Section: 心算乘十的幂 | 4.13心算乘十的幂

Section outline

According to the United States Mint, a quarter weighs 5.67 grams. Jeremy has a kitchen scale that can measure up to 5,050 grams. Susan bets Jeremy that they could probably weigh about 1,000 quarters on the kitchen scale. Jeremy thinks that a thousand quarters would be too much for the scale. Who is correct?
::根据美国Mint,四分之一重5.67克。Jeremy的厨房规模可以达到5,050克。Susan打赌,他们可能在厨房规模上达到1,000克。Jeremy认为,一千4/4比这个规模大。谁是正确的?

In this concept, you will learn to multiply decimals by powers of ten using mental math.
::在这个概念中,你会学会使用心理数学以十分之十的功率乘以十分位数。

Powers of Ten
::十国权力的十国权力

You have already learned to multiply decimal by whole numbers , however, there is a pattern that you can follow when you multiply decimals by powers of ten .
::您已经学会了将小数点乘以整数, 但是, 当将小数点乘以十位数乘以十位数时, 您可以遵循一种模式。

Powers of ten are numbers where 10 is multiplied by itself a certain number of times. It can be written in exponent form where the base is 10 and the exponent , $\begin{aligned} n \end{aligned}$ , is the number of times 10 is multiplied by itself.
::十的权数是十的权数,其中十的权数本身乘以一定次数。它可以以引言形式写成,其中基数为十,引言,n,是十的权数本身乘以十。

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

Here are some examples of power of ten numbers.
::以下是十个数字的力量的例子。

$\begin{aligned} \begin{array}{rcl} 10^{1} & = & 1 \times 10 = 10 \\ 10^{2} & = & 1 \times 10 \times 10 = 100 \\ 10^{3} & = & 1 \times 10 \times 10 \times 10 = 1, 000 \\ 10^{4} & = & 1 \times 10 \times 10 \times 10 \times 10 = 10, 000 \end{array} \end{aligned}$

Notice that the exponent is the same as the number of zeros in the number. Let’s look at a decimal number multiplied by power of ten numbers.
::注意前言与数字中的零数相同。让我们看看小数数乘以十位数的功率。

$\begin{aligned} \begin{array}{rcl} 3.4 \times 1 \underline{0} = 3 \underset{\to}{4} \\ 3.4 \times 1 \underline{0} \underline{0} = 3 \underset{\to}{40} \\ 3.4 \times 1 \underline{0} \underline{0} \underline{0} = 3, \underset{\to}{400} \end{array} \end{aligned}$

The decimal point is moved to the right once for every 10 multiplied or the number of zeros in the power of ten number. In the first example, 10 has one zero and the decimal point is moved one place to the right. In the second example, 100 has two zeros and the decimal point is moved two places to the right. The third example has three zeros and the decimal point is moved three places to the right.
::小数点每十倍向右移动一次,或者以十位数的功率向右移动一次。在第一个例子中,十位数为零,十位数为零,小数点向右移动一处。在第二个例子中,100位数为两处零,小数点向右移动两处。第三个例子为三处零,小数点向右移动三处。

Powers of ten also include decimal values. These powers of ten are also called negative powers of ten . It is written like a power of ten except that the number in the exponent has a negative sign (-). The negative sign tells you to divide by 10 a total of $\begin{aligned} n \end{aligned}$ number of times. Here are some decimal values for negative powers of ten.
::十的功率也包含十进制值。这些十的功率也称为负十进制数。它写成为十进制数,但指数中的数字有一个负符号(-)。负符号指示你除以10,总数值为n倍。这里是负十进制数的十进制数。

$\begin{aligned} \begin{array}{rcl} 10^{- 1} & = & \frac{1}{10} = 0.1 \\ 10^{- 2} & = & \frac{1}{10 \times 10} = 0.01 \\ 10^{- 3} & = & \frac{1}{10 \times 10 \times 10} = 0.001 \\ 10^{- 4} & = & \frac{1}{10 \times 10 \times 10 \times 10} = 0.0001 \end{array} \end{aligned}$

Let’s look at a decimal number multiplied by decimal power of ten numbers.
::让我们看看十位数乘以十位数的小数数乘以十位数的十位数。

$\begin{aligned} \begin{array}{rcl} 3.4 \times 0. \underline{1} = 0 . \underset{\leftarrow}{3} 4 \\ 3.4 \times 0. \underline{0} \underline{1} = 0 . \underset{\leftarrow}{03} 4 \\ 3.4 \times 0. \underline{0} \underline{0} \underline{1} = 0 . \underset{\leftarrow}{003} 4 \end{array} \end{aligned}$

The decimal point in the product is moved to the left for every decimal place in the power of ten decimal number. In the first example, there is one decimal place in 0.1 and the decimal point is moved one place to the left. In the second example, there are two decimal places in 0.01 and the decimal point is moved two places to the left. The third example has three decimal places in 0.001 and the decimal point is moved three places to the left.
::产品中的小数点移动到左方, 在十进制十进制数的每个小数点位数中, 小数点点位移到左方。在第一个例子中, 0.1 位为小数点位, 小数点位移到左方一位。在第二个例子中, 0.01 位为小数点位, 小数点位移到左方两位。第三个例子在0.001 位为小数点位数,小数点位移到左方三位。

Depending on what type of power of ten is being multiplied, you can calculate the product using mental math.
::取决于10的功率乘以何种类型,您可以用心理数学计算出产品。

When multiplying by whole number powers of ten, the number will become larger. Move the decimal to the right once for every zero in the power of ten.
::当乘以十的整数功率乘以十时, 数字会变大。在十时的功率为每零时, 将小数点向右移动一次。

When multiplying by decimal powers of ten, the number will become smaller. Move the decimal to the left once for every decimal place value .
::当乘以十位数的十位数乘以十位数时,数字会变小。将小数数位数移到左位数时,每个小数点位数的位数都会移动一次。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Jeremy and Susan and their bet.
::早些时候,你得到一个问题杰里米和苏珊和他们的赌注。

Susan and Jeremy bet on if the kitchen scale can weigh 1,000 quarters. To find the weight of 1,000 quarters, multiply the weight of a quarter by 1,000. A quarter weighs 5.67 grams.
::Susan和Jeremy打赌,如果厨房的重量比重是1 000个季度。要找到1 000个季度的重量,重量乘以1/4乘以1 000。四分之一重5.67克。

First, write an equation to find the weight of 1,000 quarters.
::首先,写一个方程找到1000个季度的重量

$\begin{aligned} 5.67 \times 1, 000 = \underline{} \end{aligned}$

Then, use mental math to find the product. Count the number of zeros in the power of ten number. There are three.
::然后,用心理数学来寻找产品。在十位数的功率中计算零数。有三个。

Next, move the decimal point three places to the right.
::接下来,将小数点三移到右边。

$\begin{aligned} 5.67 \to 5, 670 . \end{aligned}$

Finally, compare the weight of 1,000 quarter to the maximum weight the scale can measure. The kitchen scale can measure up to 5,050 grams.
::最后,将1 000个季度的重量与比额表能够测量的最大重量作比较。厨房的重量可以测量到5 050克。

$\begin{aligned} \begin{array}{rcl} \underline{Quarters} & \underline{Scale} \\ 5, 670 g & > & 5, 050 g \end{array} \end{aligned}$

The scale will not be able to weigh the quarters. Jeremy is correct.
::Jeremy是对的

Example 2
::例2

Use mental math to find the product.
::运用心理数学找到产品

$\begin{aligned} 4.567 \times 1000 = \underline{} \end{aligned}$

First, count the number of zeros in the power of ten number. There are three.
::首先,在十位数的功率下计数零数。有三位。

Then, move the decimal point three places to the right.
::然后,将小数点三移到右边。

$\begin{aligned} 4.567 \to 4, 567 . \end{aligned}$

The product of 4.567 times 1000 is 4,567.
::4.567乘以1000是4 567的产物。

$\begin{aligned} 6.7 \times 0.1 = \underline{} \end{aligned}$

First, count the number of decimal place values in the power of ten number. There is one.
::首先,以十位数的功率计算十位数中的小数位数。有一个。

Then, move the decimal point once to the left.
::然后,将小数点移到左边。

$\begin{aligned} 6.7 \to 0.67 \end{aligned}$

The product of 6.7 and 0.1 is 0.67.
::6.7和0.1的产品为0.67。

Use mental math to find the product for the following problems.
::使用心理数学来寻找产品,解决以下问题。

Example 3
::例3

$\begin{aligned} 0.23 \times 100 = \underline{} \end{aligned}$

First, count the number of zeros in the power of ten number. There are two.
::首先,在十位数的功率下计数零数。有两个。

Then, move the decimal point two places to the right.
::然后,将小数点两点移到右边。

$\begin{aligned} 0.23 \to 23 . \end{aligned}$

The product of 0.23 and 100 is 23.
::0.23和100的产品为23。

Example 4
::例4

$\begin{aligned} 213.5 \times 0.01 = \underline{} \end{aligned}$

First, count the number of decimal place values in the power of ten number. There are two.
::首先,以十位数的功率计算十位数的小数位数。有两个。

Then, move the decimal point two places to the left.
::然后,将小数点两点移到左边。

$\begin{aligned} 213.5 \to 2.135 \end{aligned}$

The product of 213.5 times 0.01 is 2.135.
::产品213.5乘以0.01为2.135。

Example 5
::例5

$\begin{aligned} 127.3 \times 1, 000 = \underline{} \end{aligned}$

First, count the number of zeros in the power of ten number. There are three.
::首先,在十位数的功率下计数零数。有三位。

Then, move the decimal point three places to the right.
::然后,将小数点三移到右边。

$\begin{aligned} 127.3 \to 127, 300 . \end{aligned}$

The product of 127.3 times 1,000 is 127,300.
::127.3乘以1 000的产品为127 300。

Review
::回顾

Use mental math to find the product for the following problems.
::使用心理数学来寻找产品,解决以下问题。

$\begin{aligned} 3.4 \times 10 = \underline{} \end{aligned}$

$\begin{aligned} 0.56 \times 10 = \underline{} \end{aligned}$

$\begin{aligned} 1.234 \times 1000 = \underline{} \end{aligned}$

$\begin{aligned} 87.9 \times 100 = \underline{} \end{aligned}$

$\begin{aligned} 98.32 \times 10 = \underline{} \end{aligned}$

$\begin{aligned} 7.2 \times 1000 = \underline{} \end{aligned}$

$\begin{aligned} 12.5 \times 10 = \underline{} \end{aligned}$

$\begin{aligned} 18.91 \times 10 = \underline{} \end{aligned}$

$\begin{aligned} 5 \times 0.01 = \underline{} \end{aligned}$

$\begin{aligned} 7.7 \times 0.001 = \underline{} \end{aligned}$

$\begin{aligned} 12.4 \times 0.1 = \underline{} \end{aligned}$

$\begin{aligned} 456.8 \times 0.001 = \underline{} \end{aligned}$

$\begin{aligned} 0.8 \times 0.01 = \underline{} \end{aligned}$

$\begin{aligned} 31.5 \times 0.0001 = \underline{} \end{aligned}$

$\begin{aligned} 4.56 \times 0.1 = \underline{} \end{aligned}$

$\begin{aligned} 8.68 \times 0.01 = \underline{} \end{aligned}$

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

Powers of Ten ::十国权力的十国权力

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

Powers of Ten
::十国权力的十国权力

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