Course: 1.7 整数指数 | The Way To Learn

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整数指数
Sheila's parents are encouraging her to save her money. Sheila currently has $3. Sheila's parents tell her that for every month she saves her money instead of spending it, they will double the amount of money she has! Sheila decides she will save her money and not spend it for 6 months. How could Sheila write an expression in exponential form and evaluate to determine how much money she will have in 6 months?
::希拉的父母鼓励她存钱。希拉的父母现在有3美元。希拉的父母告诉她,她每个月省钱而不是花钱,她们会把钱翻一番!希拉决定,她会省钱,而不会花6个月。希拉怎么能够写出指数式的表情,评估6个月里她会有多少钱呢?

In this concept, you will learn how to write and evaluate expressions in exponential form.
::在此概念中,您将学习如何以指数形式书写和评估表达式。

Writing and Evaluating Whole Number Exponents
::撰写和评估整体数量指数

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。

$\begin{aligned} 4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4 \end{aligned}$ is 4 multiplied by itself 7 times.
::4x 4x 4x 4x 4x 4x 4x 4x 4x 4乘以 4x 7 次。

To avoid having to write out the 4 again and again, you can use an exponent . Whole number are shorthand for repeated multiplication of a number by itself.
::为了避免一次又一次地写出 4, 您可以使用一个引号。整个数字是重复乘以数字的简称。

$\begin{aligned} 4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4 = 4^{7} \end{aligned}$

In this example, 7 is the exponent and 4 is the base . The exponent indicates how many times the base is being multiplied by itself.
::在这个例子中,7是指数,4是基数,说明基数本身乘以多少倍。

Using an exponent can also be called "raising to a power". The exponent represents the power.
::使用标语也可以称为“向强权挺进”。标语代表强权。

For example, $\begin{aligned} 4^{7} \end{aligned}$ could be read "4 to the seventh power".
::例如,4 7 可以读为“4至7强”。

There are two exponents that have special names. A base raised to the power of 2 is said to be squared . A base raised to the power of 3 is said to be cubed .
::有两个有特殊名称的表率。被提升到 2 位数的基数据说是平的。被提升到 3 位数的基数据说是立方的。

Here is an example.
::举一个例子。

$\begin{aligned} 4^{2} \end{aligned}$ could be read "4 to the second power" or "4 squared".
::4 2 可读为“4对二功率”或“4平方”。

$\begin{aligned} 4^{3} \end{aligned}$ could be read "4 to the third power" or "4 cubed".
::3 可读为“4对第三电源”或“4立方”。

When you use an exponent to write an expression you are using exponential form . $\begin{aligned} 4^{7} \end{aligned}$ is exponential form. When you write out the expression using multiplication without an exponent you are using expanded form . $\begin{aligned} 4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4 \end{aligned}$ is expanded form.
::当您使用一个引言来写一个表达式时,您正在使用指数形式。 4 7 是指数形式。当您用乘法来写出表达式时, 没有引言, 您正在使用扩展的表达式。 4 x 4 4 x 4 4 x 4 4 x 4 x 4 4 x 4 4 4 4 是扩展的表达式。

Here is an example.
::举一个例子。

Write the following in exponential form: $\begin{aligned} 6 \times 6 \times 6 \times 6 \end{aligned}$
::以指数形式写如下: 6× 6× 6× 6x 6

First, notice that 6 is being multiplied by itself 4 times. 6 will be your base and 4 will be your exponent.
::第一,注意6乘以4倍。 6将是你的基地,4将是你的表率。

The answer is $\begin{aligned} 6 \times 6 \times 6 \times 6 = 6^{4} \end{aligned}$ .
::答案是 6 × 6 × 6 × 6 = 6 4 。

Here is another example.
::下面是另一个例子。

Write the following in expanded form and evaluate the expression: $\begin{aligned} 5^{3} \end{aligned}$
::以扩大的形式写作并评价表达方式:5 3

First, read the expression. This expression could be read as "5 to the third power" or "5 cubed".
::首先,读读这个表达式。这个表达式可以读为“ 5 到第三权力” 或“ 5 立方”。

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。

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

Then, multiply.
::然后,乘数。

$\begin{aligned} 5 \times 5 \times 5 = 125 \end{aligned}$

The answer is $\begin{aligned} 5^{3} = 5 \times 5 \times 5 = 125 \end{aligned}$ .
::答案是5 3 = 5 × 5 × 5 = 125。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Sheila and her $3.
::早些时候,你得到一个问题关于希拉和她3美元。

Her parents told her they would double the amount of money she has for every month she saves her money instead of spending it. Sheila decides she will save her money for 6 months and wonders how much money she will have at this point.
::她的父母告诉她,她每月储蓄的钱将翻一番,而不是花掉这笔钱。 Sheila决定,她将节省6个月的钱,并想知道现在她有多少钱。

First, write an expression to represent how much money Sheila will have after 6 months. Start with how much money she will have after one month and work your way up to 6 months. Remember that to double means to multiply by 2.
::首先, 写一个表达方式来代表 Sheila 6个月后会有多少钱。从她一个月后会有多少钱开始, 工作到6个月。记住要双倍地乘以 2 。

Sheila starts with $3.
::希拉从3美元开始

After 1 month she will have $\begin{aligned} $ 3 \times 2 \end{aligned}$ .
::一个月后,她将有3x2美元。

After 2 months she will have $\begin{aligned} $ 3 \times 2 \times 2 \end{aligned}$ or $\begin{aligned} $ 3 \times 2^{2} \end{aligned}$ .
::两个月后,她将有3x2x2或3x22美元。

After 3 months she will have $\begin{aligned} $ 3 \times 2 \times 2 \times 2 \end{aligned}$ or $\begin{aligned} $ 3 \times 2^{3} \end{aligned}$ .
::三个月后,她将有3x2x2x2x2或3x2x3美元。

Continuing in this pattern you can see that
::继续此模式您可以看到

After 6 months she will have $\begin{aligned} $ 3 \times 2^{6} \end{aligned}$ .
::6个月后,她将有3x2 6美元。

Next, evaluate the expression in order to figure out how much money she will have in 6 months. First, write the expression in expanded form.
::接下来,评估表达式,以便弄清楚她6个月后会有多少钱。首先,以扩展的形式写出表达式。

$\begin{aligned} 3 \times 2^{6} = 3 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \end{aligned}$

Now, multiply.
::现在,乘数。

$\begin{aligned} 3 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 192 \end{aligned}$

The answer is that after 6 months of saving Sheila will have $192. Not bad!
::答案是,在拯救希拉六个月后, 将拥有192美元。不错!

Example 2
::例2

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

First, write the expression in expanded form.
::首先,以扩展形式写出表达式。

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

Next, multiply each part of the expression.
::下一步,将表达式的每个部分乘以。

$\begin{aligned} 2 \times 2 \times 2 + 4 \times 4 = 8 + 16 \end{aligned}$

Then, add.
::然后,添加。

$\begin{aligned} 8 + 16 = 24 \end{aligned}$

The answer is $\begin{aligned} 2^{3} + 4^{2} = 24 \end{aligned}$ .
::答案是2 3 + 4 2 = 24。

Example 3
::例3

Write the following in exponential form: $\begin{aligned} 3 \times 3 \times 3 \times 3 \times 3 \end{aligned}$
::以指数形式写如下: 3x 3x 3x 3x 3x 3

First, remember that to write an expression in exponential form you need a base and an exponent. The base is the number that is being multiplied by itself. The exponent is the number of times the base is being multiplied by itself.
::首先, 请记住要以指数形式写出表达式, 您需要一个基数和一个指数。基数是自己乘以的数。引号是基数乘以的数。引号是基数乘以的数。

The base is 3.
::基数是3

The exponent is 5.
::指数是5

Now, write in exponential form.
::现在,以指数形式写作。

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

The answer is $\begin{aligned} 3 \times 3 \times 3 \times 3 \times 3 = 3^{5} \end{aligned}$ .
::答案是3x3x3x3x3x3x3x3=3x5。

Example 4
::例4

Write the following in expanded form and evaluate the expression: $\begin{aligned} 6^{3} \end{aligned}$
::以扩大的形式写作并评价表达方式:6 3

First, write the expression in expanded form without an exponent.
::首先,用扩展的表达式写出表达式,而没有缩略语。

$\begin{aligned} 6^{3} = 6 \times 6 \times 6 \end{aligned}$

Next, multiply.
::下一个,乘数。

$\begin{aligned} 6 \times 6 \times 6 = 216 \end{aligned}$

The answer is $\begin{aligned} 6^{3} = 6 \times 6 \times 6 = 216 \end{aligned}$ .
::答案是6 3 = 6 × 6 × 6 = 216。

Example 5
::例5

Evaluate: $\begin{aligned} 4^{3} - 5^{2} \end{aligned}$
::评价:4 3 - 5 2

First, write the expression in expanded form.
::首先,以扩展形式写出表达式。

$\begin{aligned} 4^{3} - 5^{2} = 4 \times 4 \times 4 - 5 \times 5 \end{aligned}$

Next, multiply each part of the expression.
::下一步,将表达式的每个部分乘以。

$\begin{aligned} 4 \times 4 \times 4 - 5 \times 5 = 64 - 25 \end{aligned}$

Then, subtract.
::然后,减。

$\begin{aligned} 64 - 25 = 39 \end{aligned}$

The answer is $\begin{aligned} 4^{3} - 5^{2} = 39 \end{aligned}$ .
::答案是4 3 - 5 2 = 39 。

Review
::回顾

Name the base and exponent in the following expressions. Then, write each in expanded form.
::以下列表达式命名基数和引号。然后,以扩展的形式写出每个字。

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

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

$\begin{aligned} 5^{8} \end{aligned}$

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

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

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

$\begin{aligned} 1^{10} \end{aligned}$

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

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

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

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

$\begin{aligned} 8^{10} \end{aligned}$

$\begin{aligned} 9^{3} \end{aligned}$

$\begin{aligned} 12^{2} \end{aligned}$

$\begin{aligned} 13^{3} \end{aligned}$

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。

Resources
::资源

Section outline

General

整数指数

Writing and Evaluating Whole Number Exponents ::撰写和评估整体数量指数

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)

Resources ::资源

Writing and Evaluating Whole Number Exponents
::撰写和评估整体数量指数

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)

Resources
::资源