8.7 评价指数表达式

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  评价指数表达式

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  Evaluating Exponential Expressions 
  ::评价指数表达式

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  When evaluating expressions we must keep in mind the . You must remember PEMDAS :
  ::在评价表达式时,我们必须记住:

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  1. Evaluate inside the " data-term="Parentheses" role="term" tabindex="0"> Parentheses .
     ::在括号内部进行评估。
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  2. Evaluate E xponents.
     ::评价指数。
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  3. Perform M ultiplication and D ivision operations from left to right.
     ::执行从左到右的乘法和分解操作。
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  4. Perform A ddition and S ubtraction operations from left to right.
     ::执行从左到右的加法和减法操作。

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  Evaluating Expressions 
  ::评价表达式

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  1. Evaluate the following expressions.
  ::1. 评价以下表述。

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  a)  $\begin{array}{r}{5}^0\end{array}$
  :a) 50

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  $\begin{array}{r}{5}^0=1\end{array}$ A number raised to the power 0 is always 1.
  ::50=1 0号电源的号码总是1

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  b)  $\begin{array}{r}{\left(\frac{2}{3}\right)}^3\end{array}$
  :b) (23)3

  $\begin{array}{r}{\left(\frac{2}{3}\right)}^3=\frac{2^3}{3^3}=\frac{8}{27}\end{array}$

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  c)  $\begin{array}{r}{16}^{\frac{1}{2}}\end{array}$
  :c) 1612(2005)

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  $\begin{array}{r}{16}^{\frac{1}{2}}=\sqrt{16}=4\end{array}$ Remember that an exponent of $\begin{array}{r}\frac{1}{2}\end{array}$ means taking the square root .
  ::16=16=4 记住,12个字的出处意味着用平方根。

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  d)  $\begin{array}{r}{8}^{-\frac{1}{3}}\end{array}$
  :d) 8-13

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  $\begin{array}{r}{8}^{-\frac{1}{3}}=\frac{1}{8^{\frac{1}{3}}}=\frac{1}{\sqrt[3]{8}}=\frac{1}{2}\end{array}$ Remember that an exponent of $\begin{array}{r}\frac{1}{3}\end{array}$ means taking the cube root .
  ::8-13=1813=183=12 记住,13个字的出处意味着取下立方根。

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  2. Evaluate the following expressions.
  ::2. 评价以下表述。

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  a)  $\begin{array}{r}3\cdot 5^2-10\cdot 5+1\end{array}$
  ::a) 3-52-10-5+1

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  Evaluate the exponent: $\begin{array}{r}3\cdot 5^2-10\cdot 5+1=3\cdot 25-10\cdot 5+1\end{array}$
  ::评估指数:352-105+1=325-105+1

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  Perform multiplications from left to right: $\begin{array}{r}3\cdot 25-10\cdot 5+1=75-50+1\end{array}$
  ::从左到右的执行乘数: 325- 105+1=75- 50+1

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  Perform additions and subtractions from left to right: $\begin{array}{r}75-50+1=26\end{array}$
  ::从左到右的演练增减:75-50+1=26

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  b)  $\begin{array}{r}\frac{2\cdot 4^2-3\cdot 5^2}{3^2-2^2}\end{array}$
  :b) 242-35232-22

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  Treat the expressions in the numerator and denominator of the fraction like they are in parentheses: $\begin{array}{r}\frac{(2\cdot 4^2-3\cdot 5^2)}{(3^2-2^2)}=\frac{(2\cdot 16-3\cdot 25)}{(9-4)}=\frac{(32-75)}{5}=\frac{-43}{5}\end{array}$
  ::将分数的分子和分母中的表达式处理成括号中的表达式: (2)42-352 (32-22) = (2)16-325) (9-4) = (32-755)5435

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  c)  $\begin{array}{r}{\left(\frac{3^3}{2^2}\right)}^{-2}\cdot \frac{3}{4}\end{array}$
  :c) (3322)-234

  $\begin{array}{r}{\left(\frac{3^3}{2^2}\right)}^{-2}\cdot \frac{3}{4}={\left(\frac{2^2}{3^3}\right)}^2\cdot \frac{3}{4}=\frac{2^4}{3^6}\cdot \frac{3}{4}=\frac{2^4}{3^6}\cdot \frac{3}{2^2}=\frac{2^2}{3^5}=\frac{4}{243}\end{array}$

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  Evaluating Expressions for Given Values 
  ::给定值评价表达式

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  Evaluate the following expressions for $\begin{array}{r}x=2,y=-1,z=3\end{array}$ .
  ::x=2, y1,z=3 的下列表达式进行评价 。

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  a)  $\begin{array}{r}2x^2-3y^3+4z\end{array}$
  :a) 2x2--3y3+4z

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  $$\begin{array}{rl}2x^2-3y^3+4z& =2\cdot 2^2-3\cdot (-1{)}^3+4\cdot 3\\ & =2\cdot 4-3\cdot (-1)+4\cdot 3=8+3+12\\ & =23\end{array}$$

  
  ::2x2-3y3+4z=222-3(-1)3+43=24-3(-1)3+4}(-1)+43=8+3+12=23

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  b)  $\begin{array}{r}(x^2-y^2{)}^2\end{array}$
  ::b) (x2-y2)2

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  $\begin{array}{r}(x^2-y^2{)}^2=(2^2-(-1{)}^2{)}^2=(4-1{)}^2=3^2=9\end{array}$
  :x2-y2)2=(22-(-1))2=(4-1)2=(4-1)2=32=9)

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  c)  $\begin{array}{r}{\left(\frac{3x^2{y}^5}{4z}\right)}^{-2}\end{array}$
  :c) (3x2y54z)-2

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  $$\begin{array}{rl}{\left(\frac{3x^2{y}^5}{4z}\right)}^{-2}& ={\left(\frac{3\cdot 2^2\cdot (-1{)}^5}{4\cdot 3}\right)}^{-2}\\ & ={\left(\frac{3\cdot 4\cdot (-1)}{12}\right)}^{-2}\\ & ={\left(\frac{-12}{12}\right)}^{-2}\\ & ={\left(\frac{-1}{1}\right)}^{-2}\\ & ={\left(\frac{1}{-1}\right)}^2\\ & =(-1{)}^2\\ & =1\end{array}$$

  
  :3x2y54z)-2=(322(-1)54/3)-2=(34(--1)12)-2=(-1212)-2=(-11)-2=(-1)-2=(1-1)2=(-1)2=(-1)2=1

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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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  Evaluate the following expression for $\begin{array}{r}x=3,y=-2,z=-1\end{array}$ .
  ::x=3,y2,z1 的以下表达式进行评价 。

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  $\begin{array}{r}2z((x+1{)}^{\frac{1}{2}}-y^3{)}^2\end{array}$
  ::2z( (x+1) 12- y3) 2

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  $$\begin{array}{rl}2z((x+1{)}^{\frac{1}{2}}-y^3{)}^2& =2(-1)(((3)+1{)}^{\frac{1}{2}}-(-2{)}^3{)}^2\\ & =-2(4^{\frac{1}{2}}+8{)}^2\\ & =-2(2+8{)}^2\\ & =-2(10{)}^2\\ & =-200\end{array}$$

  
  ::2z(x+1)12-y3)2=2-1(((3)+1)12-(-)3)2(2(412+8)2(2)2(2)+8)2(2)+8)2(2)+8)2(2)_(10)2(2)_(10)2_(200)

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  Review 
  ::回顾

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  Evaluate the following expressions to a single number.
  ::评估单个数字的下列表达式。

  1. $\begin{array}{r}{3}^{-2}\end{array}$
  2. $\begin{array}{r}{-4}^{-3}\end{array}$
  3. $\begin{array}{r}(6.2{)}^0\end{array}$
  4. $\begin{array}{r}{8}^{-4}\cdot 8^6\end{array}$
  5. $\begin{array}{r}{\left({16}^{\frac{1}{2}}\right)}^3\end{array}$
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  6. $\begin{array}{r}{x}^2\cdot 4x^3\cdot y^4\cdot 4y^2\end{array}$ , if $\begin{array}{r}x=2\end{array}$ and $\begin{array}{r}y=-1\end{array}$
     ::x24x3y44y2, 如果 x=2 和 y1
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  7. $\begin{array}{r}{a}^4(b^2{)}^3+2ab\end{array}$ , if $\begin{array}{r}a=-2\end{array}$ and $\begin{array}{r}b=1\end{array}$
     ::a4(b2)3+2ab,如果 a%2 和 b=1
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  8. $\begin{array}{r}5x^2-2y^3+3z\end{array}$ , if $\begin{array}{r}x=3,y=2,\end{array}$ and $\begin{array}{r}z=4\end{array}$
     ::5x2-2y3+3z, 如果 x=3,y=2, z=4
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  9. $\begin{array}{r}{\left(\frac{a^2}{b^3}\right)}^{-2}\end{array}$ , if $\begin{array}{r}a=5\end{array}$ and $\begin{array}{r}b=3\end{array}$
     :a2b3)-2,如果 a=5和 b=3
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  10. $\begin{array}{r}{\left(\frac{x^{-2}}{y^4}\right)}^{\frac{1}{2}}\end{array}$ , if $\begin{array}{r}x=-3\end{array}$ and $\begin{array}{r}y=2\end{array}$
      :x2y4) 12, 如果 x%3 和 y=2

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  Review (Answers)
  ::回顾(答复)

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  ::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 。