Section: 简化单一变量表达式的产品或引号 | 7.5 简化单一变量表达式的产品或引号

Section outline

Laine is helping his mother decorate the patio by putting tiles on a rectangular wall that is 20 tiles wide and 32 tiles high. If each tile is a square with side s, then the total area is $\begin{aligned} 20 s \times 30 s \end{aligned}$ . How can Laine figure out the total area to be tiled in simplified form?
::莱恩帮助他的母亲装饰院子,把瓷砖放在一个长方形的墙上,长20瓦宽,高32瓦。如果每张瓷砖是带有侧面的方形,那么总面积是20sx30s。莱恩如何以简化的形式确定要铺设的总面积 ?

In this concept, you will learn to simplify products or quotients of single variable expressions.
::在此概念中,您将学会简化单变量表达式的产品或商数。

Simplifying Products or Quotients of Single Variable Expressions
::简化单一变量表达式的产品或引号

With expressions, terms are separated from each other by addition or subtraction , while factors are separated by multiplication or division . For example, the expression $\begin{aligned} 3 a b - 7 b a \end{aligned}$ is composed of two terms, $\begin{aligned} 3 a b \end{aligned}$ and $\begin{aligned} 7 b a \end{aligned}$ , and each of those terms is composed of three factors.
::使用用语时,用语因增加或减法而彼此分离,而因数因乘数或除法而分离,例如,3ab-7ba的用语由两个用语组成,即3ab和7ba,这些用语各由三个因素组成。

When adding or subtracting terms in an expression , you can only combine like terms , which are composed of only the same variables. However, you can multiply or divide terms whether they are like terms or not.
::当在表达式中添加或减去术语时,您只能将类似术语组合在一起,这些术语只由相同的变量组成。然而,无论是否类似术语,您都可以乘或除。

For example, $\begin{aligned} 3 a b \end{aligned}$ and $\begin{aligned} 7 b a \end{aligned}$ are like terms - both terms include only the variables of a and b, regardless of the order in which they appear in each term . However, $\begin{aligned} 3 a \end{aligned}$ and $\begin{aligned} 7 b \end{aligned}$ are not like terms, since one contains the variable a, and the other contains the variable b . Since $\begin{aligned} 3 a \end{aligned}$ and $\begin{aligned} 7 b \end{aligned}$ are not like terms, they can't be added together:
::例如,3ab 和 7ba 等词相似----这两个术语仅包括a和b的变量,而不论这两个词在每个术语中出现的先后顺序。然而,3a和7b 等词不同,因为一个包含变量a,另一个包含变量b。由于3a和7b 等词不同,因此不能将它们加在一起:

$\begin{aligned} 3 a + 7 b = 3 a + 7 b \end{aligned}$

::3a+7b=3a+7b

There is nothing you can do to simplify the expression.
::您无法简化表达式。

However, $\begin{aligned} 3 a \end{aligned}$ and $\begin{aligned} 7 b \end{aligned}$ can be multiplied by each other:
::然而,3a和7b可相互乘以:

$\begin{aligned} 3 a \times 7 b = 21 a b \end{aligned}$

::3ax7b=21ab

Clearly, $\begin{aligned} 21 a b \end{aligned}$ is simpler than $\begin{aligned} 3 a \times 7 b \end{aligned}$ .
::显然,21ab比3ax7b简单。

Two rules will help you multiply expressions that contain variables. The Commutative Property of Multiplication states that two terms can be multiplied in any order. The Associative Property of Multiplication states that the grouping of terms does not change your answer.
::两个规则会帮助您乘以包含变量的表达式。乘法的交流属性表示两个词可以任意乘以。乘法的连带属性表示,术语组合不会改变您的答复。

It is also helpful to remember that when multiplying like variables together, you add the exponents.
::记住一点也是有益的, 当像变量一样的变量一起乘时, 您会添加引号。

For example, remember that $\begin{aligned} x \end{aligned}$ is the same as $\begin{aligned} x^{1} \end{aligned}$ :
::例如, 记住 x 与 x1 相同 :

$\begin{aligned} \begin{array}{rcl} x (x) & = & x^{2} \\ x (x) (x) & = & x^{3} \\ x^{2} (x) & = & x^{3} \end{array} \end{aligned}$

:xx)=x2x(x)(x)=x3x2(x)=x3

Look at the following example.
::以下面的例子为例。

Simplify $\begin{aligned} 6 a (3 a) \end{aligned}$ .
::简化 6a( 3a) 。

First, multiply the number parts.
::首先,乘以数字部件。

$\begin{aligned} 6 \times 3 = 18 \end{aligned}$

Next, multiply the variables.
::下一步,乘以变量。

$\begin{aligned} a \cdot a = a^{2} \end{aligned}$

::a-a=a2

The answer is $\begin{aligned} 18 a^{2} \end{aligned}$ .
::答案是18a2

Here is another example.
::下面是另一个例子。

Simplify $\begin{aligned} 5 \times (8 y) \end{aligned}$ .
::简化 5x( 8y) 。

These are not like terms, since they contain different variables, but they can still be multiplied.
::这些与术语不同,因为它们包含不同的变量,但仍可以乘以乘数。

First, multiply the numbers.
::首先,乘以数字。

$\begin{aligned} 5 \times 8 = 40 \end{aligned}$

Next, multiply the variables.
::下一步,乘以变量。

$\begin{aligned} x \cdot y = x y \end{aligned}$

::xy=xy

The answer is $\begin{aligned} 40 x y \end{aligned}$ .
::答案是40xy。

Here is one more example.
::这里还有一个例子。

Find the product $\begin{aligned} 4 z \times \frac{1}{2} \end{aligned}$ .
::找到产品 4zx12。

$\begin{aligned} 4 z \end{aligned}$ and $\begin{aligned} \frac{1}{2} \end{aligned}$ are not like terms, however, you can multiply terms even if they are not like terms.
::4z 和 12 和 4z 和 12 并不像条件, 但是, 你可以乘数的条件 , 即使与 4z 和 12 的条件不同。

Use the commutative and associative properties to rearrange the factors to make it easier to see how they can be multiplied.
::利用通量属性和关联属性重新排列各种因素,以便更容易看出它们是如何乘以的。

According to the commutative property , $\begin{aligned} z (\frac{1}{2}) = \frac{1}{2} (z) \end{aligned}$ .
::根据通量财产,z(12)=12(z)。

$\begin{aligned} 4 z \times \frac{1}{2} = 4 \times \frac{1}{2} (z) \end{aligned}$

::4zx12=4x12(z)

According to the associative property, the grouping of the factors does not change the answer. Group the factors so that the numbers are multiplied first.
::根据连带财产,将各种因素分组并不改变答案。将各种因素分组,以便先乘以数字。

$\begin{aligned} 4 \times \frac{1}{2} (z) = 4 \times \frac{1}{2} \times z = (4 \times \frac{1}{2}) \times z \end{aligned}$

::4x12(z)=4x12xz=(4x12)xz

Now, multiply.
::现在,乘数。

$\begin{aligned} (4 \times \frac{1}{2}) \times z = (2) \times z = 2 z \end{aligned}$

:4x12)xz=(2)xz=2z

The answer is $\begin{aligned} 2 z \end{aligned}$ .
::答案是 2z 。

Here is an example using division.
::以下是一个使用“划分”的例子。

Find the quotient $\begin{aligned} 42 c \div 7 \end{aligned}$ .
::查找商数 42c7。

First, rewrite the problem like this $\begin{aligned} \frac{42 c}{7} \end{aligned}$ .
::首先,重写问题像这样 42c7 。

Then separate out the numbers and variables like this.
::然后将数字和变数分开,像这样。

$\begin{aligned} \frac{42 c}{7} = \frac{42 \cdot c}{7} = \frac{42}{7} \cdot c \end{aligned}$

::427c=427c=427c=427c

Now, divide 42 by 7 to find the quotient.
::现在,将42除以7 以找到商数。

$\begin{aligned} \frac{42}{7} \cdot c = 6 \cdot c = 6 c \end{aligned}$

::427c=6c=6c

The answer is $\begin{aligned} 6 c \end{aligned}$ .
::答案是6c

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Laine, who is decorating the patio by gluing tiles on a rectangular wall 20 tiles tall (long) and 30 tiles wide.
::早些时候,你曾遇到关于Laine的问题, Laine正在装饰院子,在长长20瓦和宽30瓦的长长长长长长长的长长长长长长长长长的长长长长长长长长长长长的长长长长长长长长长的长长长长长长长的长长长长长的长长长长长长长长的长长长长长长的长长长长长长长的长长长长长的长长长长长长长长长的长长长长长长长长的长长长长长长的长长长长长长的长长长长长长的长长长长长长长长长的长长长长的长长长长长长长的长长长长的长长长长的长长长长长长长长长长长的长长长长长长长的长长长长的长长长长长的长长长长长长长长的长长长长的长长长长长长的长长长的长长长的长的长长长长长长长长长的长的长长长长长的长长长长的长的长长长长长的长长长长长的长的长长长长长的长长长长长的长的长的长长长长的长长长长的长长长的长的长的长的长的长的长的长的长长长长长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的长的

Each tile is a square with side $\begin{aligned} s \end{aligned}$ . Since the area of a rectangle is length times width, the total area is $\begin{aligned} 20 s \times 30 s \end{aligned}$ .
::由于矩形区域是长度乘以宽度,总区域为20sx30s。

Laine needs to know what the total area is to be tiled in simplified form. So he will need to multiply 20s by 30s.
::莱恩需要知道总面积是多少以简化的形式拼凑。所以他需要乘以 20 乘以 30 。

First, multiply the numbers.
::首先,乘以数字。

$\begin{aligned} 20 \times 30 = 600 \end{aligned}$

Next, multiply the variables.
::下一步,乘以变量。

$\begin{aligned} s \times s = s^{2} \end{aligned}$

::sxs=s2 键

The answer is $\begin{aligned} 600 s^{2} \end{aligned}$ .
::答案是600秒2

Example 2
::例2

Find the quotient of $\begin{aligned} 50 g \div 10 g \end{aligned}$ .
::找到50千兆克的商数

First, rewrite the problem.
::首先,重写问题。

$\begin{aligned} \frac{50 g}{10 g} \end{aligned}$

::50克10克

Then, separate the factors.
::然后,将因素分开。

$\begin{aligned} \frac{50 \cdot g}{10 \cdot g} \end{aligned}$

::50g10g

Next, reduce and cancel.
::下一个,减少和取消。

$\begin{aligned} \frac{50 5 \cdot g}{10 1 \cdot g} = \frac{5}{1} = 5 \end{aligned}$

::50 5g10 1g=51=5

The answer is 5.
::答案是5。

Example 3
::例3

Use the commutative and associative properties of multiplication to simplify $\begin{aligned} 6 a (9 a) \end{aligned}$ .
::利用乘法的通量和连带性来简化 6a(9a)

First, apply the associative property to separate the a and the 9.
::首先,适用连带财产将a和9分开。

$\begin{aligned} 6 a (9 a) = 6 (a 9) a \end{aligned}$

::6a(9a)=6(a9)a

Next, apply the commutative property to put the numbers and variables next to each other.
::下一步,应用通量属性将数字和变量相邻。

$\begin{aligned} 6 (a 9) a = 6 (9 a) a \end{aligned}$

::6(a9)a=6(a)a

Then, apply the associative property again to group the similar factors.
::然后,将连带财产再次用于将类似因素组合在一起。

$\begin{aligned} 6 (9 a) a = (6 \cdot 9) (a \cdot a) \end{aligned}$

::6(9a)a=(6)9(a)a

Finally, multiply the similar factors.
::最后,将类似因素乘以。

$\begin{aligned} (6 \cdot 9) (a \cdot a) = 54 a^{2} \end{aligned}$

:69)(aa)=54a2

The answer is $\begin{aligned} 54 a^{2} \end{aligned}$ .
::答案是54a2。

Example 4
::例4

Use the commutative and associative properties of multiplication to simplify $\begin{aligned} 15 b \div 5 b \end{aligned}$ .
::使用乘法的平流和关联性来简化 15b_5b。

First, rewrite the problem in vertical format.
::首先, 以垂直格式重写问题。

$\begin{aligned} \frac{15 b}{5 b} \end{aligned}$

::15b5b

Next, separate the factors.
::接下来,分别列出各种因素。

$\begin{aligned} \frac{15 \cdot b}{5 \cdot b} \end{aligned}$

::15比b5bb

Then, identify and cancel similar factors.
::然后找出并取消类似因素。

$\begin{aligned} \frac{5 \cdot 3 \cdot b}{51 \cdot b} \end{aligned}$

::5_____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Finally, simplify to get the answer.
::最后,简化以获得答案。

$\begin{aligned} \frac{3}{1} = 3 \end{aligned}$

The answer is 3.
::答案是3

Example 5
::例5

Simplify $\begin{aligned} \frac{20 c}{4} \end{aligned}$ .
::简化 20c4。

First, separate the factors.
::首先,将因素分开。

$\begin{aligned} \frac{20 \cdot c}{4} \end{aligned}$

::20c4

Next, identify and cancel similar factors.
::其次,查明并取消类似因素。

$\begin{aligned} \frac{5 \cdot 4 \cdot c}{4} \end{aligned}$

::54c4

Finally, simplify to get the answer.
::最后,简化以获得答案。

$\begin{aligned} \frac{5 c}{1} = 5 c \end{aligned}$

::5c1=5c

The answer is $\begin{aligned} 5 c \end{aligned}$ .
::答案是5c。

Review
::回顾

Simplify each product or quotient.
::简化每种产品或商数。

$\begin{aligned} 6 a (4 a) \end{aligned}$
::6a(4a)

$\begin{aligned} 9 x (2) \end{aligned}$
::9x(2)

$\begin{aligned} 14 y (2 y) \end{aligned}$
::14y(2y)

$\begin{aligned} 16 a (a) \end{aligned}$
::16(a) 16(a)

$\begin{aligned} 22 x (2 x) \end{aligned}$
::22x(2x)

$\begin{aligned} 18 b (2) \end{aligned}$
::18b(2) 18b(2)

$\begin{aligned} 21 a \div 7 \end{aligned}$
::21a+7

$\begin{aligned} 22 b \div 2 b \end{aligned}$
::22b%2b

$\begin{aligned} 25 x \div x \end{aligned}$
::25x%xx

$\begin{aligned} 45 a \div 5 a \end{aligned}$
::45a5a

$\begin{aligned} 15 x \div 3 x \end{aligned}$
::15x3x

$\begin{aligned} 18 y \div 9 \end{aligned}$
::18y9

$\begin{aligned} 22 y \div 11 y \end{aligned}$
::22y11y

$\begin{aligned} \frac{15 x}{3 y} \end{aligned}$
::15x3y 15x3y

$\begin{aligned} \frac{82 x}{2 x} \end{aligned}$
::82x2x

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

Simplifying Products or Quotients of Single Variable Expressions ::简化单一变量表达式的产品或引号

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

Simplifying Products or Quotients of Single Variable Expressions
::简化单一变量表达式的产品或引号

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