9.8 科学标记中的乘数和分数-interactive

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• 选择节科学标记中的乘数与发展数字

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  科学标记中的乘数与发展数字

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  Continental Shift
  ::大陆转移

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  In the lesson  , scientific notation  was used to represent distances. However, can also be used to represent times, , masses, and any quantifiable amount . In this lesson, the focus is on numbers in scientific notation, but you will also explore scientific notation in a variety of real-world contexts. The first context you will look at involves a rate .
  ::在课程中,科学符号用于代表距离。 但是, 也可以用来代表时间、 、 质量和任何量化的数量。 在这个课程中, 重点是科学符号中的数字, 但是您也会在各种现实世界背景下探索科学符号。 您要查看的第一个背景包含一个速率 。

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  The seven continents are formed by continental plates which are constantly moving. Over a century ago the scientist Alfred Wegener proposed that all of the continents originally fit together to form one supercontinent called Pangea. Evidence points toward this continent existing approximately 200 million years ago. One continent meant that there was only one ocean, called the Panthalassa Ocean. Strong volcanic activity caused the continents to break apart and move into their current positions. As the continents continue to move, imagine what the world will look like in another 200 million years.
  ::七大洲由不断移动的大陆板块组成。 一百多年前,科学家阿尔弗雷德·韦格纳(Alfred Wegener)提出所有大陆最初都适合组成一个称为“潘热”的超级大陆。 证据指向这个大陆大约两亿年前就已存在的两亿年前。 一个大陆意味着只有一个海洋,叫做“潘塔拉萨海 ” 。 强大的火山活动导致各大洲分裂并移动到目前的位置。 随着各大洲的继续移动,想象世界在另外两亿年中会是什么样子。

  

   

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  Multiplying Numbers In Scientific Notation
  ::科学符号中的乘数

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  The rules for multiplying expressions in scientific notation come from the exponent rules and the commutative property .
  ::科学符号中乘法表达法的规则来自引言规则和通货财产。

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  Example
  ::示例示例示例示例

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  Simplify :  $\begin{array}{r}(3\times {10}^6)(5.2\times {10}^3)\end{array}$
  ::简化: (3x106)(5.2x103)

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  Using the commutative property, you can arrange this to read as follows:
  ::使用通货财产,您可以安排如下:

  $\begin{array}{r}3\cdot 5.2\cdot {10}^6\cdot {10}^3\end{array}$

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  Multiply 3 and 5.2 to get 15.6. To multiply  $\begin{array}{r}{10}^6\end{array}$  and $\begin{array}{r}{10}^3\end{array}$ , use the Product of Powers Property to keep the base and add the to get $\begin{array}{r}{10}^9\end{array}$ . When these two parts are put together, you get $\begin{array}{r}15.6\times {10}^9\end{array}$ . However,  $\begin{array}{r}15.6\times {10}^9\end{array}$  is not in scientific notation because 15.6 is not between 1 and 10. To fix this, move the decimal one place to the left which will result in adding one to the exponent.
  ::乘以 3 和 5.2 获得 15.6. 乘以 106 和 103 。 乘以 106 和 103 , 使用 Power Property 产品来保持基数, 并增加109 。 当这两个部分合在一起时, 你得到 15.6x109 。 但是, 15.6x 109 没有科学标记, 因为 15.6 不在 1 和 10 之间 。 要解决这个问题, 将小数点之一移到左边, 从而导致向引号加一个 。

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  Answer:  $\begin{array}{r}1.56\times {10}^{10}\end{array}$
  ::答复:1.56x1010

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  To multiply a number in scientific notation by a number in standard form, multiply the number in standard form by the coefficient and keep the exponent. As stated earlier, any whole number could be thought of as being multiplied by $\begin{array}{r}{10}^0\end{array}$ . The rules for multiplying two numbers in scientific notation are as follows:
  ::将科学符号数乘以标准格式的数字,将标准格式的数字乘以系数,并保留指数。如前所述,任何整数都可以视为乘以100。 在科学符号中,将两个数字乘以两个数字的规则如下:

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  1. Multiply the coefficients.
     ::乘以系数。
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  2. Keep the base and add the exponents.
     ::保留基数并添加前列词 。

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  A galactic year is the amount of time it takes for the solar system to orbit around the black hole at the center of the Milky Way Galaxy. The solar system travels at 830,000 km/hr and the entire trip takes approximately 2,200,000,000,000 hours, or about 250,000,000 years.
  ::银河系年是太阳系绕银河系中心黑洞运行所需的时间,太阳系每小时830 000公里,整个行程大约需要2 200 000 000 000小时,大约25万年。

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  Use the interactive below to answer the following questions about the Solar System’s Galactic Year.
  ::使用以下互动方式回答以下关于太阳系银河年的问题。

   

   

   

   

   

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  Dividing Numbers In Scientific Notation
  ::科学标注中的分数

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  The following rules, similar to the steps for multiplying, are used to divide numbers in scientific notation.
  ::以下规则与乘法步骤相似,用于在科学符号中进行数字分配。

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  1. Divide the coefficients.
     ::除以系数。
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  2. Keep the base and subtract the exponents.
     ::保留基数,减去引言。

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  Example
  ::示例示例示例示例

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  Divide:  $\begin{array}{r}(1.6\times {10}^5)÷(4\times {10}^3)\end{array}$
  ::除数: (1.6x105) (4x103)

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  When dividing the coefficients, you get $\begin{array}{r}1.6÷4=\mathrm{0.4.}\end{array}$  When you keep the base and subtract the exponents, you get $\begin{array}{r}{10}^{5-3}={10}^2\end{array}$ . This  quotient comes out to be  $\begin{array}{r}0.4\times {10}^2\end{array}$  which you can convert to scientific notation by moving the decimal 1 place to the right and subtracting 1 from the exponent.
  ::当分隔系数时,您可以得到1.64=0.4.。当您保持基数并减去指数时,您可以得到105-3=102。这个商数出来为0.4×102,您可以通过将小数点 1 移到右边,并从指数中减去1 来转换为科学符号。

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  Answer:  $\begin{array}{r}4\times {10}^1\end{array}$
  ::答复:4x101

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  The interactive below displays the speed at which each continental plate is currently moving. The map displays the direction. When plates are moving toward each other, it causes mountain ranges to form like the Himalayas, which run along the northern border of India. When plates are sliding away from each other, it causes trenches. When plates slide along each other, it causes seismic activity like the San Andreas Fault in California. You can use this table to predict what the continents will look like in the future.
  ::下面交互显示每个大陆板块目前移动的速度。 地图显示方向。 当板块向对方移动时, 它会让山脉形成像喜马拉雅山一样, 喜马拉雅山脉沿印度北部边界运行。 当板块相互滑离时, 它会形成战壕。 当板块相互滑动时, 它会引发像加州圣安德烈亚斯断层一样的地震活动。 您可以使用这张表格来预测各大洲未来会是什么样子 。

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  Use the activity below to use what you have learned so far to answer questions about the movement of the Earth’s continents.
  ::利用下面的活动,利用你迄今所学到的知识,回答关于地球大陆移动的问题。

   

   

   

   

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    Discussion Question
  ::讨论问题

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  Why does dividing the coefficients, keeping the base, and subtracting the exponent's work when dividing numbers in scientific notation?
  ::为何在科学符号中除以数字时区分系数、保持基数和减去指数的工作?

    

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  Scientific Notation Word Problems
  ::科学名词问题

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  As nanotechnology continues to advance, computer chips are becoming increasingly small. Computer companies are currently building computer chips smaller than the human eye can see. Nano-sized computer chips and parts will lead to the ability to process information almost instantly, flexible and even wearable computers, higher resolution screens, and longer battery life. Nano-sized computer chips can also be used in other nanotechnology like RFID chips.
  ::随着纳米技术的继续进步,计算机芯片变得越来越小。 计算机公司目前正在制造比人类眼目所见的更小的计算机芯片。 纳米规模的计算机芯片和部件将导致能够几乎即时、灵活甚至穿戴计算机、高分辨率屏幕和更长的电池寿命等信息处理能力。 纳米规模的计算机芯片也可以用于其他纳米技术,如RFID芯片。

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  Example
  ::示例示例示例示例

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  A company is designing a computer chip that is  $\begin{array}{r}5\times {10}^{-9}\text{meters}\end{array}$  by $\begin{array}{r}7\times {10}^{-9}\end{array}$   $\begin{array}{r}\text{meters.}\end{array}$  Find the area and perimeter of the computer chip.
  ::一家公司正在设计一个5x10-9公尺乘7x10-9公尺的计算机芯片,寻找计算机芯片的面积和周界。

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  To find the area of a rectangle, you can use the formula  $\begin{array}{r}\text{Area = length}\times \text{width.}\end{array}$
  ::要找到矩形区域,可以使用公式区域 = 长度 xwidth。

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    •  The $\begin{array}{r}\text{length}\end{array}$  is $\begin{array}{r}5\times {10}^{-9}\end{array}$
  ::• 长度为5x10-9

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    •  The $\begin{array}{r}\text{width}\end{array}$  is $\begin{array}{r}7\times {10}^{-9}\end{array}$
  ::• 宽度为7x10-9

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  $\begin{array}{rl}\text{Area}& =(5\times {10}^{-9})\times (7\times {10}^{-9})\\ \text{Area}& =35\times {10}^{-18}\end{array}$
  ::区域=( 5x10- 9) ×( 7x10- 9) ×( 7x10- 9) 区域= 35x10- 18

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  Since $\begin{array}{r}35\times {10}^{-18}\end{array}$  is not in scientific notation, you will need to move the decimal 1 place to the left and add 1 to the exponent.
  ::由于35x10-18不在科学标记中,您需要将小数点 1 位移到左侧,然后在引号中添加 1 位。

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  $\begin{array}{r}\text{Area}=3.5\times {10}^{-17}{\text{ meters}}^2\end{array}$
  ::面积=3.5x10-17米2

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  To find the perimeter, use the formula $\begin{array}{r}\text{Perimeter}=2\cdot \text{length}+2\cdot \text{width}\end{array}$ . The length and width are the same as when you found the area.
  ::要找到周界, 请使用 Periter = 2- 长+ 2- 宽宽。 长度和宽度与您发现区域时相同 。

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  $\begin{array}{r}\text{Perimeter}=2(5\times {10}^{-9})+2(7\times {10}^{-9})\end{array}$
  ::周边计=2(5x10-9)+2(7x10-9)

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  Recall that to multiply a number in scientific notation by a number in standard form, multiply the number in standard form by the coefficient and keep the exponent.
  ::回顾在科学符号中将数字乘以标准格式的数字,将标准格式的数字乘以系数,并保留引号。

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  $\begin{array}{r}\text{Perimeter}=(10\times {10}^{-9})+(14\times {10}^{-9})\end{array}$
  ::周边计=( 10x10- 9) +( 14x10- 9)

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  You can add these two expressions together since they have the same exponent.
  ::您可以将这两个表达式加在一起, 因为它们具有相同的引号 。

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  $\begin{array}{r}\text{Perimeter}=24\times {10}^{-9}\end{array}$
  ::周边=24x10-9

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  Finally, you must convert to scientific notation by moving the decimal 1 place to the left and adding 1 to the exponent.
  ::最后,您必须转换为科学标记,将小数点 1 移到左侧,并在引号中添加 1 。

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  $\begin{array}{r}\text{Perimeter}=2.4\times {10}^{-8}\text{ meters}\end{array}$
  ::半径=2.4x10-8米

   

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  Discussion Question
  ::讨论问题

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  If an atom has an average diameter of $\begin{array}{r}5\times {10}^{-10}\end{array}$   meters, how many atoms could fit along the length and width of the computer chip?
  ::如果原子的平均直径为5×10-10米,那么有多少原子可以与计算机芯片的长度和宽度相匹配?

  Summary

  播放段落 When multiplying two numbers in scientific notation: 播放段落 Use the Commutative Property to rewrite the expression so the coefficients are next to each other and the bases are next to each other. ::使用通俗属性重写表达式,使系数相邻,基数相邻。 播放段落 Multiply the coefficients and add the exponents (keeping the base 10). ::乘以系数并加上指数(保持基准10)。 ::当乘以科学符号中的两个数字时:使用通俗属性重写表达式,使系数相邻,基数相邻。乘以系数并添加引号(保持基数 10) 播放段落 When dividing two numbers in scientific notation: 播放段落 Use the Commutative Property to rewrite the expression so the coefficients are next to each other and the bases are next to each other. ::使用通俗属性重写表达式,使系数相邻,基数相邻。 播放段落 Divide the coefficients and subtract the exponents (keeping the base 10). ::除以系数,减去指数(保持基准10)。 ::当在科学符号中除去两个数字时:使用通俗属性重写表达式,使系数相邻,基数相邻。除以系数,减去指数(保持基数10)。