1.3 带有指数的代数表达式

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• 选择节带有指数的代数表达式

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  带有指数的代数表达式

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  Exponents 
  ::指数

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  Many formulas and equations in mathematics contain exponents. Exponents are used as a short-hand notation for repeated multiplication . For example:
  ::数学中的许多公式和方程式含有指数。 指数被用作重复乘法的短手符号。 例如 :

  $$\begin{array}{rl}2\cdot 2& =2^2\\ 2\cdot 2\cdot 2& =2^3\end{array}$$

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  The exponent stands for how many times the number is used as a factor (multiplied). When dealing with integers , it's usually easiest to simplify the expression . For example, as shown here, we commonly just write 4 instead of  $\begin{array}{r}{2}^2,\end{array}$  and 8 instead of  $\begin{array}{r}{2}^3.\end{array}$
  ::引号表示数字是作为系数( 倍数) 的多少倍。 在处理整数时, 通常最容易简化表达式。 例如, 如此处所示, 我们通常只写 4, 而不是 22, 8 而不是 23 。

  $$\begin{array}{rl}{2}^2& =4\\ 2^3& =8\end{array}$$

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  However, we generally keep the exponents when we work with variables, because it is much easier to write $\begin{array}{r}{x}^8\end{array}$ than $\begin{array}{r}x\cdot x\cdot x\cdot x\cdot x\cdot x\cdot x\cdot x.\end{array}$
  ::然而,我们通常在与变量合作时保留指数,因为写x8比写xxxxxxxxxxxxxxxxxxxxxxxxxx要容易得多。

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  To evaluate expressions with exponents, substitute the values you are given for each variable and simplify. It is especially important when working with exponents to substitute using " data-term="Parentheses" role="term" tabindex="0"> parentheses in order to make sure that the simplification is done correctly.
  ::要用表率来评价表达式, 请替换您给每个变量提供的值, 并简化。 在与表率一起使用括号时, 使用括号替换特别重要, 以确保简化正确进行 。

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  Evaluating an Expression Containing Exponents 
  ::评估含有指数的表达式

  

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  The area of a circle is given by the formula $\begin{array}{r}A=\pi r^2.\end{array}$  Find the area of a circle with radius $\begin{array}{r}r=17inches.\end{array}$
  ::公式Ar2给定一个圆区域,以半径 r=17 英寸查找圆区域。

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  Substitute values into the equation .
  ::等式中的替代值 。

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  $$\begin{array}{rl}A& =\pi r^2\text{Substitute 17 for }r.\\ & =\pi (17{)}^2\\ & =3.14(17)(17)\\ A& \approx 907.92\text{ Rounded to 2 decimal places.}\end{array}$$

  
  ::r2 替代 17 r. (17)2= 3.14(17)(17)A 907.92四舍五入到小数点后两位。

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  The area of the circle is approximately 907.92 square inches.
  ::圆的面积约为907.92平方英寸。

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  Evaluating a Multi-Variable Expression Containing Exponents 
  ::评估含有多变量指数的多功能表达式

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  $\begin{array}{r}\text{Find the value of }\frac{x^2{y}^3}{x^3+y^2},\text{ when }x=2\text{ and }y=-4.\end{array}$
  ::当 x=2 和 y4 时, 查找 x2y3x3+y2 的值 。

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  Substitute the given values for  $\begin{array}{r}x\end{array}$ and $\begin{array}{r}y\end{array}$ .
  ::替换 x 和 y 的给定值。

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  $$\begin{array}{rl}\frac{x^2{y}^3}{x^3+y^2}& =\frac{(2{)}^2(-4{)}^3}{(2{)}^3+(-4{)}^2}\text{Substitute}2\text{for}x\text{and}-4\text{for}y.\\ & =\frac{4(-64)}{8+16}(2{)}^2=4,(-4{)}^3=-64,(2{)}^3=8,(-4{)}^2=16.\\ & =\frac{-256}{24}\\ & =\frac{-32}{3}\end{array}$$

  
  ::x2y3x3+y2=(2)2(-4)3(2)3+(-4)2x和y.=4(-648+16(2)2=4,(-4)3x3+y2=2,(-4)3(2)3+(-4)3+(-4)2 y.=4(-648+16(2)2=4,(-4)3x3+y2=8,(-4)2=16.*25624=323

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  Real World Application 
  ::真实世界应用

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  The height "> $\begin{array}{r}(h)\end{array}$ of a ball in flight is given by the formula $\begin{array}{r}h=-32t^2+60t+20\end{array}$ , where the height is given in feet and the time $\begin{array}{r}(t)\end{array}$ is given in seconds. Find the height of the ball at time $\begin{array}{r}t=2\text{ seconds.}\end{array}$
  ::飞行中的球的高度 由公式 h32t2+60t+20给定, 高度以足为单位, 时间(t) 以秒为单位。 在时间 t=2 秒时查找球的高度 。

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  $$\begin{array}{rl}h& =-32t^2+60t+20\\ & =-32(2{)}^2+60(2)+20\text{Substitute}2\text{for}t.\\ & =-32(4)+60(2)+20\\ & =12\end{array}$$

  
  ::há32t2+60t+2032(2)+60(2)+20替代品2, t. @32(4)+60(2)+20=12

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  The height of the ball is 12 feet.
  ::球的高度是12英尺

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  Example
  ::示例示例示例示例

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  Example 1
  ::例1

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  Find the value of $\begin{array}{r}\frac{a^2+b^2}{a^2-b^2}\end{array}$ , for $\begin{array}{r}a=-1\end{array}$ and $\begin{array}{r}5\end{array}$ .
  ::查找 a2+b2a2-b2 和 a%1 和 5 的 a2+b2a2-b2 值。

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  Substitute the values of $\begin{array}{r}x\end{array}$ and $\begin{array}{r}y\end{array}$ in the following.
  ::以 x 和 y 的值替换以下数值。

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  $$\begin{array}{r}\frac{a^2+b^2}{a^2-b^2}=\frac{(-1{)}^2+(5{)}^2}{(-1{)}^2-(5{)}^2}\text{Substitute}-1\text{for}a\text{and}5\text{for}b.\\ \frac{1+25}{1-25}=\frac{26}{24}=\frac{13}{12}\text{Evaluate and simplify expressions.}\end{array}$$

  
  ::a2+b2a2-b2=(-1)2+(5)2(-2-1)-2-(5)2替代品-1和b.1+251-25=2624=1312

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  Review 
  ::回顾

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  Evaluate 1-8 using $\begin{array}{r}x=-1,y=2,z=-3,\end{array}$ and $\begin{array}{r}w=4\end{array}$ .
  ::使用 x1, y=2, z3 和 w=4 来评价 1-8 。

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  1. $\begin{array}{r}8x^3\end{array}$
     ::8x3
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  2. $\begin{array}{r}\frac{5x^2}{6z^3}\end{array}$
     ::5x26z3
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  3. $\begin{array}{r}3z^2-5w^2\end{array}$
     ::3z2-5w2
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  4. $\begin{array}{r}{x}^2-y^2\end{array}$
     ::x2 - y2
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  5. $\begin{array}{r}\frac{z^3+w^3}{z^3-w^3}\end{array}$
     ::z3+w3z3-w3 z3 -w3
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  6. $\begin{array}{r}2x^3-3x^2+5x-4\end{array}$
     ::2x3-3x2+5x-4
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  7. $\begin{array}{r}4w^3+3w^2-w+2\end{array}$
     ::4w3+3w2-2-2+2
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  8. $\begin{array}{r}3+\frac{1}{z^2}\end{array}$
     ::3+1z2 3+1z2

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  For 9-10, use the fact that the volume of a box without a lid is given by the formula $\begin{array}{r}V=4x(10-x{)}^2\end{array}$ , where $\begin{array}{r}x\end{array}$ is a length in inches and $\begin{array}{r}V\end{array}$ is the volume in cubic inches.
  ::9-10时,使用公式V=4x(10-x)2给出无盖子的盒体体积这一事实,x是一个长度英寸,V是立方英寸的体积。

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  9. What is the volume when $\begin{array}{r}x=2\end{array}$ ?
     ::x=2 时的音量是多少 ?
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  10. What is the volume when $\begin{array}{r}x=3\end{array}$ ?
      ::x=3 时的音量是多少 ?

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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 。