课程： 1.4 行动次序 | The Way To Learn

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行动命令令
Substitution and the Order of Operations
::替代和作业命令

Consider this expression : $\begin{aligned} a + (b \times c) - 1 \end{aligned}$ . Suppose you were asked to simplify this expression, given $\begin{aligned} a = 2, b = 4, and c = 7. \end{aligned}$ This is a problem requiring substitution . To simplify the expression, the first thing you need to do is substitute the values you are given into the expression, replacing the variables (the letters). It may help to put each substituted value in " data-term="Parentheses" role="term" tabindex="0"> parentheses , just to keep things organized. This will be particularly useful in other problems, if the given value happens to be negative.
::考虑此表达式 : a+ (bxxc) - 1. 假设您被要求简化此表达式, 给 a=2, b=4, 和 c=7 。这是一个需要替换的问题。要简化表达式, 您首先需要做的是替换您在表达式中给出的值, 替换变量( 字母) 。它可以帮助将每个替代值放在括号中, 仅保持排列。如果给定值为负值, 这对于其它问题特别有用。

$\begin{aligned} a + (b \times c) - 1 & given a = 2, b = 4, c = 7 \\ (2) + [(4) \times (7)] - 1 & Replace the variables with numbers \\ 2 + (4 \times 7) - 1 \end{aligned}$

::a+(bxxc)--1given a=2, b=4, c=7(2)+[(4)x(7)]-1 将变量替换为2+(4x7)-1

Now evaluate the expression inside the parentheses: $\begin{aligned} 4 \times 7 = 28. \end{aligned}$ The expression becomes:
::现在评价括号中的表达式: 4x7=28。表达式改为:

$\begin{aligned} 2 + (28) - 1 \end{aligned}$

Now you have only addition and subtraction . Start at the left and work across:
::现在您只有附加和减法。从左侧开始, 并交叉工作 :

$\begin{aligned} 2 + 28 - 1 \\ = 30 - 1 \\ = 29 \end{aligned}$

Many people use the word “PEMDAS” to help remember the priority order of the mathematical operations : P arentheses, E xponents, M ultiplication and D ivision, A ddition and S ubtraction. Just remember that multiplication and division have the same priority, and so do addition and subtraction, so those operations should be completed left to right.
::许多人使用“PEMDAS”一词来帮助人们记住数学操作的优先顺序:括号、指数、乘法和除法、加法和减法。只要记住乘法和除法具有同样的优先,加法和减法也具有同样的优先,所以这些操作应该从右到右完成。

Order of Operations
::行动命令令

First evaluate expressions within Parentheses, including brackets [ ] and braces { }.
::括号内的第一次评价表达式,包括括号 [ ] 和括号 {}。

Next evaluate all Exponents ( terms such as $\begin{aligned} 3^{2} \end{aligned}$ or $\begin{aligned} x^{3} \end{aligned}$ ).
::下次评估所有指数(术语如32或x3) 。

Then Multiplication and Division - work from left to right completing both multiplication and division in the order that they appear.
::然后是乘法和除法 -- -- 从左到右的工作按其出现的顺序完成乘法和除法。

Finally, evaluate Addition and Subtraction - work from left to right completing both addition and subtraction in the order that they appear.
::最后,评估增加和减法----从左到右的工作,按其出现的顺序完成增加和减法。

Evaluating Expressions
::评价表达式

Each of the three expressions below, a), b), and c), has the same numbers and the same mathematical operations in the same order. The placement of the grouping symbols is not the same, which means that we must evaluate everything in a different order each time. Consider the effect of the parentheses in each example.
::下面的三个表达式中,a、b、c)的每个表达式都有相同的数字和相同的数学操作。组合符号的位置不同,这意味着我们每次必须按不同的顺序来评估一切。请考虑每个示例中括号的效果。

a) $\begin{aligned} 4 - 7 - 11 + 2 \end{aligned}$
:a) 4-7-11+2

This expression doesn't have parentheses, exponents, multiplication, or division. Treat addition and subtraction as they appear, starting at the left and working right (NOT completing all of the addition then all of the subtraction).
::此表达式没有括号、引号、乘法或除法。从左侧和工作右侧开始, 处理附加和减法。 (NOT 完成所有加法, 然后全部减法 )

$\begin{aligned} 4 - 7 - 11 + 2 \\ = - 3 - 11 + 2 \\ = - 14 + 2 \\ = - 12 \end{aligned}$

b) $\begin{aligned} 4 - (7 - 11) + 2 \end{aligned}$
:b) 4-(7-11)+2

This expression has parentheses, so first evaluate $\begin{aligned} 7 - 11 = - 4 \end{aligned}$ . Remember that subtracting a negative is equivalent to adding a positive:
::此表达式有括号, 所以首先评价 7- 11= 4。请记住, 减去负等于添加正数 :

$\begin{aligned} 4 - (7 - 11) + 2 \\ = 4 - - 4 + 2 \\ = 8 + 2 \\ = 10 \end{aligned}$

c) $\begin{aligned} 4 - [7 - (11 + 2)] \end{aligned}$
:c) 4-[7-(11+2)

An expression can contain any number of sets of parentheses. Sometimes expressions will have sets of parentheses inside other sets of parentheses. When faced with these nested parentheses , start at the innermost parentheses and work outward.
::表达式可以包含任意数组括号。有时表达式会在其他括号内包含数组括号。当面对这些嵌套括号时, 从括号内部开始, 向外工作。

Brackets are commonly used to group expressions already containing parentheses. This expression has both brackets and parentheses. Start with the innermost group: $\begin{aligned} 11 + 2 = 13 \end{aligned}$ . Then complete the resulting operation in the brackets.
::括号通常用于已经包含括号的表达式组。此表达式有括号, 有括号, 有括号, 也有括号。从最核心组组开始 : 11+2=13。然后在括号中完成相应的操作。

$\begin{aligned} 4 - [7 - (11 + 2)] \\ = 4 - [7 - 13] \\ = 4 - - 6 \\ = 10 \end{aligned}$

Evaluating Expressions with Variables
::带有变量的评价表达式

Use the order of operations to evaluate the following:
::使用操作顺序来评价:

a) $\begin{aligned} 2 - (3 x + 2) \end{aligned}$ when $\begin{aligned} x = 2 \end{aligned}$
:a) x=2 时 2 - (3x+2)

The first step is to substitute the value for $\begin{aligned} x \end{aligned}$ into the expression. It may help to put the substituted value in parentheses to clarify the resulting expression.
::第一步是将 x 的值替换为表达式。它可能有助于将替代值置于括号中,以澄清由此产生的表达式。

$\begin{aligned} 2 - [3 (2) + 2] \end{aligned}$

(Note: $\begin{aligned} 3 (2) \end{aligned}$ is the same as $\begin{aligned} 3 \times 2 \end{aligned}$ .)
:注:3(2)与3×2相同。 )

Follow PEMDAS : parentheses are first. Inside the parentheses, follow PEMDAS again.
::紧随PEMDAS:括号是第一位的。括号中,请再次跟随PEMDAS。

$\begin{aligned} 2 - (3 \times 2 + 2) Inside the parentheses, multiply first. \\ = 2 - (6 + 2) Next, add inside the parentheses. \\ = 2 - 8 Finally, subtract. \\ = - 6 \end{aligned}$

::2 - (3x2+2) 括号内括号内,先乘2=2 - (6+2) 下一个,在括号内增加。=2-8 最后,减去 =6

b) $\begin{aligned} 3 y^{2} + 2 y + 1 \end{aligned}$ when $\begin{aligned} y = - 3 \end{aligned}$
:b) y=-3时为3y2+2y+1

The first step is to substitute the value for $\begin{aligned} y \end{aligned}$ into the expression. Leave the parentheses around the negative numbers to clarify the problem. Since there are no operations within the parentheses they will not affect the order of operations, but will help avoid confusion when multiplying the negative numbers.
::第一步是将 y 的值替换为表达式。留下负数周围的括号来澄清问题。由于括号内没有操作, 它们不会影响操作顺序, 但是在乘以负数时会帮助避免混乱。

$\begin{aligned} 3 \times (- 3)^{2} + 2 \times (- 3) + 1 \end{aligned}$

Follow PEMDAS : we cannot simplify the expressions in parentheses, so exponents come next.
::跟随 PEMDAS : 我们无法简化括号中的表达式, 因此引言者会接着来。

$\begin{aligned} 3 \times (- 3)^{2} + 2 \times - 3 + 1 Evaluate exponents: (- 3)^{2} = 9 \\ = 3 \times 9 + 2 \times - 3 + 1 Evaluate multiplication: 3 \times 9 = 27; 2 \times - 3 = - 6 \\ = 27 + - 6 + 1 Add and subtract from left to right \\ = 27 - 6 + 1 \\ = 22 \end{aligned}$

::3x(-3)2+2+2x-3+1 评价指数-3) 2=9=3x9+2x-3+1 评价乘数: 3x9=27; 2x-3=6=27+6+1 添加并从左到右减去=27-6+1=22

Technology Note: Graphing Calculators
::技术注释:图表计算器

A graphing calculator is a very useful tool when evaluating algebraic expressions. Like a scientific calculator, a graphing calculator follows PEMDAS .
::在评估代数表达式时,图形计算器是一种非常有用的工具。像科学计算器一样,图形计算器跟随PEMDAS。

Evaluate $\begin{aligned} [3 (x^{2} - 1)^{2} - x^{4} + 12] + 5 x^{3} - 1 \end{aligned}$ when $\begin{aligned} x = - 3 \end{aligned}$ .
::评价[[(x2-1)2-x4+12]+5x3-1]x=3。

Method 1: Substitute for the variable first. Then evaluate the numerical expression with the calculator.
::方法1:首先替代变量。然后用计算器来评价数字表达式。

Substitute the value $\begin{aligned} x = - 3 \end{aligned}$ into the expression.
::将 x=3 值替换为表达式。

$\begin{aligned} [3 ((- 3)^{2} - 1)^{2} - (- 3)^{4} + 12] + 5 (- 3)^{3} - 1 \end{aligned}$

Type this into the calculator just as it is and press [ENTER] . (Note: use $\begin{aligned} \land \end{aligned}$ to enter exponents)
::将此输入计算器, 按 [ENTER] 键即可。 (注意: 使用 {} 来输入引号)

The answer is -13.
::答案是 -13

Method 2: Type the original expression into the calculator first and then evaluate.
::方法2:先将原表达式输入计算器,然后进行评估。

First, store the value $\begin{aligned} x = - 3 \end{aligned}$ in the calculator. Type -3 [STO] $\begin{aligned} x \end{aligned}$ (The letter $\begin{aligned} x \end{aligned}$ can be entered using the $\begin{aligned} x - \end{aligned}$ [VAR] button or [ALPHA] + [STO] ). Then type the original expression in the calculator and press [ENTER] .
::首先, 在计算器中存储值 x3 。类型 - 3 [STO] x( 字母 x 可以使用 x- [VAR] 按钮或 [ALPHA] + [STO] 输入。然后键入计算器和按 [ENTER] 中的原始表达式。

The answer is -13.
::答案是 -13

The second method is better because you can easily evaluate the same expression for any value you want. For example, evaluate the same expression using the values $\begin{aligned} x = 2 \end{aligned}$ and $\begin{aligned} x = \frac{2}{3} : \end{aligned}$
::第二种方法是更好的,因为您可以方便地对任意值的相同表达式进行评价。例如,使用 x=2 和 x=23 来评价相同的表达式:

For $\begin{aligned} x = 2 \end{aligned}$ , store the value of $\begin{aligned} x \end{aligned}$ in the calculator: $\begin{aligned} 2 \end{aligned}$ [STO] $\begin{aligned} x \end{aligned}$ . Press [2nd] [ENTER] twice to get the previous expression you typed in on the screen without having to enter it again. Press [ENTER] to evaluate the expression.
::对于 x=2, 在计算器中存储 x 的值 : 2 [STO]x. 按 [ 2 [ ENTER] 两次, 以获得屏幕上您输入的前一个表达式, 而不必再输入它。按 [ENTER] 来评价表达式。

The answer is 62.
::答案是62。

For $\begin{aligned} x = \frac{2}{3} \end{aligned}$ , store the value of $\begin{aligned} x \end{aligned}$ in the calculator: $\begin{aligned} \frac{2}{3} \end{aligned}$ [STO] $\begin{aligned} x \end{aligned}$ . Press [2nd] [ENTER] twice to get the expression on the screen without having to enter it again. Press [ENTER] to evaluate.
::对于 x=23, 在计算器中存储 x 的值 : 23 [STO]x。按 [ 2 [ ENTER] 两次, 在屏幕上获取表达式, 不必再输入它。按 [ENTER] 进行评审。

The answer is 13.21, or $\begin{aligned} \frac{1070}{81} \end{aligned}$ in fraction form.
::回答是13.21,或107081分数。

Note: On graphing calculators there is a difference between the minus sign and the negative sign. When you stored the value 'negative three', you needed to use the negative sign, which is to the left of the [ENTER] button on the calculator. On the other hand, to perform the subtraction operation in the expression you use the minus sign. The minus sign is directly above the plus sign on the right.
::注意 : 在图形计算器中, 负号与负号之间有差。当存储值“ 负3 ” 时, 您需要使用负号, 即计算器上[ 按钮左侧的负号。另一方面, 要在表达式中执行减号操作, 您则使用减号。减号直接高于右侧的加号。

Examples
::实例

Use the order of operations to evaluate the following:
::使用操作顺序来评价:

Example 1
::例1

$\begin{aligned} (3 \times 5) - (7 \div 2) \end{aligned}$

First evaluate the expressions inside parentheses: $\begin{aligned} 3 \times 5 = 15 \end{aligned}$ and $\begin{aligned} 7 \div 2 = 3.5 \end{aligned}$ . Then work from left to right:
::首先评价括号内的表达式: 3x5=15 和 72=3.5。然后从左到右工作 :

$\begin{aligned} (3 \times 5) - (7 \div 2) \\ = 15 - 3.5 \\ = 11.5 \end{aligned}$

Note that adding parentheses didn’t change the expression in part c, but did make it easier to read. Parentheses can be used to change the order of operations in an expression, but they can also be used simply to make it easier to understand.
::请注意,加括号并没有改变C部分的表达方式,但确实使阅读更加容易。括号可以用来改变一个表达式的操作顺序,但也可以简单地用来使人们更容易理解。

Example 2
::例2

$\begin{aligned} 2 - (t - 7)^{2} \times (u^{3} - v) \end{aligned}$ when $\begin{aligned} t = 19, u = 4, \end{aligned}$ and $\begin{aligned} v = 2 \end{aligned}$
::2-(t-7)2x(u3-v) t=19, u=4, v=2

The first step is to substitute the values for $\begin{aligned} t, u, \end{aligned}$ and $\begin{aligned} v \end{aligned}$ into the expression.
::第一步是将 t、 u 和 v 的值替换为表达式。

$\begin{aligned} 2 - (19 - 7)^{2} \times (4^{3} - 2) \end{aligned}$

Follow PEMDAS :
::跟随PEMDAS:

$\begin{aligned} 2 - (19 - 7)^{2} \times (4^{3} - 2) & First follow PEMDAS within parentheses: \\ (19 - 7) = 12 and (4^{3} - 2) = (64 - 2) = 62 \\ = 2 - 12^{2} \times 62 & Evaluate exponents: 12^{2} = 144 \\ = 2 - 144 \times 62 & Multiply: 144 \times 62 = 8928 \\ = 2 - 8928 & Subtract. \\ = - 8926 \end{aligned}$

::2-(19-7)2x(43-2)首先在括号内跟随PEMDAS19-7)=12和(43-2)=(64-2)=(64-2)=62=2-122×62Evaluate exporates:122=144=2-144×62Blitly:144×62=8928=2-8928Text.=-8926

Notice that even when an expression is inside parentheses, the order of operations still applies.
::请注意,即使括号内有表达式,操作顺序仍然适用。

Review
::回顾

Evaluate the following expressions involving variables.

$\begin{aligned} 2 y^{2} \end{aligned}$ when $\begin{aligned} x = 1 \end{aligned}$ and $\begin{aligned} y = 5 \end{aligned}$
::x=1和y=5时 2y2

$\begin{aligned} 3 x^{2} + 2 x + 1 \end{aligned}$ when $\begin{aligned} x = 5 \end{aligned}$
::x=5 时 3x2+2x+1

::评估下列变量的表达式。当 x=1 和 y= 5 3x2+2x+1 时, 当 x=5 时, 2y2 和 y= 5 3x2+2x+1 时, 2y2 和 y=5 3x2+2x+1

Use the order of operations to evaluate the expression.

$\begin{aligned} 2 + 7 \times 11 - 12 \div 3 \end{aligned}$

::使用操作顺序来评价表达式。 2+7x11- 123

Evaluate the expression by substituting for the variables.

$\begin{aligned} (y^{2} - x)^{2} \end{aligned}$ when $\begin{aligned} x = 2 \end{aligned}$ and $\begin{aligned} y = 1 \end{aligned}$
:y2-x) 2 当 x=2 和 Y=1 时

::替换变量来评价表达式。 x=2 和 y=1 时为 (y2- x) 2

For 4-6, use the order of operations to evaluate the expressions.
::4-6,使用操作顺序来评价表达式。

$\begin{aligned} 8 - (19 - (2 + 5) - 7) \end{aligned}$

$\begin{aligned} (3 + 7) \div (7 - 12) \end{aligned}$

$\begin{aligned} (4 - 1)^{2} + 3^{2} \cdot 2 \end{aligned}$

For 7-10, insert parentheses in each expression to make a true equation.
::7-10时,在每个表达式中插入括号,以形成一个真实的等式。

$\begin{aligned} 5 - 2 \times 6 - 5 + 2 = 5 \end{aligned}$

$\begin{aligned} 12 \div 4 + 10 - 3 \times 3 + 7 = 31 \end{aligned}$

$\begin{aligned} 22 - 32 - 5 \times 3 - 5 = 30 \end{aligned}$

$\begin{aligned} 12 - 8 - 4 \times 5 = - 8 \end{aligned}$

For 11-12, evaluate each expression using a graphing calculator.
::对于 11-12, 使用图形计算器对每个表达式进行评价。

$\begin{aligned} x^{2} + 2 x - x y \end{aligned}$ when $\begin{aligned} x = 250 \end{aligned}$ and $\begin{aligned} y = - 120 \end{aligned}$
::x2+2x-xy x=250和y=-120时

$\begin{aligned} (x y - y^{4})^{2} \end{aligned}$ when $\begin{aligned} x = 0.02 \end{aligned}$ and $\begin{aligned} y = - 0.025 \end{aligned}$
:xy-y4) 2x=0.02和y=- 0.025时

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。

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