Course: 12.8 承认和应用产品产权权

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承认并应用产品产权权
Simone is building a platform for the stage of the new band stand. She needs to determine the area of the platform so she can order the wood she needs. She knows the platform which has a side length of $\begin{align*}6a^2\end{align*}$ will be square.
::Simone正在为新乐队架的舞台建立一个平台。她需要确定平台的面积, 以便她可以订购她需要的木柴。她知道平台的侧长为 6a2 将会是正方形。

How can she find the area of the platform?
::她如何找到平台的面积?

In this concept, you will learn to recognize and apply the power of a product property .
::在此概念中,您将学会承认和应用产品产权的力量。

Power of Product Property
::产品产权的功率

When multiplying monomials, an exponent is applied to the constant , variable , or quantity that is directly to its left.
::当乘单数时,对左侧的常数、可变数或数量应用一个引号。

Let’s look at an example where the exponents can be applied to products using " data-term="Parentheses" role="term" tabindex="0"> parentheses .
::让我们看看一个例子, 说明出价者可以用括号来应用到产品中。

$\begin{aligned} (5 x)^{4} \end{aligned}$
$\begin{align*}(5x)^4\end{align*}$
:5x)4

If you apply the exponent 4 to whatever is directly to its left, apply it to the parentheses, not just to the $\begin{align*}x\end{align*}$ .
::如果您将前言 4 应用到左侧的任何直接内容上,请将其应用到括号上,而不是仅仅应用到 x 。

The parentheses are directly to the left of the 4. This indicates that the entire product in the parentheses is taken to the 4 ^th power.
::括号直接放在4号左侧,表示括号中的全部产品被带至第4号电源。

First, write $\begin{align*}(5x)^4\end{align*}$ in expanded form .
::首先,以扩展形式(5x)4书写(5x)4。

$\begin{aligned} (5 x)^{4} = (5 x) (5 x) (5 x) (5 x) \end{aligned}$
$\begin{align*}(5x)^4 = (5x)(5x)(5x)(5x)\end{align*}$
:5x)4=(5x)5x(5x)5x(5x)5x(5x)5x)

Next, multiply the monomials by placing like factors next to each other, multiplying the coefficients, and simplifying using exponents.
::其次,乘以单数,将类似因素放在彼此旁边,乘以系数,并简化使用引号。

$\begin{aligned} \begin{array}{rcl} (5 x)^{4} & = & (5 x) (5 x) (5 x) (5 x) \\ = & 5 \cdot 5 \cdot 5 \cdot 5 \cdot x \cdot x \cdot x \cdot x \\ = & 625 x^{4} \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} (5x)^4 &=& (5x)(5x)(5x)(5x) \\ &=& 5 \cdot 5 \cdot 5 \cdot 5 \cdot x \cdot x \cdot x \cdot x \\ &=& 625x^4 \end{array}\end{align*}$
:5x)4=(5x)5x(5x)5x(5x)5x)=5=5=5=5=5=5=5=5=5=5}5=5=5=5=5=5=5=5=5=5x}xxxxxxxxxxxxx=625x4

This is the Power of a Product Property which says, for any nonzero numbers $\begin{align*}a\end{align*}$ and $\begin{align*}b\end{align*}$ and any integer $\begin{align*}n\end{align*}$ :
::这是产品属性的功率,它表示,对于任何非零数a和b以及任何整数n:

$\begin{aligned} (a b)^{n} = a^{n} b^{n} \end{aligned}$
$\begin{align*}(ab)^n = a^n b^n\end{align*}$
:ab)n=anbn

Let’s look at an example.
::让我们举个例子。

Use the Power of a Product Property to expand $\begin{align*}(7h)^3\end{align*}$ .
::利用产品产权权扩大(7h)3。

First, expand the parentheses by multiplying $\begin{align*}7h\end{align*}$ times itself, three times.
::首先,扩大括号,将本身乘以7小时,乘以3倍。

$\begin{aligned} (7 h)^{3} = (7 h) (7 h) (7 h) \end{aligned}$
$\begin{align*}(7h)^3 = (7h)(7h)(7h)\end{align*}$
:7h)3=(7h)7(7h)7(7h)7

Next, multiply the monomials by placing like factors next to each other, multiplying the coefficients, and simplifying using exponents.
::其次,乘以单数,将类似因素放在彼此旁边,乘以系数,并简化使用引号。

$\begin{aligned} \begin{array}{rcl} (7 h)^{3} & = & (7 h) (7 h) (7 h) \\ = & 7 \cdot 7 \cdot 7 \cdot h \cdot h \cdot h \\ = & 343 h^{3} \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} (7h)^3 &=& (7h)(7h)(7h) \\ &=& 7 \cdot 7 \cdot 7 \cdot h \cdot h \cdot h \\ &=& 343 h^3 \end{array}\end{align*}$
:7h)3 = (7h)7(7h)7(7h)7= 7_7(7h)7_7h= 343h3)3

The answer is $\begin{align*}343 h^3\end{align*}$ .
::答案是343小时3

There is a definite pattern between the exponents and the final product. When you multiply like bases, there is a shortcut-add the exponents of like bases. Another way of saying it is:
::指数和最终产品之间有明确的模式。当你像基数一样乘以时,就会有一条捷径加插类似基数的指数。另一种说法是:

$\begin{aligned} a^{m} \times a^{n} = a^{m + n} \end{aligned}$
$\begin{align*}a^m \times a^n = a^{m+n}\end{align*}$
::AMXan= am+n

Let’s look at another problem.
::让我们来看看另一个问题。

Use the Power of a Product Property to expand $\begin{align*}(-2x^4)^5\end{align*}$ .
::利用产品产权的力量扩大(-2x4)5。

First, expand the parentheses by multiplying the base of $\begin{align*}-2x^4\end{align*}$ by itself, five times.
::首先,扩大括号,将-2x4的基数本身乘以5倍。

$\begin{aligned} (- 2 x^{4})^{5} = (- 2 x^{4}) (- 2 x^{4}) (- 2 x^{4}) (- 2 x^{4}) (- 2 x^{4}) \end{aligned}$
$\begin{align*}(-2x^4)^5 = (-2x^4)(-2x^4)(-2x^4)(-2x^4)(-2x^4)\end{align*}$
:-2x4)5=(-2x4)(-2x4)(-2x4)(-2x4)(-2x4)(-2x4)(-2x4)(-2x4)(-2x4)(-2x4)(-2x4)

Next, multiply the monomials by placing like factors next to each other, multiplying the coefficients, and simplifying using exponents.
::其次,乘以单数,将类似因素放在彼此旁边,乘以系数,并简化使用引号。

$\begin{aligned} \begin{array}{rcl} (- 2 x^{4})^{5} & = & (- 2 x^{4}) (- 2 x^{4}) (- 2 x^{4}) (- 2 x^{4}) (- 2 x^{4}) \\ = & - 2 \cdot - 2 \cdot - 2 \cdot - 2 \cdot - 2 \cdot x^{4} \cdot x^{4} \cdot x^{4} \cdot x^{4} \cdot x^{4} \\ = & - 2 \cdot - 2 \cdot - 2 \cdot - 2 \cdot - 2 \cdot x^{4 + 4 + 4 + 4 + 4} \\ = & - 32 x^{20} \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} (-2x^4)^5 &=& (-2x^4)(-2x^4)(-2x^4)(-2x^4)(-2x^4) \\ &=& -2 \cdot -2 \cdot -2 \cdot -2 \cdot -2 \cdot x^4 \cdot x^4 \cdot x^4 \cdot x^4 \cdot x^4 \\ &=& -2 \cdot -2 \cdot -2 \cdot -2 \cdot -2 \cdot x^{4+4+4+4+4} \\ &=& -32 x^{20} \end{array}\end{align*}$
: - 2x4) 5= (-2x4)(-2x4)(-2x4)(-2x4)(-2x4)(-2x4)(-2x4)(-2x4) 222222x4*x4*x4*x4*4*x4*4*x4*4*4*2}22}22}2×4+4+4+4+4+4*3x20)

The answer is $\begin{align*}(-32x)^{20}\end{align*}$ .
::答案是(-32x)20。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Simone and the square platform. She needs to figure out the area of the platform to order the needed wood.
::早些时候,有人给了你Simone和广场平台的问题。她需要弄清楚平台的面积,以便订购所需的木柴。

The side length of the square platform is $\begin{align*}6a^2\end{align*}$ .
::平方平台的侧长为6a2。

First, set up the area of the square platform.
::首先,设置广场平台的面积。

$\begin{aligned} \begin{array}{rcl} A & = & s^{2} \\ A & = & (6 a^{2})^{2} \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} A &=& s^2 \\ A &=& (6a^2)^2 \end{array}\end{align*}$
::A=S2A=(6a2)2

Next, expand the parentheses by multiplying the monomial times itself two times.
::接下来,通过将单倍数本身乘以两次来扩大括号。

$\begin{aligned} (6 a^{2})^{2} = (6 a^{2}) (6 a^{2}) \end{aligned}$
$\begin{align*}(6a^2)^2 = (6a^2)(6a^2)\end{align*}$
:6a2)2=(6a2)6a2)

Then, multiply the monomials by placing like factors next to each other, multiplying the coefficients, and simplifying using exponents.
::然后,通过将类似因素放在彼此相邻的位置,乘以系数,并简化使用引言,使单一因素成倍增加。

$\begin{aligned} \begin{array}{rcl} (6 a^{2})^{2} & = & (6 a^{2}) (6 a^{2}) \\ = & 6 \cdot 6 \cdot a^{2} \cdot a^{2} \\ = & 6 \cdot 6 \cdot a^{2 + 2} \\ = & 36 a^{4} \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} (6a^2)^2 &=& (6a^2)(6a^2) \\ &=& 6 \cdot 6 \cdot a^2 \cdot a^2 \\ &=& 6 \cdot 6 \cdot a^{2+2} \\ &=& 36a^4 \end{array}\end{align*}$
:6a2)2=(6a2)(6a2)=666a2a2=666a2+2=36a4)

The answer is $\begin{align*}36a^4\end{align*}$ .
::答案是36a4

The area of the square platform is $\begin{align*}36a^4\end{align*}$ units squared.
::广场平台面积为36a4平方。

Example 2
::例2

Use the Power of a Product Property to expand $\begin{align*}(3x^5)^3\end{align*}$ .
::利用产品产权的力量扩大(3x5)3。

First, expand the parentheses by multiplying $\begin{align*}3x^5\end{align*}$ times itself, three times.
::首先,扩大括号,将3x5乘以本身3x5乘以3乘以3乘以3乘以3。

$\begin{aligned} (3 x^{5})^{3} = (3 x^{5}) (3 x^{5}) (3 x^{5}) \end{aligned}$
$\begin{align*}(3x^5)^3 = (3x^5)(3x^5)(3x^5)\end{align*}$
:3x5)3=(3x5)(3x5)(3x5)

Next, multiply the monomials by placing like factors next to each other, multiplying the coefficients, and simplifying using exponents.
::其次,乘以单数,将类似因素放在彼此旁边,乘以系数,并简化使用引号。

$\begin{aligned} \begin{array}{rcl} (3 x^{5})^{3} & = & (3 x^{5}) (3 x^{5}) (3 x^{5}) \\ = & 3 \cdot 3 \cdot 3 \cdot x^{5} \cdot x^{5} \cdot x^{5} \\ = & 3 \cdot 3 \cdot 3 \cdot x^{5 + 5 + 5} \\ = & 27 x^{15} \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} (3x^5)^3 &=& (3x^5)(3x^5)(3x^5) \\ &=& 3 \cdot 3 \cdot 3 \cdot x^5 \cdot x^5 \cdot x^5 \\ &=& 3 \cdot 3 \cdot 3 \cdot x^{5+5+5} \\ &=& 27x^{15} \end{array}\end{align*}$
:3x5)3=(3x5)(3x5)(3x5)=(3x5)3}(3x5)3=(3}3})3=(3x5)3x5=(3x5)3=(3x5)3=(3x5)3=(3x5)5+5+5+5=(27x15)

The answer is $\begin{align*}27x^{15}\end{align*}$ .
::答案是27x15

Simplify each monomial.
::简化每个单声道。

Example 3
::例3

Simplify the monomial $\begin{align*}(6x^3)^2\end{align*}$ .
::简化单声道( 6x3) 2。

First, expand the parentheses by multiplying $\begin{align*}6x^3\end{align*}$ times itself, two times.
::首先,扩大括号,将6x3乘以本身,乘以2倍。

$\begin{aligned} (6 x^{3})^{2} = (6 x^{3}) (6 x^{3}) \end{aligned}$
$\begin{align*}(6x^3)^2 = (6x^3)(6x^3)\end{align*}$
:6x3)2=(6x3)(6x3)

Next, multiply the monomials by placing like factors next to each other, multiplying the coefficients, and simplifying using exponents.
::其次,乘以单数,将类似因素放在彼此旁边,乘以系数,并简化使用引号。

$\begin{aligned} \begin{array}{rcl} (6 x^{3})^{2} & = & (6 x^{3}) (6 x^{3}) \\ = & 6 \cdot 6 \cdot x^{3} \cdot x^{3} \\ = & 6 \cdot 6 \cdot x^{3 + 3} \\ = & 36 x^{6} \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} (6x^3)^2 &=& (6x^3)(6x^3) \\ &=& 6 \cdot 6 \cdot x^3 \cdot x^3 \\ &=& 6 \cdot 6 \cdot x^{3+3} \\ &=& 36 x^6 \end{array}\end{align*}$
:6x3)2=(6x3)(6x3)=(6x3)=(6x6)6x3x3=(6x6)6x3=(6x6)6x3+3=(36x6)

The answer is $\begin{align*}36x^6\end{align*}$ .
::答案是 36x6 。

Example 4
::例4

Simplify the monomial $\begin{align*}(2x^3 y^3)^3\end{align*}$ .
::简化单声道(2x3y3) 3。

First, expand the parentheses by multiplying the monomial times itself, three times.
::首先,扩大括号,将单倍数本身乘以三次。

$\begin{aligned} (2 x^{3} y^{3})^{3} = (2 x^{3} y^{3}) (2 x^{3} y^{3}) (2 x^{3} y^{3}) \end{aligned}$
$\begin{align*}(2x^3 y^3)^3 = (2x^3 y^3)(2x^3 y^3)(2x^3 y^3)\end{align*}$
:2x3y3)3=(2x3y3)(2x3y3)(2x3y3)(2x3y3)

Next, multiply the monomials by placing like factors next to each other, multiplying the coefficients, and simplifying using exponents.
::其次,乘以单数,将类似因素放在彼此旁边,乘以系数,并简化使用引号。

$\begin{aligned} \begin{array}{rcl} (2 x^{3} y^{3})^{3} & = & (2 x^{3} y^{3}) (2 x^{3} y^{3}) (2 x^{3} y^{3}) \\ = & 2 \cdot 2 \cdot 2 \cdot x^{3} \cdot x^{3} \cdot x^{3} \cdot y^{3} \cdot y^{3} \cdot y^{3} \\ = & 2 \cdot 2 \cdot 2 \cdot x^{3 + 3 + 3} \cdot y^{3 + 3 + 3} \\ = & 8 x^{9} y^{9} \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} (2x^3 y^3)^3 &=& (2x^3 y^3)(2x^3 y^3)(2x^3 y^3) \\ &=& 2 \cdot 2 \cdot 2 \cdot x^3 \cdot x^3 \cdot x^3 \cdot y^3 \cdot y^3 \cdot y^3 \\ &=& 2 \cdot 2 \cdot 2 \cdot x^{3+3+3} \cdot y^{3+3+3} \\ &=& 8x^9 y^9 \end{array} \end{align*}$
:2x3y3)3 = (2x3y3)(2x3y3)(2x3y3) = 2-2}2x2x3x3x3x3x3x3}3}3}3=2x2}2x3x3+3+3+3+3×3+3=8x9y9)

The answer is $\begin{align*}8x^9y^9\end{align*}$ .
::答案是 8x9y9。

Example 5
::例5

Simplify the monomial $\begin{align*}(-3x^2 y^2 z)^4\end{align*}$ .
::简化单项 (- 3x2y2z) 4 。

First, expand the parentheses by multiplying the monomial times itself, four times.
::首先,扩大括号,将单倍数本身乘以四倍。

$\begin{aligned} (- 3 x^{2} y^{2} z)^{4} = (- 3 x^{2} y^{2} z) (- 3 x^{2} y^{2} z) (- 3 x^{2} y^{2} z) (- 3 x^{2} y^{2} z) \end{aligned}$
$\begin{align*}(-3x^2 y^2 z)^4 = (-3x^2 y^2 z)(-3x^2 y^2 z)(-3x^2 y^2 z)(-3x^2 y^2 z)\end{align*}$
:-3x2y2z)4=(-3x2y2z) (-3x2y2z) (-3x2y2z) (-3x2y2z) (-3x2y2z)

Next, multiply the monomials by placing like factors next to each other, multiplying the coefficients, and simplifying using exponents.
::其次,乘以单数,将类似因素放在彼此旁边,乘以系数,并简化使用引号。

$\begin{aligned} \begin{array}{rcl} (- 3 x^{2} y^{2} z)^{4} & = & (- 3 x^{2} y^{2} z) (- 3 x^{2} y^{2} z) (- 3 x^{2} y^{2} z) (- 3 x^{2} y^{2} z) \\ = & - 3 \cdot - 3 \cdot - 3 \cdot - 3 \cdot x^{2} \cdot x^{2} \cdot x^{2} \cdot x^{2} \cdot y^{2} \cdot y^{2} \cdot y^{2} \cdot y^{2} \cdot z \cdot z \cdot z \cdot z \\ = & - 3 \cdot - 3 \cdot - 3 \cdot - 3 \cdot x^{2 + 2 + 2 + 2} \cdot y^{2 + 2 + 2 + 2} \cdot z^{1 + 1 + 1 + 1} \\ = & 81 x^{8} y^{8} z^{4} \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} (-3x^2 y^2 z)^4 &=& (-3x^2 y^2 z)(-3x^2 y^2 z)(-3x^2 y^2 z)(-3x^2 y^2 z) \\ &=& -3 \cdot -3 \cdot -3 \cdot -3 \cdot x^2 \cdot x^2 \cdot x^2 \cdot x^2 \cdot y^2 \cdot y^2 \cdot y^2 \cdot y^2 \cdot z \cdot z \cdot z \cdot z \\ &=& -3 \cdot -3 \cdot -3 \cdot -3 \cdot x^{2+2+2+2} \cdot y^{2+2+2+2} \cdot z^{1+1+1+1} \\ &=& 81 x^8 y^8 z^4 \end{array}\end{align*}$
:- 3x2y2z) 4=(- 3x2y2z)(- 3x2y2z)(-3x2y2z)(-3x2y2z)(- 3x2y2z) 3333×3x2x2x2x2x2x2y2z) *}}}}}}}}}4= (- 3x2y2z)(- 3x2y2z)(- 3x2y2z)(-3x2x2y2z)(-3x2x2z2)(- 3x2x2x2z2)(-3x2x2x2x2z1+1+1+1=81x8Y8z4)

The answer is $\begin{align*}81 x^8 y^8 z^4\end{align*}$ .
::答案是81x8Y8z4

Review
::回顾

Simplify.
::简化。

$\begin{align*}(6x^5)^2\end{align*}$
:6x5)2

$\begin{align*}(-13 d^5)^2\end{align*}$
:-13d5)2

$\begin{align*}(-3 p^3 q^4)^3\end{align*}$
:-3p3q4)3

$\begin{align*}(1 0 xy^2)^4\end{align*}$
:10xy2)4

$\begin{align*}(-4t^3)^5\end{align*}$
:-43)5

$\begin{align*}(18 r^2 s^3)^2\end{align*}$
:18r2s3)2

$\begin{align*}(2r^{11} s^3 t^2)^3\end{align*}$
:2r11s3t2)3

$\begin{align*}(7x^2)^2\end{align*}$
:7x2)2

$\begin{align*}(2y^2)^3\end{align*}$
:2y2)3

$\begin{align*}(5x^2)^3\end{align*}$
:5x2)3

$\begin{align*}(12 y^3)^2\end{align*}$
:12y3)2

$\begin{align*}(5x^5)^5\end{align*}$
:5x5)5

$\begin{align*}(2x^2 y^2 z)^3\end{align*}$
:2x2y2z)3

$\begin{align*}(3x^4 y^3 z^2)^3\end{align*}$
:3x4y3z2)3

$\begin{align*}(-5x^4 y^3 z^3)^3\end{align*}$
:-5x4y3z3)3

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

承认并应用产品产权权

Power of Product Property ::产品产权的功率

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Simplify each monomial. ::简化每个单声道 。

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Review ::回顾

Review (Answers) ::回顾(答复)

Power of Product Property
::产品产权的功率

Examples
::实例

Example 1
::例1

Example 2
::例2

Simplify each monomial.
::简化每个单声道。

Example 3
::例3

Example 4
::例4

Example 5
::例5

Review
::回顾

Review (Answers)
::回顾(答复)