1.3 指数

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  指数

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  Introduction
  ::导言

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  Exponents are used to express quantities in a variety of fields. Economists use exponents to calculate compound interest. Biologists use exponents to model population growth. Chemists use exponents to express the kinetics involved in chemical reactions. Physicists use exponents to model wave behavior. Computer scientists use exponents in public-key cryptography. The properties and laws of exponents enable us to efficiently simplify exponential expressions. In this section, you will develop an understanding of the properties and laws of exponents through notes, videos, and practice examples. 
  ::指数用于表达各个领域的数量。经济学家使用指数来计算复合利息。生物学家使用指数来模拟人口增长。化学家使用指数来表达化学反应所涉及的动能。物理家使用指数来模拟波的行为。计算机科学家在公用钥匙加密中使用指数。引言的属性和法律使我们能够有效地简化指数表达方式。在本节,你将通过笔记、视频和实践实例来了解指数的属性和法律。

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  Exponents
  ::指数

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  If   $\begin{align*}a \in \mathbb{R}\end{align*}$  (read as " $\begin{align*}a\end{align*}$  is an element in the set of real numbers") and $\begin{align*}n \in \mathbb{N}\end{align*}$   (" $\begin{align*}n\end{align*}$  is an element in the set of n atural n umbers") , then the $\begin{align*}n\end{align*}$ th power of  $\begin{align*}a\end{align*}$  is written as
  ::aR(读为“a是一组实际数字中的一个元素”)和nN(“n是一组自然数字中的一个元素”),那么,a的 n 次功率以下列方式写成:

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  $$\begin{align*}& a^n = \underleftrightarrow{a \times a \times a \times \ldots \times a}\\ & \qquad \qquad \qquad {\color{red}\downarrow}\\ & \qquad \qquad \ {\color{red}n \ \text{factors}},\end{align*}$$
  ::a=a=axaxaxxxx...xa+n因数,

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  where $\begin{align*}n\end{align*}$ is called the exponent and $\begin{align*}a\end{align*}$ is called the base. The term $\begin{align*}a^n\end{align*}$ is known as a power. In other words,  $\begin{align*}4^3 = 4 \times 4 \times 4 = 64\end{align*}$ and $\begin{align*}2^6=2 \times 2 \times 2 \times 2 \times 2 \times 2=64.\end{align*}$
  ::n 的名称是前奏, a 的名称是基数。该词称为功率。 换句话说, 43=4x4x4x4=64, 26=2x2x2x2x2x2xx2=64。

   

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  Multiplying and Dividing Exponential Terms
  ::乘数和分数指数术语

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  To multiply two powers with the same base ,  add the exponents:
  ::乘以相同基数的两种权力时,加上引号:

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  $$\begin{align*}& a^m \times a^n = \underleftrightarrow{(a \times a \times \ldots \times a)} \ \underleftrightarrow{(a \times a \times \ldots \times a)}\\ & \qquad \qquad \qquad \qquad \ {\color{red}\downarrow} \qquad \qquad \qquad \quad \ {\color{red}\downarrow}\\ & \qquad \qquad \qquad {\color{red} m \ \text{factors}} \qquad \qquad {\color{red} n \ \text{factors}}\\ & a^m \times a^n = \underleftrightarrow{(a \times a \times a \ldots \times a)}\\ & \qquad \qquad \qquad \qquad \ {\color{red}\downarrow}\\ & \qquad \qquad \qquad {\color{red} m+n \ \text{factors}}\\ & a^m \times a^n=a^{{\color{red}m+n}}\end{align*}$$
  ::amxan= (axxaxxx...xa) \\ (axaxxx...xa) \ \ \ \ \ \ \ \ \ \ imcern imcusamxan= (axaxxa...a) \ \ \ \ m+n imcusamxan= am+n

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  To divide two powers with the same base , $\begin{align*}a \neq 0\end{align*}$ , ​  subtract the exponents:
  ::以相同基数划分两个权力单位, a0, 减去前言 :

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  $$\begin{align*}& \qquad \qquad \ {\color{red} m \ \text{factors}}\\ & \qquad \qquad \qquad {\color{red}\uparrow}\\ & \frac{a^m}{a^n}=\frac{\overleftrightarrow{(a \times a \times \ldots \times a)}}{\underleftrightarrow{(a \times a \times \ldots \times a)}} \ m>n;a \neq 0\\ & \qquad \qquad \qquad {\color{red}\downarrow}\\ & \qquad \qquad \ {\color{red} n \ \text{factors}}\\ & \frac{a^m}{a^n}=\underleftrightarrow{(a \times a \times \ldots \times a)}\\ & \qquad \qquad \qquad {\color{red}\downarrow}\\ & \qquad \qquad \ {\color{red} m-n \ \text{factors}}\\ & \frac{a^m}{a^n}=a^{\color{red}m-n}\end{align*}$$
  ::ma (axaxxx...xa)(axaxxxx...xa)\\ m>n;a0\ n因子aman=(axaxxx...xa)\\\ m-n因子aman=am-n

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  Raising Terms to an Exponent
  ::提高指数的比值

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  To raise a power to a new exponent, multiply the exponents:
  ::将权力加到一个新的推手, 使推手倍增:

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  $$\begin{align*}& (a^m)^n = \underleftrightarrow{(a \times a \times \ldots \times a)^n}\\ & \qquad \qquad \qquad \quad {\color{red}\downarrow}\\ & \qquad \qquad \quad {\color{red}m} \ \color{red}{\text{factors}}\\ & (a^m)^n=\underleftrightarrow{(a \times a \times \ldots \times a)} \times \underleftrightarrow{(a \times a \times \ldots \times a)} \ \underleftrightarrow{(a \times a \times \ldots \times a)}\\ & \qquad \qquad \qquad \quad \ {\color{red}\downarrow} \qquad \qquad \qquad {\color{red}\downarrow} \qquad \qquad \quad {\color{red}\downarrow}\\ & \qquad \quad \quad \underleftrightarrow{\quad {\color{red}m} \ \color{red}{\text{factors}} \qquad \qquad \ \ {\color{red}m} \ \color{red}{\text{factors}} \qquad \quad \ {\color{red}m} \ \color{red}{\text{factors}} \ \ }\\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad {\color{red}n \ times}\\ & (a^m)^n=\underleftrightarrow{a \times a \times a \ldots \times a}\\ & \qquad \qquad {\color{red}mn \ \text{factors}}\\ & (a^m)^n = a^{\color{red}mn}\end{align*}$$
  :axxxxxxxxxxxxa) n=(axxaxxxxxxxxxxxxa) *(axxxxxxxxxxxxxa) */m因数 m因数 m因数 n=axxxxaxaxxaxxxxxxaxxxn) *xxxxxxxxxaxxxn因数 (axxxxxxxxxxxxxxa) n=amn

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  To raise a product to an exponent, raise each of the factors to the exponent:
  ::将产品带给推手,将每一个因素都带给推手:

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  $$\begin{align*}&(ab)^n=\underleftrightarrow{(ab) \times (ab) \times \ldots \times (ab)}\\ & \qquad \qquad \qquad \qquad {\color{red}\downarrow}\\ & \qquad \qquad \qquad {\color{red}n} \ {\color{red}\text{factors}}\\ & (ab)^n=\underleftrightarrow{(a \times a \times \ldots \times a)} \times \underleftrightarrow{(b \times b \times \ldots \times b)}\\ & \qquad \qquad \qquad \quad {\color{red}\downarrow} \qquad \qquad \qquad \qquad {\color{red}\downarrow}\\ & \qquad \qquad \quad \ {\color{red}n} \ {\color{red}\text{factors}} \qquad \qquad \ {\color{red}n} \ {\color{red}\text{factors}}\\ & (ab)^n=a^{{\color{red}n}} b^{{\color{red}n}}\end{align*}$$
  :ab)n = (ab)xxxxxxxxxxxxxxxxxxxxxxxxb) n因数 n因数 n(ab)n=anbn

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  To raise a quotient to an exponent, raise both the numerator and the denominator to the exponent:
  ::将分子和分母都加到指数上:

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  $$\begin{align*}& \left(\frac{a}{b} \right)^n= \underleftrightarrow{\frac{a}{b} \times \frac{a}{b} \times \ldots \times \frac{a}{b}}\\ & \qquad \qquad \qquad \quad \ {\color{red}\downarrow}\\ & \qquad \qquad \quad \quad {\color{red}n} \ {\color{red}\text{factors}}\\ & \qquad \qquad \quad \quad {\color{red}n} \ {\color{red}\text{factors}}\\ & \qquad \qquad \qquad \quad \ {\color{red}\uparrow}\\ & \left(\frac{a}{b}\right)^n=\frac{\overleftrightarrow{(a \times a \times \ldots \times a)}}{\underleftrightarrow{(b \times b \times \ldots \times b)}}\\ & \qquad \qquad \qquad \quad \ {\color{red}\downarrow}\\ & \qquad \qquad \quad \quad {\color{red}n} \ {\color{red}\text{factors}}\\ & \left(\frac{a}{b} \right)^n=\frac{a^{\color{red}n}}{b^{\color{red}n}} \ (b \neq 0)\end{align*}$$
  :ab)n=abxabxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxbxxxxxxxxxxxxxxxxbxxxxxxxxxxxxxxxxxxxbxxxxxxxxxxxxxxxxbxxxxxxxxxxxxxbxxbxbxaxaxbxbxxxxxxxbxaxbxbxbxaxaxbxbxbxbxbxbxbxbxbxbxbxbxbxbxxbxnxnxnxbxbxxbxbxbxxbxbxbxxxxxxnxbxbxbxbxbxbxbxbxbxnxbxbxnxnxbxbxbxbxbxnxbxbxbxbxbxbxbxbxbxbxbxbxbxbxbxbxbxbxb

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  Laws of Exponents
  ::名人法

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  The following are basic rules or laws that govern powers and exponents:
  ::以下是管辖权力和指挥者的基本规则或法律:

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     Basic Laws of Exponents
  ::生物物物法基本法

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  If $\begin{align*}a, \ b, \ m,\ \mathrm{and} \ n\end{align*}$  are real numbers, then:
  ::如果a、b、m和n为实际数字,则:

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  1. Multiplying Power s with the Same Base
     $\begin{align*}a^m \times a^n = a^{m+n}\end{align*}$
     ::以同一基地 amxan=am+n 乘以乘以功率
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  2. Dividing Power s with the Same Base
     $\begin{align*}\tfrac{a^m}{a^n}=a^{m-n}\end{align*}$ ( $\begin{align*}a \neq 0\end{align*}$ )
     ::拥有同一基地的分权国a_0
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  3. Raising a Power to a New Exponent
     $\begin{align*}(a^m)^n=a^{mn}\end{align*}$
     ::将权力提升至新人,
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  4. Raising a Product to an Exponent
     $\begin{align*}(ab)^n=a^n b^n\end{align*}$
     ::将产品提高到指数(ab)n=anbn
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  5. Raising a Quotient to an Exponent
     $\begin{align*}\left(\tfrac{a}{b}\right)^n=\tfrac{a^n}{b^n} \quad (b \neq 0)\end{align*}$
     ::将引数提高到指数(ab)n=anbn(b0)

   

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  Properties of Exponents
  ::指数属性

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  Watch the following video for examples of using the basic rules of exponents and an introduction to the zero exponent:
  ::使用推手的基本规则及零推手的介绍,

   

   

   

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  Zero Exponent
  ::零指数

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  The procedure for reducing exponents with like bases in a fraction is as follows:
  

  $$\begin{align*}\frac{a^m}{a^n}=a^{m-n}\end{align*}$$ If $\begin{align*}m = n\end{align*}$ , then the following would be true:
  ::将具有类似基数的指数分数减少的程序如下:aman=am-nifm=n,然后是:

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  $$\begin{align*}\frac{a^m}{a^n}&=a^{{\color{red}m-n}}=a^{\color{red}0}\\ \frac{3^3}{3^3} &= 3^{{\color{red}3-3}}=3^{\color{red}0}\end{align*}$$
  ::aman=am-n=a0333=33-3=30

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  Any quantity divided by itself is equal to 1. For example,   $\begin{align*}\frac{3^3}{3^3}=1\end{align*}$   so  $\begin{align*}3^{\color{red}0}={\color{red}1}\end{align*}$ .
  ::任何数量除以本身等于 1. 例如, 3333=1 so 30=1。

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     Zero as an Exponent
  ::零作为指数

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  In general,  $\begin{align*}{a}^{\color{red}0}=1 \ \mathrm{if} \ a\neq 0\end{align*}$ . 
  ::一般说来,如果是0,0=1。

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  Note that if  $\begin{align*}a=0\mathrm{,} \ {0}^{\color{red}0}\end{align*}$  is not defined.
  ::请注意,如果 a=0,则未定义 00。

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  Negative Exponents
  ::负指数

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  If we apply the general rule for multiplying exponents with a common base,  $\begin{align*}{a}^{m}\times {a}^{n}={a}^{m+n}\end{align*}$ , we can determine the property of negative exponents.   Review  the following example:
  ::如果我们适用通用规则,将指数乘以USXan=am+n这一共同基点,我们可以确定负指数的属性。

  $$\begin{align*}4^2 \times 4^{-2}=4^{\color{red}2+(-2)}=4^{\color{red}0}={\color{red}1}\end{align*}$$

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  Therefore,
  ::因此,

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  $$\begin{align*}& 4^2 \times 4^{-2}=1\\ & \frac{4^2 \times 4^{-2}}{4^2}=\frac{1}{4^2} && \text{Divide both sides by} \ 4^2.\\ & \frac{\cancel{4^2} \times 4^{-2}}{\cancel{4^2}}=\frac{1}{4^2} && \text{Simplify the equation.}\\ & 4^{{\color{red}-2}}=\frac{1}{4^{\color{red}2}}\end{align*}$$
  ::42x4-2=142x4-2=142=142 以42.42x4-2-4-242=142为两侧的方差为42.4242x4-242=142 简化方程 4-2=142

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  This is true in general, and creates the following laws for negative exponents:
  ::一般来说,这是事实,为消极的推手制定了下列法律:

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     Laws for Negative Exponents
  ::负指数法律

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  If  $\begin{align*}a \neq 0\end{align*}$ , then: 
  ::如果a0,那么:

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  1.   $\begin{align*}{a}^{\color{red}-m} = \frac{1}{{a}^{\color{red}m}}\end{align*}$  
     ::a-m=1毫米
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  2.   $\begin{align*}\frac{1}{{a}^{\color{red}-m}} ={a}^{\color{red}m}\end{align*}$
     ::1-米=立方米

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  These laws for negative exponents can be expressed in many ways:
  ::这些法律对消极的推手可以以多种方式表述:

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  • If a term has a negative exponent, write it as 1 over the term with a positive exponent. For example,  $\begin{align*}a^{\color{red}-m}=\frac{1}{a^{\color{red}m}}\end{align*}$ and $\begin{align*}\frac{1}{a^{\color{red}-m}}=a^{\color{red}m}\end{align*}$ .
    ::如果一个术语有负表情,请以正表情写为一,例如,a-m=1am和1a-m=am。
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  • If a term has a negative exponent, write the reciprocal with a positive exponent. For example,  $\begin{align*}\left(\frac{2}{3}\right)^{{\color{red}-2}}=\left(\frac{3}{2}\right)^{\color{red}2}\end{align*}$ and $\begin{align*}a^{{\color{red}-m}}=\frac{a^{-m}}{1}=\frac{1}{a^{\color{red}m}}\end{align*}$ .
    ::如果一个术语有负指数,请用正指数写对等指数,例如(23)-2=(32)2和a-m=a-m1=1am。
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  • If the term is a factor in the numerator with a negative exponent, write it in the denominator with a positive exponent. For example,  $\begin{align*}3x^{{\color{red}-3}}y=\frac{3y}{x^{\color{red}3}}\end{align*}$ and $\begin{align*}a^{{\color{red}-m}}b^n=\frac{1}{a^{\color{red}m}}(b^n)=\frac{b^n}{a^{\color{red}m}}\end{align*}$ .
    ::如果该词在分子中是一个系数,带有负引号,则用正引号写在分母中。例如,3x-3y=3yx3和a-mbn=1am(bn)=bnam。
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  • If the term is a factor in the denominator with a negative exponent, write it in the numerator with a positive exponent. For example,  $\begin{align*}\frac{2x^3}{x^{-2}}=2x^3(x^2)\end{align*}$ and $\begin{align*}\frac{b^n}{a^{{\color{red}-m}}}=b^n \left(\frac{a^{{\color{red}m}}}{1}\right)=b^na^{\color{red}m}\end{align*}$ .
    ::如果该词是分母中的负引号的一个系数, 请用正引号写在分子中。 例如, 2x3x-2=2x3(x2) 和 bna- m=bn( am1) = bn。

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  These ways of understanding negative exponents provide algorithms  for arriving at solutions without doing multiple steps of  calculations. The results will be the same.
  ::这些理解负面指数的方法提供了算法,用以在不进行多步计算的情况下达成解决方案。结果是一样的。

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  Watch the following video for examples of simplifying expressions that involve negative exponents:
  ::观看下列视频,以举例说明涉及负面推手的简化表达方式:

   

   

  
   

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  Writing a Number as a Power of Another Number
  ::a 以另一个数字的功率写入数字

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  In many of the previous examples, we evaluated powers with numerical bases by expanding the power into its factors and determining the product of the factors.
  ::在许多先前的例子中,我们评估了具有数字基础的权力,将权力扩大到各种因素,并确定各种因素的产物。

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  $\begin{align*}2^4\end{align*}$ was expanded to $\begin{align*}2 \times 2 \times 2 \times 2\end{align*}$ .
  ::24 扩大为 2x2x2x2x2。

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  The product was determined:

  $$\begin{align*}2 \times 2&={\color{red}4} \\ {\color{red}4} \times 2&={\color{red}8} \\ {\color{red}8} \times 2&={\color{red}16}\end{align*}$$
  ::确定的产品: 2x2=44x2=88x2=16

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  Therefore,  $\begin{align*}\boxed{2^4=16}\end{align*}$
  ::因此,24=16

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  This concept can also be reversed. Write 32 as a power of 2:
  ::也可以颠倒这一概念。

  $$\begin{align*}32={\color{red}2} \times {\color{red}2} \times {\color{red}2} \times {\color{red}2} \times {\color{red}2}=2^5\end{align*}$$

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  There are 5 twos. Therefore,  $\begin{align*}\boxed{32=2^{\color{red}5}}\end{align*}$
  ::共有5个2个,因此32=25

   

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  Examples   
  ::实例

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  Example 1
  ::例1

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  Evaluate the following using the laws of exponents:
  ::使用指数法评估下列情况:

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  a)  $\begin{align*}\left(\frac{3}{4}\right)^{-2}\end{align*}$
  :a) (34)-2

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  Solution:
  ::解决方案 :

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  Method 1: Apply the negative exponent rule.
  ::方法1:适用负引号规则。

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  $$\begin{align*}& \left(\frac{3}{4}\right)^{-2}=\frac{1}{{\color{red}\left(\frac{3}{4}\right)^2}} && \text{Write the expression with a positive}\\ & && \text{exponent by applying} \ \boxed{a^{-m}=\frac{1}{a^m}}.\\ & \frac{1}{\left(\frac{3}{4}\right)^2}=\frac{1}{{\color{red}\frac{3^2}{4^2}}} && \text{Apply the law of exponents for}\\ & && \text{raising a quotient to a power} \ \boxed{\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}}.\\ & \frac{1}{\frac{3^2}{4^2}}=\frac{1}{{\color{red}\frac{9}{16}}} && \text{Evaluate the powers.}\\ & \frac{1}{\frac{9}{16}}=1 \div \frac{9}{16} && \text{Divide.}\\ & 1 \div \frac{9}{16}=1 \times \frac{16}{9}={\color{red}\frac{16}{9}}\\ & \boxed{\left(\frac{3}{4}\right)^{-2}=\frac{16}{9}}\end{align*}$$
  :34) -2=1(3434)2 应用 a-m=1am.1(342)=13242,以正速写方式写出该表达式,应用 a-m=1am.1(342)=13242 应用强权(ab)n=anbn.13242=1916)推举商数法 使强权(ab)n=anbn. 13242=16Evaluation the poweries(1916)=1916Divide.1916=1x169=169(34)-2=169

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  Method 2: Apply a shortcut and write the reciprocal with a positive exponent.
  ::方法2:采用快捷键,用正表征写对等词。

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  $$\begin{align*}& \left(\frac{3}{4}\right)^{-2}={\color{red}\left(\frac{4}{3}\right)^2} && \text{Write the reciprocal with a positive exponent.}\\ & \left(\frac{4}{3}\right)^2={\color{red}\frac{4^2}{3^2}} && \text{Apply the law of exponents for}\\ & && \text{raising a quotient to a power } \boxed{\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}}.\\ & \frac{4^2}{3^2}={\color{red}\frac{16}{9}} && \text{Simplify.}\\ & \boxed{\left(\frac{3}{4}\right)^{-2}=\frac{16}{9}}\end{align*}$$
  :34)-2=(43)2 以正面的推理写对等词。 (43)2=4232=4232 适用向权力(ab)n=anbn.4232=169 简化(34)-2=169)的推理人法。

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  Applying this shortcut facilitates the process for obtaining the solution.
  ::应用此快捷键有利于获取解决方案的进程 。

   

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  b)  $\begin{align*}{3}^{2} \times {3}^{3}\end{align*}$
  :b) 32×33

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  Solution:
  ::解决方案 :

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  $$\begin{align*}& 3^2 \times 3^3 && \text{The base is} \ 3.\\ & 3^{2+3} && \text{Keep the base of} \ 3 \ \text{and add the exponents.}\\ & 3^{\color{red}5} && \text{This answer is in exponential form.}\end{align*}$$
  ::32x33 基数是 3. 32+3 保持基数为 3, 并添加引号。 35 此回答以指数形式出现 。

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  The answer can be taken one step further. The base is numerical so the term can be evaluated:
  ::答案可以再往前一步。基数是数字,这样可以评价该词:

  $$\begin{align*}& 3^5=3 \times 3 \times 3 \times 3 \times 3\\ & {\color{red}3^5}={\color{red}243}\\ & \boxed{3^2 \times 3^3 = 3^5=243}\end{align*}$$

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  c)  $\begin{align*}{2}^{7} \div {2}^{3}\end{align*}$
  :c) 2723

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  Solution:
  ::解决方案 :

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  $$\begin{align*}& 2^7 \div 2^3 && \text{The base is} \ 2.\\ & 2^{7-3} && \text{Keep the base of} \ 2 \ \text{and subtract the exponents}.\\ & 2^{\color{red}4} && \text{The answer is in exponential form}.\end{align*}$$
  ::2723 基数是 2.27 - 3, 保持基数为 2, 减去引号 24 答案为指数形式 。

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  The answer can be taken one step further. The base is numerical so the term can be evaluated:
  ::答案可以再往前一步。基数是数字,这样可以评价该词:

  $$\begin{align*}& 2^4 = 2 \times 2 \times 2 \times 2\\ &{\color{red}2^4} = {\color{red}16}\\ & \boxed{2^7 \div 2^3 =2^4=16}\end{align*}$$

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  d)  $\begin{align*}{({2}^{3} \times{3}^{2})}^{2}\end{align*}$
  :d) (23x32)2

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  Solution:
  

  $$\begin{align*}& (2^3 \times 3^2)^2 && \text{The base is} \ 2^3 \times 3^2.\\ & (2)^{3 \times 2} \cdot (3)^{2 \times 2} && \text{Keep the base of} \ 2^3 \times 3^2 \ \text{and raise each}\\ & && \text{factor of the base to the power of} \ 2.\\ & 2^{\color{red}6} \times 3^{\color{red}4} && \text{Simplify. Apply the exponent to each factor of the base.}\\ & 2^{\color{red}6} \times 3^{\color{red}4} && \text{The answer is in exponential form.}\end{align*}$$
  ::解析度 : (23x32) 2 基数为 23x32. (2)3x2x2(3)2x2 保持基数 23x32 , 并将基数的每个因子提升到 2.26x34 的功率 简化 。 对基数的每个因数应用引号 26x34 。 答案以指数形式显示 。

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  The answer can be taken one step further. The base of each factor is numerical so each term can be evaluated. The final answer will be the product of the two answers:
  ::答案可以再往前一步。每个因素的基数是数字,这样每个术语都可以被评估。最后答案将是两个答案的产物:

  $$\begin{align*}& 2^6=2 \times 2 \times 2 \times 2 \times 2 \times 2\\ & 2^6 = 64\\ & 3^4=3 \times 3 \times 3 \times 3\\ & 3^4 = 81\\ & {\color{red}64 \times 81}={\color{red}5184}\\ & \boxed{(2^3 \times 3^2)^2=2^6 \times 3^6=5184}\end{align*}$$

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  e) $\begin{align*}{\left(\tfrac{2}{3}\right)}^{2}\end{align*}$
  ::e(23)2

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  Solution:
  ::解决方案 :

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  $$\begin{align*}& \left(\frac{2}{3}\right)^2 && \text{The base is} \ \frac{2}{3}.\\ & \frac{2^{1 \times 2}}{3^{1 \times 2}} && \text{Keep the base of} \ \frac{2}{3} \ \text{and multiply the exponents}\\ & && \text{of both the numerator and the denominator by} \ 2.\\ & \frac{2^{\color{red}2}}{3^{\color{red}2}} && \text{The answer is in exponential form}.\end{align*}$$
  :23)2 基数为 2321x231x2, 保持基数为 23, 并将分子和分母的指数乘以 2.2232 乘以 2.2232 。 答案为指数形式 。

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  The answer can be taken one step further. The base is numerical so each term can be evaluated:
  ::答案可以再向前一步。基数是数字,这样每个术语都可以评估:

  $$\begin{align*}& 2^2 = 2 \times 2 \qquad \ 3^2=3 \times 3\\ & 2^2=4 \qquad \qquad 3^2=9\\ & \frac{{\color{red}4}}{{\color{red}9}}\\ & \boxed{\left(\frac{2}{3}\right)^2=\frac{2^2}{3^2}=\frac{4}{9}}\end{align*}$$

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  Example 2
  ::例2

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  Simplify the following (using only positive exponents and, if possible,  shortcuts):
  ::简化以下内容(仅使用正表征,如有可能使用快捷键):

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  a) $\begin{align*}y^{-6}\end{align*}$
  :a)y-6

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  Solution:
  

  $$\begin{align*}& y^{-6} && \text{Write the expression with a positive exponent by}\\ & && \text{applying } \boxed{a^{-m}=\frac{1}{a^m}}.\\ & \boxed{y^{-6}=\frac{1}{y^6}}\end{align*}$$  
  ::解决方案 : y- 6Write the 表达式使用 a- m= 1 am.y-6= 1y6 来以正表情表示 。

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  b) $\begin{align*}\left(\frac{a}{b}\right)^{-3}\end{align*}$
  :b) (ab)-3

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  Solution:  

  $$\begin{align*}& \left(\frac{a}{b}\right)^{-3} && \text{Write the reciprocal with a positive exponent.}\\ & \left(\frac{a}{b}\right)^{-3}={\color{red}\left(\frac{b}{a}\right)^3} && \text{Apply the law of exponents for raising a}\\ & && \text{quotient to a power } \boxed{\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}}.\\ & \left(\frac{b}{a}\right)^3={\color{red}\frac{b^3}{a^3}}\\ & \boxed{\left(\frac{a}{b}\right)^{-3}=\frac{b^3}{a^3}}\end{align*}$$
  ::解决办法ab)-3Write the economy with a presententent. (ab)-3=(ba)3=(ba)3 apply the fagentients law of expenters 提高权力(ab)n=anbn.(ba)3=b3a3(ab)-3=b3a3

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  c)  $\begin{align*}{(\mathrm{-}3x)}^{2}\end{align*}$
  :c) (-3x)2

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  Solution:
  ::解决方案 :

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  $$\begin{align*}& (-3x)^2 && \text{The base is} \ -3x.\\ & (-3)^{1 \times 2} \cdot (x)^{1 \times 2} && \text{Keep the base of} \ -3x \ \text{and raise each factor of}\\ & && \text{the base to the power of} \ 2.\\ & (-3)^{\color{red}2} \cdot (x)^{\color{red}2} && \text{Simplify. Apply the exponent to each factor of the base.}\\ & {\color{red}9}x^{\color{red}2} && \text{The answer is in exponential form.}\\ & \boxed{(-3x)^2=9x^2}\end{align*}$$
  :- 3x) 2 基数为- 3x. (- 3) 1x2, 基数保持 3x 基数, 并将基数的每个因数提高到 2. (- 3) 2x( x) 2 的功率, 简化。 将引号应用到 base.9x2 的每个因数 。 答案为指数形式 。 (- 3x) 2= 9x2

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  d)  $\begin{align*}\frac{{y}^{3}}{{y}^{\mathrm{-}5}}\end{align*}$
  :d) y3y-5

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  Solution:
  ::解决方案 :

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  $$\begin{align*}& \frac{y^3}{y^{-5}} && \text{The base is} \ y.\\ & y^{3 -(- 5)} && \text{Keep the base of} \ y \ \text{and subtract the exponents.}\\ & y^{\color{red}8} && \text{The answer is in exponential form.}\\ & \boxed{\frac{y^3}{y^{-5}} = y^8}\end{align*}$$
  ::y3- 5 基数为y.y3- (- 5) , 保持 y 的基数, 减去 expronents. y8 。 答案为指数形式 .y3y-5=y8 。

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  e)  $\begin{align*}{\left(\frac{4{a}^{5}{b}^{3}}{6ab}\right)}^{3}\end{align*}$
  :e) (4a5b36ab)3

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  Solution:
  ::解决方案 :

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  $$\begin{align*}& \left(\frac{4a^5b^3}{6ab}\right)^3 && \text{The base is} \ \frac{4a^5b^3}{6ab}. \ \text{Begin by simplifying the base.}\\ & \frac{2}{3}a^{{\color{red}5-1}}b^{{\color{red}3-1}}=\frac{2a^{\color{red}4}b^{\color{red}2}}{3} && \text{Write the problem with the simplified base.}\\ & \left(\frac{2a^4b^2}{3}\right)^3\\ & \frac{2^{1 \times 3} a^{4 \times 3} b^{2 \times 3}}{3^{1 \times 3}} && \text{Keep the base of} \ \frac{2a^4b^2}{3} \ \text{and multiply the exponents of}\\ & && \text{both the numerator and the denominator by} \ 3.\\ & \frac{2^{\color{red}3}a^{{\color{red}12}}b^{\color{red}6}}{3^{\color{red}3}} && \text{The answer is in exponential form.}\end{align*}$$
  :4a5b36ab) 3 基数为 4a5b36ab。 开始简化基数 23a5-1b3-1=2a4b23) 。 (2a4b23) 321x3a4x3x3b3x331x3保持基数 2a4b23, 乘以分子和分母的出处3. 23a12b633。 答案以指数形式显示 。

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  The answer can be taken one step further. The denominator and the numerator both have numerical coefficients to be evaluated:
  ::答案可以再向前一步。 分母和分子都有需要评估的数字系数:

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  $$\begin{align*}& 2^3=2 \times 2 \times 2 \quad 3^3 = 3 \times 3 \times 3\\ & 2^3=8 \qquad \qquad \ \ 3^3=27\\ & \frac{{\color{red}8}a^{{\color{red}12}}b^{\color{red}6}}{{\color{red}27}}\\ & \boxed{\left(\frac{4a^5b^3}{6ab}\right)^3=\left(\frac{2a^4b^2}{3}\right)^3=\frac{2^3a^{12}b^6}{3^3}=\frac{8a^{12}b^6}{27}}\end{align*}$$
  ::23=2x2x233=3x3x3x323=8 33=278a12b627(4a5b36ab)3=2a4b233=23a12b633=8a12b627

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  Example 3
  ::例3

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  Use the above concept to answer the following:
  ::使用上述概念回答以下问题:

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  a) Write 81 as a power of 3.
  :a) 将81写成3的功率。

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  Solution:
  ::解决方案 :

    $\begin{align*}81={\color{red}3} \times {\color{red}3} \times {\color{red}3} \times {\color{red}3}=3^4\end{align*}$

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  There are 4 threes. Therefore,  $\begin{align*}\boxed{81=3^{\color{red}4}}\end{align*}$
  ::四个三,所以,81=34

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  b) Write $\begin{align*}(9)^3\end{align*}$ as a power of 3.
  :b) 书写(9)3功率为3。

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  Solution:
  ::解决方案 :

  $\begin{align*}9={\color{red}3} \times {\color{red}3}=9\end{align*}$

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  There are 2 threes. Therefore,  $\begin{align*}\boxed{9=3^{\color{red}2}}\end{align*}$
  ::两三三,所以,9=32

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  $\begin{align*}(3^2)^3\end{align*}$ Apply the law of exponents for power to a power—multiply the exponents.
  :32)3 对强权者适用强权者法则——将强权者乘以强权者法则。

  $\begin{align*}3^{2 \times 3}=3^{\color{red}6}\end{align*}$

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  Therefore,  $\begin{align*}\boxed{(9)^3=3^{\color{red}6}}\end{align*}$
  ::因此,(9)3=36

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  c) Write $\begin{align*}(4^3)^2\end{align*}$ as a power of 2.
  :c) 写(43)2,作为2的功率。

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  Solution:
  ::解决方案 :

  $\begin{align*}4={\color{red}2} \times {\color{red}2}=4\end{align*}$

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  There are 2 twos. Therefore,  $\begin{align*}\boxed{4=2^{\color{red}2}}\end{align*}$
  ::有两个两个,所以,4=22

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  $\begin{align*}\left((2^2)^3\right)^2\end{align*}$ Apply the law of exponents for power to a power—multiply the exponents:
  :(22)3)2 将权力的推手法适用于强者——将强者乘以强者:

  $\begin{align*}\boxed{2^{2 \times 3}=2^{\color{red}6}}\end{align*}$

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  $\begin{align*}(2^6)^2\end{align*}$ Apply the law of exponents for power to a power—multiply the exponents:
  :262) 将权力的推手法适用于强者——将强者乘以强者:

  $\begin{align*}\boxed{2^{6 \times 2}=2^{\color{red}12}}\end{align*}$

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  Therefore,  $\begin{align*}\boxed{(4^3)^2=2^{\color{red}12}}\end{align*}$
  ::因此,(43)2=212

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  Example 4
  ::例4

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  a) Use the laws of exponents to simplify the following: $\begin{align*}(-3x^2)^3 (9x^4y)^{-2}\end{align*}$
  :a) 使用指数法简化-3x2)3(9x4y)-2

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  Solution:
  ::解决方案 :

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  Apply the law of exponents to simplify:
  ::为简化下列手续,适用推事的法律:

  $$\begin{align*}(-3x^2)^3(9x^4y)^{-2} &=(-3)^{\color{red}3}x^{\color{red}6} \cdot \frac{1}{(9x^4y)^2}\ \\ & ={\color{red}-27}x^6 \cdot \frac{1}{9^{\color{red}2} x^{\color{red}8} y^{\color{red}2}} \\ & =\frac{-27x^6}{{\color{red}81}x^8y^2} \\ & ={\color{red}-\frac{1x^{-2}}{3y^2}} \\ &=-\frac{1}{3x^2y^2}\end{align*}$$

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  b) Rewrite the following using only positive exponents:  $\begin{align*}(x^2 y^{-1} -1)^2\end{align*}$
  :b) 仅使用正指数重写以下文字x2y-1-1)2

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  Solution:
  ::解决方案 :

  $$\begin{align*}& (x^2y^{-1}-1)^2 && \text{Begin by expanding the binomial.}\\ & (x^2y^{-1}-1)(x^2y^{-1}-1) && \text{Then multiply the terms by distribution. Remember}\\ & && \text{to apply the product rule for exponents} \\ & && \boxed{a^m \times a^n=a^{m+n}}.\\ & (x^{{\color{red}2+2}} y^{{\color{red}-1+(-1)}} -1x^2y^{-1}-1x^2y^{-1}+1) && \text{Simplify.}\\ & (x^{\color{red}4}y^{{\color{red}-2}}-{\color{red}2}x^2y^{-1}+1) && \text{Apply the negative exponent rule}\\ & && \boxed{a^{-m}=\frac{1}{a^m}}.\\ & \left(\frac{x^4}{{\color{red}y^2}}-\frac{2x^2}{{\color{red}y}}+1 \right)\\ & \boxed{(x^2y^{-1}-1)^2=\left(\frac{x^4}{y^2}-\frac{2x^2}{y}+1\right)}\end{align*}$$

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  c) Use the laws of exponents to evaluate the following: $\begin{align*}[5^{-4} \times (25)^3]^2\end{align*}$
  :c) 使用指数法评估下列事项: [5-4x(25)3]2

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  Solution:
  ::解决方案 :

  $$\begin{align*}& [5^{-4} \times (25)^3]^2 && \text{Try to do this one by}\\ & && \text{applying the laws of exponents.}\\ & [5^{-4} \times (25)^3]^2=[5^{-4} \times ({\color{red}5^2})^3]^2\\ & [5^{-4} \times ({\color{red}5^2})^3]^2=[5^{-4} \times 5^{\color{red}6}]^2\\ & [5^{-4} \times 5^{\color{red}6}]^2=(5^{\color{red}2})^2\\ & (5^{\color{red}2})^2=5^{\color{red}4}\\ & 5^4={\color{red}625}\\ & \boxed{[5^{-4} \times (25)^3]^2=5^4=625}\end{align*}$$

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  Summary
  ::摘要

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  • In an algebraic expression, the base is the variable, number, product, or quotient to which the exponent refers. 
    ::在代数表达式中,基数是引言人所指的变量、数字、产品或商数。
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  • In an algebraic expression, the exponent is the number to the upper right of the base that tells how many times to multiply the base by itself.
    ::在代数表达式中,引号是基底右上方的数值,该数值表示自己乘以基数的次数。
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  • In the expression $\begin{align*}2^5\end{align*}$ , "2" is the base and "5" is the exponent. This means multiply 2 by itself 5 times, as shown here:  $\begin{align*}2\times 2\times 2\times 2\times 2.\end{align*}$
    ::在表达式 25 中, “ 2” 是基数, “ 5” 是引号 。 这意味着乘以 2 本身乘以 5 次, 如此处所示 : 2x2x2x2x2x2x2x2x2。
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  • A power is simply the name given to an algebraic expression that is raised to an exponent. $\begin{align*}2^5\end{align*}$ and $\begin{align*}(-3y)^4\end{align*}$ are both examples of a power.
    ::电源仅仅是一个代数表达式的名称,该代数表达式是向表率提出的。 25 和(-3y)4 两者都是电源的例子。
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  • The laws of exponents are the algebraic rules and formulas that tell us the operation to perform on the exponents when dealing with exponential expressions. These laws are :
    ::引言者的法律是代数规则和公式,它告诉我们在处理指数表达式时要对引言者进行操作。

   

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     Laws of Exponents
  ::名人法

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  Multiplication, Division, and Power Rules:
  ::乘法、分法和电力规则:

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  •   $\begin{align*}\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}\end{align*}$    ( $\begin{align*}b \neq 0\end{align*}$ )
    :ab)n=anbn (b0)
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  •   $\begin{align*}(ab)^n=a^nb^n\end{align*}$  
    :ab)n=anbn
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  •   $\begin{align*}(a^m)^n=a^{mn}\end{align*}$  
    :am)n=amn(上午)
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  •   $\begin{align*}\frac{a^m}{a^n}=a^{m-n}\end{align*}$       ( $\begin{align*}a \neq 0\end{align*}$ )  
    ::aman=am-n (a0)
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  • $\begin{align*}{a^m}\cdot {a^n}=a^{m+n}\end{align*}$
    ::AMan=am+n

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  Zero and Negative Exponent Rules:
  ::零和负指数规则:

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  •   $\begin{align*}a^{-n}=\frac{1}{a^n}\end{align*}$       ( $\begin{align*}a \neq 0\end{align*}$ )   ​
    ::a-n=1 (a0)
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  •   $\begin{align*}{a}^{0} = 1\end{align*}$           ( ​ $\begin{align*}a \neq 0\end{align*}$ ) 
    ::a0=1 (a0)

   

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  Review
  ::回顾

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  Evaluate each of the following expressions:
  ::评估以下每一种表达方式:

  1. $\begin{align*}(-3)^{-3}\end{align*}$
  2. $\begin{align*}6 \times \left(\frac{1}{2}\right)^{-2}\end{align*}$
  3. $\begin{align*}7^{-4} \times 7^4\end{align*}$
  4. $\begin{align*}(4^0 + 4^{-1})^{-1}\end{align*}$  
  5. $\begin{align*}\mathrm{-}{({3}^{2})}^{3}\end{align*}$  
  6. $\begin{align*}{\left(\tfrac{1}{3}\right)}^{6}\div {\left(\tfrac{1}{3}\right)}^{8}\end{align*}$  

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  Simplify the expressions below. (Your answers should have only positive exponents.)
  ::简化下面的表达式 。 (您的答案应该只有积极的缩写 。 )

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  7. $\begin{align*}(x^{-1}+y^{-1})^2\end{align*}$
     :-1+y-1)2
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  8. $\begin{align*}(4wx^{-2}y^3z^{-4})^3\end{align*}$
     :4wx-2y3z-4)3
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  9. $\begin{align*}\frac{a^2b^3c^{-2}}{d^{-2}bc^{-6}}\end{align*}$
     ::a2b3c-2d-2d-2bc-6
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  10. $\begin{align*}m^4(m^2+m-5m^{-2})\end{align*}$
      ::m4(m2+m-5m-2)
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  11.   $\begin{align*}(x^3 y^2) (xy^3) (x^5 y)\end{align*}$
      :x3Y2)(x3)(x5Y)
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  12.   $\begin{align*}\frac{x^6 y^8}{x^4 y^{-2}}\end{align*}$
      ::x6Y8x4y-2
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  13. $\begin{align*}\left(\frac{2x^{10}}{3y^{20}}\right)^3\end{align*}$
      :2x103y20)3
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  14. 播放段落

      $\begin{align*}(10x^8) \div (2x^4)\end{align*}$
      :10x8)(2x4)

      
      :10x8)(2x4)
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  15. 播放段落

      $\begin{align*}(-2x)^5 (2x^2)\end{align*}$
      :-2x)5(2x2)

      
      :-2x)5(2x2)
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  16. 播放段落

      $\begin{align*}(16x^{10}) \left(\frac{3}{4}x^5\right)\end{align*}$
      :16x10)(34x5)

      
      :16x10)(34x5)
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  17. 播放段落

      $\begin{align*}\frac{(x^{15})(x^{24})(x^{25})}{(x^7)^8}\end{align*}$
      :x15(x24)(x25)(x7)8)

      
      :x15(x24)(x25)(x7)8)

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  Express each of the following as a power of 3. Do not evaluate. 
  ::以3.的威力来表达以下各点,不要评价。

  18. $\begin{align*}(3^3)^5\end{align*}$
  19. $\begin{align*}9^4\end{align*}$
  20. $\begin{align*}(9)(27^2)\end{align*}$

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  Review (Answers)
  ::回顾(答复)

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  Please see the Appendix.
  ::请参看附录。