幂和分组符号的表达式求值

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For her best friend's birthday, Jean bought a huge birthday hat. It's too big to actually wear, so her plan is to add a base to it, buy fabric, and cover it with fabric. Jean needs to know how much fabric to buy. Jean's older brother told her that to find the approximate area of the surface of the birthday hat she can use the following  formula :
::Jean的哥哥告诉她,要找到生日帽表面的大致面积,她可以使用以下公式:

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$\begin{array}{r}3.14{\left(\frac{d}{2}\right)}^2+1.57dl\end{array}$
::3.14 (d 2) 2 + 1.57 d l

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where  $\begin{array}{r}d\end{array}$  is the diameter of the base of the hat and  $\begin{array}{r}l\end{array}$  is the slant height of the hat.
::D是帽子底部的直径 而我是帽子的倾斜高度

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Jean measured her birthday hat and found that the diameter is 15 inches and the slant height is 18 inches. How can Jean use the formula that her brother gave her to find the surface area of the birthday hat so that she can buy the right amount of fabric?
::Jean测量了她的生日帽子,发现直径是15英寸,倾斜高度是18英寸。 Jean如何使用她兄弟给她的配方,找到生日帽子的表面面积,以便买到合适的布料?

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In this concept, you will learn how to  evaluate   variable  expressions involving exponents and  " data-term="Parentheses" role="term" tabindex="0"> parentheses .
::在此概念中,您将学会如何评价包含引言和括号的变量表达式。

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Evaluating Expressions with Powers and Grouping Symbols
::与权力和组群的表情评价

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An  expression  is a mathematical phrase that contains numbers and  operations . A  variable expression  is an expression that contains variables.
::表达式是一个包含数字和操作的数学短语。变量表达式是一个包含变量的表达式。

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To  evaluate  a variable expression means to find the value of the expression for given values of the variables. To evaluate,  substitute  the given values for the variables in the expression and simplify using the   (PEMDAS).
::要评价变量表达式, 需要为变量的指定值查找表达式的值。 要评估, 请将给定值替换为表达式中的变量, 并使用( PEMDAS) 简化 。

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Remember that the order of operations are:
::记住行动顺序是:

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1. P arentheses: Start by simplifying any part of the expression in parentheses using the order of operations.
   ::括号:从使用操作顺序简化括号中的任何部分开始。
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2. E xponents: Rewrite any  terms  that contain exponents without exponents.
   ::指数:重写任何含有引言但无引言的术语。
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3. M ultiplication/ D ivision: Do any  multiplication  and/or  division  in  order  from left to right.
   ::倍数/司:从左到右做任何倍数和(或)除法。
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4. A ddition/ S ubtraction: Do any  addition  and/or  subtraction  in order from left to right.
   ::加/减:按从左到右顺序增加和(或)减。

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Here is an example.
::举一个例子。

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Evaluate the expression  $\begin{array}{r}2x^2-(x+7)\end{array}$  if  $\begin{array}{r}x=4\end{array}$ .
::如果 x = 4,评价表达式 2 x 2 - ( x + 7 ) 。

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First, substitute 4 in for  $\begin{array}{r}x\end{array}$  in the expression.
::首先,以X代替4。

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Next, notice that this expression has multiplication, exponents, subtraction, parentheses, and addition. According to the order of operations, you must start by simplifying the parts of the expression in parentheses. (4) in the first part of the expression is already simplified.  $\begin{array}{r}(4+7)\end{array}$  in the second part of the expression still needs to be simplified.
::其次,请注意该表达式有乘法、引号、减法、括号、括号和添加。根据操作顺序,您必须首先简化括号中的表达式部分。(4) 表达式第一部分的表达式部分已经简化。( 4+7) 该表达式第二部分的表达式第二部分的( 4+7) 仍然需要简化。

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Next, continue to follow the order of operations and evaluate the  exponent . Note that only the 4 is squared. Remember that  $\begin{array}{r}{4}^2\end{array}$  means  $\begin{array}{r}4\cdot 4\end{array}$  which is 16. 
::下一步, 继续按照操作顺序并评估指数 。 请注意, 只有 4 是正方形 。 记住 , 4 2 意味着 4 4 等于 16 表示 4 4 等于 16 。

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Now, multiply.
::现在,乘数。

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Finally, subtract.
::最后,减去。

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The answer is 21.
::答案是21岁

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Examples
::实例

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Example 1
::例1

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Earlier, you were given a problem about Jean and the birthday hat she bought for her friend's birthday.
::早些时候,你得到一个问题 关于让 和她的生日帽子 她买了她的朋友的生日。

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She wants to find the surface area of the hat so she can buy the correct amount of fabric to cover it. She got the following formula from her brother:
::她想找到帽子的表面区域,这样她就可以买到正确数量的织物来覆盖它。

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where  $\begin{array}{r}d\end{array}$  is the diameter of the base of the hat and  $\begin{array}{r}l\end{array}$  is the slant height of the hat. Jean measured the birthday hat and found that the diameter is 15 inches and the slant height is 18 inches.
::Jean测量了生日帽子,发现直径为15英寸,斜高为18英寸。

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To find the surface area, she should first substitute 15 in for  $\begin{array}{r}d\end{array}$  and 18 in for  $\begin{array}{r}l\end{array}$  in the formula. 
::为了找到表面积,她应首先在公式中以15取代d,以18替代l。

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Next, notice that this expression has multiplication, division, exponents, parentheses, and addition. According to the order of operations, you must start by simplifying the parts of the expression in parentheses that need to be simplified. Start by simplifying the  $\begin{array}{r}\left(\frac{15}{2}\right)\end{array}$  at the beginning of the expression.
::其次, 请注意此表达式有乘法、 分隔符、 引号、 括号、 括号和添加。 根据操作顺序, 您必须首先简化括号中需要简化的表达式部分。 开始简化表达式开头的( 15 2 ) 。

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Next, continue to follow the order of operations and evaluate the exponent. Use your calculator to help you calculate  $\begin{array}{r}{7.5}^2=7.5\cdot 7.5\end{array}$ .
::下一步, 继续遵循操作顺序, 并评估指数 。 使用您的计算器来帮助您计算 7. 5 2 = 7. 5 = 7. 7 5 。

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Now, multiply from left to right. You can use your calculator to help.
::现在,从左向右乘以乘法。 您可以使用您的计算器来帮助 。

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Finally, add. 
::最后,增加。

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The answer is that the surface area of the hat (including the base that she added on) is about 600.525 square inches.
::答案是帽子的表面面积(包括她添加的底座)约为600.525平方英寸。

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With that knowledge, Jean knows exactly how much fabric she needs to buy to help decorate the birthday hat!
::凭着这种知识,让知道 她需要买多少布料 来帮助装饰生日帽子!

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Example 2
::例2

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Evaluate  $\begin{array}{r}6x^2-2x+(x+3)\end{array}$  if  $\begin{array}{r}x\end{array}$  is 4.
::如果x为4,评价6 x 2 - 2 x + (x + 3)

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First, substitute 4 in for  $\begin{array}{r}x\end{array}$  in the expression.
::首先,以X代替4。

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Next, notice that this expression has multiplication, exponents, subtraction, parentheses, and addition. According to the order of operations, you must start by simplifying the parts of the expression in parentheses that need to be simplified. Start by simplifying the  $\begin{array}{r}(4+3)\end{array}$  at the end of the expression. 
::接下来请注意, 此表达式有乘法、 引号、 减法、 括号以及添加。 根据操作顺序, 您必须首先简化括号中需要简化的表达式部分。 开始简化表达式结尾处的( 4+ 3) 。

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Next, continue to follow the order of operations and evaluate the exponent.
::其次,继续按行动顺序进行,并评估出处。

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Now, multiply from left to right.
::现在,从左到右成倍增长。

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Finally, add and subtract from left to right. You will subtract first since that comes first in the expression.
::最后,从左向右增减。自此之后,您将首先减减。

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The answer is 95.
::答案是95

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Example 3
::例3

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Evaluate  $\begin{array}{r}3x^2-2+(x+3)\end{array}$  for  $\begin{array}{r}x=2\end{array}$ .
::评价 3 x 2 - 2 + (x + 3) x = 2。

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First, substitute 2 in for  $\begin{array}{r}x\end{array}$  in the expression.
::首先,在表达式中以 x 取代 2 。

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Next, notice that this expression has multiplication, exponents, subtraction, parentheses, and addition. According to the order of operations, you must start by simplifying the parts of the expression in parentheses that need to be simplified. Start by simplifying the  $\begin{array}{r}(2+3)\end{array}$  at the end of the expression.
::接下来请注意, 此表达式有乘法、 引号、 减法、 括号以及添加。 根据操作顺序, 您必须首先简化括号中需要简化的表达式部分。 开始简化表达式结尾处的 (2+ 3) 。

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Next, continue to follow the order of operations and evaluate the exponent.
::其次,继续按行动顺序进行,并评估出处。

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Now, multiply.
::现在,乘数。

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Finally, add and subtract from left to right. You will subtract first since that comes first in the expression.
::最后,从左向右增减。自此之后,您将首先减减。

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The answer is 15.
::答案是15岁

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Example 4
::例4

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Evaluate  $\begin{array}{r}\frac{24}{x}+(9-x)+y^2\end{array}$  for  $\begin{array}{r}x=3\end{array}$  and  $\begin{array}{r}y=4\end{array}$ .
::评价24x+(9-xx)+y2x=3和y=4。

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First, substitute 3 in for  $\begin{array}{r}x\end{array}$  and 4 in for  $\begin{array}{r}y\end{array}$  in the expression.
::首先,以 x 3 和 4 代替 y 在表达式中。

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Next, notice that this expression has division, addition, parentheses, subtraction, and exponents. According to the order of operations, you must start by simplifying the parts of the expression in parentheses that need to be simplified. Start by simplifying the  $\begin{array}{r}(9-3)\end{array}$  in the middle of the expression.
::接下来请注意, 此表达式有分割、 添加、 括号、 减号、 缩放和引号 。 根据操作顺序, 您必须首先简化括号中需要简化的表达式部分。 开始简化表达式中间的( 9 - 3 ) 。

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Next, continue to follow the order of operations and evaluate the exponent.
::其次,继续按行动顺序进行,并评估出处。

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Now, divide. Remember that a  fraction bar  means division so  $\begin{array}{r}\frac{24}{3}\end{array}$  is the same as  $\begin{array}{r}24÷3\end{array}$ .
::现在,分裂。记住分数条意味着分数, 所以24 3和24 Q 3相同。

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Finally, add from left to right.
::最后,加上从左到右。

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The answer is 30.
::答案是30岁

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Example 5
::例5

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Evaluate  $\begin{array}{r}5x^2-2+(3+3)\end{array}$  if  $\begin{array}{r}x\end{array}$  is 5.
::如果x为5,评价5 x 2 - 2 + (3+ 3) + (3+ 3)

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First, substitute 5 in for the  $\begin{array}{r}x\end{array}$  in the expression.
::首先,将X改为5。

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Next, notice that this expression has multiplication, exponents, subtraction, parentheses, and addition. According to the order of operations, you must start by simplifying the parts of the expression in parentheses that need to be simplified. Start by simplifying the  $\begin{array}{r}(3+3)\end{array}$  at the end of the expression.
::接下来请注意, 此表达式有乘法、 引号、 减法、 括号以及添加。 根据操作顺序, 您必须首先简化括号中需要简化的表达式部分。 开始简化表达式结尾处的( 3+ 3) 。

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Next, continue to follow the order of operations and evaluate the exponent.
::其次,继续按行动顺序进行,并评估出处。

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Now, multiply.
::现在,乘数。

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Finally, add and subtract from left to right. You will subtract first since that comes first in the expression.
::最后,从左向右增减。自此之后,您将首先减减。

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The answer is 129.
::答案是129个。

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Review
::回顾

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Evaluate the following variable expressions for  $\begin{array}{r}x=4,y=2,z=3\end{array}$ .
::评估 x = 4, y = 2, z = 3 的下列变量表达式。

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1. $\begin{array}{r}{x}^2+y\end{array}$
   ::2 + yx 2 + y
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2. $\begin{array}{r}2y^2+5-2\end{array}$  
   ::2 y 2 + 5 - 2
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3. $\begin{array}{r}{x}^2-y^2+z\end{array}$  
   ::x 2 - y 2 + zx 2 - y 2 + z
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4. $\begin{array}{r}3x^2+2x^2\end{array}$  
   ::3 x 2 + 2 x 2
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5. $\begin{array}{r}8+x^2-4y\end{array}$  
   ::8 + x 2 - 4 y
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6. $\begin{array}{r}14÷2+z^2-y\end{array}$  
   ::14 + 2 + z 2 - y
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7. $\begin{array}{r}20+z^2-y\end{array}$  
   ::20 + z 2 - y
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8. $\begin{array}{r}5x-2y+3z\end{array}$  
   ::5 x - 2 y + 3 z
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9. $\begin{array}{r}5+(x-z)+5(6)\end{array}$  
   ::5+(x-z)+5(6)
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10. $\begin{array}{r}8+x-y^2+z\end{array}$  
    ::8 + x = y 2 + z 8 + x = y 2 + z
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11. $\begin{array}{r}(x+y)+5\cdot 2-3\end{array}$  
    :x + y ) + 5 + 2 - 3
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12. $\begin{array}{r}4x^2+3z^3+2\end{array}$  
    ::4 x 2 + 3 z 3 + 2

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Decide whether each of the following statements are true or false.
::决定下列声明是否属实。

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13. Parentheses are a grouping symbol.
    ::括号是一个组合符号。
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14. Exponents can't be evaluated unless the exponent is equal to 3.
    ::除非指数等于3,否则无法评价指数。
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15. If there is multiplication and division in a problem you always do the multiplication first.
    ::如果问题中存在乘法和分法,您总是首先进行乘法和分法。

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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 。

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Resources
::资源