Section: 使用指数属性写入等价表达式 | 9.4 使用指数属性写入等价表达式-interactive

Section outline

Equivalent Expressions
::对等表达式

In this lesson , you will explore the idea of writing using properties of exponents. The idea of writing an equivalent expression is similar to the idea of simplifying a fraction . The point is to write these expressions in simplest terms . For example, it is better to write $\begin{aligned} {3 a}^{7} \end{aligned}$ than $\begin{aligned} 3 a^{2} \cdot a^{5} \end{aligned}$ . Why do you think it might be better to write an expression in simplest terms?
::在此课中, 您将探索使用 expents 属性写字的想法。写一个等同表达式的想法与简化一个分数的想法相似。要点是用最简单的术语写这些表达式。例如, 写3a7 要比 3a2aa5 更好。为什么你认为写一个最简单的表达式会更好 ?

Negative Exponents
::负指数

Example
::示例示例示例示例

Simplify the expression $\begin{aligned} {(3 x y^{- 5})}^{2} \end{aligned}$ .
::简化表达式( 3xy-5) 2。

When writing equivalent expressions, there are often several possible orders in which to simplify them. However, they will all take you to the same result as long as you do not make a mistake when using the properties. In this example, you will distribute the outer exponent first using the Power of a Product Property .
::当写入等同表达式时, 通常会有几种可能的简化命令。但是, 只要您使用该属性时没有犯错, 它们都会将您带到相同的结果。在此示例中, 您将首先使用产品属性的权势分配外部名人。

\begin{aligned} {(3 x y^{- 5})}^{2} = 3^{2} x^{2} {(y^{- 5})}^{2} \end{aligned}

3xy-5-5)2=32x2(y-5)2

Next, use the Power of a Power Property .
::接下来,使用电力财产的权力。

\begin{aligned} 3^{2} x^{2} {(y^{- 5})}^{2} = 3^{2} x^{2} y^{- 10} \end{aligned}

::32x2(y-5)2=32x2y-10

Write 3 ² as 9. Change the negative exponent to a positive by moving the exponent and its base to the denominator. The base of the negative exponent is $\begin{aligned} y, \end{aligned}$ so this is the only factor that will be moved.
::以 9 写为 32 。将负表示数改为正。负表示数的基数是 y , 所以这是唯一要移动的因数。

\begin{aligned} \frac{9 x^{2}}{y^{10}} \end{aligned}

::9x2y10

Discussion Question
::讨论问题

Could you have simplified this expression in a different order ?
::您能按不同的顺序简化这个表达式吗 ?

Fractional Expressions
::小数表达式

Example
::示例示例示例示例

Simplify the expression $\begin{aligned} \frac{30 r^{4} s^{5}}{5 r^{6} s} \end{aligned}$ .
::简化表达式30r4s55r6s。

In this example, the only real option is to combine the similar factors in the numerator and denominator. You can do this by using the Quotient of Powers Property or by dividing out like factors using expanded form . In this case, you will use the Quotient of Powers Property.
::在此示例中,唯一真正的选择是将分子和分母中的类似因素结合起来。您可以使用“ 权力财产的报价” 来做到这一点, 或者使用扩展形式来区分类似因素。在这种情况下, 您将使用“ 权力财产的报价 ” 。

\begin{aligned} \frac{30 r^{4} s^{5}}{5 r^{6} s} = \frac{30}{5} r^{4 - 6} s^{5 - 1} = \frac{30}{5} r^{- 2} s^{4} \end{aligned}

::30r4s55r6s=305r4-6s5-1=305r-2s4

Simplify $\begin{aligned} \frac{30}{5} \end{aligned}$ to 6. Change the negative exponent to a positive by moving the exponent and its base to the denominator.
::简化305至6. 将负表征改为正表征,将表征及其基础提升到分母。

\begin{aligned} \frac{30}{5} r^{- 2} s^{4} = \frac{6 s^{4}}{r^{2}} \end{aligned}

::305r-2s4=6s4r2

Use the interactive below to explore further the concept of simplifying a fractional expression.
::利用以下互动方式进一步探讨简化片面表达方式的概念。

INTERACTIVE

Integer Powers

Kris wants to know how to simplify an expression that includes variables raised to different powers.
::Kris想知道如何简化表达式,该表达式包括向不同权力机关提出的变量。

Drag the colored points to manipulate the exponent values of each variable in the numerator and denominator.
::拖曳彩色点,以操纵分子和分母中每个变量的指数值。

Observe the changing exponent values in the simplified expression.
::观察简化表达式中变化中的指数值。

Fractional Expressions With Negative Exponents
::带有负指数的小数表达式

Example
::示例示例示例示例

Simplify $\begin{aligned} \frac{15 a^{- 4} b^{2} c}{25 a^{2} b^{7} c^{0}} \end{aligned}$
::简化 15a- 4b2c25a2b7c0

Begin by using the Quotient of Powers Property to combine similar factors.
::首先,使用“权力财产报价”将类似因素结合起来。

\begin{aligned} \frac{15 a^{- 4} b^{2} c}{25 a^{2} b^{7} c^{0}} = \frac{15}{25} a^{- 4 - 2} b^{2 - 7} c^{1 - 0} = \frac{15}{25} a^{- 6} b^{- 5} c^{1} \end{aligned}

::15-4b2c25a2b7c0=1525a-4-2b2-7c1-0=1525a-6b-5c1

Finally, reduce $\begin{aligned} \frac{15}{25} \end{aligned}$ to $\begin{aligned} \frac{3}{5} \end{aligned}$ and write the negative exponents as positive exponents by moving them and their bases to the denominator.
::最后,将1525至35人减为1525至35人,并将负指数写成正面指数,将它们及其基地移到分母上。

\begin{aligned} \frac{15}{25} a^{- 6} b^{- 5} c = \frac{3 c}{5 a^{6} b^{5}} \end{aligned}

::1525a-6b-5c=3c5a6b5

Discussion Question
::讨论问题

Could we have done anything to simplify the expression before using the Quotient of a Power Property ?
::在使用电力财产的报价之前,我们是否可以做任何事情来简化表达方式?

Distribution
::分发分发情况

Example
::示例示例示例示例

Simplify $\begin{aligned} \frac{{(8 u v^{- 2})}^{2}}{16 u^{- 3} v^{- 1}} \end{aligned}$
::简化( 8v- 2) 216u- 3v- 1

Before you can use the Quotient of Powers Property, you must distribute the outer exponent to the numerator.
::在使用“权力财产引号”之前,您必须先向分子分发外部引号。

\begin{aligned} \frac{{(8 u v^{- 2})}^{2}}{16 u^{- 3} v^{- 1}} = \frac{8^{2} u^{2} {(v^{- 2})}^{2}}{16 u^{- 3} v^{- 1}} \end{aligned}

8v-2)216u-3v-1=82u2(v-2)216u-3v-1

Then, after you simplify $\begin{aligned} 8^{2} \end{aligned}$ to 64 and $\begin{aligned} {(v^{- 2})}^{2} \end{aligned}$ to $\begin{aligned} v^{- 4} \end{aligned}$ using the Power of a Power Property, you can use the Quotient of Powers Property to simplify the fraction.
::然后,在使用电力财产的权力简化82至64和(v-2-2)2至v-4之后,您可以使用权力财产报价来简化部分。

\begin{aligned} \frac{{64 u^{2} v}^{- 4}}{16 u^{- 3} v^{- 1}} = \frac{64}{16} u^{2 - (- 3)} v^{- 4 - (- 1)} = \frac{64}{16} u^{5} v^{- 3} \end{aligned}

::64u2v-416u-3v-1=6416u2-(-3)v-4-(-1)=6416u5v-3)

Finally, reduce $\begin{aligned} \frac{64}{16} \end{aligned}$ to 4 and write the negative exponent as a positive exponent by moving it, and its base to the denominator.

\begin{aligned} \frac{4 u^{5}}{v^{3}} \end{aligned}

::最后,将6416减为4, 将负表情写成正面的表情, 将其移到底部, 并将其底部移到分母上。 4u5v3

Discussion Questions
::讨论问题讨论问题

Why did we have to distribute the outer exponent in the numerator before using the quotient of a power rule?
::为什么我们要在使用权力规则的商数之前在分子中分配外表?

Is there another way we could have simplified this expression?
::我们还有别的办法可以简化这个表达方式吗?

Is there a general order to simplify expressions? What order fo you prefer?
::是否有简化表达方式的一般顺序?您喜欢什么顺序?

Use the interactive below to explore further.
::利用下面的交互方式进一步探讨。

INTERACTIVE

Simplifying Exponents

When dividing like terms, you can use the commutative and associative properties of division to group and simplify the coefficients and variables separately.

::当像用词一样的分隔时,您可以使用除法的通和和关联属性来分组和分别简化系数和变量。

Slide the colored points at the left of each expression to change the colored values.
::将每个表达式左侧的彩色点幻灯片,以更改彩色值。

See how changing the coefficients and powers of variables changes how the expressions are simplified.
::看看变数的系数和功率的变化如何改变表达式的简化。

Summary

Use these exponent property to simplify exponential expressions:
- $\begin{aligned} (a^{m})^{n} = a^{m n} \end{aligned}$
  :am)n=amn(上午)
- $\begin{aligned} (a b)^{n} = a^{n} \times b^{n} \end{aligned}$
  :ab)n=anxbn
- $\begin{aligned} a^{- m} = \frac{1}{a^{m}} \end{aligned}$
  ::a-m=1毫米
::使用这些引号属性简化指数表达式am)n=amn (ab)n=anxbn a-m=1am

Section outline

Equivalent Expressions ::对等表达式

Negative Exponents ::负指数

Fractional Expressions ::小数表达式

Fractional Expressions With Negative Exponents ::带有负指数的小数表达式

Distribution ::分发分发情况

Equivalent Expressions
::对等表达式

Negative Exponents
::负指数

Fractional Expressions
::小数表达式

Fractional Expressions With Negative Exponents
::带有负指数的小数表达式

Distribution
::分发分发情况