Course: 1.7 利用行动命令评估权力

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利用行动命令评估权力
Shirley is a math tutor and is working with middle-schooler Marcus once a week. One day, Marcus arrived with a problem that not even Shirley knew how to answer.
::雪莉是数学老师,每周和中学生马库斯合作一次。有一天,马库斯遇到一个连雪莉都不知道如何回答的问题。

$\begin{aligned} (- 11)^{2} + 7 y^{2} + 3 x - 19 for x = 2 and y = - 1 \end{aligned}$

:- 11) 2+7y2+3x-19 x=2 和y1

How can Shirley and Marcus work out the correct answer to this problem? Are there any rules they need to follow?
::雪莉和马库斯如何解决这个问题?

In this concept, you will learn to evaluate variable expressions with powers by applying the .
::在此概念中,您将学会通过应用来用权力来评价变量表达式。

PEMDAS
::PEMDAS 话语

The order of operations is a necessary tool to use when evaluating expressions. Remember the order in which the mathematical operations are performed affect your answer. You can recall the order of operations using the shorthand form PEMDAS .
::操作顺序是评价表达式时使用的一种必要工具。记住执行数学操作的顺序会影响您的回答。您可以使用 PEMDAS 的速记格式提醒操作顺序。

P " data-term="Parentheses" role="term" tabindex="0"> parentheses or grouping symbols
::P 括号或分组符号

E exponents
::发记记记记记记记记记记记记记记记记记记记记记记记记记记记记记记记记记记记记记记记记记记记记记记

MD multiplication and division in order from left to right
::从左到右的乘数和除法

AS addition and subtraction in order from left to right
::按顺序从左到右加上和减法

Let’s look at an example.
::让我们举个例子。

Evaluate the following variable expression .
::评估以下变量表达式。

$\begin{aligned} 8 h^{2} + [51 \div (4 \cdot 4.25)] - 5^{2} \div 5 \end{aligned}$ when $\begin{aligned} h = 4 \end{aligned}$
::8h2+[51(44.25)]-525时=4

First, substitute the value $\begin{aligned} h = 4 \end{aligned}$ into the expression .
::首先,将值 h=4 替换为表达式。

$\begin{aligned} 8 (4)^{2} + [51 \div (4 \cdot 4.25)] - 5^{2} \div 5 \end{aligned}$
Next, expand the power: $\begin{aligned} (4)^{2} = 4 \times 4 \end{aligned}$ and write the new expression.
::8(4)2+[51{(44.25)]-525 下一步,扩展功率4)2=4×4并写入新表达式。

$\begin{aligned} 8 (4 \times 4) + [51 \div (4 \cdot 4.25)] - (5 \times 5) \div 5 \end{aligned}$

Remember $\begin{aligned} - 5^{2} = - (5 \times 5) \end{aligned}$ since the base is 5. However, $\begin{aligned} (- 5)^{2} = (- 5 \times - 5) \end{aligned}$ since the base is $\begin{aligned} - 5 \end{aligned}$ .
::记住 -52(5×5),因为基数是5. 但是,(-5)2=(-5)5,因为基数是-5。

Next, perform the operations inside the grouping symbols.
::接下来,在分组符号内执行操作。

First, multiply: $\begin{aligned} (4 \cdot 4.25) = 17 \end{aligned}$ and write the new expression.
::首先,乘以: (44.25) = 17, 并写新表达式。

$\begin{aligned} 8 (4 \times 4) + [51 \div 17] - (5 \times 5) \div 5 \end{aligned}$

Then, divide: $\begin{aligned} 51 \div 17 = 3 \end{aligned}$ and write the new expression.
::然后,分开: 5117=3, 并写下新的表达式。

$\begin{aligned} 8 (4 \times 4) + 3 - (5 \times 5) \div 5 \end{aligned}$

Next, multiply: $\begin{aligned} (4 \times 4) = 16 \end{aligned}$ and write the new expression.
::下一个, 乘以 : (4x4) = 16, 并写入新表达式。

$\begin{aligned} 8 (16) + 3 - (5 \times 5) \div 5 \end{aligned}$

Then, multiply: $\begin{aligned} (5 \times 5) = 25 \end{aligned}$ and write the new expression.
::然后,乘以: (5x5) =25 然后写新表达式。

$\begin{aligned} 8 (16) + 3 - 25 \div 5 \end{aligned}$

Then, multiply: $\begin{aligned} 8 (16) = 128 \end{aligned}$ to clear the parenthesis. Write the new expression.
::然后,乘以: 8(16) = 128 来清除括号。写入新的表达式。

$\begin{aligned} 128 + 3 - 25 \div 5 \end{aligned}$

Next, divide: $\begin{aligned} 25 \div 5 = 5 \end{aligned}$ and write the new expression.
::下一个分隔: 255=5, 并写下新表达式。

$\begin{aligned} 128 + 3 - 5 \end{aligned}$

Next, add: $\begin{aligned} 128 + 3 = 131 \end{aligned}$ and write the new expression.
::下加: 128+3=131, 并写下新表达式。

$\begin{aligned} 131 - 5 \end{aligned}$

Then, subtract:
::然后,减去:

$\begin{aligned} 131 - 5 = 126 \end{aligned}$

The answer is 126.
::答案是126个

Let’s look at one more example. This example will be a variable expression with two different variables. There will be a value given for each of the variables. You simply substitute the given values for each variable into the expression and evaluate the new numerical expression.
::让我们再看看一个示例。这个示例将是一个变量表达式, 有两个不同的变量。每个变量都将有一个给定值。您只需在表达式中替换每个变量的给定值, 并评估新的数字表达式。

Evaluate the following variable expression:
::评估以下变量表达式:

$\begin{aligned} 4 x^{3} - (3 y \div 9) + 12 \end{aligned}$ when $\begin{aligned} x = 3 \end{aligned}$ and $\begin{aligned} y = 3 \end{aligned}$
::4x3 - (3y9)+12 x=3和y=3

First, substitute $\begin{aligned} x = 3 \end{aligned}$ and $\begin{aligned} y = 9 \end{aligned}$ into the expression. Write the new expression.
::首先,将 x=3 和 y=9 替换为表达式。写入新的表达式。

$\begin{aligned} 4 (3)^{3} - (3 \cdot 9 \div 9) + 12 \end{aligned}$

Next, expand the power: $\begin{aligned} (3)^{3} = (3 \times 3 \times 3) \end{aligned}$ . Write the new expression.
::下一步, 扩展功率 : (3)3 = (3x3x3x3) 。写入新表达式。

$\begin{aligned} 4 (3 \times 3 \times 3) - (3 \cdot 9 \div 9) + 12 \end{aligned}$

Then, perform the operations inside the parenthesis.
::然后在括号内执行操作

First, multiply: $\begin{aligned} 3 \cdot 9 = 27 \end{aligned}$ and write the new expression.
::首先,乘以: 39=27 并写下新表达式。

$\begin{aligned} 4 (3 \times 3 \times 3) - (27 \div 9) + 12 \end{aligned}$

Then, divide: $\begin{aligned} 27 \div 9 = 3 \end{aligned}$ and write the new expression.
::然后,分开: 279=3, 并写下新的表达式。

$\begin{aligned} 4 (3 \times 3 \times 3) - 3 + 12 \end{aligned}$

Next, multiply: $\begin{aligned} (3 \times 3 \times 3) = 27 \end{aligned}$ and write the new expression.
::下一个, 乘数 : (3x3x3) = 27 并写入新表达式。

$\begin{aligned} 4 (27) - 3 + 12 \end{aligned}$

Then, multiply: $\begin{aligned} 4 (27) = 108 \end{aligned}$ to clear the parenthesis. Write the new expression.
::然后,乘以: 4(27) = 108 来清除括号。写入新的表达式。

$\begin{aligned} 108 - 3 + 12 \end{aligned}$

Next, subtract: $\begin{aligned} 108 - 3 = 5 \end{aligned}$ and write the new expression.
::下一步,减去:108-3=5,并写下新表达式。

$\begin{aligned} 105 + 12 \end{aligned}$

Then, add:
::然后,加上:

$\begin{aligned} 105 + 12 = 117 \end{aligned}$

The answer is 117.
::答案是117。

When you have powers included in variable and numerical expressions, you must use the order of operations to evaluate the expressions. No matter how complicated the problem seems to be, applying the order of operations as defined by PEMDAS will guide you to a correct answer as you evaluate the expression.
::当您在变量和数字表达式中有权力时,必须使用操作顺序来评价表达式。不管问题看起来有多复杂,应用 PEMDAS 定义的操作顺序将指导您在评估表达式时找到正确的答案。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Shirley and Marcus and their brain-bending problem.
::早些时候,你得到一个问题关于雪莉和马库斯以及他们的大脑制造问题。

They need to figure out the correct answer to the following problem.
::他们需要找出对以下问题的正确答案。

$\begin{aligned} (- 11)^{2} + 7 y^{2} + 3 x - 19 \end{aligned}$ for $\begin{aligned} x = 2 \end{aligned}$ and $\begin{aligned} y = - 1 \end{aligned}$
:- 11) 2+7y2+3x-19 x=2和y1

Shirley and Marcus must apply the order of operations to evaluate the problem correctly.
::雪莉和马库斯必须使用操作顺序来正确评估问题。

First, substitute $\begin{aligned} x = 2 \end{aligned}$ and $\begin{aligned} y = - 1 \end{aligned}$ into the expression.
::首先,在表达式中替换 x=2andy1。

$\begin{aligned} (- 11)^{2} + 7 (- 1)^{2} + 3 (2) - 19 \end{aligned}$

Next, expand the powers.
::下一个,扩大权力。

$\begin{aligned} (- 11 \times - 11) + 7 (- 1 \times - 1) + 3 (2) - 19 \end{aligned}$

Next, do the multiplication inside the parentheses.
::接下来,在括号内进行乘法。

$\begin{aligned} 121 + 7 (1) + 3 (2) - 19 \end{aligned}$

Next, do the multiplication to clear the parentheses.
::下一步,做乘法来清除括号。

$\begin{aligned} 121 + 7 + 6 - 19 \end{aligned}$

Then, in order from left to right, perform the addition and subtraction.
::然后,按照从左到右的顺序,执行增加和减法。

$\begin{aligned} 121 + 7 = 128 + 6 = 134 - 19 = 115 \end{aligned}$

The answer is 115.
::答案是115

Example 2
::例2

Evaluate the following variable expression:
::评估以下变量表达式:

$\begin{aligned} 2^{3} + 4 y + 12 \end{aligned}$ when $\begin{aligned} y = 3 \end{aligned}$ .

First, substitute $\begin{aligned} y = 3 \end{aligned}$ into the variable expression.

$\begin{aligned} 2^{3} + 4 (3) + 12 \end{aligned}$

Next, expand the power: $\begin{aligned} 2^{3} = 2 \times 2 \times 2 \end{aligned}$ and write the new expression.

$\begin{aligned} 2 \times 2 \times 2 + 4 (3) + 12 \end{aligned}$

Next, multiply: $\begin{aligned} 4 (3) = 12 \end{aligned}$ to clear the parenthesis. Write the new expression.

$\begin{aligned} 2 \times 2 \times 2 + 12 + 12 \end{aligned}$

Next, multiply: $\begin{aligned} 2 \times 2 \times 2 = 8 \end{aligned}$ and write the new expression.
::下一个, 乘以: 2x2x2x2=8, 并写入新表达式。

$\begin{aligned} 8 + 12 + 12 \end{aligned}$

Then, add:
::然后,加上:

$\begin{aligned} 8 + 12 = 20 + 12 = 32 \end{aligned}$

The answer is 32.
::答案是32个

Example 3
::例3

Evaluate the following variable expression.
::评估以下变量表达式。

$\begin{aligned} - 5^{3} + 7 y - 30 \end{aligned}$ when $\begin{aligned} y = 9 \end{aligned}$ .
::- 53+7y-30时=9。

First, substitute $\begin{aligned} y = 9 \end{aligned}$ into the variable expression.
::首先,将 Y=9 替换为变量表达式。

$\begin{aligned} - 5^{3} + 7 (9) - 30 \end{aligned}$

Next, expand the power: $\begin{aligned} 5^{3} = 5 \times 5 \times 5 \end{aligned}$ and write the new expression.
::下一步, 扩展电源: 53=5x5x5, 并写入新表达式。

$\begin{aligned} - 5 \times 5 \times 5 + 7 (9) - 30 \end{aligned}$

Next, multiply: $\begin{aligned} 7 (9) = 63 \end{aligned}$ to clear the parenthesis. Write the new expression.
::下一个, 乘: 7( 9) = 63 来清除括号。写入新表达式。

$\begin{aligned} - 5 \times 5 \times 5 + 63 - 30 \end{aligned}$

Next, multiply: $\begin{aligned} 5 \times 5 \times 5 = 125 \end{aligned}$ and write the new expression.
::下一个, 乘以: 5x5x5=125, 并写入新表达式。

$\begin{aligned} - 125 + 63 - 30 \end{aligned}$

Then, subtract: $\begin{aligned} - 125 + 63 = - 62 \end{aligned}$ and write the new expression.

$\begin{aligned} - 62 - 30 \end{aligned}$

Then, add:
::然后,加上:

$\begin{aligned} - 62 - 30 = - 92 \end{aligned}$

The answer is -92.
::- 62 -30 -92 答案是 -92

Example 4
::例4

Evaluate the following variable expression:
::评估以下变量表达式:

$\begin{aligned} 6 x + 7 y + 3^{2} \end{aligned}$ when $\begin{aligned} x = 4 \end{aligned}$ and $\begin{aligned} y = 6 \end{aligned}$
::6x+7y+32 x=4andy=6

First, substitute $\begin{aligned} x = 4 \end{aligned}$ and $\begin{aligned} y = 6 \end{aligned}$ into the variable expression.
::首先,将 x=4andy=6 替换为变量表达式。

$\begin{aligned} 6 (4) + 7 (6) + 3^{2} \end{aligned}$

Next, expand the power: $\begin{aligned} 3^{2} = 3 \times 3 \end{aligned}$ and write the new expression.
::下一步, 扩展电源: 32=3x3, 并写入新表达式。

$\begin{aligned} 6 (4) + 7 (6) + 3 \times 3 \end{aligned}$

Next, multiply: $\begin{aligned} 6 (4) = 24 \end{aligned}$ to clear the parenthesis. Write the new expression.
::下一个, 乘: 6(4)=24 清除括号。写入新表达式。

$\begin{aligned} 24 + 7 (6) + 3 \times 3 \end{aligned}$

Next, multiply: $\begin{aligned} 7 (6) = 42 \end{aligned}$ to clear the parenthesis. Write the new expression.
::下一个, 乘: 7(6)=42 清除括号。写入新表达式。

$\begin{aligned} 24 + 42 + 3 \times 3 \end{aligned}$

Next multiply: $\begin{aligned} 3 \times 3 = 9 \end{aligned}$ and write the new expression.
::下一个乘数: 3x3=9, 并写入新表达式。

$\begin{aligned} 24 + 42 + 9 \end{aligned}$

Then, add:
::然后,加上:

$\begin{aligned} 24 + 42 = 66 + 9 = 75 \end{aligned}$

The answer is 75.
::答案是75

Review
::回顾

Evaluate each numerical expression. Remember to follow the order of operations.
::评估每个数字表达式。记住要遵循操作顺序。

1. $\begin{aligned} 3^{2} + [(5 \times 2) - 3] - 8 \times 2 \end{aligned}$

2. $\begin{aligned} 5^{2} + (3 + 5) - 6^{2} + 2 \end{aligned}$

3. $\begin{aligned} 6^{3} + 5^{2} + 25 \end{aligned}$

4. $\begin{aligned} 16 (12)^{3} \end{aligned}$

5. $\begin{aligned} 8^{2} - (2 (3^{3}) \div 2) + (16 \times 5) \end{aligned}$

Evaluate each variable expression by substituting the given value into each expression. Remember to follow the order of operations.
::通过将给定值替换为每个表达式来评价每个变量表达式。记住要遵循操作顺序。

6. $\begin{aligned} - 2^{3} + 7 y + 1 \end{aligned}$ for $\begin{aligned} y = 6 \end{aligned}$ .
::y=6 = 23+7y+1。

7. $\begin{aligned} - 12 + 7 x^{2} - 8 \end{aligned}$ for $\begin{aligned} x = 6 \end{aligned}$ .
::7. x=6 = 12+7x2-8。

8. $\begin{aligned} 14 + 7 y^{2} + 22 \end{aligned}$ for $\begin{aligned} y = 3 \end{aligned}$ .
::8. y=3, 14+7y2+22。

9. $\begin{aligned} 18 x + 7 y + 12 \end{aligned}$ for $\begin{aligned} x = 3, y = 6 \end{aligned}$ .
::9. x=3,y=6, 18x+7y+12

10. $\begin{aligned} - 6^{3} + 7 x^{2} - 18 \end{aligned}$ for $\begin{aligned} x = 5 \end{aligned}$ .
::10. - 63+7x2 - 18for x=5

11. $\begin{aligned} 45 + 8 y + 3^{3} \end{aligned}$ for $\begin{aligned} y = 5 \end{aligned}$ .
::11. y=5 45+8y+33。

12. $\begin{aligned} - 3^{3} + 8 x - 2^{2} \end{aligned}$ for $\begin{aligned} x = 7 \end{aligned}$ .
::12. - 33+8x-22x=7。

13. $\begin{aligned} (- 12)^{2} + 7 y - 4^{2} \end{aligned}$ for $\begin{aligned} y = 6 \end{aligned}$ .
::13. (-12)2+7y-42y-y=6。

14. $\begin{aligned} - 4^{3} + 9 x + 11 \end{aligned}$ for $\begin{aligned} x = 4 \end{aligned}$ .
::14. - 43+9x+11for x=4。

15. $\begin{aligned} (- 7)^{2} + 7 x^{2} + 12^{2} \end{aligned}$ for $\begin{aligned} y = 2 \end{aligned}$ .
::15. (-7)2+7x2+122y=2。

16. $\begin{aligned} - 45 + 7^{2} - x^{3} \end{aligned}$ for $\begin{aligned} x = 4 \end{aligned}$ .
::16. - 45+72-x3x=4。

Review (Answers)
::回顾(答复)

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。

Section outline

General

利用行动命令评估权力

PEMDAS ::PEMDAS 话语

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Review ::回顾

Review (Answers) ::回顾(答复)

PEMDAS
::PEMDAS 话语

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Review
::回顾

Review (Answers)
::回顾(答复)