Course: 1.8带指数的代数表达式

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带指数的代数表达式
Sam is preparing to have an epic water balloon fight with his best friend Josh. He is at home making his water balloons and wonders what the total volume of water is in his 50 balloons. He knows his balloons are approximately spheres. He also learned that the volume of a sphere is approximately $\begin{aligned} 4.19 r^{3} \end{aligned}$ where $\begin{aligned} r \end{aligned}$ is the radius of the sphere. How could Sam determine the total volume of water in his balloons if the radius of each balloon is approximately 2 inches?
::Sam正准备和他最好的朋友Josh进行一场水球大战。他正在家里制造水球,并想知道他50个气球中的水量是多少。他知道他的气球大致是球体。他还了解到球体的体积大约是4.19 r 3,球体的半径是球体的半径。如果每个气球的半径约为2英寸,萨姆如何确定气球中的水总量?

In this concept, you will learn how to simplify exponential expressions using exponent rules.
::在此概念中,您将学会如何使用引言规则简化指数表达式。

Simplifying Algebraic Expressions with Exponents
::使用指数简化代数表达式

Sometimes you need to multiply a number or a variable by itself many times.
::有时您需要多次乘以一个数字或变量。

Here is an example.
::举一个例子。

$\begin{aligned} x \cdot x \cdot x \cdot x \cdot x \end{aligned}$ is $\begin{aligned} x \end{aligned}$ multiplied by itself 5 times.
::xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

To avoid having to write out the $\begin{aligned} x \end{aligned}$ again and again, you can use an exponent . Whole number exponents are shorthand for repeated multiplication of a number or a variable by itself.
::为了避免一次又一次地写出 x, 您可以使用一个引号。整个数字的引号是用于重复乘用数字或变量本身的简称。

$\begin{aligned} x \cdot x \cdot x \cdot x \cdot x = x^{5} \end{aligned}$

In this example, 5 is the exponent and $\begin{aligned} x \end{aligned}$ is the base . The exponent indicates how many times the base is being multiplied by itself.
::在此示例中, 5 是指数, x 是基数。指数表示基数本身乘以多少倍。

When you use an exponent to write an expression you are using exponential form . $\begin{aligned} x^{5} \end{aligned}$ is exponential form . When you write out the expression using multiplication without an exponent you are using expanded form . $\begin{aligned} x \cdot x \cdot x \cdot x \cdot x \end{aligned}$ is expanded form.
::当您使用一个引号来写一个表达式时,您正在使用指数形式。 x 5 是指数形式。当您用乘法来写出表达式时,没有指数,您正在使用扩张形式。 x = x x x x x x x x x x x x x x x x x x 是扩展形式。

Here is an example.
::举一个例子。

Write the following in expanded form: $\begin{aligned} y^{12} \end{aligned}$
::以扩展形式写作:y 12

First, notice that in this expression $\begin{aligned} y \end{aligned}$ is the base and 12 is the exponent. $\begin{aligned} y^{12} \end{aligned}$ means $\begin{aligned} y \end{aligned}$ is being multiplied by itself 12 times.
::首先,请注意在此表达式中 y 是基数, 12 是指数。 y 12 表示y 本身乘以 12 倍。

Next, write the expression in expanded form without an exponent.
::下一步,用扩展的表达式写出表达式,没有引言。

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

It's much easier to write the expression in exponential form!
::用指数形式写这个表达方式容易得多!

The answer is $\begin{aligned} y^{12} = y \cdot y \cdot y \cdot y \cdot y \cdot y \cdot y \cdot y \cdot y \cdot y \cdot y \cdot y \end{aligned}$ .
::答案是y 12 y y y y y y y y y y y y y y y y y y y y y y y。

When you have a variable expression with more than one exponent, you can often simplify.
::当您有一个带有多个引号的变量表达式时,您可以经常简化。

Here is an example.
::举一个例子。

Simplify $\begin{aligned} m^{3} \cdot m^{2} \end{aligned}$ .
::简化 m 3 m 2 。

First, write the expression in expanded form by expanding each part of the expression.
::首先,通过扩展表达式的每个部分,以扩展的形式写出表达式。

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

Next, rewrite your expression in exponential form using just one base and one exponent.
::接下来,用指数形式重写您的表达方式, 仅使用一个基数和一个表情。

$\begin{aligned} m \cdot m \cdot m \cdot m \cdot m = m^{5} \end{aligned}$

The answer is $\begin{aligned} m^{3} \cdot m^{2} = m^{5} \end{aligned}$ .
::答案是m 3 m 2 = m 5 。

Notice that the exponent in the answer is the sum of the original two exponents. This example helps to illustrate the first rule of exponents.
::请注意, 答案中的引言者是两个原始引言者的总和。此示例有助于说明引言者的第一个规则。

Rule 1: When multiplying two powers that have the same base, you can add the exponents. In symbols, $\begin{aligned} x^{a} \cdot x^{b} = x^{a + b} \end{aligned}$ .
::规则1:当乘以两个具有相同基数的功率时,可以添加引号。在符号中, x x x x x b = x a + b。

Let's apply this rule to the next example.
::让我们在下一个例子中应用这一规则。

Simplify $\begin{aligned} (x^{6}) (x^{3}) \end{aligned}$ .
::简化 (x 6) (x 3) 。

First, remember that when " data-term="Parentheses" role="term" tabindex="0"> parentheses are right next to one another it means multiplication.
::首先,记住当括号相邻时,它意味着乘法。

$\begin{aligned} (x^{6}) (x^{3}) = x^{6} \cdot x^{3} \end{aligned}$

Next, notice that the two bases are the same. This means you can apply Rule 1 and simplify by adding the exponents.
::接下来,请注意这两个基点是相同的。这意味着您可以应用规则1, 并通过添加引号来简化。

$\begin{aligned} x^{6} \cdot x^{3} = x^{6 + 3} = x^{9} \end{aligned}$

If you are ever not sure, remember you can always write the expression in expanded form and then convert back to exponential form.
::如果您永远不确定, 请记住您可以总是以扩展格式写出表达式, 然后转换为指数形式。

$\begin{aligned} x^{6} \cdot x^{3} = x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x = x^{9} \end{aligned}$

The answer is $\begin{aligned} (x^{6}) (x^{3}) = x^{9} \end{aligned}$ .
::答案是 (x 6) (x 3) = x 9。

You can also have an exponential term raised to a power.
::您也可以使用指数化词汇来表达权力。

Here is an example.
::举一个例子。

Simplify $\begin{aligned} (x^{2})^{3} \end{aligned}$ .
::简化 (x 2) 3 。

First, focus on the exponent of 3 and start expanding the expression. $\begin{aligned} x^{2} \end{aligned}$ is being multiplied by itself 3 times.
::首先,关注3 的引言并开始扩展表达式。 x 2 本身乘以 3 倍。

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

Now, you are multiplying three powers that each have the same base so you can use Rule 1 and add the exponents.
::现在,你正在乘以三个功率, 每一个功率都有相同的基数, 这样你就可以使用规则1, 并添加引号。

$\begin{aligned} x^{2} \cdot x^{2} \cdot x^{2} = x^{2 + 2 + 2} = x^{6} \end{aligned}$

The answer is $\begin{aligned} (x^{2})^{3} = x^{6} \end{aligned}$ .
::答案是 (x 2) 3 = x 6 。

Notice that the exponent in the answer is the product of the original two exponents. This example helps to illustrate the second rule of exponents.
::请注意, 答案中的引言者是两位原始引言者的产物。此示例有助于说明引言者的第二项规则。

Rule 2: When raising a power to a power, you can multiply the exponents. In symbols, $\begin{aligned} (x^{a})^{b} = x^{a b} \end{aligned}$ .
::第2条:当向一强权施加权力时,您可以乘以引力。在符号中,( x a) b = x a b 。

Let's apply this rule to the next example.
::让我们在下一个例子中应用这一规则。

Simplify $\begin{aligned} (y^{4})^{3} \end{aligned}$ .
::简化 (y 4) 3 。

First, notice that this expression is a power raised to another power. There are two exponents, but only one variable base. This means you can use Rule 2 and simplify by multiplying the exponents.
::首先, 请注意, 此表达式是向另一个电源生成的电源。有两个推算符, 但只有一个变量基数。这意味着您可以使用规则2 , 并且通过乘法推算符来简化。

$\begin{aligned} (y^{4})^{3} = y^{4 \cdot 3} = y^{12} \end{aligned}$

Again, remember that you can always write your expression in expanded form and then convert back to exponential form if you want to check your answer.
::再次记住,您可以总是以扩展格式写出表达式,然后转换为指数形式,如果您想要检查您的回答。

$\begin{aligned} (y^{4})^{3} = y^{4} \cdot y^{4} \cdot y^{4} = y \cdot y \cdot y \cdot y \cdot y \cdot y \cdot y \cdot y \cdot y \cdot y \cdot y \cdot y = y^{12} \end{aligned}$

The answer is $\begin{aligned} (y^{4})^{3} = y^{12} \end{aligned}$ .

The final exponent rule that you need to know right now has to do with an exponent of 0.
::答案是 (y 4) 3 = y 12 。您现在需要知道的最后提示规则与 0 的提示有关。

Rule 3: Any nonzero number raised to the power of 0 is equal to 1. In symbols, $\begin{aligned} x^{0} = 1 \end{aligned}$ as long as $\begin{aligned} x \neq 0 \end{aligned}$ .
::规则3:任何非零数调到0的功率等于1. 在符号中,x 0 = 1 只要x = 0 。

Here is an example.
::举一个例子。

Simplify $\begin{aligned} 15^{0} \end{aligned}$ .
::简化 15 0 。

First, notice that the exponent is 0. This means you can use Rule 3.
::首先,请注意前言为0。这意味着你可以使用第3条。

$\begin{aligned} 15^{0} = 1 \end{aligned}$

The answer is $\begin{aligned} 15^{0} = 1 \end{aligned}$ .
::答案是15 0=1

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Sam and his upcoming epic water balloon fight.
::之前,你被问及萨姆和他即将到来的史诗水球比赛的问题。

He has 50 water balloons that each have a volume of approximately $\begin{aligned} 4.19 r^{3} \end{aligned}$ where $\begin{aligned} r \end{aligned}$ is the radius of each balloon. He measured his balloons and found that each has a radius of approximately 2 inches. Now he wonders what the total volume of water is in his balloons.
::他拥有50个水气球,每个气球的体积大约为4.19 r 3,其中R是每个气球的半径。他测量了气球,发现每个气球的半径大约为2英寸。现在他想知道气球中的水总量是多少。

First, Sam can figure out the volume of water in one water balloon. He can substitute 2 in for the $\begin{aligned} r \end{aligned}$ in his volume formula .
::首先,山姆可以在一个水气球中找到水的量。他可以用体积配方中的 r 代替 r 2 。

$\begin{aligned} 4.19 (2^{3}) \end{aligned}$

Next, he can expand the expression. 2 is being multiplied by itself 3 times.
::接下来,他可以扩展表达式。 2 本身乘以 3 次。

$\begin{aligned} 4.19 (2^{3}) = 4.19 \cdot 2 \cdot 2 \cdot 2 \end{aligned}$

Then, he can multiply.
::然后,他可以成倍增长。

$\begin{aligned} 4.19 \cdot 2 \cdot 2 \cdot 2 = 33.52 \end{aligned}$

So each water balloon has 33.52 cubic inches of water.
::每个水气球都有33.52立方英寸的水

Now, since he has 50 water balloons, to find the total volume of water in his balloons he can multiply the volume of water in one balloon by 50.
::现在,既然他有50个水气球, 在他的气球中找到总的水量, 他就可以将一个气球中的水量乘以50。

$\begin{aligned} 50 \cdot 33.52 = 1676 \end{aligned}$

The answer is that in total, Sam has 1676 cubic inches of water in his water balloons.
::答案是萨姆的水气球里总共有1 676立方英寸的水

Example 2
::例2

Simplify $\begin{aligned} (x^{6}) (x^{2}) \end{aligned}$ .
::简化 (x 6) (x 2) 。

First, remember that when parentheses are right next to one another it means multiplication.
::首先,记住当括号相邻时,它意味着乘法。

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

Next, notice that the two bases are the same. This means you can apply Rule 1 and simplify by adding the exponents.
::接下来,请注意这两个基点是相同的。这意味着您可以应用规则1, 并通过添加引号来简化。

$\begin{aligned} x^{6} \cdot x^{2} = x^{6 + 2} = x^{8} \end{aligned}$

The answer is $\begin{aligned} (x^{6}) (x^{2}) = x^{8} \end{aligned}$ .
::答案是 (x 6) (x 2) = x 8 。

Example 3
::例3

Write the following in exponential form: $\begin{aligned} a a a a a a a \end{aligned}$
::以指数形式写下: a a a a

First, notice that $\begin{aligned} a \end{aligned}$ is being multiplied by itself 7 times. $\begin{aligned} a \end{aligned}$ will be your base and 7 will be your exponent.
::首先,请注意a本身乘以7倍。a将是你们的基地,7将是你们的表率。

The answer is $\begin{aligned} a a a a a a a = a^{7} \end{aligned}$ .
::答案是 a a a = a 7。

Example 4
::例4

Simplify $\begin{aligned} (a^{3}) (a^{8}) \end{aligned}$ .
::简化(a 3) (a 8) 。

First, remember that when parentheses are right next to one another it means multiplication.
::首先,记住当括号相邻时,它意味着乘法。

$\begin{aligned} (a^{3}) (a^{8}) = a^{3} \cdot a^{8} \end{aligned}$

Next, notice that the two bases are the same. This means you can apply Rule 1 and simplify by adding the exponents.
::接下来,请注意这两个基点是相同的。这意味着您可以应用规则1, 并通过添加引号来简化。

$\begin{aligned} a^{3} \cdot a^{8} = a^{3 + 8} = a^{11} \end{aligned}$

The answer is $\begin{aligned} (a^{3}) (a^{8}) = a^{11} \end{aligned}$ .
::答案是(a 3) (a 8) = 11。

Example 5
::例5

Simplify $\begin{aligned} (x^{4})^{2} \end{aligned}$ .
::简化( x 4) 2 。

First, notice that this expression is a power raised to another power. This means you can use Rule 2 and simplify by multiplying the exponents.
::首先, 请注意, 此表达式是向另一权力生成的能量。这意味着您可以使用规则2 , 并通过乘以引号来简化。

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

The answer is $\begin{aligned} (x^{4})^{2} = x^{8} \end{aligned}$ .
::答案是 (x 4) 2 = x 8 。

Review
::回顾

Evaluate each expression.
::评估每个表达方式。

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

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

$\begin{aligned} 5^{2} \end{aligned}$

$\begin{aligned} 9^{0} \end{aligned}$

$\begin{aligned} 5^{3} \end{aligned}$

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

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

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

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

$\begin{aligned} 6^{2} + 2^{3} \end{aligned}$

$\begin{aligned} 6^{2} - 5^{2} \end{aligned}$

$\begin{aligned} 2^{4} - 2^{2} \end{aligned}$

$\begin{aligned} 7^{2} + 3^{3} + 2^{2} \end{aligned}$

Simplify each expression.
::简化每个表达式。

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

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

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

$\begin{aligned} (b^{7}) (b^{2}) \end{aligned}$
:b) 7 (b) (b) 2 (b)

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

$\begin{aligned} (x^{9}) (x^{3}) \end{aligned}$
:x 9) (x 3)

$\begin{aligned} (y^{4}) (y^{5}) \end{aligned}$
:y 4) (y 5) (y 5)

$\begin{aligned} (x^{2})^{4} \end{aligned}$
:x 2 ) 4

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

$\begin{aligned} (a^{5})^{4} \end{aligned}$
:a) 5 ) 4

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

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

Review (Answers)
::回顾(答复)

Section outline

General

带指数的代数表达式

Simplifying Algebraic Expressions with Exponents ::使用指数简化代数表达式

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Review ::回顾

Review (Answers) ::回顾(答复)

Simplifying Algebraic Expressions with Exponents
::使用指数简化代数表达式

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Example 5
::例5

Review
::回顾

Review (Answers)
::回顾(答复)