1.12 名人财产和科学称号

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  指数属性和科学名词

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  The magnitude of an earthquake,  m,  is the exponent in the expression $\begin{align*}{10}^{m}\end{align*}$ . ¹  
  ::地震的大小, m,是表达式 10m1 中的缩略语。

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  Valdivia, Chile has suffered two major earthquakes. The 1575 Valdivia earthquake had a magnitude of 8.5. The world's largest earthquake was the 1960 Valdivia earthquake at a magnitude of 9.5. ²
  ::智利的巴尔迪维亚发生了两次重大地震,1575年的巴尔迪维亚地震发生8.5次,世界最大的地震是1960年的巴尔迪维亚地震,地震规模为9.5.2次。

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  How much bigger  was the 9.5 earthquake than the 8.5 earthquake? We discuss how to evaluate exponents and use exponents rules in this section.
  ::9.5的地震比8.5的地震大多少?我们讨论如何评估标语和在本节使用标语规则。

  

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  Product Rule of Exponents
  ::指数值规则

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  To review, the power or exponent of a number is the number that is the superscript. The number that is being raised to the power is called the base . The exponent indicates how many times the base is multiplied by itself.
  ::要审查,一个数字的功率或引号是上标的数。向该功率提出的数称为基数。引号表示基数乘以多少倍。

  

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  Investigation: Product Rule
  ::调查:产品规则

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  1. Expand $\begin{align*}{3}^{4}\cdot {3}^{5}\end{align*}$ .
  
  $\begin{align*}\underbrace{3 \cdot 3 \cdot 3 \cdot 3}_{3^4} \cdot \underbrace{3 \cdot 3 \cdot 3 \cdot 3 \cdot 3}_{3^5}\end{align*}$
  ::1. 展开34-35. 3-3-3-3 3-3 3-3-3 3-3 3-3 3-3 3-3 3-3 3-3 3-3 3-3 3-3 3

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  2. Rewrite this expansion as one power of three. 
  ::2. 将这种扩大重新改成三分之一的功率。

  $\begin{align*}3\cdot 3\cdot 3\cdot 3\cdot 3\cdot 3\cdot 3\cdot 3\cdot 3={3}^{9}\end{align*}$

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  3. What is the sum of the exponents?
  ::3. 出纳者的总和是多少?

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  The sum of the exponents is 9 since 4 + 5 = 9.
  ::从4+5=9起,指数总数为9。

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   Product Rule of Exponents
  ::指数值规则

    $$\begin{align*}{a}^{m}\cdot {a}^{n}={a}^{m+n}\end{align*}$$  

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  Rather than expand the exponents every time or find the powers separately, we can use this property to simplify the product of two terms  with the same base.
  ::与其每次扩大引言者,或单独找到权力,不如使用这种财产来简化两个条件的产物,其基础相同。

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  Example 1
  ::例1

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  Simplify:
  ::简化 :

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  a.  $\begin{align*}{x}^{3}\cdot {x}^{8}\end{align*}$
  ::a. x3x8

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  b.  $\begin{align*}{xy}^{2} {x}^{2}{y}^{9}\end{align*}$
  ::b. b. xy2x2y9

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  Solution: Use the Product Rule above.
  ::解决办法:使用上述产品规则。

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  a.  $\begin{align*}{x}^{3}\cdot {x}^{8}={x}^{(3+8)} = {x}^{11}\end{align*}$
  ::a. x3x8=x(3+8)=x11

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  b. If a number does not have an exponent written explicitly, then the exponent is 1. Reorganize this expression so the $\begin{align*}x\end{align*}$ ’s are together and $\begin{align*}y\end{align*}$ ’s are together.
  ::b. 如果一个数字没有明白写出一个引言,则引言为1。 重新组织这个表达式,使 x 的组合,y 的组合。

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  $$\begin{align*}{xy}^{2}{x}^{2}{y}^{9}={x}^{1}{x}^{2}{y}^{2}{y}^{9}={x}^{(1+2)}{y}^{(2+9)}={x}^{3}{y}^{11}\end{align*}$$
  ::xy2x2y9=x1x2y2y9=x(1+2)y(2+9)=x3y11

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  Quotient Rule of Exponents
  ::指数规则

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  Investigation: Quotient  Rule
  ::调查:报价规则

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  1. Expand $\begin{align*}{2}^{8}\div {2}^{3}\end{align*}$ . Also, rewrite this as a fraction .
  ::1. 扩展 28\\\23. 另外, 重写为分数 。

  $\begin{align*}\frac{2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2}{2\cdot 2\cdot 2}\end{align*}$

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  2. Cancel out the common factors and write the answer one power of 2.
  ::2. 取消共同因素并写出回答的二分之一的力量。

  $\begin{align*}\frac{\cancel{2} \cdot \cancel{2} \cdot \cancel{2} \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2}{\cancel{2} \cdot \cancel{2} \cdot \cancel{2}} = {2}^{5}\end{align*}$

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  3. What is the difference of the exponents?
  ::3. 出纳者有什么区别?

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  The difference in the exponents is 5 since 8 - 3 = 5.
  ::从8 - 3 = 5起,指数的差异为5。

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    Quotient Rule of Exponents
  ::指数规则

    $$\begin{align*}\frac{{a}^{m}}{{a}^{n}}={a}^{m-n}\end{align*}$$  

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  Example 2
  ::例2

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  Simplify:
  ::简化 :

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  a. $\begin{align*}\frac{{5}^{9}}{{5}^{7}}\end{align*}$
  ::a. 5957

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  b. $\begin{align*}\frac{{x}^{4}}{{x}^{2}}\end{align*}$
  ::b.x4x2

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  c. $\begin{align*}\frac{{x}^{10}{y}^{5}}{{x}^{6}{y}^{2}}\end{align*}$
  ::c.x10y5x6y2

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  Solution: Use the Quotient Rule  from above.
  ::解决办法:使用上面的引号规则。

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  a. $\begin{align*}\frac{{5}^{9}}{{5}^{7}}={5}^{(9-7)}={5}^{2}= 25\end{align*}$  
  ::a. 5957=5(9-7)=52=25

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  b. $\begin{align*}\frac{{x}^{4}}{{x}^{2}}={x}^{(4-2)}={x}^{2}\end{align*}$
  ::b.x4x2=x(4-2)=x2

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  c.  $\begin{align*}\frac{{x}^{10}{y}^{5}}{{x}^{6}{y}^{2}}= {x}^{(10-6)}{y}^{(5-2)}={x}^{4}{y}^{3}\end{align*}$
  ::c. x10y5x6y2=x(10-6)y(5-2)=x4y3

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  Example 3
  ::例3

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  The magnitude of an earthquake represents the exponent m in the expression  $\begin{align*}10^m\end{align*}$ . ¹  
  ::地震的大小代表了10m1表达式中的缩写m。

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  Valdivia, Chile has suffered two major earthquakes. The 1575 Valdivia earthquake had a magnitude of 8.5. The world's largest earthquake was the 1960 Valdivia earthquake at a magnitude of 9.5. ²
  ::智利的巴尔迪维亚发生了两次重大地震,1575年的巴尔迪维亚地震发生8.5次,世界最大的地震是1960年的巴尔迪维亚地震,地震规模为9.5.2次。

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  What was the size of the 1575 earthquake compared to the 1960 one?
  ::与1960年相比,1575年地震的规模是多少?

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  Solution: Set each earthquake's magnitude up as an exponential expression and divide.
  ::解决方案: 将每场地震的大小设置为指数表达和分隔 。

  $$\begin{align*}\frac{{10}^{8.5}}{{10}^{9.5}}={10}^{-1}=\frac{1}{10^1}=\frac{1}{10}\end{align*}$$

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  Therefore , the amount of energy released during the 1575 earthquake was $\begin{align*}\frac{1}{10}\end{align*}$  of the amount of energy released during  the 1960 earthquake.
  ::因此,1575年地震期间释放的能量是1960年地震期间释放的能量的110。

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  Zero and Negative Exponents
  ::零和负指数

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  We can determine rules for by using the quotient rule. First, let’s address a zero in the exponent through an investigation.
  ::我们可以通过使用商数规则来决定规则。 首先,让我们通过调查在引言中解决零问题。

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  Investigation: Zero Exponents
  ::调查:零指数

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  1. Evaluate $\begin{align*}\frac{{5}^{6}}{{5}^{6}}\end{align*}$   by using the quotient rule .
  ::1. 使用商数规则评价5656。

  $\begin{align*}\frac{{5}^{6}}{{5}^{6}}= {5}^{(6-6)}={5}^{0}\end{align*}$

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  2. What is a number divided by itself? Apply this to #1.
  ::2. 数字本身除以多少?

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  A number divided by itself is 1. Therefore, $\begin{align*}\frac{{5}^{6}}{{5}^{6}}= 1\end{align*}$  .
  ::数字本身除以1。 因此,5656=1。

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   Zero Exponent
  ::零指数

    $$\begin{align*}{a}^{0}= 1\end{align*}$$  

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  Investigation: Negative Exponents
  ::调查:负指数

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  1. Expand $\begin{align*}\frac{3^2}{3^7}\end{align*}$  and cancel out the common 3’s and write your answer with positive exponents.
  
  $\begin{align*}\frac{3^2}{3^7} = \frac{\cancel{3} \cdot \cancel{3}}{\cancel{3} \cdot \cancel{3} \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3} = \frac{1}{3^5}\end{align*}$
  ::1. 扩大3237,取消3号公文,并用正面的提示书写你的答复。 3237=33333}3}3}3}3}3}3}3}3}3}3}3}3}3}3 3 =135

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  2. Evaluate $\begin{align*}\frac{3^2}{3^7}\end{align*}$  by using the quotient rule .
  $\begin{align*}\frac{3^2}{3^7} = 3^{2-7} = 3^{-5}\end{align*}$
  ::2. 使用商数规则评价3237。 3237=32-7=3-5

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  3. Are the answers from #1 and #2 equal? Write them as a single statement.
  
  $\begin{align*}\frac{1}{3^5} = 3^{-5}\end{align*}$
  ::3. 答案来自#1和#2的答案是否相等?把它们写成单句。 135=3-5

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   Negative Exponents
  ::负指数

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  $$\begin{align*}\frac{1}{a^m} = a^{-m} \ \text{and} \ \ \frac{1}{a^{-m}} = a^m\end{align*}$$
  ::1日上午1时=a-m和1日上午1时=m

   

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  Example 4
  ::例4

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  Simplify the following expressions. Express your answer with positive exponents.
  ::简化以下表达式。 以正面的提示表达您的答复 。

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  a.  $\begin{align*}\frac{5^2}{5^5}\end{align*}$
  ::a. 5255

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  b.  $\begin{align*}\frac{x^7 yz^{12}}{x^{12} yz^7}\end{align*}$
  ::b. b. x7yz12x12yz7

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  c.  $\begin{align*}\frac{a^4 b^0}{a^8 b}\end{align*}$
  ::c. a4b0a8b

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  Solution: Use the two properties from above.
  ::解决方案:使用上面的两个属性。

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  a.  $\begin{align*}\frac{5^2}{5^5} = 5^{-3} = \frac{1}{5^3} = \frac{1}{125}\end{align*}$
  ::a. 5255=5-3=153=1125

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  b.  $\begin{align*}\frac{x^7 yz^{12}}{x^{12} yz^7} = \frac{y^{1-1} z^{12-7}}{x^{12-7}} = \frac{y^0 z^5}{x^5} = \frac{z^5}{x^5}\end{align*}$
  ::b. x7yz12x12yz7=y1-1z12-7x12-7=y0z5x5=z5x5

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  c.  $\begin{align*}\frac{a^4 b^0}{a^8 b} = a^{4-8} b^{0-1} = a^{-4} b^{-1} = \frac{1}{a^4 b}\end{align*}$
  ::c. a4b0a8b=a4-8b0-1=a-4b-1=1a-4b-1=1a4b

   

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  Power Rule of Exponents
  ::生物体的权力

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  The last set of properties to explore are the power properties. Let’s investigate what happens when a power is raised to another power.
  ::最后一组要探索的属性是功率属性。让我们来调查当一个权力被提升到另一个权力时会发生什么。

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  Investigation:  Power Rule
  ::调查:权力规则

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  1. Rewrite $\begin{align*}(2^3)^5\end{align*}$  as $\begin{align*}2^3\end{align*}$  five times.
  ::1. 重写(23)5为23次5次。

    $\begin{align*}(2^3)^5 = 2^3 \cdot 2^3 \cdot 2^3 \cdot 2^3 \cdot 2^3\end{align*}$

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  2. Expand each $\begin{align*}2^3\end{align*}$ . How many 2’s are there?
  ::2. 扩大每个23个,有多少2个?

  $\begin{align*}(2^3)^5 = \underbrace{2 \cdot 2 \cdot 2}_{2^3} \cdot \underbrace{2 \cdot 2 \cdot 2}_{2^3} \cdot \underbrace{2 \cdot 2 \cdot 2}_{2^3} \cdot \underbrace{2 \cdot 2 \cdot 2}_{2^3} \cdot \underbrace{2 \cdot 2 \cdot 2}_{2^3} = 2^{15}\end{align*}$

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  3. What is the product of the powers?
  ::3. 权力的产物是什么?

  $\begin{align*}3 \cdot 5 = 15\end{align*}$

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   Power Rule of Exponents
  ::生物体的权力

    $$\begin{align*}(a^m)^n = a^{mn}\end{align*}$$  

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     Other Power Properties
  ::其他电源属性

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  Power of a Product Property :  $\begin{align*}(ab)^m = a^m b^m\end{align*}$  
  ::产品产权的功率ab)m=ambm

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  Power of a Quotient Property :  $\begin{align*}\left( \frac{a}{b} \right)^m = \frac{a^m}{b^m}\end{align*}$  
  ::引号属性的功率ab)m=ambm

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  Example 5
  ::例5

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  Simplify the following.
  ::简化如下。

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  a.  $\begin{align*}(3^4)^2\end{align*}$
  ::a. (34)2

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  b.  $\begin{align*}{({x}^2 y)}^{5}\end{align*}$
  ::b. (x2y)5

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  Solution: Use the properties from above.
  ::解决方案:使用上面的属性。

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  a.  $\begin{align*}(3^4)^2 = 3^{4 \cdot 2} = 3^8 = 6561\end{align*}$
  ::a. (342=342=342=38=6561)

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  b.  $\begin{align*}(x^2 y)^5 = x^{2 \cdot 5} y^5 = x^{10} y^5\end{align*}$
  ::b. (x2y)5=x2=x2=5y5=x10y5

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  Example 6
  ::例6

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  Simplify $\begin{align*}{\left( \frac{3a^{-6}}{2^2 a^2} \right)}^{4}\end{align*}$  without negative exponents.
  ::简化(3a-622a2)4,无负指数。

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  Solution:  Apply the power outside the " data-term="Parentheses" role="term" tabindex="0"> parentheses first and then move the negative power of from the numerator to the denominator.
  ::解答: 先应用括号外的功率, 然后将分子的负功率从分子移动到分母 。

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  $$\begin{align*}\left( \frac{3a^{-6}}{2^2 a^2} \right)^4 = \frac{3^4 a^{-6 \cdot 4}}{2^{2 \cdot 4} a^{2 \cdot 4}} = \frac{81a^{-24}}{2^8 a^8} = \frac{81}{256a^{8+24}} = \frac{81}{256a^{32}}\end{align*}$$
  :3a-622a2)4=34a-64224a24=81a-2428a8=81256a8+24=81256a32

   

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  by Mathispower4u explains the properties of exponents .
  ::Mathispower4u 解释出名者的属性。

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  Scientific Notation
  ::科学符号

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  For numbers that are quite small or quite large, we use a shorthand notation for convenience. This notation is called scientific notation . Numbers written in scientific notation are a product of a number between 1 and 10,  $\begin{align*}1 \le x <10\end{align*}$ , and a power of 10.
  ::对于非常小或相当大的数字,为了方便起见,我们使用一个短手标记。这个标记称为科学标记。在科学标记中写下的数字是数字1至10、1x < 10和10的产物,其功率是10。

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  Example 7
  ::例7

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  Write the distance from the Earth to the Sun, 92,960,000 miles, in scientific notation. ³
  ::在科学符号中写上地球到太阳的距离92 960 000英里。 3

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  Solution:
  ::解决方案 :

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  We consider the significant digits: 9296. We need to write this part as a number between 1 and 10: 9.296. The first 9 is in the ten millions place or $\begin{align*}10^7\end{align*}$ , so 
  ::9296,我们需要把这部分写成1到10之间的数字:9.296。前9位在1000万个或107个地方。

  $$\begin{align*}92,960,000=9.296 \times 10^7\end{align*}$$

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  (A quick way to determine the exponent for the power of ten is to count the digits after the first 9. There are seven of them, so $\begin{align*}10^7\end{align*}$  is the power of ten.)
  :快速确定十人功率指数的捷径是计算前九人后的数字, 其中七人, 所以107是十人功率。 )

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  Example 8
  ::例8

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  Write the size of a 0.000000014 meter chip in scientific notation.
  ::在科学符号中写一个0.000014米芯片的大小。

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  Solution: The 14 are the significant digits. Writing them as a number between 1 and 10, we have 1.4. To figure out the power of ten, 1 is eight places behind the decimal point or the hundred millionths place. So $\begin{align*}10^{-8}\end{align*}$  . Our result
  ::解决方案 : 14 是重要数字。 把它们写成 1 至 10 之间的数字, 我们有 1. 4 。 要弄清楚十分的功率, 1 是小数点后8 位数, 或是十亿位数后8 位数。 所以 10 - 8 。 我们的结果是 10 - 8 。

  $$\begin{align*}0.000000014=1.4 \times 10^{-8}\end{align*}$$

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  Summary
  ::摘要

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  • The product rule of exponents says that if you multiply two expressions with the same base, you can add the exponents. That is: $\begin{align*}{a}^{m}\times {a}^{n} = {a}^{m+n}\end{align*}$ .
    ::引言者的产物规则指出,如果用同一基数乘以两个表达式,您可以添加引言。即: AMxan=am+n。
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  • The quotient rule of exponents says that if you divide two expressions with the same base, you can subtract the exponents. That is: $\begin{align*} \frac{{a}^{m}}{{a}^{n}}= {a}^{m-n}\end{align*}$ .
    ::引言人的商数规则指出,如果用同一基数将两个表达式分开,可以减去引言。这就是:aman=am-n。
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  • The power rule of exponents says that if you raise a base to a power to another power, you can multiply the exponents. That is: $\begin{align*}{({a}^{m}})^{n}={a}^{mn}\end{align*}$ .
    ::远征者的权力规则说,如果你把一个基地加到另一个力量上,你就可以使先征者倍增。这就是说am)n=amn。
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  •   $\begin{align*}b^0 = 1, b \ne 0\end{align*}$   
    ::b0=1,b0
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  •   $\begin{align*}b^{-n} = \frac{1}{b^n}, b \ne 0\end{align*}$   
    ::b_b_b_b_b_b_b_b_b_b__b_b_b_b_b_b_b_b_b_b_b_b_b_b_b_b_b_b_b_b_b_b_b_b_b_b_b_b_b_b_b_b_b_b_b_b_b_b_b_b_b{\b_b_b_b_b_b____b>%0
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  • To write a number in scientific notation, write the significant digits as a number $\begin{align*}1 \le x < 10\end{align*}$  and the place value as a power of 10.
    ::要在科学符号中写出数字,请将重要数字写成 1x < 10,将位置值写成10。

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  Review
  ::回顾

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  Simplify the following expressions. Write your answer with positive  exponents.
  ::简化以下表达式。 以正表征写入您的答复 。

  1.  $\begin{align*}6 \cdot 6^3 \cdot 6^2\end{align*}$

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  2.  $\begin{align*}\frac{x^6}{x^{15}}\end{align*}$
  ::2. x6x15

  3.  $\begin{align*}\frac{7^{-3}}{7^{-2}}\end{align*}$

  4.  $\begin{align*}\frac{2^4 \cdot 3^5}{2 \cdot 3^2}\end{align*}$

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  5.  $\begin{align*}d^5 f^3 d^9 f^7\end{align*}$
  ::5.5f3d9f7

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  6.  $\begin{align*}\frac{x^0 y^5}{xy^7}\end{align*}$
  ::6. x0Y5xy7

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  7.  $\begin{align*}\frac{a^{-1} b^8}{a^5 b^7}\end{align*}$
  ::7. a-1b8a5b7

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  8.  $\begin{align*}\frac{14c^{10} d^{-4}}{21c^6 d^{-3}}\end{align*}$
  ::8. 14c10d-421c6d-3

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  9.  $\begin{align*}\frac{8g^0 h}{30g^{-9} h^2}\end{align*}$
  ::9. 8g0h30g-9h2

  10.  $\begin{align*}(2^5)^3\end{align*}$

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  11.  $\begin{align*}(3x)^4\end{align*}$
  ::11. (3x)4

  12.  $\begin{align*}\left( \frac{4}{5} \right)^2\end{align*}$

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  13.  $\begin{align*}(4x^8)^{-2}\end{align*}$
  ::13. (4x8)-2

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  14.  $\begin{align*}\left( \frac{2x^4 y^2}{5x^{-3} y^5} \right)^3\end{align*}$
  ::14.(2x4y25x-3y5)3

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  15.  $\begin{align*}\left( \frac{9m^5 n^{-7}}{27 m^6 n^5} \right)^{-4}\end{align*}$
  ::15. (9m5n-7727m6n5)-4

   

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  Explore More
  ::探索更多

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  1. There are 1,000 bacteria present in a culture. A microbiological culture, or microbial culture, is a method of multiplying microbial organisms by letting them reproduce under controlled laboratory conditions. Microbial cultures are used to determine the type of organism, its abundance in the sample being tested, or both. When the culture is treated with an antibiotic, the bacteria count is halved every 4 hours. How many bacteria remain 24 hours later?
  ::1. 一种文化中有1 000种细菌,微生物文化或微生物文化是一种在受控制的实验室条件下允许微生物生物繁殖的方法,微生物文化用来确定生物的种类、试验样本中的丰富性,或两者兼而有之。当这种文化受到抗生素治疗时,细菌数量每4小时减少一半,有多少细菌24小时之后还存在?

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  2. Simplify  $\begin{align*}\frac{a^7 b^{10}}{4a^{-5} b^{-2}} \cdot \left[ \frac{(6ab^{12})^2}{12a^9 b^{-3}} \right]^2 \div (3a^5 b^{-4})^3\end{align*}$ . 
  ::2. 简化a7b104a-5b-2[(6ab12)212a9b-33]2(3a5b-4)3]。

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  3. Miguel says that the expression $\begin{align*}\frac{2^5 \cdot 2^4}{2^2}\end{align*}$  equals $\begin{align*}2^{10}\end{align*}$ . Elise says that it is equal to $\begin{align*}2^7\end{align*}$ . Carlos says that because the exponents of the terms are different, the expression cannot be simplified. Which one of them is correct?
  ::3. Miguel说,252422的表达方式等于210。Elise说,它等于27。Carlos说,由于术语的表述方式不同,不能简化。其中哪一个是正确的?

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  4. Investigation Evaluate the powers of negative numbers.
  ::4. 调查评估负数的力量。

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  Find:
  $\begin{align*}(-2)^1\end{align*}$
  ::查找-2)1

  $\begin{align*}(-2)^2\end{align*}$

  $\begin{align*}(-2)^3\end{align*}$

  $\begin{align*}(-2)^4\end{align*}$

  $\begin{align*}(-2)^5\end{align*}$

  $\begin{align*}(-2)^6\end{align*}$

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  Make a conjecture about even versus odd powers with negative numbers.
  ::想象一下,即使与负数的奇异力量相比, 也会出现一些猜想。

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  5. Is $\begin{align*}(-2)^4\end{align*}$  different from $\begin{align*}-2^4\end{align*}$ ? Why or why not?
  ::5. (-2)4 是否与-24不同?为什么或为什么没有?

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

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  Please see the Appendix. 
  ::请参看附录。

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  PLIX
  ::PLIX

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  Try this interactive that reinforces the concepts explored in this section:
  ::尝试这一互动,强化本节所探讨的概念:

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  References
  ::参考参考资料

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  1. “Richter Magnitude Scale,” last edited June 22, 2017, .
  ::1. " Richter磁尺度 " ,上次编辑于2017年6月22日。

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  2.  “Lists of Earthquakes,” last edited June 21, 2017, .
  ::2. " 地震历程 " ,上次编辑于2017年6月21日,2017年6月21日。

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  3.  “Earth-Sun Distance Measurement Redefined,” by Nola Taylor Redd, September 24, 2012, .
  ::3. 2012年9月24日Nola Taylor Redd撰写的“重新定义地球-太阳距离测量”,