课程： 1.6 评价涉及强权的数值和变量表达方式

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选择节评价涉及强国的数值和变数表达式

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评价涉及强国的数值和变数表达式
Casey missed three days of school this week because of a nasty cold. When he returned to school on Thursday he asked his math teacher what he had missed. Miss Brown wrote a problem on a sheet of paper and handed it to Casey. When he arrived home from school Casey looked at his homework which was the following problem:
::凯西因为感冒而错过了本周的三天学校。当他星期四回到学校时,他问数学老师他错过了什么。布朗小姐在一张纸上写了一个问题,并把它交给凯西。当他从学校回家的时候,凯西看了他的作业,这是下面的问题:

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

Casey didn’t know what to do with $\begin{aligned} 5^{4} \end{aligned}$ and $\begin{aligned} {(- 2)}^{4} \end{aligned}$ . How can he rewrite the problem so that he understands what operations to perform?
::凯西不知道与54和(-2)4有什么关系。他如何重写问题,以便他了解要执行什么行动?

In this concept, you will learn how to evaluate numerical expressions involving powers.
::在此概念中,您将学会如何评价涉及权力的数字表达式。

Evaluating Expressions Involving Powers
::评价涉及权力方的表达方式

A numerical expression is an expression made up only of numbers and operations but an expression written with a variable in it, is called a variable expression .
::数字表达式是一个仅由数字和操作组成的表达式,但其中写有变量的表达式则称为变量表达式。

Both a numerical expression and a variable expression can include powers.
::数字表达式和变量表达式都可以包括权力。

A power is the result of multiplying a number by itself one or more times. The number 16 is the fourth power of 2.
::一个功率是数字本身乘以一个或多个倍的结果。数字 16 是 2 的第四功率。

$\begin{aligned} 2 \times 2 \times 2 \times 2 = 16 \end{aligned}$ . The ‘fourth power of 2’ or ‘2 to the power of 4’ can be written as $\begin{aligned} 2^{4} \end{aligned}$ . The number raised to the power is called the base and the number expressing the power is called the exponent . The exponent tells you how many times to multiply the base times itself. $\begin{aligned} 6^{3} = 6 \times 6 \times 6 \end{aligned}$ .
::2x2x2xx2=16。 ` 2 或 ' 2 到 4 ' 的第四功率可以写为 24. 向该功率提出的数字称为基数,表示该功率的号码称为推算数。推算数告诉你乘基数本身多少倍。 63= 6x6x6x6 。

Let’s look at an example.
::让我们举个例子。

Evaluate the following expression:
::评价以下表述:

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

First, expand the powers to see the operations you need to perform.
::首先,扩大权力范围,看您需要执行的操作。

$\begin{aligned} 6^{3} = 6 \times 6 \times 6 and 5^{2} = 5 \times 5 \end{aligned}$
Next, write the expression in expanded form .
::63=6x6x6 和 52=5x5 next, 以扩展格式写入表达式。

$\begin{aligned} 6 \times 6 \times 6 + 5 \times 5 + 25 \end{aligned}$

Next, apply the order to operations (PEMDAS) to evaluate the expanded expression.
::下一步,对操作( PEMDAS) 应用此命令来评价扩展表达式。

Then, in order from left to right, multiply: $\begin{aligned} 6 \times 6 = 36 \times 6 = 216 \end{aligned}$ and write the new expression.
::然后,按照从左到右的顺序,乘以:6x6=36x6=216,并写下新的表达式。

$\begin{aligned} 216 + 5 \times 5 + 25 \end{aligned}$

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。

$\begin{aligned} 216 + 25 + 25 \end{aligned}$

Next, from left to right, add: $\begin{aligned} 216 + 25 = 241 \end{aligned}$ and write the new expression.
::下, 从左到右, 添加: 216+25=241, 并写入新表达式。

$\begin{aligned} 241 + 25 \end{aligned}$

Then, add: $\begin{aligned} 241 + 25 = 266 \end{aligned}$
::然后加上:241+25=266

The answer is 266.
::答案是266

Let’s look at another example.
::让我们再看看另一个例子。

$\begin{aligned} 6^{2} + 15 + 3^{3} - 11 \end{aligned}$

First, expand the powers to see the operations you need to perform.
::首先,扩大权力范围,看您需要执行的操作。

$\begin{aligned} 6^{2} = 6 \times 6 and 3^{3} = 3 \times 3 \times 3 \end{aligned}$

::62=6x6和33=3x3x3x3

Next, write the expression in expanded form.
::下一步,用扩展格式写出表达式。

$\begin{aligned} 6 \times 6 + 15 + 3 \times 3 \times 3 - 11 \end{aligned}$

Next, apply the order to operations (PEMDAS) to evaluate the expanded expression.
::下一步,对操作( PEMDAS) 应用此命令来评价扩展表达式。

Then, in order from left to right multiply: $\begin{aligned} 6 \times 6 = 36 \end{aligned}$ and write the new expression.
::然后,按照从左到右的顺序乘以: 6x6=36, 并写下新的表达式。

$\begin{aligned} 36 + 15 + 3 \times 3 \times 3 - 11 \end{aligned}$

Then, multiply: $\begin{aligned} 3 \times 3 \times 3 = 27 \end{aligned}$ and write the new expression.
::然后,乘以:3x3x3x3=27, 并写下新表达式。

$\begin{aligned} 36 + 15 + 27 - 11 \end{aligned}$

Next, from left to right, add: $\begin{aligned} 36 + 15 = 51 \end{aligned}$ and write the new expression.
::接下来,从左到右,加上:36+15=51,并写下新表达式。

$\begin{aligned} 51 + 27 - 11 \end{aligned}$

Then, add: $\begin{aligned} 51 + 27 - 78 \end{aligned}$ and write the new expression.
::然后,加上:51+27-78 并写下新表达式。

$\begin{aligned} 78 - 11 \end{aligned}$

Next, subtract: $\begin{aligned} 78 - 11 = 67 \end{aligned}$
::下一步,减去:78-11=67

The answer is 67.
::答案是67。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Casey and his confusing homework.
::早些时候,你得到一个问题凯西和他令人困惑的作业。

Casey needs to expand the powers so he can see what operations he has to do to evaluate the expression.
::凯西需要扩大权力范围这样他才能看清他必须做什么行动来评价表达方式

First, evaluate the two powers given in the problem.
::首先,评估这一问题中赋予的两项权力。

$\begin{aligned} 5^{4} = 5 \times 5 \times 5 \times 5 = 625 and (- 2)^{4} = - 2 \times - 2 \times - 2 \times - 2 = 16 \end{aligned}$

::54=5x5x5x5=625和(-2)42222×2=16

Next, replace the powers in the expression with their values.
::接下来,将表达式中的权力替换为它们的值。

$\begin{aligned} 625 + 16 + 12 \end{aligned}$
Then, add.
::625+16+12 然后,添加。

$\begin{aligned} 625 + 16 = 641 + 12 = 653 \end{aligned}$

The answer is 653.
::答案是653

Example 2
::例2

Evaluate the following variable expression when $\begin{aligned} x = 4 \end{aligned}$ .
::评估以下变量表达式时x=4 。

$\begin{aligned} 2 x^{3} - 12 \end{aligned}$
::2x3-12

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

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

Next, expand the power.
::下一个,扩展电源。

$\begin{aligned} (4)^{3} = (4 \times 4 \times 4) \end{aligned}$
Next, write the expression in expanded form.
:4)3=( 4x4x4) 下一步, 以扩展格式写入表达式。

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

Then, perform the operation in the parenthesis.
::然后在括号里做手术

Multiply: $\begin{aligned} 4 \times 4 = 16 \times 4 = 64 \end{aligned}$ and write the new expression.
::乘以: 4x4=16x4=64, 并写入新表达式。

$\begin{aligned} 2 (64) - 12 \end{aligned}$

Next, multiply: $\begin{aligned} 2 (64) = 128 \end{aligned}$ and write the new expression.
::下一步,乘以:2(64)=128,并写下新表达式。

$\begin{aligned} 128 - 12 \end{aligned}$

Then, subtract.
::然后,减。

$\begin{aligned} 128 - 12 = 116 \end{aligned}$

The answer is 116.
::答案是116

Example 3
::例3

Evaluate the following numerical expression.
::评估以下数字表达式。

$\begin{aligned} 20 - 2^{4} + 1 + 3^{3} \end{aligned}$

First, expand the powers.
::首先,扩大权力。

$\begin{aligned} 2 \times 2 \times 2 \times 2 and 3 \times 3 \times 3 \end{aligned}$

::2x2x2x2和3x3x3x3

Next, write the expression in expanded form.
::下一步,用扩展格式写出表达式。

$\begin{aligned} 20 - 2 \times 2 \times 2 \times 2 + 1 + 3 \times 3 \times 3 \end{aligned}$

Then, in order from left to right multiply: $\begin{aligned} 2 \times 2 \times 2 \times 2 = 16 \end{aligned}$ and write the new expression.
::然后,按照从左到右的顺序乘以: 2x2xx2xx2xx2=16, 并写下新的表达式。

$\begin{aligned} 20 - 16 + 1 + 3 \times 3 \times 3 \end{aligned}$

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

$\begin{aligned} 20 - 16 + 1 + 27 \end{aligned}$

Then, in order from left to write, subtract: $\begin{aligned} 20 - 16 = 4 \end{aligned}$ and write the new expression.
::然后,按照左边的顺序写,减去:20-16=4,并写下新的表达式。

$\begin{aligned} 4 + 1 + 27 \end{aligned}$

Next, add: $\begin{aligned} 4 + 1 = 5 \end{aligned}$ and write the new expression.
::下加: 4+1=5, 并写下新表达式。

$\begin{aligned} 5 + 27 \end{aligned}$

Then add: $\begin{aligned} 5 + 27 = 32 \end{aligned}$
::然后加上:5+27=32

The answer is 32.
::答案是32个

Example 4
::例4

Evaluate the following variable expression when $\begin{aligned} m = 3 \end{aligned}$ and $\begin{aligned} n = 2 \end{aligned}$ .
::当 m=3andn=2 时, 评估以下变量表达式。

$\begin{aligned} n^{5} + 3 m^{2} - 15 \end{aligned}$

::n5+3m2-15

First, substitute $\begin{aligned} m = 3 \end{aligned}$ and $\begin{aligned} n = 2 \end{aligned}$ into the variable expression.
::首先,将 m=3andn=2 替换为变量表达式。

$\begin{aligned} 2^{5} + 3 (3)^{2} - 15 \end{aligned}$
Next, expand the powers:
$\begin{aligned} 2^{5} = (2 \times 2 \times 2 \times 2 \times 2) and (3)^{2} = (3 \times 3) \end{aligned}$

::25+3(3)2- 15 next, 扩展权力范围: 25=( 2x2x2x2x2x2) 和 (3)2=( 3x3)

Then, write the expression in expanded form.
::然后,用扩展的形式写出表达式。

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

Then, perform the operation in the parenthesis.
::然后在括号里做手术

First, multiply: $\begin{aligned} (3 \times 3) = (9) \end{aligned}$ and write the new expression.
::首先,乘以( 3x3) =( 9) 并写入新表达式。

$\begin{aligned} 2 \times 2 \times 2 \times 2 \times 2 + 3 (9) - 15 \end{aligned}$

Next, multiply: $\begin{aligned} 3 (9) = 27 \end{aligned}$ to clear the parenthesis. Write the new expression.
::下一个, 乘以 3( 9) = 27 来清除括号。写入新表达式。

$\begin{aligned} 2 \times 2 \times 2 \times 2 \times 2 + 27 - 15 \end{aligned}$

Next, multiply: $\begin{aligned} 2 \times 2 = 4 \times 2 = 8 \times 2 = 16 \times 2 = 32 \end{aligned}$ Write the new expression.
::下一个, 乘以: 2x2=4x2=8x2=8x2=16x2=32 写下新表达式。

$\begin{aligned} 32 + 27 - 15 \end{aligned}$

Next, add: $\begin{aligned} 32 + 27 = 59 \end{aligned}$ and write the new expression.
::下加: 32+27=59 并写下新表达式。

$\begin{aligned} 59 - 15 \end{aligned}$
Then, subtract: $\begin{aligned} 59 - 15 = 44 \end{aligned}$
::59- 15 然后, 减去: 59- 15= 44

The answer is 44.
::答案是44

Review
::回顾

Expand and evaluate each power.
::扩大并评估每个力量。

1. $\begin{aligned} 3^{3} \end{aligned}$

2. $\begin{aligned} 4^{2} \end{aligned}$

3. $\begin{aligned} (- 2)^{4} \end{aligned}$

4. $\begin{aligned} (- 8)^{2} \end{aligned}$

5. $\begin{aligned} 5^{3} \end{aligned}$

6. $\begin{aligned} 2^{6} \end{aligned}$

7. $\begin{aligned} (- 9)^{2} \end{aligned}$

8. $\begin{aligned} (- 2)^{6} \end{aligned}$

Evaluate each numerical expression. Remember to apply PEMDAS to evaluate the expression accurately.
::评估每个数字表达式。记住应用 PEMDAS 来准确评价表达式。

9. $\begin{aligned} 6^{2} + 22 \end{aligned}$

10. $\begin{aligned} (- 3)^{3} + 18 \end{aligned}$

11. $\begin{aligned} 2^{3} + 16 - 4 \end{aligned}$

12. $\begin{aligned} (- 5)^{2} - 19 \end{aligned}$

13. $\begin{aligned} (- 7)^{2} + 52 - 2 \end{aligned}$

14. $\begin{aligned} 18 + 9^{2} - 3 \end{aligned}$

15. $\begin{aligned} 22 - 3^{3} + 7 \end{aligned}$

Evaluate each variable expression using the given values.
::使用给定值评价每个变量表达式。

16. $\begin{aligned} 6 a + 4^{2} - 2 \end{aligned}$ , when $\begin{aligned} a = 3 \end{aligned}$
::16.a+42-2,a=3时

17. $\begin{aligned} a^{3} + 14, when a = 6. \end{aligned}$
::a3+14, 当 a=6时。

18. $\begin{aligned} 2 a^{2} - 16, when a = 4 \end{aligned}$
::18. A=4时2a2-16

19. $\begin{aligned} 5 b^{3} + 12, when b = - 2 \end{aligned}$
::19. 5b3+12, b=-2

20. $\begin{aligned} 2 x^{2} + 52, when x = 4 \end{aligned}$
::20. 2x2+52,x=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。

Resources
::资源

章节大纲

常规

评价涉及强国的数值和变数表达式

Evaluating Expressions Involving Powers ::评价涉及权力方的表达方式

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)

Resources ::资源

Evaluating Expressions Involving Powers
::评价涉及权力方的表达方式

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)

Resources
::资源