Course: 8.1 涉及产品的指数属性

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涉及产品的指数属性
Exponential Properties Involving Products
::涉及产品的指数属性

In expressions involving exponents, like $\begin{aligned} 3^{5} \end{aligned}$ or $\begin{aligned} x^{3} \end{aligned}$ . the number on the bottom is called the base and the number on top is the power or exponent . The whole expression is equal to the base multiplied by itself a number of times equal to the exponent; in other words, the exponent tells us how many copies of the base number to multiply together.
::在涉及表率的表达式中,例如35 或 x3, 底部的数字被称为基数, 顶部的数字是权数或表率数。整个表达式等于基数乘以等于指数数的乘数; 换句话说, 该表率告诉我们基数的乘数是多少。

Writing Expressions in Exponential Form
::指数表单中的写法表达式

Write in exponential form .
::以指数形式写入。

a) $\begin{aligned} 2 \cdot 2 \end{aligned}$
:a) 2.%2

$\begin{aligned} 2 \cdot 2 = 2^{2} \end{aligned}$ because we have 2 factors of 2
::22=22 因为我们有两个系数为 2

b) $\begin{aligned} (- 3) (- 3) (- 3) \end{aligned}$
:b) (-3)(-3)(-3)(-3)

$\begin{aligned} (- 3) (- 3) (- 3) = (- 3)^{3} \end{aligned}$ because we have 3 factors of (-3)
:-3)(-3)(-3)(-3)=(-3)3,因为我们有三个因素(-3)

c) $\begin{aligned} y \cdot y \cdot y \cdot y \cdot y \end{aligned}$
::c)yyyyyyyy yyyyy yyy yyyy yy yy yy yy yy yy yy yy yy yy yy yy yy yy yy yy yyy yyy y yyy yyy yyy yy yyy yy y y y y y yyy yy yyyyy y y y y y y y y y y y y y y y y y y y y y y y y yyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyy yyyy y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Y Y Y YY Y Y YYYY YYYYYY YYYYYYY Y Y YY Y

$\begin{aligned} y \cdot y \cdot y \cdot y \cdot y = y^{5} \end{aligned}$ because we have 5 factors of $\begin{aligned} y \end{aligned}$
::yyy'y=y5 因为我们有5个y因数

d) $\begin{aligned} (3 a) (3 a) (3 a) (3 a) \end{aligned}$
:d) (3a)(3a)(3a)(3a)(3a)

$\begin{aligned} (3 a) (3 a) (3 a) (3 a) = (3 a)^{4} \end{aligned}$ because we have 4 factors of $\begin{aligned} 3 a \end{aligned}$
:3a)(3a)(3a)(3a)(3a)=(3a)4,因为我们有4个因素3a

When the base is a variable , it’s convenient to leave the expression in exponential form; if we didn’t write $\begin{aligned} x^{7} \end{aligned}$ , we’d have to write $\begin{aligned} x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \end{aligned}$ instead. But when the base is a number, we can simplify the expression further than that; for example, $\begin{aligned} 2^{7} \end{aligned}$ equals $\begin{aligned} 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \end{aligned}$ , but we can multiply all those 2’s to get 128.
::当基点是一个变量时, 将表达式以指数形式留下是方便的; 如果我们不写 x7 , 我们就必须反之写 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx。但是当基点是一个数字时, 我们可以将表达式简化得比这个数字更简单; 例如, 27 等于 22&2, 222- 22x2xxxxxxxxxxxxxxxxxxxxxxxxxx, 但我们可以将所有2 乘以128 。

Let’s simplify the expressions from Example A.
::让我们简化例A中的表达方式。

Simplifying Expressions
::简化表达式

Simplify
::简化

a) $\begin{aligned} 2^{2} \end{aligned}$
:a) 22

$\begin{aligned} 2^{2} = 2 \cdot 2 = 4 \end{aligned}$

b) $\begin{aligned} (- 3)^{3} \end{aligned}$
:b) (-3)3

$\begin{aligned} (- 3)^{3} = (- 3) (- 3) (- 3) = - 27 \end{aligned}$

c) $\begin{aligned} y^{5} \end{aligned}$
::c) y5

$\begin{aligned} y^{5} \end{aligned}$ is already simplified
::y5 已简化

d) $\begin{aligned} (3 a)^{4} \end{aligned}$
::d) (3a)4

$\begin{aligned} (3 a)^{4} = (3 a) (3 a) (3 a) (3 a) = 3 \cdot 3 \cdot 3 \cdot 3 \cdot a \cdot a \cdot a \cdot a = 81 a^{4} \end{aligned}$
:3a)4=(3a)(3a)(3a)(3a)(3a)=333333a)aa=81a4

Be careful when taking powers of negative numbers. Remember these rules:
::使用负数的权势时要小心。记住这些规则 :

$\begin{aligned} (n e g a t i v e n u m b e r) \cdot (p o s i t i v e n u m b e r) = n e g a t i v e n u m b e r \\ (n e g a t i v e n u m b e r) \cdot (n e g a t i v e n u m b e r) = p o s i t i v e n u m b e r \end{aligned}$

:正数)=正数(正数)=正数(正数)=正数(负数)=正数(负数)=正数

So even powers of negative numbers are always positive. Since there are an even number of factors, we pair up the negative numbers and all the negatives cancel out.
::因此,即使是负数的功率也总是正数。由于有数量偶数的因素,我们把负数加在一起,所有的负数都取消。

$\begin{aligned} (- 2)^{6} & = (- 2) (- 2) (- 2) (- 2) (- 2) (- 2) \\ = \underset{+ 4}{\underset{⏟}{(- 2) (- 2)}} \cdot \underset{+ 4}{\underset{⏟}{(- 2) (- 2)}} \cdot \underset{+ 4}{\underset{⏟}{(- 2) (- 2)}} \\ = + 64 \end{aligned}$

And odd powers of negative numbers are always negative. Since there are an odd number of factors, we can still pair up negative numbers to get positive numbers, but there will always be one negative factor left over, so the answer is negative:
::负数的奇异功率总是负数。由于有奇数的因素,我们仍然可以将负数对齐,以获得正数,但总是会留下一个负数,所以答案是否定的:

$\begin{aligned} (- 2)^{5} & = (- 2) (- 2) (- 2) (- 2) (- 2) \\ = \underset{+ 4}{\underset{⏟}{(- 2) (- 2)}} \cdot \underset{+ 4}{\underset{⏟}{(- 2) (- 2)}} \cdot \underset{- 2}{\underset{⏟}{(- 2)}} \\ = - 32 \end{aligned}$

Use the Product of Powers Property
::使用权力产品财产

So what happens when we multiply one power of $\begin{aligned} x \end{aligned}$ by another? Let’s see what happens when we multiply $\begin{aligned} x \end{aligned}$ to the power of 5 by $\begin{aligned} x \end{aligned}$ cubed . To illustrate better, we’ll use the full factored form for each:
::那么当我们乘以一个x的功率乘以另一个时会发生什么呢?让我们看看乘以x乘以5乘以x立方。为了更好地说明问题,我们将使用每个功率的全因数表:

$\begin{aligned} \underset{x^{5}}{\underset{⏟}{(x \cdot x \cdot x \cdot x \cdot x)}} \cdot \underset{x^{3}}{\underset{⏟}{(x \cdot x \cdot x)}} = \underset{x^{8}}{\underset{⏟}{(x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x)}} \end{aligned}$

:xx)xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx3=x8xxxxxxx

So $\begin{aligned} x^{5} \times x^{3} = x^{8} \end{aligned}$ . You may already see the pattern to multiplying powers, but let’s confirm it with another example. We’ll multiply $\begin{aligned} x \end{aligned}$ squared by $\begin{aligned} x \end{aligned}$ to the power of 4 :
::所以 x5xxx3=x8. 您可能已经看到乘以功率的模式, 但让我们用另一个例子来确认它。我们将乘以 x x 乘以 x 乘以 x 至 4 的功率 4 :

$\begin{aligned} \underset{x^{2}}{\underset{⏟}{(x \cdot x)}} \cdot \underset{x^{4}}{\underset{⏟}{(x \cdot x \cdot x \cdot x)}} = \underset{x^{6}}{\underset{⏟}{(x \cdot x \cdot x \cdot x \cdot x \cdot x)}} \end{aligned}$

:xxxxxxxxxxxxxxxxxx4=xxxxxxxxxxxxxxxxxxxxxxxxxxx6)

So $\begin{aligned} x^{2} \times x^{4} = x^{6} \end{aligned}$ . Look carefully at the powers and how many factors there are in each calculation. $\begin{aligned} 5 x \end{aligned}$ ’s times $\begin{aligned} 3 x \end{aligned}$ ’s equals $\begin{aligned} (5 + 3) = 8 x \end{aligned}$ ’s. $\begin{aligned} 2 x \end{aligned}$ ’s times $\begin{aligned} 4 x \end{aligned}$ ’s equals $\begin{aligned} (2 + 4) = 6 x \end{aligned}$ ’s.
::So x2xx4=x6. 仔细看一看每次计算中的功率和因数。 5x乘以3x等于( 5+3)=8x。 2x乘以4x等于(2+4)=6x。

You should see that when we take the product of two powers of $\begin{aligned} x \end{aligned}$ , the number of $\begin{aligned} x \end{aligned}$ ’s in the answer is the total number of $\begin{aligned} x \end{aligned}$ ’s in all the terms you are multiplying. In other words, the exponent in the answer is the sum of the exponents in the product.
::您应该看到,当我们使用x的两种功率的产物时,答案中的x的数是按您正在乘以的所有条件计算的x的总数。换句话说,答案中的引号是产品中推手的总和。

Product Rule for Exponents: $\begin{aligned} x^{n} \cdot x^{m} = x^{(n + m)} \end{aligned}$
::指数产品规则:xnxm=x(n+m)

There are some easy mistakes you can make with this rule, however. Let’s see how to avoid them.
::然而,这一规则可以犯一些容易的错误。让我们看看如何避免。

Multiplying Exponents
::乘数指数

1. Multiply $\begin{aligned} 2^{2} \cdot 2^{3} \end{aligned}$ .
::1. 乘2223。

$\begin{aligned} 2^{2} \cdot 2^{3} = 2^{5} = 32 \end{aligned}$

Note that when you use the product rule you don’t multiply the bases . In other words, you must avoid the common error of writing $\begin{aligned} 2^{2} \cdot 2^{3} = 4^{5} \end{aligned}$ . You can see this is true if you multiply out each expression: 4 times 8 is definitely 32, not 1024.
::请注意, 当使用产品时, 您不会乘以基准值。换句话说, 您必须避免写入 22_ 23 = 45 的常见错误。如果您将每个表达式都乘以4 乘以 8 表示: 4 乘以 8 绝对是 32 , 而不是 1024 。

2. Multiply $\begin{aligned} 2^{2} \cdot 3^{3} \end{aligned}$ .
::2. 乘以22+33。

$\begin{aligned} 2^{2} \cdot 3^{3} = 4 \cdot 27 = 108 \end{aligned}$

In this case, we can’t actually use the product rule at all, because it only applies to terms that have the same base . In a case like this, where the bases are different, we just have to multiply out the numbers by hand—the answer is not $\begin{aligned} 2^{5} \end{aligned}$ or $\begin{aligned} 3^{5} \end{aligned}$ or $\begin{aligned} 6^{5} \end{aligned}$ or anything simple like that.
::在此情况下,我们根本无法实际使用产品规则,因为它只适用于具有相同基础的术语。在这样的情况中,如果基础不同,我们只需要亲手将数字乘以数字 — — 答案不是25、35、65或类似简单的东西。

Examples
::实例

Simplify the following exponents:
::简化下列引言:

Example 1
::例1

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

$\begin{aligned} (- 2)^{5} = (- 2) (- 2) (- 2) (- 2) (- 2) = - 32 \end{aligned}$

Example 2
::例2

$\begin{aligned} (10 x)^{2} \end{aligned}$
:10x)2

$\begin{aligned} (10 x)^{2} = 10^{2} \cdot x^{2} = 100 x^{2} \end{aligned}$
:10x)2=102x2=100x2

Review
::回顾

Write in exponential notation:
::以指数符号写入 :

$\begin{aligned} 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \end{aligned}$

$\begin{aligned} 3 x \cdot 3 x \cdot 3 x \end{aligned}$
::3x3x3x3x3xx

$\begin{aligned} (- 2 a) (- 2 a) (- 2 a) (- 2 a) \end{aligned}$
:-2a)(-2a)(-2a)(-2a)(-2a)(-2a)(-2a)(-2a))

$\begin{aligned} 6 \cdot 6 \cdot 6 \cdot x \cdot x \cdot y \cdot y \cdot y \cdot y \end{aligned}$
::6... 6... 6... 6... 6xxxxxy...y...yyyyyyyy

$\begin{aligned} 2 \cdot x \cdot y \cdot 2 \cdot 2 \cdot y \cdot x \end{aligned}$
::2x2xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

Find each number.
::查找每个号码。

$\begin{aligned} 5^{4} \end{aligned}$

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

$\begin{aligned} (0.1)^{5} \end{aligned}$

$\begin{aligned} (- 0.6)^{3} \end{aligned}$

$\begin{aligned} (1.2)^{2} + 5^{3} \end{aligned}$

$\begin{aligned} 3^{2} \cdot (0.2)^{3} \end{aligned}$

Multiply and simplify:
::乘数和简化:

$\begin{aligned} 6^{3} \cdot 6^{6} \end{aligned}$

$\begin{aligned} 2^{2} \cdot 2^{4} \cdot 2^{6} \end{aligned}$

$\begin{aligned} 3^{2} \cdot 4^{3} \end{aligned}$

$\begin{aligned} x^{2} \cdot x^{4} \end{aligned}$
::x2x4

$\begin{aligned} (- 2 y^{4}) (- 3 y) \end{aligned}$
:-2y4(--3y))

$\begin{aligned} (4 a^{2}) (- 3 a) (- 5 a^{4}) \end{aligned}$
:4a2)(-3a)(-5a4)

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

涉及产品的指数属性

Exponential Properties Involving Products ::涉及产品的指数属性

Writing Expressions in Exponential Form ::指数表单中的写法表达式

Simplifying Expressions ::简化表达式

Use the Product of Powers Property ::使用权力产品财产

Multiplying Exponents ::乘数指数

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Review ::回顾

Review (Answers) ::回顾(答复)

Exponential Properties Involving Products
::涉及产品的指数属性

Writing Expressions in Exponential Form
::指数表单中的写法表达式

Simplifying Expressions
::简化表达式

Use the Product of Powers Property
::使用权力产品财产

Multiplying Exponents
::乘数指数

Examples
::实例

Example 1
::例1

Example 2
::例2

Review
::回顾

Review (Answers)
::回顾(答复)