Course: 9.3 乘数和分裂的理性表达式 | The Way To Learn

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乘法和分裂逻辑表达式
Resistors are a part of most electrical systems. An important law for a circuit is Ohm's Law , which says that the voltage in a circuit is directly proportional to the resistance of the circuit and the current in the circuit ¹ . Consider a circuit where the voltage is 12 volts, and two resistors— one 5 ohms more than the other—are connected in parallel, which means the voltage across the resistors is equal. An expression for the current is $\begin{aligned} \frac{12}{\frac{1}{x} + \frac{1}{x + 5}} \end{aligned}$ . This type of expression is called a complex fraction , and we discuss expressions like this and other rational expressions that involve multiplication and division below.
::电源是大多数电源系统的一部分。电路的重要法律是《 Ohm 法》,其中规定电路的电压与电路的抗力和电路的电流直接成比例。考虑电压为12伏的电路和电路1 的电路,两个阻力器——一个比另一个高5奥姆——平行连接,这意味着阻力器的电压是相等的。电流的表达法是121x1x1x+5。这种表达法被称为复杂部分,我们讨论这种表达法和其他合理表达法,这些表达法在下面涉及乘数和分裂。

Multiplying Rational Expressions
::乘数逻辑表达式

Multiplying rational expressions is just like multiplying fractions—we multiply the numerators together and the denominators together. However, it will be helpful to factor out the rational expressions before multiplying, because factors could cancel out.
::乘法理性表达法和乘法分数一样——我们把分子和分母一起乘。然而,在乘法之前将理性表达法考虑在内是有助益的,因为因素可以取消。

Multiplying Rational Expressions
::乘数逻辑表达式

For any expressions, A, B, C, and D :
::任何表示式A、B、C和D:

$\begin{aligned} \frac{A}{B} \cdot \frac{C}{D} = \frac{A C}{B D} . \end{aligned}$

::ABCD=《生物多样性公约》。

Example 1
::例1

Multiply $\begin{aligned} \frac{4 x^{2} y^{5} z}{6 x y z^{6}} \cdot \frac{15 y^{4}}{35 x^{4}} \end{aligned}$ .
::乘以 4x2y5z6xyz6}15y435x4 。

Solution: These rational expressions are monomials with more than one variable . The easiest way to find this product is to multiply the two fractions together 1st, and then s implify since the terms are monomials .
::解决方案 : 这些理性表达式是单项, 含有多个变量。最容易找到此产品的方法是将两个分数相加, 然后简化, 因为术语是单项的。

$\begin{aligned} \frac{4 x^{2} y^{5} z}{6 x y z^{6}} \cdot \frac{15 y^{4}}{35 x^{4}} = \frac{60 x^{2} y^{9} z}{210 x^{5} y z^{6}} = \frac{60^{2} \cdot x \cdot x \cdot y \cdot y \cdot y \cdot y \cdot y \cdot y \cdot y \cdot y \cdot y \cdot z}{210^{7} \cdot x \cdot x \cdot x \cdot x \cdot x \cdot y \cdot z \cdot z \cdot z \cdot z \cdot z \cdot z} = \frac{2 y^{8}}{7 x^{3} z^{5}} \end{aligned}$

::4x2y5z6xyz6}[15]15y435x4=60x2y9z2-210x5yz6=602xxx}yyyyyyyyy z2107xxxxxxxxxxxxxxxxxxzzzzzzzzz=2y87x3z5

Remember to add the exponents when multiplying and subtract the exponents when dividing. Note that y ou can reverse the order and cancel any common factors 1st, and then multiply.
::记住, 当分隔时, 当乘数和减数时, 要添加引号。注意您可以将顺序颠倒, 取消任何共同系数 1, 然后乘数。

Example 2
::例2

Multiply $\begin{aligned} \frac{x^{2} - 4 x}{x^{3} - 9 x} \cdot \frac{x^{2} + 8 x + 15}{x^{2} - 2 x - 8} \end{aligned}$ .
::乘以 x2 - 4x23 - 9xx2+8x+15x2 - 2x-8。

Solution: Rather than multiply together each numerator and denominator to get very complicated , it is much easier to factor 1st and then cancel out any common factors.
::解答: 与其将每个分子和分母相乘, 以变得非常复杂,

$\begin{aligned} \frac{x^{2} - 4 x}{x^{3} - 9 x} \cdot \frac{x^{2} + 8 x + 15}{x^{2} - 2 x - 8} = \frac{x (x - 4)}{x (x - 3) (x + 3)} \cdot \frac{(x + 3) (x + 5)}{(x + 2) (x - 4)} \end{aligned}$

::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。

$\begin{aligned} \frac{x (x - 4)}{x (x - 3) (x + 3)} \cdot \frac{(x + 3) (x + 5)}{(x + 2) (x - 4)} = \frac{x + 5}{(x - 3) (x + 2)} = \frac{x + 5}{x^{2} - x - 6} \end{aligned}$

::x( x- 4) x( x- 4) x( x-3)( x+3)( x+3)( x+3)( x+5)( x+2)( x-4) =x+5( x-3( x+2) =x+5( x-3)( x+2) =x+5x2 -x6)

by CK-12 demonstrates how to multiply rational expressions.
::以 CK-12 显示如何乘以理性表达式。

Example 3
::例3

Multiply $\begin{aligned} \frac{4 x^{2} + 4 x + 1}{2 x^{2} - 9 x - 5} \cdot (3 x - 2) \cdot \frac{x^{2} - 25}{6 x^{2} - x - 2} \end{aligned}$ .
::乘以 4x2+4x+12x2-9x-5(3x-2)x2-256x2-x-2。

Solution: We can write the middle expression over 1 to help us keep the numerators and denominators together. Then, we factor, cancel, and multiply what remains.
::解答: 我们可以写一个以上的中间表达式, 帮助我们保持分子和分母的组合。然后, 我们考虑, 取消, 并乘以剩下的部分。

$\begin{aligned} \frac{4 x^{2} + 4 x + 1}{2 x^{2} - 9 x - 5} \cdot (3 x - 2) \cdot \frac{x^{2} - 25}{6 x^{2} - x - 2} \\ = \frac{(2 x + 1) (2 x + 1)}{(2 x + 1) (x - 5)} \cdot \frac{3 x - 2}{1} \cdot \frac{(x - 5) (x + 5)}{(3 x - 2) (2 x + 1)} \\ = x + 5 \end{aligned}$

::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

Dividing Rational Expressions
::Divide 理性表达式

Recall that when you divide fractions, you need to multiply by the reciprocal . (Sometimes said as, "Keep, switch, flip.") The same rule applies to dividing rational expressions.
::记得当分数分数时,你需要乘以对等法。(有时说,“保持、开关、翻转”)。同样的规则也适用于分解理性表达方式。

Dividing Rational Expressions
::Divide 理性表达式

For any expressions, A, B, C , and D :
::任何表示式A、B、C和D:

$\begin{aligned} \frac{A}{B} \div \frac{C}{D} = \frac{A}{B} \cdot \frac{D}{C} = \frac{A D}{B C} . \end{aligned}$

::ABCD=ABDC=亚银。

Example 4
::例4

Divide $\begin{aligned} \frac{5 a^{3} b^{4}}{12 a b^{8}} \div \frac{15 b^{6}}{8 a^{6}} \end{aligned}$ .
::5a3b412ab815b68a6除以5a3b412ab8

Solution: Change the $\begin{aligned} \div \end{aligned}$ sign to multiplication, ta ke the reciprocal of the 2nd fraction , and simplify.
::解决方案: 将 {} 符号更改为乘法, 接受第二分数的对等, 并简化。

$\begin{aligned} \frac{5 a^{3} b^{4}}{12 a b^{8}} \div \frac{15 b^{6}}{8 a^{6}} = \frac{5 a^{3} b^{4}}{12 a b^{8}} \cdot \frac{8 a^{6}}{15 b^{6}} = \frac{40 a^{9} b^{4}}{180 a b^{14}} = \frac{2 a^{8}}{9 b^{10}} \end{aligned}$

::5a3b412AB815b68a6=5a3b412ab8_8a615b6=40a9b4180AB14=2a89b10

Example 5
::例5

Divide $\begin{aligned} \frac{x^{4} - 3 x^{2} - 4}{2 x^{2} - x - 10} \div \frac{x^{3} - 3 x^{2} + x - 3}{x - 2} \end{aligned}$ .
::除以 x4 - 3x2 - 42x2 - x- 10x3 - 3x2+x-3x-2。

Solution: Change the $\begin{aligned} \div \end{aligned}$ sign to multiplication, ta ke the reciprocal of the 2nd fraction, and simplify .
::解决方案: 将 {} 符号更改为乘法, 接受第二分数的对等, 并简化。

$\begin{aligned} \frac{x^{4} - 3 x^{2} - 4}{2 x^{2} - x - 10} \div \frac{x^{3} - 3 x^{2} + x - 3}{x - 2} & = \frac{x^{4} - 3 x^{2} - 4}{2 x^{2} - x - 10} \cdot \frac{x - 2}{x^{3} - 3 x^{2} + x - 3} \\ = \frac{(x^{2} - 4) (x^{2} + 1)}{(2 x - 5) (x + 2)} \cdot \frac{x - 2}{(x^{2} + 1) (x - 3)} \\ = \frac{(x - 2) (x + 2) (x^{2} + 1)}{(2 x - 5) (x + 2)} \cdot \frac{x - 2}{(x^{2} + 1) (x - 3)} \\ = \frac{(x - 2)^{2}}{(2 x - 5) (x - 3)} \end{aligned}$

::x4 - 3x2 - 3x2 - 422 - 3x2 - 3x2 - 3x2=x4 - 3x2 - 3x2 - 3x2 - 4x2 - 3x2 - 3x2 - 3x2 - 3x2 - 3x2 - 3x2 - 3x2 - 3x2 - 422 - 3x2 - 3x2 - 4x2 - 3x2 - 3x3=(x2 - 4x2 - 4x2 - 4x2 - 4x2 - 4x2 - 4x2 - 4x2 - 4x2 - 4x2+1(x1)(2x - 5) =(x-2 - 5(x - 3)

by CK-12 demonstrates how to divide rational expressions.
::CK-12 演示如何区分合理表达方式。

Example 6
::例6

Perform the indicated operations : $\begin{aligned} \frac{x^{3} - 8}{x^{2} - 6 x + 9} \div (x^{2} + 3 x - 10) \cdot \frac{x^{2} + x - 12}{x^{2} + 11 x + 30} \end{aligned}$ .
::执行指定的操作 : x3- 8x2- 6x+9( x2+3x- 10)}x2+x- 12x2+11x+30 。

Solution: Since we have two operations, we need to use the . The order of operations tells us to multiply and divide from left to right. Therefore , we 1st multiply by the reciprocal of the 2nd expression. Now that it is multiplication throughout, we factor all the expressions. Next we cancel common factors. Finally, we multiply the numerators by the numerators and the denominators by the denominators.
::解决方案 : 由于我们有两个操作, 我们需要使用。操作的顺序告诉我们从左向右进行倍增和分割。因此, 我们首先乘以第二个表达式的对等。现在它正在乘以所有表达式。下一步我们取消共同因素。最后, 我们将数字乘以数数和分母乘以分母。

$\begin{aligned} \frac{x^{3} - 8}{x^{2} - 6 x + 9} \div (x^{2} + 3 x - 10) \cdot \frac{x^{2} + x - 12}{x^{2} + 11 x + 30} \\ = \frac{x^{3} - 8}{x^{2} - 6 x + 9} \cdot \frac{1}{x^{2} + 3 x - 10} \cdot \frac{x^{2} + 2 x - 15}{x^{2} + 11 x + 30} \\ = \frac{(x - 2) (x^{2} + 2 x + 4)}{(x - 3) (x - 3)} \cdot \frac{1}{(x - 2) (x + 5)} \cdot \frac{(x + 5) (x - 3)}{(x + 5) (x + 6)} \\ = \frac{x^{2} + 2 x + 4}{(x - 3) (x + 5) (x + 6)} \end{aligned}$

::x3 - 8x2 - 6x+9 (x2+3x- 10) x2+xx12x2+11x30=x12x2+30=x3x3-8x2 - 6x6x+9_1x2+3x- 10xxxx2+2x- 15x2+2x- 15x2+15x2+11xx=(x-2)(x2+2+2xx4+4)(x3)(x-3)(x3)(x-3)(x3)(x3)(x3)(x3)(x+5)(x+6) =x2+2x4(x-3)(x+5)(x+6)

Complex Fractions
::复杂分数

A complex fraction is a fraction that has fractions in the numerator and/or denominator.
::复杂分数是指分子和/或分母中含有分数的分数。

How to Simplify
::如何简化

For any expression, A, B, C, and D :
::任何一种表达方式,A、B、C和D:

$\begin{aligned} \frac{A / B}{C / D} = \frac{A}{B} \div \frac{C}{D} = \frac{A}{B} \cdot \frac{D}{C} . \end{aligned}$

::A/BC/D=ABCD=ABDC。

1. If the complex fraction has one fraction in the numerator and one fraction in the denominator, rewrite as a division problem and solve as above.
::1. 如果复杂分数在分子中有一分数,分母中有一分数,则重写为分解问题,并按上述方式解决。

2. If there is addition or subtraction between the fractions in the numerator or the denominator, you can find a common denominator for all the fractions and multiply the numerator and denominator by that common denominator.
::2. 如果分子或分母中的分数有增减,您可以为所有分数找到一个共同的分母,并将分子和分母乘以该共同分母。

3. If there is addition or subtraction between the fractions in the numerator or the denominator, you can find a common denominator for the top and a common denominator for the bottom, and then proceed as in step 1.
::3. 如果分子或分母中的分数有增减,您可以找到顶部的共同分母和底部的共同分母,然后按第1步进行。

Example 7
::例7

Simplify $\begin{aligned} \frac{\frac{9 x}{x + 2}}{\frac{3}{x^{2} - 4}} \end{aligned}$ .
::简化 9xx+23x2-4。

Solution: This complex fraction is a fraction divided by another fraction. Rewrite the complex fraction as a division problem.
::解析 : 此复杂分数是一个分数除以另一个分数。将复杂分数重写为分割问题。

$\begin{aligned} \frac{\frac{9 x}{x + 2}}{\frac{3}{x^{2} - 4}} = \frac{9 x}{x + 2} \div \frac{3}{x^{2} - 4} \end{aligned}$

::9xx+23x2 - 4=9xx+2}3x2 - 4

Change the $\begin{aligned} \div \end{aligned}$ sign to multiplication, ta ke the reciprocal of the 2nd fraction, and simplify .
::将 {} 符号更改为乘法, 接受第二分数的对等, 并简化。

$\begin{aligned} \frac{9 x}{x + 2} \div \frac{3}{x^{2} - 4} = \frac{9 x}{x + 2} \cdot \frac{x^{2} - 4}{3} = \frac{\overset{3}{9 x}}{x + 2} \cdot \frac{(x + 2) (x - 2)}{3} = 3 x (x - 2) \end{aligned}$

::9xx+23x2-4=9xx+2x2x2-43=9x3xxx2}(x+2)(x-2)(x-2)3=3x(x-2)

by Mathispower4u demonstrates how to simplify complex fractions.
::由 Mathispower4u 演示如何简化复杂分数。

Example 8
::例8

Simplify $\begin{aligned} \frac{\frac{1}{x} + \frac{1}{x + 1}}{4 - \frac{1}{x}} \end{aligned}$ .
::简化 1x+1x+14-1x 。

Solution: To simplify this complex fraction, we can add the fractions in the numerator and subtract the two in the denominator. The LCD of the numerator is $\begin{aligned} x (x + 1), \end{aligned}$ and the denominator is just $\begin{aligned} x \end{aligned}$ .
::解答: 要简化此复杂分数, 我们可以在分子中添加分数, 并在分母中减去两个分数。分子的液晶是 x( x+1) , 分数只是 x 。

$\begin{aligned} \frac{\frac{1}{x} + \frac{1}{x + 1}}{4 - \frac{1}{x}} = \frac{\frac{x + 1}{x + 1} \cdot \frac{1}{x} + \frac{1}{x + 1} \cdot \frac{x}{x}}{\frac{x}{x} \cdot 4 - \frac{1}{x}} = \frac{\frac{x + 1}{x (x + 1)} + \frac{x}{x (x + 1)}}{\frac{4 x}{x} - \frac{1}{x}} = \frac{\frac{2 x + 1}{x (x + 1)}}{\frac{4 x - 1}{x}} \end{aligned}$

::1x+1x1x+14-1x=1x1x1x1x1x1x1x1x1x1x1xxxxxxxx1xxxx1xxx1x1x1x1x1xxxxxxxx+1x1x1x1x1x1xxx1x1x1x1x1xx1x1x1xx1xx1x1x1xx1xxx1xxx1xxx1xx1xxx1xxxxxx1xxx1xxx1x1x1x1x1x1xxxx1xxx1x1x1x1x1x1x1x1x1x1xxx1xx1x1xxxx1x1x1x1xx1x1x1x1x1xx1x1x1x1x1x1x1xx1x1x1xx1x1x1x1xxx1x1xx1x1xxx1x1x1x1xx1xx1xx1xxxx1x1x1xx1xxx1xxxx1xxxxx1xxxx1xxxxxxxxxxx1xxxxxxxxxxx1xxxxxxxxxxxxx1xxxxxxx1xxxxxxxx1xxxxxxxxxxxxxxxxxxxxxxx

Divide and simplify if possible.
::在可能的情况下,分开和简化。

$\begin{aligned} \frac{\frac{2 x + 1}{x (x + 1)}}{\frac{4 x - 1}{x}} = \frac{2 x + 1}{x (x + 1)} \div \frac{4 x - 1}{x} = \frac{2 x + 1}{x (x + 1)} \cdot \frac{x}{4 x - 1} = \frac{2 x + 1}{(x + 1) (4 x - 1)} \end{aligned}$

::2x+1x( x+1) 4x -1x=2x1x1x( x+1) 4x -1x=2x1x( x+1) x4x -1x=2x1x( x+1) x4x -1x=2x1x1( x+1) (4x-1)

Example 9
::例9

Consider a circuit where the voltage is 12 volts, and two resistors—one 5 ohms more than the other—are connected in parallel, which means the voltage across the resistors is equal. An expression for the current is $\begin{aligned} \frac{12}{(\frac{1}{x} + \frac{1}{x + 5})} \end{aligned}$ . Simplify the expression.
::电压为 12 伏的电路, 两个阻力器—— 比另一个高1 5 ohms—— 平行连接, 这意味着阻力器的电压是相等的。电流的表达方式是 12(1x+1x+5) 。简化表达式。

Solution: We can multiply the numerator and the denominator by the LCD for all of the fractions. Make sure to multiply all the terms.
::解答: 我们可以将所有分数的分子和分母乘以液晶体, 以确保将所有的术语乘以。

$\begin{aligned} \frac{12}{(\frac{1}{x} + \frac{1}{x + 5})} \cdot \frac{x (x + 5)}{x (x + 5)} & = \frac{12 x (x + 5)}{\frac{1}{x} \cdot x (x + 5) + \frac{1}{x + 5} \cdot x (x + 5)} \\ = \frac{12 x (x + 5)}{x + 5 + x} \\ = \frac{12 x^{2} + 60 x}{2 x + 5} \end{aligned}$

::12(1x+1x+5+5)x(x+5)xx(x+5)=12x(x+5)1xxx(x+5)+1x+5x(x+5)x(x+5)=12x(x+5)x+5+5x=12x2+60x2x+5

by Mathispower4u demonstrates how to simplify complex fractions.
::由 Mathispower4u 演示如何简化复杂分数。

Example 10
::例10

Simplify $\begin{aligned} \frac{\frac{5 - x}{x^{2} + 6 x + 8} + \frac{x}{x + 4}}{\frac{6}{x + 2} - \frac{2 x + 3}{x^{2} - 3 x - 10}} \end{aligned}$ .
::简化 5 -xx2+6x+8+xx+46x+2 -2x+3x2 -3x2 -3x-10。

Solution: First, add the fractions in the numerator and subtract the ones in the denominator.
::解答:首先,在分子中添加分数,并在分母中减去分数。

$\begin{aligned} \frac{\frac{5 - x}{x^{2} + 6 x + 8} + \frac{x}{x + 4}}{\frac{6}{x + 2} - \frac{2 x + 6}{x^{2} - 3 x - 10}} = \frac{\frac{5 - x}{(x + 4) (x + 2)} + \frac{x}{x + 4} \cdot \frac{x + 2}{x + 2}}{\frac{x - 5}{x - 5} \cdot \frac{6}{x + 2} - \frac{2 x + 6}{(x + 2) (x - 5)}} = \frac{\frac{5 - x + x (x + 2)}{(x + 4) (x + 2)}}{\frac{6 (x - 5) - (2 x + 6)}{(x + 2) (x - 5)}} = \frac{\frac{x^{2} + x + 5}{(x + 4) (x + 2)}}{\frac{4 x - 36}{(x + 2) (x - 5)}} \end{aligned}$

::5 -xx2+6x8+6x8+6x8+6x6x6+6x2+6x2+2x2+2x2x2x6x2x6x8+6xxx6+6x6xx6+6x6xx6+6xx2x2x2x+6x2xx2x2x2xx2x2x6xx6xx2x6xx6x6xx6xx2x2x6xxx6x6x6x6xx2x2xxxxxx3xxx10x10x5xxxxx+4xx4x+4x2x+2xx5x5x5x5x5x6xxxx2x6x2x(x2+2x5x5xxxx2xx)6x5xxx5xxxxxx2+2x4x2(x2+2x4x5x5)xxx5xxx5xxxx6x6x6x5xxxxx6xxxx6x6x6x6x6x6xxxxxxxx6x6xxx6x6x6x6xx6x6x6xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx6x6x6x6x6x6x6x6x6x6x6x6x6

Now, rewrite as a division problem, take the reciprocal , multiply, and simplify.
::现在,重写为一个分裂问题, 采取对等, 倍增,和简化。

$\begin{aligned} \frac{\frac{x^{2} + x + 5}{(x + 4) (x + 2)}}{\frac{4 x - 36}{(x + 2) (x - 5)}} \frac{x^{2} + x + 5}{(x + 4) (x + 2)} \div \frac{4 x - 36}{(x + 2) (x - 5)} \\ = \frac{x^{2} + x + 5}{(x + 4) (x + 2)} \cdot \frac{(x + 2) (x - 5)}{4 (x - 9)} \\ = \frac{(x^{2} + x + 5) (x - 5)}{4 (x + 4) (x - 9)} \end{aligned}$

::x2+x5(x+4)(x+4)(x+2)(x+2)(x-5)(x+2)(x-5)(x+4)(x+2)(x+2)4x-36(x+2)(x-5)(x-2)(x-5)=x2+5(x+4(x+4)(x+4)(x+2)(x+2)(x-5)(x-5)(x+9)=(x2+x+5)(x+5)(x-5)(x-5)4(x+4)(x-9)

Feature: Resistance Is Futile
::特点:抵抗是发泡

by Deirdre Mundy
::由Deirdre Mundy 编辑

Every electronic device you own is full of tiny parts called resistors. These resistors help protect the sensitive chips inside your phone, computer, and camera. They are the reason why your devices become hot to the touch as they are charging. Without resistors, none of the electronics in your life would work.
::你拥有的每个电子设备都装满了微小的部件,叫做抵抗器。这些抵抗器帮助保护你的手机、电脑和相机中的敏感芯片。这就是为什么你的装置在充电时会变得热到触碰。没有抵抗器,你生活中的电子设备都不会起作用。

Currents and Volts
::洋流和伏

Ohm's Law describes how electricity flows through a wire. According to German physicist Georg Ohm, the current, or the rate of the flow of electric charge, is equal to the voltage divided by the resistance. Resistance slows down the current. In most devices, engineers use multiple resistors to slow down the current as it races through the wires. The resistors reduce the current by turning some of the electrical energy into heat energy. When a resistor is working, it becomes hot. As engineers add resistors to a circuit, the rational expression for voltage divided by resistance can get quite complicated. Regardless, the current has to be calculated accurately in order to protect sensitive parts.
::Ohm's Law 描述电力如何通过电线流动。根据德国物理学家Georg Ohm, 电流或电流速度相当于抵抗力量所分离的电压。抵抗运动放慢了电流的速度。在大多数装置中, 工程师使用多种抵抗器来减慢电流, 当电流通过电线运动时, 抵抗者通过将部分电能转化为热能来减少电流。当抵抗者起作用时, 电流会变得热。随着工程师在电路中添加阻力, 抵抗力量所分裂的电压的合理表达方式会变得相当复杂。不管怎样, 电流必须精确计算, 以保护敏感部分。

When you charge your phone, its resistors go into action. They make sure the charge does not damage sensitive hardware. If a resistor overheats, it breaks. However, this break also breaks the circuit and stops the flow of electricity. This means that an overloaded resistor will not damage the rest of your phone. Without resistors, we would not be able to use technology like touch screens or LED lights. We would not be able to listen to music or even use a blender. Even our cars depend on resistors. By understanding how resistors work, you can learn to troubleshoot the machines that surround us.
::当你充电时,它的阻力器就会启动,它们确保充电不会损坏敏感的硬件。如果阻力器过热,它就会破裂。但是,这个断线器也会断电,停止电流。这意味着一个超载的阻力器不会破坏你手机的其余部分。没有阻力器,我们就无法使用触摸屏幕或LED灯等技术。我们无法听音乐,甚至不能使用搅拌器。即使我们的汽车也依赖阻力器。通过理解阻力器如何运作,你可以学会打乱我们周围的机器。

by The Verge demonstrates why phone batteries like the Note 7's explode.
::Verge展示了为什么像Note 7这样的电话电池爆炸的原因。

Summary
::摘要

To multiply rational expressions, we multiply the numerators together and the denominators together. Factor the rational expressions before multiplying to make simplifying easier.
::乘以理性表达方式, 我们将分子和分母相乘。乘以理性表达方式后再乘以简化。

To divide rational expressions, multiply by the reciprocal.
::将合理表达方式分开,乘以对等。

If the complex fraction has one fraction in the numerator and one fraction in the denominator, rewrite as a division problem.
::如果复杂分数在分子中有一个分数,分母中有一个分数,则重写为分裂问题。

If there is addition or subtraction between the fractions in the numerator or the denominator, you can find a common denominator for all the fractions and multiply the numerator and denominator by that common denominator; or, you can find a common denominator for the top and a common denominator for the bottom, and then rewrite as a division problem.
::如果在分子或分母中的分数之间有增减,您可以为所有分数找到一个共同的分母,并将分子和分母乘以该共同分母;或者,您可以为顶部找到一个共同的分母,为底部找到一个共同的分母,然后重写为分裂问题。

Review
::回顾

Perform the operations below. Simplify your answers.
::执行下面的操作。简化您的答案。

1. $\begin{aligned} \frac{11 x^{3} y^{9}}{2 x^{4}} \cdot \frac{6 x^{7} y^{2}}{33 x y^{3}} \end{aligned}$
::1. 11x3y92x4x6x7y233xy3

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

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

4. $\begin{aligned} \frac{8 x^{2} - 10 x - 3}{4 x^{3} + x^{2} - 36 x - 9} \cdot \frac{5 x + 3}{x - 1} \cdot \frac{x^{3} + 3 x^{2} - x - 3}{5 x^{2} + 8 x + 3} \end{aligned}$
::4. 8x2-10x-34x3+3x2-36x-9=5x+3x-1x3+3x2-x35x2+8x+3

5. $\begin{aligned} \frac{16 x^{3} y^{9} z^{3}}{15 x^{5} y^{2} z} \div \frac{42 x y^{7} z^{2}}{45 x^{2} y z^{5}} \end{aligned}$
::5. 16x3y9z315x5y2z @42xy7z245x2yz5

6. $\begin{aligned} \frac{x^{2} + 6 x + 2}{12 - 3 x} \div \frac{6 x^{2} - 13 x - 5}{x^{2} - 4 x} \end{aligned}$
::6. x2+6x+212-3x=6x2-13x-5x2-4x

7. $\begin{aligned} \frac{x^{2} - 5 x}{x^{2} + x - 6} \div \frac{x^{2} - 2 x - 15}{x^{3} + 3 x^{2} - 4 x - 12} \end{aligned}$
::7. x2 - 5xx2+x - 6x2 - 2x - 15x3+3x2 - 4x - 12

8. $\begin{aligned} \frac{x^{2} + 2 x - 15}{2 x^{3} + 7 x^{2} - 4 x} \div (5 x + 3) \div \frac{21 - 10 x + x^{2}}{5 x^{3} + 23 x^{2} + 12 x} \end{aligned}$
::8. x2+2x - 152x3+7x2 - 4x( 5x+3) 21 - 10x+x25x3+23x2+12x

9. $\begin{aligned} \frac{\frac{4}{x^{2} - 9}}{\frac{6 x}{x + 3}} \end{aligned}$
::9. 4x2-96xxx+3

10. $\begin{aligned} \frac{\frac{7 x^{3}}{x^{2} + 5 x + 6}}{\frac{35 x^{2}}{x + 2}} \end{aligned}$
::10. 7x3x2+5x+635x2x2+2

11. $\begin{aligned} \frac{\frac{24 x + 3}{3 x + 1}}{\frac{16 x + 2}{6 x^{2} - 13 x - 5}} \end{aligned}$
::11. 24x+33x+116x+26x2-13x-5

12. $\begin{aligned} \frac{\frac{4}{x - 1} + \frac{1}{x}}{\frac{1}{x} - 5} \end{aligned}$
::12. 4x-1+1x1x-5

13. $\begin{aligned} \frac{\frac{x}{x + 3} - \frac{4}{2 x + 1}}{\frac{3}{2 x + 1} + \frac{6}{x^{2} - 9}} \end{aligned}$
::13.xx+3-42x+132x+1+6x2-9

14. $\begin{aligned} \frac{\frac{x + 3}{x} + \frac{2 x}{5 - x}}{\frac{3}{2 x} - \frac{4 x}{x - 5}} \end{aligned}$
::14. x+3x+2x5-x32x-4xx-5

15. $\begin{aligned} \frac{\frac{3 x}{x^{2} - 4} + \frac{x + 4}{x^{2} + 3 x + 2}}{\frac{x + 1}{x^{2} - x - 2} - \frac{2 x}{x^{2} + 2 x + 1}} \end{aligned}$
::15. 3xx2-4+x4x2+3x2x2+2x1x2-x-2-2-2x2+2x1+1

Explore More
::探索更多

1. Error Analysis: Identify the mistake and correct it. $\begin{aligned} \frac{(4 x + 12)}{12} = 4 x . \end{aligned}$
::1. 错误分析:查明错误并纠正错误。 (4x+12)12=4x。

2. Ohm’s Law ¹ states that in an electrical circuit, $\begin{aligned} I = \frac{V}{R_{c}} \end{aligned}$ . The total resistance for resistors placed in parallel is given by $\begin{aligned} \frac{1}{R_{t o t}} = \frac{1}{R_{1}} + \frac{1}{R_{2}} \end{aligned}$ . Write the formula for the electric current in terms of the component resistances $\begin{aligned} R_{1} \end{aligned}$ and $\begin{aligned} R_{2} \end{aligned}$ .
::2. Ohm ' s Law1 规定,在电路中,I=VRc。 1Rtot=1R1+1R2给出了平行置放阻力的完全抵抗力。用R1和R2的部件阻力写电流的公式。

3. The U.S. Department of Commerce likes to keep track of the amount of money spent in different countries each year by Americans. Mathematicians have helped the department figure out a pattern that best represents this amount ² .
::3. 美国商务部喜欢追踪美国人每年在不同国家花费的钱款数额,数学家帮助该部找出了最能代表这一数额的模式。

For the sake of this exercise, the function that represents the total amount of money that Americans have spent on foreign travel every year since 2000 is $\begin{aligned} T (x) = (3 x^{2} + 15 x) / (3 x^{2} + 12 x) \end{aligned}$ , where x represents the number of years that have passed since 1990.
::为了开展这项工作,2000年以来美国人每年用于国外旅行的金额总额,即T(x)=(3x2+15x)/(3x2+12x),X代表1990年以来的年数。

A function has also been determined to represent the number of people who have left the United States to travel each year: $\begin{aligned} P (x) = (x^{2} - x - 30) / (x^{2} - 7 x + 6) \end{aligned}$ , where x also represents the number of years that have passed since 1990.
::还确定了一个函数,以代表每年离开美国旅行的人数:P(x)=(x2-x-30-30)/(x2-7x+6),其中x还代表1990年以来的年数。

Divide the total amount of money spent ( $\begin{aligned} T (x) \end{aligned}$ ) by the total number of Americans who have traveled ( $\begin{aligned} P (x) \end{aligned}$ ) to find the average amount of money spent by each American per year. Express your answer in simplest form.
::将花费的金额总额(T(x))除以为寻找每个美国人每年平均花费金额而旅行的美国人总数(P(x)),并以最简单的形式表达你的答复。

3. Determine if the following statements are true or false. If false, explain why.
::3. 确定以下陈述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。

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

4. a. The length of a rectangle is $\begin{aligned} \frac{2 x y^{3} z}{5 x y z^{2}} \end{aligned}$ . The width of the rectangle is $\begin{aligned} \frac{3 x^{2} y z^{3}}{4 x^{3} y^{2} z^{2}} \end{aligned}$ . What is the area of the rectangle?
::4. a. 矩形的长度为 2xy3z5xyz2. 矩形的宽度为 3x2yz34x3y2z2. 矩形的面积是 3x2y34x3y2z2. 矩形的面积是多少?

b. The area of a rectangle is $\begin{aligned} \frac{12 x^{2} y z^{3}}{5 x y^{2} z} \end{aligned}$ . The length of the rectangle is $\begin{aligned} \frac{2 x y}{z^{2}} \end{aligned}$ . What is the width of the rectangle?
::b. 矩形区域为 12x2yz35xy2z。矩形的长度为 2xyz2。矩形的宽度是多少?

c. Gupta knows the area and width of a rectangle. He comes up with this equation for the length of the rectangle: $\begin{aligned} \frac{\frac{2}{x^{2} - 1}}{\frac{2 x}{x + 1}} \end{aligned}$ . What is the length of the rectangle in simplified form?
::c. Gupta知道矩形的面积和宽度,他提出矩形长度的这个方程式: 2x2- 12xx+1. 简化形式的矩形长度是多少?

5. We all know that when you divide fractions, you take the 2nd fraction, take the reciprocal, and change it to a multiplication problem. But do you know why? Let's investigate the why here.
::5. 我们都知道,当分分数分数时,你拿第二分数,拿对等分数,然后把它变成乘法问题,但你知道原因吗?让我们在这里调查原因。

What is $\begin{aligned} 6 \div 2 \end{aligned}$ ? What about $\begin{aligned} \frac{1 \div 1}{6 \div 2} \end{aligned}$ ? Are these two problems the same ? Why or why not?
::62是什麽?1162是什麽?这两个问题是一样的吗?为什么或为什么没有?

6. Use the following pattern to answer the four questions below.
::6. 使用以下模式回答以下四个问题。

$\begin{aligned} 2 + \frac{1}{1 + \frac{1}{2}}, 2 + \frac{1}{1 + \frac{1}{2 + \frac{2}{3}}}, 2 + \frac{1}{1 + \frac{1}{2 + \frac{2}{3 + \frac{3}{4}}}} \end{aligned}$

Find the next two terms in the pattern.
::查找图案中的下两个术语。

Using your graphing calculator, simplify each term in the pattern to a decimal.
::使用您的图形计算计算器,将模式中的每个词简化为小数点后的一个词。

Make a conjecture about this pattern and the number the terms appear to be approaching.
::对这种模式和术语似乎即将到来的数量进行猜测。

Find the 6th term in the pattern. Does it support your conjecture?
::在模式中查找第6个学期。它是否支持您的猜测 ?

Answers for Review and Explore More Problems
::回顾和探讨更多问题的答复

Please see the Appendix.
::请参看附录。

PLIX
::PLIX

Try this interactive that reinforces the concepts explored in this section.
::尝试一下这种互动关系,加强本节所探讨的概念。

References
::参考参考资料

1. "Ohm’s Law," last edited May 17, 2017,
::2017年5月17日,

2. "Research Spotlight: Estimates of Categories of Personal Consumption Expenditures Adjusted for Net Foreign Travel Spending," by Michael Armah and Teresita Teensma, April 2012,
::2. " 研究焦点:按外国旅行支出净额调整的个人消费支出类别估计数 " ,Michael Armah和Teresita Teensma著,2012年4月。

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