3.2 指数属性

章节大纲

• 选择节常规

  折叠 展开

  常规

  全部折叠 展开全部

  • 选择活动新闻通告

    

    新闻通告 讨论区
• 选择节指数属性

  折叠 展开

  指数属性

  播放段落

  It is important to quickly and effectively manipulate algebraic expressions involving exponents.  One simplification that comes up often is that expressions and numbers raised to the 0 power are always equal to 1.  Why is this true and is it always true? 
  ::快速和有效地操控涉及引言的代数表达式非常重要。 经常出现的一个简单简化是,向0强的表达式和数字总是等于1。 为什么这是正确的,而且它也总是真实的?

  播放段落

  Exponent Properties
  ::指数属性

  播放段落

  Consider the following exponential expressions with the same base and what happens through the algebraic operations .  You should feel comfortable with all of these types of manipulations. Let   $\begin{array}{r}{b}^y,b^x\end{array}$  be exponential terms .
  ::考虑以下以相同基数的指数表达式, 以及通过代数操作发生的情况。 您应该对所有这些类型的操纵感到自在。 bx是指数术语 。

  播放段落

  Addition and Subtraction
  ::加和减

  播放段落

    $\begin{array}{r}{b}^x\pm b^y=b^x\pm b^y\end{array}$
  ::bxby=bxby

  播放段落

  Only in the special case when  $\begin{array}{r}x=y\end{array}$ can the terms be combined.  This is a basic property of combining like terms .
  ::只有在特殊情况下, x=y 才能合并术语。 这是将类似术语合并的基本属性 。

  播放段落

  Multiplication
  ::乘法乘法

  播放段落

  $\begin{array}{r}{b}^x\cdot b^y=b^{x+y}\end{array}$
  ::bxby=bx+y

  播放段落

  When the bases are the same then exponents can be added.
  ::当基数相同时, 引号可以添加 。

  播放段落

  Division
  ::司司司

  播放段落

  $\begin{array}{r}\frac{b^x}{b^y}=b^{x-y}\end{array}$
  ::bxby=bx-y

  播放段落

  The division rule is an extension of the multiplication rule with the possibility of a negative in the exponent .
  ::分法规则是乘法规则的延伸,在引言中有可能出现负数。

  播放段落

  Negative exponent
  ::负表列负数

  播放段落

    $\begin{array}{r}{b}^{-x}=\frac{1}{b^x}\end{array}$
  ::b-x=1bx

  播放段落

  A negative exponent means reciprocal .
  ::负指数意味着对等。

  播放段落

  Fractional exponent
  ::小数前列

  播放段落

  $\begin{array}{r}(b{)}^{\frac{1}{x}}=\sqrt[x]{b}\end{array}$
  
  :b) 1x=bx

  播放段落

  Square roots are what most people think of when they think of roots, but roots can be taken with any real number using .
  ::平方根是多数人认为是根时所想到的,

  播放段落

  Powers of Powers
  ::强 权 权 权 权 权 权

  播放段落

    $\begin{array}{r}(b^x{)}^y=b^{x\cdot y}\end{array}$
  :xx)y=bxy

  播放段落

  Examples
  ::实例

  播放段落

  Example 1
  ::例1

  播放段落

  Earlier, you were asked why expressions and numbers raised to the 0 power are always equal to 1. Consider the following pattern and decide what the next term in the should be:
  ::早些时候,有人问到,为什么向0强的表达方式和数字总是等于1。 考虑以下模式,并决定下一个任期应该是什么:

  16, 8, 4, 2, ___

  播放段落

  It makes sense that the next term is 1 because each successive term is half that of the previous term.  These numbers correspond to powers of 2. 
  ::下一个任期为1,这是有道理的,因为每个连续任期为上一个任期的一半,这些数字相当于2个任期的权力。

  $\begin{array}{r}{2}^4,2^3,2^2,2^1,\underset{\_}{}\end{array}$

  播放段落

  In this case you could decide that the next term must be $\begin{array}{r}{2}^0\end{array}$ .  This is a useful technique for remembering what happens when a number is raised to the 0 power. 
  ::在此情况下,你可以决定下一个任期必须是20年,这是一个有用的方法,用于记起当一个数字被提升到0强时会发生什么情况。

  播放段落

  One question that extends this idea is what is the value of $\begin{array}{r}{0}^0\end{array}$ ?  People have argued about this for centuries.  Euler argued that it should be 1 and many other mathematicians like Cauchy and Möbius argued as well.  If you search today you will still find people discussing what makes sense. In practice, many mathematicians note this value as undefined .
  ::扩大这个想法的一个问题是,00的价值是什么? 几个世纪以来,人们一直在争论这个价值。 欧勒认为它应该是一个数学家和许多其他数学家,如卡乌西和莫比乌斯也这样认为。 如果你今天搜索,你仍然会发现人们在讨论什么是有道理的。 实际上,许多数学家认为这个价值是没有定义的。

  播放段落

  Example 2
  ::例2

  播放段落

  Simplify the following expression until all exponents are positive.
  ::简化以下表达式,直到所有提示为正。

  播放段落

  $$\begin{array}{r}\frac{(a^{-2}{b}^3{)}^{-3}}{a{b}^2{c}^0}\end{array}$$

  
  :a)-2b3)-3ab2c0

  播放段落

  $$\begin{array}{r}\frac{(a^{-2}{b}^3{)}^{-3}}{a{b}^2{c}^0}=\frac{a^6{b}^{-9}}{a{b}^2\cdot 1}=\frac{a^5}{b^{11}}\end{array}$$

  
  :a)-2b3)-3ab2c0=a6b-9ab21=a5b11

  播放段落

  Example 3
  ::例3

  播放段落

  Simplify the following expression until all exponents are positive. 
  ::简化以下表达式,直到所有提示为正。

  播放段落

  $$\begin{array}{r}(2x{)}^5\cdot \frac{4^2}{2^{-3}}\end{array}$$

  
  :2x)5422-3

  播放段落

  $$\begin{array}{r}(2x{)}^5\cdot \frac{4^2}{2^{-3}}=\frac{2^5{x}^5{2}^4}{2^{-3}}=\frac{2^9{x}^5}{2^{-3}}=2^{12}{x}^5\end{array}$$

  
  :2x)5422-3=25x5242-3=29x52-3=212x5)

  播放段落

  Example 4
  ::例4

  播放段落

  Simplify the following expression using positive exponents.
  ::使用正表征简化以下表达式 。

  $$\begin{array}{r}\frac{(2^6\cdot 8^3{)}^{-3}}{4^2{\left(\frac{1}{2}\right)}^4{64}^{\frac{1}{3}}}\end{array}$$

  播放段落

  Rewrite every exponent as a power of 2. 
  ::将每个表情重写为2的功率。

  播放段落

  For example $\begin{array}{r}{8}^3=(2^3{)}^3=2^9\end{array}$ and $\begin{array}{r}{64}^{\frac{1}{3}}=(2^6{)}^{\frac{1}{3}}=2^2\end{array}$
  ::例如83=(233)3=29和6413=(26)13=22

  $$\begin{array}{r}\frac{(2^6\cdot 8^3{)}^{-3}}{4^2{\left(\frac{1}{2}\right)}^4{64}^{\frac{1}{3}}}=\frac{(2^6\cdot 2^9{)}^{-3}}{2^4{2}^{-4}{2}^2}=\frac{(2^{15}{)}^{-3}}{2^2}=\frac{2^{-45}}{2^2}=\frac{1}{2^{47}}\end{array}$$

  播放段落

  Example 5
  ::例5

  播放段落

  Solve the following equation using properties of exponents.
  ::使用指数属性解决以下方程式 。

  播放段落

  $\begin{array}{r}({32}^{0.6}{)}^2=x^3\end{array}$
  :320.6)2=x3

  播放段落

  First work with the left hand side of the equation.
  ::以方程式左侧为首的工作。

  播放段落

  $$\begin{array}{rl}({32}^{0.6}{)}^2& ={\left((2^5{)}^{\frac{3}{5}}\right)}^2=2^6\\ 2^6& =x^3\\ (2^6{)}^{\frac{1}{3}}& =(x^3{)}^{\frac{1}{3}}\\ 2^2& =x\\ 4& =x\end{array}$$

  
  :320.6)2=((25)352=2626=x3(26)13=(x3)1322=x4=x)

  Summary

  播放段落 Expressions and numbers raised to the 0 power are always equal to 1. ::0 功率的表达式和数字总是等于 1 。 播放段落 Adding or subtracting exponential terms can only be performed when the bases are equal and the exponents are equal.. ::只有在基数相等,引数相等时才能进行增减指数值。 播放段落 When multiplying exponential terms with the same base, exponents can be added. ::当以相同基数乘以指数值时,可以添加指数。 播放段落 When dividing exponential terms with the same base, exponents can be subtracted. ::当用相同基数来除以指数值时,可以减去指数值。 播放段落 A negative exponent means reciprocal. ::负指数意味着对等。 播放段落 Fractional exponents represent roots. ::分数前列代表根。 播放段落 When an exponential expression is raised to another power , the exponents can be multiplied. ::当指数表达式升到另一个电源时, 推数可以乘以 。

  播放段落

  Review
  ::回顾

  播放段落

  Simplify each expression using positive exponents.
  ::使用正表征来简化每个表达式 。

  1. $\begin{array}{r}{81}^{-\frac{1}{4}}\end{array}$

  2.  $\begin{array}{r}{64}^{\frac{2}{3}}\end{array}$

  3. $\begin{array}{r}{\left(\frac{1}{32}\right)}^{-\frac{2}{5}}\end{array}$

  4.  $\begin{array}{r}(-125{)}^{\frac{1}{3}}\end{array}$

  播放段落

  5. $\begin{array}{r}(4x^{3}y)(3x^5{y}^2{)}^4\end{array}$
  ::5. (4x3y)(3x5y2)4

  播放段落

  6.  $\begin{array}{r}(5x^3{y}^2{)}^2(7x^{3}y{)}^2\end{array}$
  ::6. (5x3y2)(2x7x3y)2

  播放段落

  7.  $\begin{array}{r}{\displaystyle \frac{8a^3{b}^{-2}}{(-4a^2{b}^4{)}^{-2}}}\end{array}$
  ::7. 8a3b-2(2-4a2b4)-2

  播放段落

  8.  $\begin{array}{r}{\displaystyle \frac{5x^2{y}^{-3}}{(-2x^3{y}^2{)}^{-4}}}\end{array}$
  ::8. 5x2y-3(-2x3y2)-4

  播放段落

  9.  $\begin{array}{r}{\left({\displaystyle \frac{3m^3{n}^{-4}}{2m^{-5}{n}^{-2}}}\right)}^{-4}\end{array}$
  ::9. (3m3n-42m-5n-2)-4

  播放段落

  10.  $\begin{array}{r}{\left({\displaystyle \frac{4m^{-3}{n}^{-4}}{5m^5{n}^{-4}}}\right)}^{-3}\end{array}$
  ::10. (4m-3n-45m5n-4)-3

  播放段落

  11.  $\begin{array}{r}{\left({\displaystyle \frac{a^{-1}b}{a^5{b}^4}}\right)}^{-3}\end{array}$
  ::11. (a-1ba5b4)-3

  播放段落

  12.  $\begin{array}{r}{\displaystyle \frac{15c^{-2}{d}^{-6}}{3c^{-4}{d}^{-2}}}\end{array}$
  ::12. 15c-2d-63c-4d-2

  播放段落

  13.  $\begin{array}{r}{\displaystyle \frac{12e^{5}f}{(-2e{f}^3{)}^{-2}}}\end{array}$
  ::13. 12e5f(-2ef3)-2

  播放段落

  Solve the following equations using properties of exponents.
  ::使用指数属性解决以下方程式 。

  播放段落

  14.  $\begin{array}{r}({81}^{0.75}{)}^2=x^3\end{array}$
  ::14. (810.752=x3)

  播放段落

  15. $\begin{array}{r}{\left({64}^{\frac{1}{6}}\right)}^{-3}=x^3\end{array}$
  ::15. (6416)-3=x3

  播放段落

  Review (Answers)
  ::回顾(答复)

  播放段落

  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 。