Course: 9.8 乘数逻辑表达式

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乘数逻辑表达式
The length of a rectangle is $\begin{align*}\frac{2xy^3z}{5xyz^2}\end{align*}$ . The width of the rectangle is $\begin{align*}\frac{3x^2yz^3}{4x^3y^2z^2}\end{align*}$ . What is the area of the rectangle?
::矩形的长度是 2xy3z5xyz2. 矩形的宽度是 3x2yz34x3y2z2. 矩形的面积是 3x2yz34x3y2。矩形的面积是多少?

Multiplying Rational Expressions
::乘数逻辑表达式

We take what you have learned previously a step further in this concept and multiply two rational expressions together. When multiplying rational expressions, it is just like multiplying fractions. However, it is much easier to factor the rational expressions before multiplying because factors could cancel out.
::我们将你们以前学到的东西在这个概念中向前推进一步,然后将两个理性表达方式相乘。当将理性表达方式乘以数分数时,它就像乘以数分数一样。然而,在乘以数之前将理性表达方式考虑在内容易得多,因为因素可以取消。

Let's multiply the following rational expressions.
::让我们乘以以下的理性表达方式。

$\begin{align*}\frac{x^2-4x}{x^3-9x} \cdot \frac{x^2+8x+15}{x^2-2x-8}\end{align*}$
::x2 - 4x23 - 9xxx2+8x+15x2 - 2x-8

Rather than multiply together each numerator and denominator to get very complicated , it is much easier to first factor and then cancel out any common factors.
::与其将每个分子和分母相乘, 以变得非常复杂, 不如先考虑一个因素, 再取消任何共同因素, 容易得多。

$\begin{align*}\frac{x^2-4x}{x^3-9x} \cdot \frac{x^2+8x+15}{x^2-2x-8} = \frac{x(x-4)}{x(x-3)(x+3)} \cdot \frac{(x+3)(x+5)}{(x+2)(x-4)}\end{align*}$
::x2-4x23-9xxx2+8x+15x2-2-2x-8=xx(x-4)xx(x-3)(x+3)(x+3)x(x+3)(x+3)(x+5)(x+2)(x-4)

At this point, we see there are common factors between the fractions.
::此时此刻,我们看到分数之间存在共同因素。

$\begin{align*}\frac{\cancel{x} \cancel{(x-4)}}{\cancel{x}(x-3) \cancel{(x+3)}} \cdot \frac{\cancel{(x+3)}(x+5)}{(x+2)\cancel{(x-4)}} = \frac{x+5}{(x-3)(x+2)}\end{align*}$
::x(x- 4) x(x-3)(x+3)(x+3)(x+3)(x+5)(x+2)(x-4)=x+5(x-3)(x+2)

At this point, the answer is in factored form and simplified. You do not need to multiply out the base.
::此时,答案是按因素计算和简化的。您不需要乘出基数。

$\begin{align*}\frac{4x^2y^5z}{6xyz^6} \cdot \frac{15y^4}{35x^4}\end{align*}$
::4x2y5z6xyz615y435x4

These rational expressions are monomials with more than one variable . Here, we need to remember the laws of exponents. Remember to add the exponents when multiplying and subtract the exponents when dividing. The easiest way to solve this type of problem is to multiply the two fractions together first and then subtract common exponents.
::这些理性表达式是一个单一的表达式, 有多个变量。这里, 我们需要记住引言者的定律。记住, 当分隔时要加上引言并减去引言。最容易解决这类问题的方法是先将两个部分相乘,然后再减去共同引言。

$\begin{align*}\frac{4x^2y^5z}{6xyz^6} \cdot \frac{15y^4}{35x^4} = \frac{60x^2y^9z}{210x^5yz^6} = \frac{2y^8}{7x^3z^5}\end{align*}$
::4x2y5z6xyz6}15y435x4=60x2y9z 210x5yz6=2y87x3z5

You can reverse the order and cancel any common exponents first and then multiply, but sometimes that can get confusing.
::您可以将命令倒转, 并取消任何共同的指数, 然后再乘数, 但有时会混淆。

$\begin{align*}\frac{4x^2+4x+1}{2x^2-9x-5} \cdot (3x-2) \cdot \frac{x^2-25}{6x^2-x-2}\end{align*}$
::4x2+4x+12x2-9x-5(3x-2)x2-2556x2-x-2

Because the middle term is a linear expression , rewrite it over 1 to make it a fraction .
::因为中期内是一个线性表达式, 重写 1 以上, 使之成为一个分数。

$\begin{align*}\frac{4x^2+4x+1}{2x^2-9x-5} \cdot (3x-2) \cdot \frac{x^2-25}{6x^2-x-2} = \frac{\cancel{(2x+1)} \cancel{(2x+1)}}{\cancel{(2x+1)} \cancel{(x-5)}} \cdot \frac{\cancel{3x-2}}{1} \cdot \frac{\cancel{(x-5)}(x+5)}{\cancel{(3x-2)} \cancel{(2x+1)}} = x+5\end{align*}$
::4x2+4x+12x2-9x+12x2-9x-5(3x-2)x2-256x2-x2=(2x+1)(2x+1)(2x+1)(2x+1)(x5)(5x)(5x+5)(3x-2)(2x+1)=x5

Examples
::实例

Example 1
::例1

Earlier, you were asked to find the area of the rectangle.
::早些时候,有人要求你找到矩形区域。

The area of the rectangle is length times width. So to find the area, multiply the two terms and simplify.
::矩形区域是长度乘以宽度。为了查找区域,将两个条件乘以,并简化。

$\begin{aligned} \frac{2 x y^{3} z}{5 x y z^{2}} \cdot \frac{3 x^{2} y z^{3}}{4 x^{3} y^{2} z^{2}} \\ \frac{6 x^{3} y^{4} z^{4}}{20 x^{4} y^{3} z^{4}} \\ \frac{3 y}{10 x} \end{aligned}$
$\begin{align*}\frac{2xy^3z}{5xyz^2} \cdot \frac{3x^2yz^3}{4x^3y^2z^2}\\ \frac{6x^3y^4z^4}{20x^4y^3z^4}\\ \frac{3y}{10x}\end{align*}$
::2x3z5xyz2}3x2yz34x3y2z26x3y4z420x4y3z43y10x

Therefore , the area of the rectangle is $\begin{align*}\frac{3y}{10x}\end{align*}$ .
::因此,矩形区域是3y10x。

Multiply the following expressions.
::乘以下列表达式。

Example 2
::例2

$\begin{align*}\frac{4x^2-8x}{10x^3} \cdot \frac{15x^2-5x}{x-2}\end{align*}$
::4x2-8x10x3x15x2-5xx-2

$\begin{align*}\frac{4x^2-8x}{10x^3} \cdot \frac{15x^2-5x}{x-2} = \frac{\cancel{2} \cdot 2 \cancel{x}\cancel{(x-2)}}{\cancel{2} \cdot \cancel{5x} \cdot \cancel{x} \cdot x} \cdot \frac{\cancel{5x}(3x-1)}{\cancel{x-2}} = \frac{2(3x-1)}{x}\end{align*}$
::4x2-8x10x3x15x2-5x-2=2x2x2x2xx2xx2x5xxxx5x(3x-1)x-2=2x-1xxxx

Example 3
::例3

$\begin{align*}\frac{x^2+6x-7}{x^2-36} \cdot \frac{x^2-2x-24}{2x^2+8x-42}\end{align*}$
::x2+6x-7x2 - 36_x2 - 2x-2x-242x2+8x-42

$\begin{align*}\frac{x^2+6x-7}{x^2-36} \cdot \frac{x^2-2x-24}{2x^2+8x-42} = \frac{\cancel{(x+7)}(x-1)}{\cancel{(x-6)}(x+6)} \cdot \frac{\cancel{(x-6)}(x+4)}{2 \cancel{(x+7)}(x-3)} = \frac{(x-1)(x+4)}{2(x-3)(x+6)}\end{align*}$
::x2+6x-7x-2-36x2-36x2-2x-2x-242x2+8x-42=(x+7x-1)(x-6)(x+6)(x-6)(x+6)(x-6)(x-6)(x-6)(x+4)(x+7)(x)(x-3)=(x-1)(x+4)(x-3)(2x-3)(x+6)

Example 4
::例4

$\begin{align*}\frac{4x^2y^7}{32x^4y^3} \cdot \frac{16x^2}{8y^6}\end{align*}$
::4x2y732x4y3x16x28y6

$\begin{align*}\frac{4x^2y^7}{32x^4y^3} \cdot \frac{16x^2}{8y^6} = \frac{64x^4y^7}{256x^4y^9} = \frac{1}{4y^2}\end{align*}$
::4x2y732x4y3}}16x28y6=64x4y7256x4y9=14y2

Review
::回顾

Determine if the following statements are true or false. If false, explain why.
::确定下列声明是真实的还是虚假的。如果是虚假的,请解释原因。

When multiplying two variables with the same base, you multiply the exponents.
::当用同一基数乘以两个变量时,会乘以指数。

When dividing two variables with the same base, you subtract the exponents.
::以相同基数分隔两个变量时,要减去指数。

When a power is raised to a power, you multiply the exponents.
::当权柄被升起的时候,你使权柄加倍。

$\begin{align*}(x+2)^2 = x^2 + 4\end{align*}$
:x+2)2=x2+4

Multiply the following expressions. Simplify your answers.
::乘以下列表达式。简化您的答案。

$\begin{align*}\frac{8x^2y^3}{5x^3y} \cdot \frac{15xy^8}{2x^3y^5}\end{align*}$
::8x2y35x3y=15xY82x3y5

$\begin{align*}\frac{11x^3y^9}{2x^4} \cdot \frac{6x^7y^2}{33xy^3}\end{align*}$
::11x3y92x4_6x7Y233xy3

$\begin{align*}\frac{18x^3y^6}{13x^8y^2} \cdot \frac{39x^{12}y^5}{9x^2y^9}\end{align*}$
::18x3y613x8Y2}39x12y59x2y9

$\begin{align*}\frac{3x+3}{y-3} \cdot \frac{y^2-y-6}{2x+2}\end{align*}$
::3x+3y-3y2-y-62x+2

$\begin{align*}\frac{6}{2x+3} \cdot \frac{4x^2+4x-3}{3x+3}\end{align*}$
::62x+3=4x2+4x-33x+3

$\begin{align*}\frac{6+x}{2x-1} \cdot \frac{x^2+5x-3}{x^2+5x-6}\end{align*}$
::6+x2x-1x2+5x-3x2+5x-6

$\begin{align*}\frac{3x-21}{x-3} \cdot \frac{-x^2+x+6}{x^2-5x-14}\end{align*}$
::3-21x-3x2+x6x2-5x-14

$\begin{align*}\frac{6x^2+5x+1}{8x^2-2x-3} \cdot \frac{4x^2+28x-30}{6x^2-7x-3}\end{align*}$
::6x2+5x+18x2-2x-34x2+28x-306x2-7x-3

$\begin{align*}\frac{x^2+9x-36}{x^2-9} \cdot \frac{x^2+8x+15}{-x^2+11x+12}\end{align*}$
::x2+9x-36x2-9x2+8x+15-x2+11x+12

$\begin{align*}\frac{2x^2+x-21}{x^2+2x-48} \cdot (4-x) \cdot \frac{2x^2-9x-18}{2x^2-x-28}\end{align*}$
::2x2+x-21x2+2x-2x-48(4-x)_2x2-9x-182x2-x-28

$\begin{align*}\frac{8x^2-10x-3}{4x^3+x^2-36x-9} \cdot \frac{5x+3}{x-1} \cdot \frac{x^3+3x^2-x-3}{5x^2+8x+3}\end{align*}$
::8x2-10x-34x3+3x2-36x-9.5x+3x-1x3+3x2-35x2+8x+3

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

乘数逻辑表达式

Multiplying Rational Expressions ::乘数逻辑表达式

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)

Multiplying Rational Expressions
::乘数逻辑表达式

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)