了指数为正和负整数的表达方式。 如果权力是分数或非理性数字,我们学到的规则是完全一样的 — — 但是,一个分数的推论甚至意味着什么?让我们看看我们是否能够通过使用我们已经知道的规则来找出答案。

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Suppose we have an expression like $\begin{array}{r}{9}^{\frac{1}{2}}\end{array}$ —how can we relate this expression to one that we already know how to work with? For example, how could we turn it into an expression that doesn’t have any fractional exponents?
::假设我们有一个像912这样的表达方式,我们如何将这个表达方式与我们已经知道如何工作的表达方式联系起来? 比如,我们如何把它变成一个没有任何分数的表率的表达方式?

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Well, the power rule tells us that if we raise an exponential expression to a power, we can multiply the exponents.
::那么,权力规则告诉我们,如果我们提高一个指数 表达的力量, 我们可以乘以推手。

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For example, if we raise $\begin{array}{r}{9}^{\frac{1}{2}}\end{array}$ to the power of 2, we get $\begin{array}{r}{\left(9^{\frac{1}{2}}\right)}^2=9^{2\cdot \frac{1}{2}}=9^1=9\end{array}$
::例如,如果我们把912提升到2的功率, 我们就会得到(912)2=9212=91=9。

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So if $\begin{array}{r}{9}^{\frac{1}{2}}\end{array}$ squared equals 9, what does $\begin{array}{r}{9}^{\frac{1}{2}}\end{array}$ itself equal? Well, 3 is the number whose square is 9 (that is, it’s the square root of 9), so $\begin{array}{r}{9}^{\frac{1}{2}}\end{array}$ must equal 3. And that’s true for all numbers and variables: a number raised to the power of $\begin{array}{r}\frac{1}{2}\end{array}$ is just the square root of the number. We can write that as $\begin{array}{r}\sqrt{x}=x^{\frac{1}{2}}\end{array}$ , and then we can see that’s true because $\begin{array}{r}{\left(\sqrt{x}\right)}^2=x\end{array}$ just as $\begin{array}{r}{\left(x^{\frac{1}{2}}\right)}^2=x\end{array}$ .
::912 平方等于 9 9 。 那么, 912 平方等于 9 等于 9 等于 9 等于 9 等于 9 等于 9 等于 9 等于 9 等于 9 等于 9 等于 9 等于 9 等于 9 等于 9 等于 9 等于 9 等于 9 等于 9 等于 9 等于 9 等于 9 等于 9 等于 9 等于 9 等于 9 等于 平方等于 9 等于 9 等于 9 等于 9 等于 9 等于 9 等于 9 等于 9 等于 9 等于 9 等于 9 等于 9 等于 9 等于 9 等于 9 等于 9 等于 9 等于 9 等于 9 等于 9 等于 9 等于 9 等于 9 等于 9 等于 9 等于 9 等 等于 等于 等于 3 等于 等于 等于 3 等于 3 等于 等于 3 。 所有数字和变量 都属实如此 : 向 12 升到 的 12 数 仅 是 数字 平方 的 根 。 我们可以写成 x= x x 12 12 12 等于 , 然后看这是 的 。 我们可以看到 的 , 因为 (x 2=x)

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Similarly, a number to the power of $\begin{array}{r}\frac{1}{3}\end{array}$ is just the cube root of the number, and so on. In general, $\begin{array}{r}{x}^{\frac{1}{n}}=\sqrt[n]{x}\end{array}$ . And when we raise a number to a power and then take the root of it, we still get a fractional exponent; for example, $\begin{array}{r}\sqrt[3]{x^4}={\left(x^4\right)}^{\frac{1}{3}}=x^{\frac{4}{3}}\end{array}$ . In general, the rule is as follows:
::类似地, 13 功率的数数, 仅仅是数字的立方根, 等。 一般来说, x1n=xn 。 当我们将数数加到一个电源, 然后从它根上, 我们仍然能得到一个分数的推算, 例如, x43=( x4)13=x43。 一般来说, 规则如下 :

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Rule for Fractional Exponents: $\begin{array}{r}\sqrt[m]{a^n}=a^{\frac{n}{m}}\end{array}$ and $\begin{array}{r}{\left(\sqrt[m]{a}\right)}^n=a^{\frac{n}{m}}\end{array}$
::小数指数规则: amm=anm 和 (am)n=anm

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We’ll examine roots and radicals in detail in a later chapter. In this section, we’ll focus on how exponent rules apply to fractional exponents.
::我们将在稍后的一章中详细研究根源和激进。 在本节中,我们将着重探讨引言规则如何适用于分数的推论者。

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Simplifying Expressions 
::简化表达式

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a)  $\begin{array}{r}{a}^{\frac{1}{2}}\cdot a^{\frac{1}{3}}\end{array}$
::a) a12-a13

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Apply the product rule: $\begin{array}{r}{a}^{\frac{1}{2}}\cdot a^{\frac{1}{3}}=a^{\frac{1}{2}+\frac{1}{3}}=a^{\frac{5}{6}}\end{array}$
::应用产品规则: a12a13=a12+13=a56

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b)  $\begin{array}{r}{\left(a^{\frac{1}{3}}\right)}^2\end{array}$
:b)(a13)2

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Apply the power rule: $\begin{array}{r}{\left(a^{\frac{1}{3}}\right)}^2=a^{\frac{2}{3}}\end{array}$
::应用权力规则a13)2=a23

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c)  $\begin{array}{r}\frac{a^{\frac{5}{2}}}{a^{\frac{1}{2}}}\end{array}$
:c) a52a12

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Apply the quotient rule: $\begin{array}{r}\frac{a^{\frac{5}{2}}}{a^{\frac{1}{2}}}=a^{\frac{5}{2}-\frac{1}{2}}=a^{\frac{4}{2}}=a^2\end{array}$
::应用商数规则: a52a12=a52-12=a42=a2

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d)  $\begin{array}{r}{\left(\frac{x^2}{y^3}\right)}^{\frac{1}{3}}\end{array}$
:d) (x2y3)13

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Apply the power rule for quotients: $\begin{array}{r}{\left(\frac{x^2}{y^3}\right)}^{\frac{1}{3}}=\frac{x^{\frac{2}{3}}}{y}\end{array}$
::对商数应用功率规则 : (x2y3)13=x23y

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Examples 
::实例

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

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Example 1
::例1

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$\begin{array}{r}{\left(x^{\frac{2}{5}}\right)}^5\end{array}$
:x25.5)5

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Apply the power rule: $\begin{array}{r}{\left(x^{\frac{2}{5}}\right)}^5=x^{\frac{2}{5}\cdot 5}=x^2\end{array}$
::应用权力规则x25)5=x25=x25=5=x2

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Example 2
::例2

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$\begin{array}{r}\frac{y^{\frac{3}{4}}}{y^{\frac{1}{8}}}\end{array}$
::y34y18 y34y18

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Apply the quotient rule: $\begin{array}{r}\frac{y^{\frac{3}{4}}}{y^{\frac{1}{8}}}=y^{\frac{3}{4}-\frac{1}{8}}=y^{\frac{6}{8}-\frac{1}{8}}=y^{\frac{5}{8}}\end{array}$
::应用商数规则: y34y18=y34- 18=y68- 18=y58

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Example 3
::例3

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$\begin{array}{r}{\left(\frac{x^{2a}}{y^{4b}}\right)}^{\frac{1}{2}}\end{array}$
:x2ay4b)12

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Apply the power rule for quotients: $\begin{array}{r}{\left(\frac{x^{2a}}{y^{4b}}\right)}^{\frac{1}{2}}=\frac{x^{2a\cdot \frac{1}{2}}}{y^{4b\cdot \frac{1}{2}}}=\frac{x^a}{y^{2b}}\end{array}$
::对商数应用功率规则 : (x2ay4b) 12=x2a+12y4b12=xa2b

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Review
::回顾

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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 。

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1. $\begin{array}{r}(x^{\frac{1}{2}}{y}^{\frac{-2}{3}})(x^2{y}^{\frac{1}{3}})\end{array}$
   :x12y-23)(x2y13)
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2. $\begin{array}{r}{x}^{-3}\cdot x^3\end{array}$
   ::x- 3x3 级
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3. $\begin{array}{r}{y}^5\cdot y^{-5}\end{array}$
   ::y5y - 5
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4. $\begin{array}{r}\frac{x^2{y}^{-3}}{x^{-4}{y}^{-2}}\cdot x^2\end{array}$
   ::x2y - 3x - 4y - 2x2

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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 。

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5. $\begin{array}{r}{x}^{\frac{1}{2}}{y}^{\frac{5}{2}}\end{array}$
   ::x12y52
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6. $\begin{array}{r}{\left(\frac{a}{b}\right)}^{3/4}\end{array}$
   :ab)3/4
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7. $\begin{array}{r}\frac{3x^2{y}^{\frac{3}{2}}}{x{y}^{\frac{1}{2}}}\end{array}$
   ::3x2y32xy12
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8. $\begin{array}{r}\frac{x^{\frac{1}{2}}{y}^{\frac{5}{2}}}{x^{\frac{3}{2}}{y}^{\frac{3}{2}}}\end{array}$
   ::x12y52x32y32
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9. $\begin{array}{r}{\left(\frac{a^2{b}^{\frac{1}{3}}}{a^{3}b}\right)}^{\frac{1}{2}}\end{array}$
   :a2b13a3b)12
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10. $\begin{array}{r}{\left(\frac{a^{\frac{1}{2}}b}{a{b}^{\frac{1}{4}}}\right)}^2\end{array}$
    :a12bab14)2

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Review (Answers)
::回顾(答复)

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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 。