Course: 6.2 负和零指数 | The Way To Learn

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负和零指数
The magnitude of an earthquake represents the exponent m in the expression $\begin{align*}10^m.\end{align*}$
::地震的强度代表了10米的表情。

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.
::智利的巴尔迪维亚遭受了两次重大地震,1575年的巴尔迪维亚地震规模为8.5,世界最大的地震是1960年的巴尔迪维亚地震,规模为9.5。

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?

Source: Wikipedia
::资料来源:维基百科。

Negative and Zero Exponents
::负和零指数

In this concept, we will introduce negative and zero exponents.
::在这个概念中,我们将提出消极和零的推论。

Zero Exponents
::零指数

Step 1: Evaluate $\begin{align*}\frac{5^6}{5^6}\end{align*}$ by using the Quotient of Powers property .
::第1步:利用权力财产引号评价5656。

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

Step 2: What is a number divided by itself? Apply this to Step 1.
::第2步:数字本身除以多少?应用到第1步中。

$\begin{align*}\frac{5^6}{5^6} = 1\end{align*}$

Step 3: Fill in the blanks. $\begin{align*}\frac{a^m}{a^m} = a^{m-m} = a^- = _-\end{align*}$
::步骤3:填补空白。 amam=am- m=a

$\begin{align*}a^0 = 1\end{align*}$
::0.0=1

Negative Exponents
::负指数

Step 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.
::第一步:扩大3237,取消共同的3, 并用正面的推文写下你的答复。

$\begin{align*}\frac{3^2}{3^7} = \frac{\cancel{3} \cdot \cancel{3}}{\cancel{3} \cdot \cancel{3} \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3} = \frac{1}{3^5}\end{align*}$

Step 2: Evaluate $\begin{align*}\frac{3^2}{3^7}\end{align*}$ by using the Quotient of Powers property.
::步骤2:使用权力财产引号评价3237。

$\begin{align*}\frac{3^2}{3^7} = 3^{2-7} = 3^{-5}\end{align*}$

Step 3: Are the answers from Step 1 and Step 2 equal? Write them as a single statement.
::第3步:第1步和第2步的答案是否相等?把它们写成单词。

$\begin{align*}\frac{1}{3^5} = 3^{-5}\end{align*}$

Step 4: Fill in the blanks. $\begin{align*}\frac{1}{a^m} = a-\end{align*}$ and $\begin{align*}\frac{1}{a^{-m}} = a-\end{align*}$
::第4步:填充空白。

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

From the the discussions above, we have learned two very important properties of exponents. First, anything to the zero power is one. Second, negative exponents indicate placement. If an exponent is negative, it needs to be moved from where it is to the numerator or denominator.
::从上述讨论中,我们学到了引言的两个非常重要的特性。首先,任何零功率的特性都是一。其次,负引言表示位置。如果引言为负,则需要将其从位置移到分子或分母。

Simplify the following expression. Your answer should only have positive exponents.
::简化下面的表达式。您的回答应该只有正面的提示。

$\begin{aligned} \frac{5^{2}}{5^{5}} \end{aligned}$
$\begin{align*}\frac{5^2}{5^5}\end{align*}$

$\begin{aligned} \frac{5^{2}}{5^{5}} = 5^{- 3} = \frac{1}{5^{3}} = \frac{1}{125} \end{aligned}$
$\begin{align*}\frac{5^2}{5^5} = 5^{-3} = \frac{1}{5^3} = \frac{1}{125}\end{align*}$

Simplify the following expression. Your answer should only have positive exponents.
::简化下面的表达式。您的回答应该只有正面的提示。

$\begin{aligned} \frac{x^{7} y z^{12}}{x^{12} y z^{7}} \end{aligned}$
$\begin{align*}\frac{x^7 yz^{12}}{x^{12} yz^7}\end{align*}$
::x7yz12x12yz7

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

Simplify the following expression. Your answer should only have positive exponents.
::简化下面的表达式。您的回答应该只有正面的提示。

$\begin{aligned} \frac{a^{4} b^{0}}{a^{8} b} \end{aligned}$
$\begin{align*}\frac{a^4 b^0}{a^8 b}\end{align*}$
::a4b0a8b (千分之4b0a8b)

$\begin{aligned} \frac{a^{4} b^{0}}{a^{8} b} = a^{4 - 8} b^{0 - 1} = a^{- 4} b^{- 1} = \frac{1}{a^{4} b} \end{aligned}$
$\begin{align*}\frac{a^4 b^0}{a^8 b} = a^{4-8} b^{0-1} = a^{-4} b^{-1} = \frac{1}{a^4 b}\end{align*}$
::a4b0a8b=a4-8b0-1=a-4b-1=1a4bb=1a4b

or
::或

$\begin{aligned} \frac{a^{4} b^{0}}{a^{8} b} = \frac{1}{a^{8 - 4} b} = \frac{1}{a^{4} b} \end{aligned}$
$\begin{align*}\frac{a^4 b^0}{a^8 b} = \frac{1}{a^{8-4} b} = \frac{1}{a^4 b}\end{align*}$
::a4b0a8b=1a8-4b=1a4b

Simplify the following expression. Your answer should only have positive exponents.
::简化下面的表达式。您的回答应该只有正面的提示。

$\begin{aligned} \frac{x y^{5}}{8 y^{- 3}} \end{aligned}$
$\begin{align*}\frac{xy^5}{8y^{-3}}\end{align*}$
::xy58y- 3

$\begin{aligned} \frac{x y^{5}}{8 y^{- 3}} = \frac{x y^{5} y^{3}}{8} = \frac{x y^{5 + 3}}{8} = \frac{x y^{8}}{8} \end{aligned}$
$\begin{align*}\frac{xy^5}{8y^{-3}} = \frac{xy^5 y^3}{8} = \frac{xy^{5+3}}{8} = \frac{xy^8}{8}\end{align*}$
::xy58y-3=xy5y38=xy5+38=xy88

Simplify the following expression. Your answer should only have positive exponents.
::简化下面的表达式。您的回答应该只有正面的提示。

$\begin{aligned} \frac{27 g^{- 7} h^{0}}{18 g} \end{aligned}$
$\begin{align*}\frac{27 g^{-7} h^0}{18 g}\end{align*}$
::27g-7h018g 27g-7h018g

$\begin{aligned} \frac{27 g^{- 7} h^{0}}{18 g} = \frac{3}{2 g^{1} g^{7}} = \frac{3}{2 g^{1 + 7}} = \frac{3}{2 g^{8}} \end{aligned}$
$\begin{align*}\frac{27 g^{-7} h^0}{18 g} = \frac{3}{2g^1 g^7} = \frac{3}{2g^{1+7}} = \frac{3}{2g^8}\end{align*}$
::27g-7h018g=32g1g7=32g1+7=32g8

Now, let's multiply the following two fractions together and simplify. The answer should only have positive exponents.
::现在,让我们将以下两个分数相乘并简化。答案应该只有正表征。

$\begin{aligned} \frac{4 x^{- 2} y^{5}}{20 x^{8}} \cdot \frac{- 5 x^{6} y}{15 y^{- 9}} \end{aligned}$
$\begin{align*}\frac{4x^{-2} y^5}{20x^8} \cdot \frac{-5x^6 y}{15y^{-9}}\end{align*}$
::4 - 2y520x85x6y15y-9

The easiest way to approach this problem is to multiply the two fractions together first and then simplify.
::解决这一问题的最简单办法是首先将两个部分相乘,然后简化。

$\begin{aligned} \frac{4 x^{- 2} y^{5}}{20 x^{8}} \cdot \frac{- 5 x^{6} y}{15 y^{- 9}} = - \frac{20 x^{- 2 + 6} y^{5 + 1}}{300 x^{8} y^{- 9}} = - \frac{x^{- 2 + 6 - 8} y^{5 + 1 + 9}}{15} = - \frac{x^{- 4} y^{15}}{15} = - \frac{y^{15}}{15 x^{4}} \end{aligned}$
$\begin{align*}\frac{4x^{-2} y^5}{20x^8} \cdot \frac{-5x^6 y}{15y^{-9}} = -\frac{20x^{-2+6} y^{5+1}}{300x^8 y^{-9}} = -\frac{x^{-2+6-8}y^{5+1+9}}{15} = -\frac{x^{-4} y^{15}}{15} = -\frac{y^{15}}{15x^4}\end{align*}$
::4 - 2y520x8=5x6y15y-920x-2+6y5+1300x8y-92+6-8y5+1+915*x-4y1515*y1515x4

Examples
::实例

Example 1
::例1

Earlier, you were asked to find the size of the 1575 earthquake compared to the 1960 earthquake.
::更早之前,你被要求找到与1960年地震相比 1575年地震的规模。

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

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

Therefore , the size of the 1575 earthquake was $\begin{align*}\frac{1}{10}\end{align*}$ the 1960 one.
::因此,1575年地震的面积是1960年的110次。

Example 2
::例2

Simplify the expression.
::简化表达式。

$\begin{align*}\frac{8^6}{8^9}\end{align*}$

$\begin{align*}\frac{8^6}{8^9} = 8^{6-9} = \frac{1}{8^3} = \frac{1}{512}\end{align*}$

Example 3
::例3

Simplify the expression.
::简化表达式。

$\begin{align*}\frac{3x^{10} y^2}{21x^7 y^{-4}}\end{align*}$
::3x10y221x7y-4

$\begin{align*}\frac{3x^{10} y^2}{21x^7 y^{-4}} = \frac{x^{10-7} y^{2-(-4)}}{7} = \frac{x^3 y^6}{7}\end{align*}$
::3x10y221x7y-4=x10-7y2-(-4)7=x3y67

Example 4
::例4

Simplify the expression.
::简化表达式。

$\begin{align*}\frac{2a^8 b^{-4}}{16a^{-5}} \cdot \frac{4^3 a^{-3} b^0}{a^4 b^7}\end{align*}$
::2a8b-416a-543a-3b0a4b7

$\begin{align*}\frac{2a^8 b^{-4}}{16a^{-5}} \cdot \frac{4^3 a^{-3} b^0}{a^4 b^7} = \frac{128a^{8-3} b^{-4}}{16a^{-5+4} b^7} = \frac{8a^{5+1}}{b^{7+4}} = \frac{8a^6}{b^{11}}\end{align*}$
::2a8b-416a-543a-3b0a4b7=128a8-3b-416a-5+4b7=8a5+1b7+4=8a6b11

Review
::回顾

Simplify the following expressions. Answers cannot have negative exponents.
::简化以下表达式。答案不能有负引号。

$\begin{align*}\frac{8^2}{8^4}\end{align*}$

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

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

$\begin{align*}\frac{y^{-9}}{y^{10}}\end{align*}$
::y- 9y10

$\begin{align*}\frac{x^0 y^5}{xy^7}\end{align*}$
::x0y5xy7 级

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

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

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

$\begin{align*}\frac{5x^4}{10y^{-2}} \cdot \frac{y^7 x}{x^{-1} y}\end{align*}$
::5x410y-2y7xx-1yy

$\begin{align*}\frac{g^9 h^5}{6gh^{12}} \cdot \frac{18h^3}{g^8}\end{align*}$
::g9h56gh1218h3g8

$\begin{align*}\frac{4a^{10} b^7}{12a^{-6}} \cdot \frac{9a^{-5} b^4}{20a^{11} b^{-8}}\end{align*}$
::4a10b712a-69a-5b420a11b-8

$\begin{align*}\frac{-g^8 h}{6g^{-8}} \cdot \frac{9g^{15} h^9}{-h^{11}}\end{align*}$
::- G8h6g-89g15h9-h11

Rewrite the following exponential pattern with positive exponents: $\begin{align*}5^{-4}, 5^{-3}, 5^{-2}, 5^{-1}, 5^0, 5^1, 5^2, 5^3, 5^4\end{align*}$ .
::以正指数重写下列指数模式:5-4、5-35-2、5-2、1、50、51、52、53、54。

Evaluate each term in the pattern from #13.
::评估13号的每个术语。

Fill in the blanks.
::填满空白。

As the numbers increase, you ______________ the previous term by 5.
::随着数字的增加,您在上学期增加了5个学期。

As the numbers decrease, you _____________ the previous term by 5.
::随着人数的减少,您在上学期将减少5个学期。

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

负和零指数

Negative and Zero Exponents ::负和零指数

Zero Exponents ::零指数

Negative Exponents ::负指数

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)

Negative and Zero Exponents
::负和零指数

Zero Exponents
::零指数

Negative Exponents
::负指数

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)