Course: 6.1 指数的产品和引号属性

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指数值的产品和引号属性
Miguel says that the expression $\begin{align*}\frac{2^5 \cdot 2^4}{2^2}\end{align*}$ equals $\begin{align*}2^{10}\end{align*}$ .
::Miguel说252422的表达方式等于210。

Alise says that it is equal to $\begin{align*}2^7\end{align*}$ .
::Alise说,它等于27。

Carlos says that because the exponents of the terms are different, the expression can't be simplified.
::Carlos说,因为术语的出处不同, 表达方式无法简化。

Which one of them is correct?
::哪一个是正确的?

Product and Quotient Properties of Exponents
::指数值的产品和引号属性

To review, the power (or exponent) of a number is the little number in 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(或表率)是上标中的小数。向该功率提出的数字称为基数。该表率表示基数是多少倍。

There are several properties of exponents. We will investigate two in this concept.
::引言者有几个属性。我们将在这个概念中调查两个属性。

Let's expand and solve $\begin{align*}5^6\end{align*}$ .
::让我们扩大和解决56。

$\begin{align*}5^6\end{align*}$ means 5 times itself six times.
::56意味着5乘以6次

$\begin{align*}5^6 = 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 = 15,625\end{align*}$

Product Property
::产品产权

Step 1: Expand $\begin{align*}3^4 \cdot 3^5\end{align*}$ .
::第1步:扩大3435。

$\begin{align*}\underbrace{3 \cdot 3 \cdot 3 \cdot 3}_{3^4} \cdot \underbrace{3 \cdot 3 \cdot 3 \cdot 3 \cdot 3}_{3^5}\end{align*}$

Step 2: Rewrite this expansion as one power of three.
::第2步:将这一扩展重写为三分之一的功率。

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

Step 3: What is the sum of the exponents?
::第3步:出处的总和是多少?

$\begin{align*}4 + 5 = 9\end{align*}$

Step 4: Fill in the blank: $\begin{align*}a^m \cdot a^n = a^{-^+-}\end{align*}$
::第4步:填写空白: aman=a________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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

Rather than expand the exponents every time or find the powers separately, we can use this property to simplify the product of two exponents with the same base.
::与其每次扩大指数或单独找到权力,不如使用这一财产来简化两个指数的产物,两个指数的基础相同。

Let's simplify the following expressions using the Product Property above.
::让我们使用上述产品属性简化以下表达式。

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

$\begin{align*}x^3 \cdot x^8 = x^{3+8} = x^{11}\end{align*}$
::x3x8=x3+8=x11

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

If a number does not have an exponent, you may assume 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.
::如果一个数字没有前言,你可以假定前言为 1。重新组织这个表达式, 使 x 在一起, y 在一起。

$\begin{align*}xy^2 x^2 y^9 = x^1 \cdot x^2 \cdot y^2 \cdot y^9 = x^{1+2} \cdot y^{2+9} = x^3 y^{11}\end{align*}$
::xy2x2y9 = x1x2x2222Y2}=x1+2y2+9=x3y11

Quotient Property
::引号属性

Step 1: Expand $\begin{align*}2^8 \div 2^3\end{align*}$ . Also, rewrite this as a fraction .
::第一步扩大 28\\\ 23。另外, 重写此部分。

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

Step 2: Cancel out the common factors and write the answer one power of 2.
::步骤2:取消共同因素,将答案写成二分之一的回答。

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

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

$\begin{align*}8 - 3 = 5\end{align*}$

Step 4: Fill in the blank: $\begin{align*}\frac{a^m}{a^n} = a^{-^--}\end{align*}$
::第4步:填写空白:aman=a___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

$\begin{align*}\frac{a^m}{a^n} = a^{m-n}\end{align*}$
::aman=am-n (aman=am-n)

Simplify the expression using the Quotient Property above.
::使用以上引号属性简化表达式。

$\begin{align*}\frac{5^9}{5^7}\end{align*}$

$\begin{align*}\frac{5^9}{5^7} = 5^{9-7} = 5^2 = 25\end{align*}$

Simplify the expression using the Quotient Property above.
::使用以上引号属性简化表达式。

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

$\begin{align*}\frac{x^4}{x^2} = x^{4-2} = x^2\end{align*}$
::x4x2=x4-2=x2

Examples
::实例

Example 1
::例1

Earlier, you were asked to find which student is correct.
::早些时候,有人要求你找出哪个学生是正确的。

Using the Product Property and then the Quotient Property, the expression can be simplified.
::使用产品属性,然后使用引号属性,该表达式可以简化。

$\begin{align*}\frac{2^5 \cdot 2^4}{2^2}\\ = \frac{2^9}{2^2}\\ = 2^7\end{align*}$

Therefore , Alise is correct.
::因此,Alise是对的。

Example 2
::例2

Simplify the expression. Evaluate any numerical answers.
::简化表达式。评估任何数字答案。

$\begin{align*}7 \cdot 7^2\end{align*}$

$\begin{align*}7 \cdot 7^2 = 7^{1+2} = 7^3 = 343\end{align*}$

Example 3
::例3

Simplify the expression. Evaluate any numerical answers.
::简化表达式。评估任何数字答案。

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

$\begin{align*}\frac{3^7}{3^3} = 3^{7-3} = 3^4 = 81\end{align*}$

Example 4
::例4

Simplify the expression. Evaluate any numerical answers.
::简化表达式。评估任何数字答案。

$\begin{align*}\frac{16x^4 y^5}{4x^2 y^2}\end{align*}$
::16x4y54x2y2

$\begin{align*}\frac{16x^4 y^5}{4x^2 y^2} = 4x^{4-2} y^{5-3} = 4x^2 y^2\end{align*}$
::16x4y54x2y2=4x4--2y5-3=4x2y2

Review
::回顾

Expand the following numbers and evaluate.
::扩大以下数字并进行评估。

$\begin{align*}2^6\end{align*}$

$\begin{align*}10^3\end{align*}$

$\begin{align*}(-3)^5\end{align*}$

$\begin{align*}(0.25)^4\end{align*}$

Simplify the following expressions. Evaluate any numerical answers.
::简化以下表达式。评价任何数字答案。

$\begin{align*}4^2 \cdot 4^7\end{align*}$

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

$\begin{align*}\frac{8^3}{8}\end{align*}$

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

$\begin{align*}b^6 \cdot b^3\end{align*}$
::b6b3

$\begin{align*}5^2 x^4 \cdot x^9\end{align*}$
::52x4xx9

$\begin{align*}\frac{y^{12}}{y^5}\end{align*}$
::y12y5

$\begin{align*}\frac{a^8 \cdot b^6}{b \cdot a^4}\end{align*}$
::a 8b6ba4

$\begin{align*}\frac{3^7 x^6}{3^3 x^3}\end{align*}$
::37x633x3

$\begin{align*}d^5 f^3 d^9 f^7\end{align*}$
::d5f3d9f7 d5f3d9f7

$\begin{align*}\frac{2^8 m^{18} n^{14}}{2^5 m^{11} n^4}\end{align*}$
::28m18n1425m11n4

$\begin{align*}\frac{9^4 p^5 q^8}{9^2 pq^2}\end{align*}$
::94p5q892pq2

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

Find:

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

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

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

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

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

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

::查找-2)1(-2)2(-2)3(-2)4(-2)5(-2)6

Make a conjecture about even vs. odd powers with negative numbers.
::想象一下,即使与负数的奇异力量比起来, 也有负数的奇异力量。

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

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

指数值的产品和引号属性

Product and Quotient Properties of Exponents ::指数值的产品和引号属性

Product Property ::产品产权

Quotient Property ::引号属性

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

16 x 4 y 5 4 x 2 y 2 = 4 x 4 − 2 y 5 − 3 = 4 x 2 y 2 \begin{align*}\frac{16x^4 y^5}{4x^2 y^2} = 4x^{4-2} y^{5-3} = 4x^2 y^2\end{align*} ::16x4y54x2y2=4x4--2y5-3=4x2y2

Review ::回顾

Review (Answers) ::回顾(答复)

Product and Quotient Properties of Exponents
::指数值的产品和引号属性

Product Property
::产品产权

Quotient Property
::引号属性

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

$\begin{align}\frac{16x^4 y^5}{4x^2 y^2} = 4x^{4-2} y^{5-3} = 4x^2 y^2\end{align}$
::16x4y54x2y2=4x4--2y5-3=4x2y2

Review
::回顾

Review (Answers)
::回顾(答复)