节: 提高至指数值的指数值术语 | 8.2 指数术语提高到指数

章节大纲

Exponential Terms Raised to an Exponent
::提高至指数值的指数值术语

What happens when we raise a whole expression to a power? Let’s take $\begin{aligned} x \end{aligned}$ to the power of 4 and cube it . Again we’ll use the full factored form for each expression:
::当我们提高整个电源的表达方式时会怎样?让我们把 x 带到 4 和立方体的表达方式。我们再次对每个表达方式使用完整因素的表达方式 :

$\begin{aligned} (x^{4})^{3} & = x^{4} \times x^{4} \times x^{4} 3 f a c t o r s o f {x t o t h e p o w e r 4} \\ (x \cdot x \cdot x \cdot x) \cdot (x \cdot x \cdot x \cdot x) \cdot (x \cdot x \cdot x \cdot x) & = x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x = x^{12} \end{aligned}$

:x4)3=x4xx4xx4x4xxxx43 乘以 {x 到功率 4} (xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx)=xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

So $\begin{aligned} (x^{4})^{3} = x^{12} \end{aligned}$ . You can see that when we raise a power of $\begin{aligned} x \end{aligned}$ to a new power, the powers multiply.
::So (x4) 3=x12。您可以看到当我们将 x 的功率提升到新功率时, 功率会倍增。

Power Rule for Exponents: $\begin{aligned} (x^{n})^{m} = x^{(n \cdot m)} \end{aligned}$
::指数规则 : (xn) m=x m

If we have a product of more than one term inside the " data-term="Parentheses" role="term" tabindex="0"> parentheses , then we have to distribute the exponent over all the factors, like distributing multiplication over addition . For example:
::如果我们在括号内有一个多个术语的产物,那么我们就必须在所有因素上分配引号,例如分配乘法乘法加法。例如:

%5E4%20%3D%20x%5E8y%5E4.">
$\begin{aligned} (x^{2} y)^{4} = (x^{2})^{4} \cdot (y)^{4} = x^{8} y^{4} . \end{aligned}$

:x2y)4=(x2)44=x8y4。

Or, writing it out the long way:
::或者,写出来很长的路要走:

$\begin{aligned} (x^{2} y)^{4} = (x^{2} y) (x^{2} y) (x^{2} y) (x^{2} y) & = (x \cdot x \cdot y) (x \cdot x \cdot y) (x \cdot x \cdot y) (x \cdot x \cdot y) \\ = x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot y \cdot y \cdot y \cdot y = x^{8} y^{4} \end{aligned}$

:x2y) 4= (x2y)(x2y)(x2y)(x2y)(x2y) = (xxxx非) (xxxx非) (xxxx非) =xxxx非) =xxxxxxxxxxxxxx非) =x*xxxxxxxxxxxxxxxxxxx

Note that this does NOT work if you have a sum or difference inside the parentheses! For example, $\begin{aligned} (x + y)^{2} \neq x^{2} + y^{2} \end{aligned}$ . This is an easy mistake to make, but you can avoid it if you remember what an exponent means: if you multiply out $\begin{aligned} (x + y)^{2} \end{aligned}$ it becomes $\begin{aligned} (x + y) (x + y) \end{aligned}$ , and that’s not the same as $\begin{aligned} x^{2} + y^{2} \end{aligned}$ . We’ll learn how we can simplify this expression in a later chapter.
::请注意, 如果您在括号内有一个总和或差数, 此操作无效。例如, (x+y) 2\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

Simplifying Expressions
::简化表达式

1. Simplify the following expressions.
::1. 简化以下表达式。

When we’re just working with numbers instead of variables, we can use the product rule and the power rule, or we can just do the multiplication and then simplify.
::当我们只是用数字而不是变量来工作时, 我们可以使用产品规则和电力规则, 或者我们可以做乘法然后简化。

a) $\begin{aligned} 3^{5} \cdot 3^{7} \end{aligned}$
:a) 35日37

We can use the product rule first and then evaluate the result: $\begin{aligned} 3^{5} \cdot 3^{7} = 3^{12} = 531441 \end{aligned}$ .
::我们可以首先使用产品规则,然后对结果进行评估:35日37=312=531441。

OR we can evaluate each part separately and then multiply them: $\begin{aligned} 3^{5} \cdot 3^{7} = 243 \cdot 2187 = 531441 \end{aligned}$ .
::我们可以分别评估每一部分,然后乘以:35日37=243日2187=531441。

b) $\begin{aligned} 2^{6} \cdot 2 \end{aligned}$
:b) 262

We can use the product rule first and then evaluate the result: $\begin{aligned} 2^{6} \cdot 2 = 2^{7} = 128 \end{aligned}$ .
::我们可以首先使用产品规则,然后对结果进行评估:262=27=128。

OR we can evaluate each part separately and then multiply them: $\begin{aligned} 2^{6} \cdot 2 = 64 \cdot 2 = 128 \end{aligned}$ .
::我们可以分别评估每一部分,然后乘以:262=642=128。

c) $\begin{aligned} (4^{2})^{3} \end{aligned}$
:c) (42)3

We can use the power rule first and then evaluate the result: $\begin{aligned} (4^{2})^{3} = 4^{6} = 4096 \end{aligned}$ .
::我们可以首先使用权力规则,然后评估结果42)3=46=4096。

OR we can evaluate the expression inside the parentheses first, and then apply the exponent outside the parentheses: $\begin{aligned} (4^{2})^{3} = (16)^{3} = 4096 \end{aligned}$ .
::OR 我们可以首先评价括号内的表达式, 然后应用括号外的引号423=(16)3=4096)。

2. Simplify the following expressions.
::2. 简化以下表达式。

When we’re just working with variables, all we can do is simplify as much as possible using the product and power rules.
::当我们只是与变数合作时, 我们所能做的就是尽量简化使用产品和电力规则。

a) $\begin{aligned} x^{2} \cdot x^{7} \end{aligned}$
::a) x2x7

$\begin{aligned} x^{2} \cdot x^{7} = x^{2 + 7} = x^{9} \end{aligned}$
::x2x7=x2+7=x9

b) $\begin{aligned} (y^{3})^{5} \end{aligned}$
:b) (y3)5

$\begin{aligned} (y^{3})^{5} = y^{3 \times 5} = y^{15} \end{aligned}$
:y3)5=y3x5=y15

3. Simplify the following expressions.
::3. 简化以下表达式。

When we have a mix of numbers and variables, we apply the rules to each number and variable separately.
::当我们混合了数字和变量时,我们将规则分别适用于每个数字和变量。

a) $\begin{aligned} (3 x^{2} y^{3}) \cdot (4 x y^{2}) \end{aligned}$
:a) (3x2y3)□(4x2)

First we group like terms together: $\begin{aligned} (3 x^{2} y^{3}) \cdot (4 x y^{2}) = (3 \cdot 4) \cdot (x^{2} \cdot x) \cdot (y^{3} \cdot y^{2}) \end{aligned}$
::首先,我们将类似术语组合为: (3x2y3) (4xy2) = (3}4) (x2xx) (y3y2)

Then we multiply the numbers or apply the product rule on each grouping: $\begin{aligned} = 12 x^{3} y^{5} \end{aligned}$
::然后我们乘以数字或对每个组别应用产品规则:=12x3y5

b) $\begin{aligned} (4 x y z) \cdot (x^{2} y^{3}) \cdot (2 y z^{4}) \end{aligned}$
:b) (4xyz)□(x2y3)□(2yz4)

Group like terms together: $\begin{aligned} (4 x y z) \cdot (x^{2} y^{3}) \cdot (2 y z^{4}) = (4 \cdot 2) \cdot (x \cdot x^{2}) \cdot (y \cdot y^{3} \cdot y) \cdot (z \cdot z^{4}) \end{aligned}$
::组式相同词组 : (4xyz) (x2y3) (2yz4) = (4}2}(xxx2) (yy3y) (zz4)

Multiply the numbers or apply the product rule on each grouping: $\begin{aligned} = 8 x^{3} y^{5} z^{5} \end{aligned}$
::乘以数字或对每个组别应用产品规则:=8x3y5z5

c) $\begin{aligned} (2 a^{3} b^{3})^{2} \end{aligned}$
:c) (2a3b3)2

Apply the power rule for each separate term in the parentheses: $\begin{aligned} (2 a^{3} b^{3})^{2} = 2^{2} \cdot (a^{3})^{2} \cdot (b^{3})^{2} \end{aligned}$
::括号内为每个单独的术语适用权力规则2a3b3)2=22(a3)22(b3)2

Multiply the numbers or apply the power rule for each term $\begin{aligned} = 4 a^{6} b^{6} \end{aligned}$
::乘以数字或应用每个术语 = 4a6b6 的权力规则

Examples
::实例

Simplify the following expressions.
::简化下列表达式。

In problems where we need to apply the product and power rules together, we must keep in mind the . Exponent operations take precedence over multiplication.
::在我们需要共同适用产品和权力规则的问题中,我们必须牢记.能见性操作优先于倍增。

Example 1
::例1

$\begin{aligned} (x^{2})^{2} \cdot x^{3} \end{aligned}$
:x2)2xx3

We apply the power rule first: $\begin{aligned} (x^{2})^{2} \cdot x^{3} = x^{4} \cdot x^{3} \end{aligned}$
::我们首先应用权力规则x2)2x3=x4x3

Then apply the product rule to combine the two terms : $\begin{aligned} x^{4} \cdot x^{3} = x^{7} \end{aligned}$
::然后应用产品规则将两个词: x4x3=x7合并

Example 2
::例2

$\begin{aligned} (2 x^{2} y) \cdot (3 x y^{2})^{3} \end{aligned}$
:2x2y)(3x2)3

Apply the power rule first: $\begin{aligned} (2 x^{2} y) \cdot (3 x y^{2})^{3} = (2 x^{2} y) \cdot (27 x^{3} y^{6}) \end{aligned}$
::首先应用权力规则 : (2x2y) = (2x2y) = (2x2y) = (27x3y6)

Then apply the product rule to combine the two terms: $\begin{aligned} (2 x^{2} y) \cdot (27 x^{3} y^{6}) = 54 x^{5} y^{7} \end{aligned}$
::然后适用产品规则,将两个术语合并2x2y)(27x3y6)=54x5y7)

Example 2
::例2

$\begin{aligned} (4 a^{2} b^{3})^{2} \cdot (2 a b^{4})^{3} \end{aligned}$
:4a2b3)2(2ab4)3

Apply the power rule on each of the terms separately: $\begin{aligned} (4 a^{2} b^{3})^{2} \cdot (2 a b^{4})^{3} = (16 a^{4} b^{6}) \cdot (8 a^{3} b^{12}) \end{aligned}$
::分别对每个条款分别适用权力规则: (4a2b3)2(2ab4)3=(16a4b6)_(8a3b12)

Then apply the product rule to combine the two terms: $\begin{aligned} (16 a^{4} b^{6}) \cdot (8 a^{3} b^{12}) = 128 a^{7} b^{18} \end{aligned}$
::然后适用产品规则,将两个术语结合起来: (16a4b6)(8a3b12)=128a7b18

Review
::回顾

Simplify:
::简化 :

$\begin{aligned} (a^{3})^{4} \end{aligned}$
:a3)4

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

$\begin{aligned} (- 5 y)^{3} \end{aligned}$
:-5y)3

$\begin{aligned} (3 a^{2} b^{3})^{4} \end{aligned}$
:3a2b3)4

$\begin{aligned} (- 2 x y^{4} z^{2})^{5} \end{aligned}$
:-2xy4z2)5

$\begin{aligned} (- 8 x)^{3} (5 x)^{2} \end{aligned}$
:-8x)3(5x)2

$\begin{aligned} (- x)^{2} (x y)^{3} \end{aligned}$
:xx)(xy)3

$\begin{aligned} (4 a^{2}) (- 2 a^{3})^{4} \end{aligned}$
:4a2)(-2a3)4

$\begin{aligned} (12 x y) (12 x y)^{2} \end{aligned}$
:12xy)(12xy)2

$\begin{aligned} (2 x y^{2}) (- x^{2} y)^{2} (3 x^{2} y^{2}) \end{aligned}$
:2x)(-x2y)(2x2x2y)(2x2x2y)

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。

章节大纲

Exponential Terms Raised to an Exponent ::提高至指数值的指数值术语

Simplifying Expressions ::简化表达式

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 2 ::例2

Review ::回顾

Review (Answers) ::回顾(答复)

Exponential Terms Raised to an Exponent
::提高至指数值的指数值术语

Simplifying Expressions
::简化表达式

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 2
::例2

Review
::回顾

Review (Answers)
::回顾(答复)