Section: 用分组符号计算数值表达式 | 1.11用分组符号计算数值表达式

Section outline

Doug is collecting all the ribbon he can find in his house. He receives three 6 ft blue ribbon rolls from his mom, finds 4 ft of orange ribbon in a drawer, and grabs 7 ft of gold ribbon from his little sister, who then comes over with scissors and takes back 2 ft for herself. Doug sits at his desk and starts to measure all his collected ribbon with a measuring tape. Is there an easier way for Doug to figure out how many feet of ribbon he has?
::道格正在收集他家所有能找到的丝带。他从他妈妈那里收到三张6英尺长的蓝丝带卷,在抽屉里找到4英尺长的橙色丝带,从他妹妹那里抢走7英尺长的金丝带,然后她拿着剪刀过来,自己拿回两英尺。道格坐在桌子上,开始用一个测量带测量他收集的丝带。道格有比较容易的办法来弄清楚他有多少英尺长的丝带吗?

In this concept, you will learn how to evaluate numerical expressions using powers and grouping symbols .
::在此概念中,您将学习如何使用功率和组合符号来评估数字表达式。

Order of Operations
::行动命令令

" data-term="Parentheses" role="term" tabindex="0"> Parentheses are symbols that group things together. This becomes very important in numerical expressions, because operations inside parentheses are always completed first when evaluating the expression . Let’s review the .
::括号是将事物组合在一起的符号。这在数字表达式中变得非常重要, 因为括号内的操作在评价表达式时总是先完成的。让我们回顾一下。

Order of Operations
::行动命令令

P - parentheses
::P - 括号括号

E - exponents
::E. 推举

MD - multiplication or division , in order from left to right
::MD - 乘数或除法,按从左到右顺序排列

AS - addition or subtraction , in order from left to right
::AS - 增加或减减,按从左到右顺序排列

You can see that, according to the order of operations, parentheses come first.
::您可以看到,根据操作顺序,括号排在前面。

Let’s see how this works.
::让我们看看这如何运作。

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

In this problem, there are four elements to consider. You have one set of parentheses, addition, subtraction and multiplication. You can evaluate this expression using the order of operations. Here is what the process looks like:.
::在此问题上, 有四个元素需要考虑。您有一组括号、添加、减法和乘法。您可以使用操作顺序来评估这个表达式。以下是这个过程的外观 :

$\begin{aligned} 2 + (3 - 1) \times 2 \\ 2 + 2 \times 2 \\ 2 + 4 \\ = 6 \end{aligned}$

The answer is 6.
::答案是6个

Let's consider a different problem, and take it step by step:
::让我们考虑一个不同的问题,一步一步地去做:

$\begin{aligned} 35 + 3^{2} - (3 \times 2) \times 7 \end{aligned}$

First, evaluate parentheses.
::首先,评价括号。

$\begin{aligned} 35 + 3^{2} - 6 \times 7 \end{aligned}$

Next, evaluate exponents.
::下一步,评价表率。

$\begin{aligned} 35 + 9 - 6 \times 7 \end{aligned}$

Then, complete multiplication in order from left to right.
::然后从左向右完全乘以乘法

$\begin{aligned} 35 + 9 - 42 \end{aligned}$

Finally, complete addition and/or subtraction in order from left to right.
::最后,按从左到右的顺序完全增加和(或)减。

$\begin{aligned} 44 - 42 = 2 \end{aligned}$

The answer is 2.
::答案是2。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Doug and his ribbon pile.
::早些时候,你得到一个问题关于道格和他的丝带堆。

Doug needs to figure out the total length of ribbon he has collected.
::道格需要弄清楚他收集的丝带总长度

He receives 3 of the 6 ft blue ribbon rolls from his mom, finds 4 ft of orange ribbon in a drawer, and grabs 7 ft of gold ribbon from his little sister, who then comes over with scissors and takes back 2 ft for herself.
::他从他妈妈那里收到6英尺蓝丝带卷中的3张,在抽屉里发现4英尺橙色丝带,从他妹妹那里抢取7英尺金丝带,然后她拿着剪刀过来,为自己拿2英尺。

First, identify the important information.
::首先,确定重要信息。

receives 3 6 ft rolls
::接收 3 6 英尺

finds 4 ft
::发现4英尺

takes 7 ft
::7英尺(7英尺)

gives back 2 ft
::向后返回 2 英尺

Next, write this as an expression.
::下一个,写成表达式。

$\begin{aligned} 3 \times 6 + 4 + 7 - 2 \end{aligned}$

Then, use order of operations to evaluate the expression.
::然后,使用操作顺序来评价表达式。

$\begin{aligned} 3 \times 6 + 4 + 7 - 2 & Multiply 3 \times 6 = 18 \\ 18 + 4 + 7 - 2 & Add 18 + 4 = 22 \\ 22 + 7 - 2 & Add 22 + 7 = 29 \\ 29 - 2 & Subtract 29 - 2 = 27 \\ 27 \end{aligned}$

The answer is 27.
::答案是27个

Doug has collected 27 feet of ribbon with which to make mischief.
::道格收集了27英尺的丝带用来捣乱

Example 2
::例2

Evaluate the following expression.
::评估以下表述。

$\begin{aligned} 7^{3} - 3^{2} + 15 \times 2 + (2 + 3) \end{aligned}$

First, follow the order of operations and evaluate the parentheses and exponents.
::首先,按操作顺序排列,并评价括号和指数。

$\begin{aligned} 7^{3} = 7 \times 7 \times 7 = & 343 \\ 3^{2} = 3 \times 3 = & 9 \\ (2 + 3) = & 5 \end{aligned}$

Next, substitute these values back into the original number sentence.
::接下来,将这些值换回原数句。

$\begin{aligned} 343 - 9 + 15 \times 2 + 5 \end{aligned}$

Then, complete the multiplication.
::然后完成乘法

$\begin{aligned} 15 \times 2 = 30 \end{aligned}$

Finally, complete the addition and subtraction in order from left to right.
::最后,按照从左到右的顺序完成增减。

$\begin{aligned} 343 - 9 + 30 + 5 \end{aligned}$

$\begin{aligned} 369 \end{aligned}$

The answer is 369.
::答案是369

Example 3
::例3

Evaluate the following expression.
::评估以下表述。

$\begin{aligned} 16 + 2^{3} - 5 + (3 \times 4) \end{aligned}$

First, evaluate the operations inside of parenthesis.
::首先,评价括号内的操作。

$\begin{aligned} 16 + 2^{3} - 5 + (3 \times 4) & Evaluate 3 \times 4 = 12 \end{aligned}$

Next, evaluate the exponents.
::接下来,评估出名者。

$\begin{aligned} 16 + 2^{3} - 5 + 12 & Evaluate 2^{3} = 8 \\ 16 + 8 - 5 + 12 \end{aligned}$

Then, complete the addition and subtraction from left to right.
::然后,完成加法和减法,然后从左向右减法。

$\begin{aligned} 16 + 8 - 5 + 12 & Add 16 + 8 = 24 \\ 24 - 5 + 12 & Subtract 24 - 5 = 19 \\ 19 + 12 & Add 19 + 12 = 31 \\ 31 \end{aligned}$

The answer is 31.
::答案是31岁

Example 4
::例4

Evaluate the following expression.
::评估以下表述。

$\begin{aligned} 9^{2} + 2^{2} - 5 \times (2 + 3) \end{aligned}$
First, evaluate the parenthesis.

$\begin{aligned} 9^{2} + 2^{2} - 5 \times (2 + 3) & Evaluate 2 + 3 = 5 \\ 9^{2} + 2^{2} - 5 \times 5 \end{aligned}$

Next, evaluate the exponents.
::接下来,评估出名者。

$\begin{aligned} 9^{2} + 2^{2} - 5 \times 5 & Evaluate 9^{2} = 81 \\ 81 + 2^{2} - 5 \times 5 & Evaluate 2^{2} = 4 \\ 81 + 4 - 5 \times 5 \end{aligned}$

Then, multiply.
::然后,乘数。

$\begin{aligned} 81 + 4 - 5 \times 5 & Multiply 5 \times 5 = 25 \\ 81 + 4 - 25 \end{aligned}$

Finally, complete the addition and subtraction operations from left to right.
::最后,完成从左向右增减作业。

$\begin{aligned} 81 + 4 - 25 & Add 81 + 4 = 85 \\ 85 - 25 & Subtract 85 - 25 = 60 \\ 60 \end{aligned}$

The solution is 60.
::解决办法是60。

Example 5
::例5

Evaluate the following expression.
::评估以下表述。

$\begin{aligned} 8^{2} \div 2 + 4 - 1 \times 6 \end{aligned}$

First, evaluate the exponent .
::首先,评估前台。

$\begin{aligned} 8^{2} \div 2 + 4 - 1 \times 6 & Evaluate 8^{2} = 64 \\ 64 \div 2 + 4 - 1 \times 6 \end{aligned}$

Then, divide and multiply from left to right.
::然后,你们从左边到右边,各行各业,各行各业,各行各业,各行各业,各行各业,各行各业,各行各业,各行各业,各行各业,各行各业,各行各业,各行各业,各行各业,各行各业,各行各业。

$\begin{aligned} 64 \div 2 + 4 - 1 \times 6 & Divide 64 \div 2 = 32 \\ 32 + 4 - 1 \times 6 & Multiply 1 \times 6 = 6 \\ 32 + 4 - 6 \end{aligned}$
Next, add and subtract from left to right.

$\begin{aligned} 32 + 4 - 6 & Add 32 + 4 = 36 \\ 36 - 6 & Subtract 36 - 6 = 30 \\ 30 \end{aligned}$

The answer is 30.
::答案是30岁

Review
::回顾

Evaluate each expression according to the order of operations.
::根据行动顺序对每种表达方式进行评估。

$\begin{aligned} 3 + (2 + 7) - 3 + 5 = \underline{} \end{aligned}$

$\begin{aligned} 2 + (5 - 3) + 7^{2} - 11 = \underline{} \end{aligned}$

$\begin{aligned} 4 \times 2 + (6 - 4) - 9 + 5 = \underline{} \end{aligned}$

$\begin{aligned} 8^{2} - 4 + (9 - 3) + 12 = \underline{} \end{aligned}$

$\begin{aligned} 7^{3} - 100 + (3 + 4) - 9 = \underline{} \end{aligned}$

$\begin{aligned} 7 + (3^{2} + 7) - 11 + 5 = \underline{} \end{aligned}$

$\begin{aligned} 2^{4} + (8 + 7) + 13 - 5 = \underline{} \end{aligned}$

$\begin{aligned} 3 \times 2 + (2^{2} + 7) - 11 + 15 = \underline{} \end{aligned}$

$\begin{aligned} 8 + (6 + 7) - 2 \times 3 = \underline{} \end{aligned}$

$\begin{aligned} 22 + (3^{4} + 7) - 73 + 15 = \underline{} \end{aligned}$

$\begin{aligned} 3^{2} + (4^{2} - 7) - 3 + 25 = \underline{} \end{aligned}$

$\begin{aligned} 6^{3} + (3^{2} + 17) - 73 + 4 = \underline{} \end{aligned}$

$\begin{aligned} 243 - (5^{3} + 27) - 83 + 9 = \underline{} \end{aligned}$

$\begin{aligned} 72 + (11^{2} + 117) - 193 + 75 = \underline{} \end{aligned}$

$\begin{aligned} 82 + (10^{2} + 130) - 303 + 115 = \underline{} \end{aligned}$

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。

Resources
::资源

Section outline

Order of Operations ::行动命令令

Order of Operations ::行动命令令

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Review ::回顾

Review (Answers) ::回顾(答复)

Resources ::资源

Order of Operations
::行动命令令

Order of Operations
::行动命令令

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Example 5
::例5

Review
::回顾

Review (Answers)
::回顾(答复)

Resources
::资源