9.7 在科学符号中添加和减法数字-interactive

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• 选择节在科学符号中添加和减法数字

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  在科学符号中添加和减法数字

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  Mapping the Universe
  ::映射宇宙

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  If you search on the internet or in a textbook, you can find the distance between any two planets , but   where did these numbers come from?  One cannot simply  use a tape measure.
  ::如果您在互联网或教科书中搜索, 您可以找到两个行星之间的距离, 但这些数字来自何处? 我们不能简单地使用磁带测量 。

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  There are many methods for determining the distance of an object from the Earth such as radar, parallax, brightness, and more. However, most of these are used to find a distance from Earth, not the distance between two other objects. Luckily, you can use your basic math skills to figure out these distances based on the distances of the objects involved from Earth.
  ::确定一个物体与地球的距离有许多方法,例如雷达、准光、亮度等等。然而,大多数这些方法都用来寻找与地球的距离,而不是另外两个物体之间的距离。幸运的是,你可以使用基本数学技能根据与地球有关的物体的距离来算出这些距离。

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  1. If you know the average distance from the Earth to the Sun and the closest distance from the Earth to Jupiter, how can you find the average distance from the Sun to Jupiter?
     ::如果你知道从地球到太阳的平均距离 以及从地球到木星的最近距离 你如何找到从太阳到木星的平均距离?
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  2. If you know the average distance the Earth to the Sun and the closest distance from the Earth to Venus, how can you find the average distance from the Sun to Venus?
     ::如果你知道地球与太阳的平均距离 以及从地球到金星的最近距离 你如何找到从太阳到金星的平均距离?

   

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  Using Scientific Notation
  ::使用科学名词

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  In this lesson, you are going to encounter situations where you need to adjust numbers to have the same exponent and where you need to adjust the answers to convert them back into . To do this, you will need to use the LARS trick used in
  ::在此课程中, 您将遇到一些需要调整数字以拥有相同指数的情况, 并且需要调整答案以将其转换为 。 要做到这一点, 您需要使用 LARS 游戏中的 LARS 游戏

  

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  Example
  ::示例示例示例示例

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  Convert the expression $\begin{array}{r}45\times {10}^6\end{array}$  back to scientific notation.
  ::将表达式 45x106 转换为科学符号 。

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  The expression  $\begin{array}{r}45\times {10}^6\end{array}$  is not in scientific notation because 45 is not between 1 and 10. To convert this expression to scientific notation, you will need to move the decimal from the end of 45 to between the 4 and the 5. When you do this what happens to the exponent? You can use the LARS trick to determine how much you should add or subtract to the decimal. Moving the decimal 1 place to the left will add 1 to the exponent.
  ::45x106 表达式不是科学符号, 因为 45 不在 1 和 10 之间。 要将这个表达式转换为科学符号, 您需要将小数点从 45 结尾移到 4 和 5 之间。 当您这样做时, 引言人会怎样发生 ? 您可以使用 LARS 技巧来确定您应该向小数点增减多少 。 将小数点 1 移到左边, 将会向引言处增加 1 。

  

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  The answer is the expression $\begin{array}{r}4.5\times {10}^7\end{array}$ . This is in scientific notation because 4.5 is between 1 and 10.
  ::答案是表达式4.5x107. 这是在科学符号中,因为4.5介于1至10之间。

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  Example
  ::示例示例示例示例

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  Change the expression $\begin{array}{r}6\times {10}^{-3}\end{array}$  to have the exponent -4.
  ::将表达式 6x10-3 更改为前言 - 4。

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  I nstead of converting an expression into scientific notation, sometimes you may have to manipulate the expression out of scientific notation so that it has a specific exponent. To go from an exponent of -3 to an exponent of -4, you need to subtract 1. When you use the LARS trick, you know that to subtract 1 you need to move the decimal one place to the right. Moving the decimal one place to the right will result in the following:
  ::与其将表达式转换为科学符号,有时你可能不得不从科学符号中操纵表达式,使其具有特定的缩略语。要从 -3 的缩略语转换为 4 的缩略语,您需要减去 1 。当使用 LARS 骗术时,您知道要减去 1 ,您就需要将小数小数点一位移到右边。将小数点一位移到右边,将产生以下结果:

  

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  Doing this will give you the expression $\begin{array}{r}60\times {10}^{-4}\end{array}$  which has the desired exponent: -4.
  ::这样做会给你们60x10-4的表达式, 这个表达式有想要的提示: - 4 。

   

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  Adding Numbers in Scientific Notation
  ::在科学标注中添加数字

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  In the introduction, you were asked how to find the average distance from the Sun to Jupiter. You know the average distance from the Earth to the Sun and the distance from the Earth to Jupiter at their closest point. Imagine that the Sun, Earth, and Jupiter were perfectly aligned. As a reminder, Earth is the third planet from the Sun and Jupiter is the fifth planet from the Sun. To find the distance, you need to add the distances from the Sun to Earth and the Earth to Jupiter together. The average distance from the Sun to the Earth is $\begin{array}{r}1.5\times {10}^8\end{array}$  km. The distance from the Earth to Jupiter at their closest point is $\begin{array}{r}5.88\times {10}^8\end{array}$  km. To add the numbers in scientific notation, you must follow the steps below:
  ::在引言中,有人问你如何找到从太阳到木星的平均距离。 你知道从地球到太阳的平均距离以及从地球到木星的最接近点的距离。 想象一下太阳、 地球和木星是完全对齐的。 作为提醒, 地球是太阳的第三个行星, 木星是太阳的第五颗行星。 要找到距离, 您需要将太阳到地球的距离加到木星。 从太阳到地球的平均距离是1.5x108公里。 从地球到木星的最接近点的距离是5.88x108公里。 要在科学标记中加上数字, 您必须遵循以下步骤:

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  1. Make sure the bases and are the same.
     ::确保基地和基地都一样
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  2. Add the coefficients and keep the exponent.
     ::添加系数并保留指数。

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  You have to make sure the exponents are the same because they need to be like terms in order the add them. You cannot add  $\begin{array}{r}x\end{array}$  and  $\begin{array}{r}{x}^2\end{array}$  for the same reason you cannot directly add 10 and $\begin{array}{r}{10}^2\end{array}$ . To add 10 and $\begin{array}{r}{10}^2\end{array}$ , you would need to convert one to be in the same form as the other. In this example, the bases and exponents are the same, so you can add the coefficient and keep the exponent.
  ::您必须确认指数是相同的, 因为他们需要相似的条件来添加它们。 您无法添加 x和 x2 的原因与您无法直接添加 10 和 102 的原因相同 。 要添加 10 和 102, 您需要转换一个和另一个格式。 在此示例中, 基数和指数是相同的, 这样您就可以添加系数并保留指数 。

  $$\begin{array}{rl}& 1.50\times {10}^8\\ +& 5.88\times {10}^8\\ & 7.38\times {10}^8\end{array}$$

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  The distance from the Sun to Jupiter is a bout   $\begin{array}{r}\mathrm{7.38.}\times {10}^8\end{array}$  km.
  ::从太阳到木星的距离约为7.38.x108公里。

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  Discussion Question
  ::讨论问题

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  How many kilometers is this in standard form?
  ::这是多少公里的标准形式?

    

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  Subtracting Numbers in Scientific Notation
  ::科学符号中的减号

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  The second question in the introduction asked you to find the distance between Venus and the Sun if you already knew the distance from the Sun to Earth and the closest distance from the Earth to Venus. Imagine the Sun, Venus, and Earth lined up. Recall that Venus is the second planet from the Sun. To find the distance, you need to subtract the distance of Venus to Earth from the distance of Earth to the Sun. The average distance from the Sun to the Earth is $\begin{array}{r}1.5\times {10}^8\end{array}$  km. The distance from the Earth to Venus at their closest point is $\begin{array}{r}3.8\times {10}^7\end{array}$  km. To subtract the numbers in scientific notation, you must follow the steps below:
  ::引言中的第二个问题要求您找到维纳斯和太阳之间的距离。 如果您已经知道太阳与地球之间的距离, 以及地球与维纳斯之间的距离。 想象一下太阳、 维纳斯和地球之间的距离。 记得维纳斯是太阳的第二颗行星。 要找到距离, 您需要减去维纳斯与地球之间的距离。 从太阳到地球的平均距离是1.5x108公里。 从地球到金星的最接近点的距离是3. 8x107公里。 要在科学符号中减去数字, 您必须遵循以下步骤:

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  1. Make sure the bases and exponents are the same.
     ::确保基地和表率是相同的。
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  2. Subtract the coefficients and keep the exponent.
     ::减去系数,保留指数。

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  The exponents are not the same, so you must convert one to have the same exponent as the other. In this case, change to have 7 as the exponent. You will need to subtract 1 from the exponent so you will need to move the decimal one place to the right.
  ::指数不同, 所以您必须转换一个, 以拥有与另一个相同的指数。 在这种情况下, 更改为以 7 为指数 。 您需要从指数中减去 1 , 因此您需要将小数点后一位移到右边 。

  $$\begin{array}{r}1.5\times {10}^8\to 15.\times {10}^7\end{array}$$

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  Now you can subtract the coefficients and keep the exponent.
  ::现在您可以减去系数,保留指数。

  $$\begin{array}{rl}& 15\times {10}^7\\ -& 3.8\times {10}^7\\ & 11.2\times {10}^7\end{array}$$

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  The answer is $\begin{array}{r}11.2\times {10}^7\end{array}$ , but it is not in scientific notation because 11.2 is not between 1 and 10. To convert the expression to scientific notation, you must move the decimal one place to the left which will increase the exponent by 1.
  ::答案是11.2x107,但这不是科学标记,因为11.2不介于1和10之间。 要将表达式转换为科学标记,您必须将小数点后一位移到左边,这样会增加1个引号。

  $$\begin{array}{r}11.2\times {10}^7\to 1.12\times {10}^8\end{array}$$

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  The distance from the Sun to Venus is approximately $\begin{array}{r}1.12\times {10}^8\text{ km}.\end{array}$
  ::从太阳到金星的距离约为112×108公里。

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  Discussion Questions
  ::讨论问题 讨论问题

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  1. If we had changed the distance from Earth to Venus to have the exponent 8 would our answer be in scientific notation?
     ::如果我们从地球到金星的距离 换成8号指数 我们的答案会是科学符号吗?
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  2. Our distances were very close to the real average distances but why weren’t they exact?
     ::我们的距离非常接近实际平均距离,

    

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  Use the interactive below independently to practice numbers written in scientific notation. 
  ::独立使用下面的交互式数字来练习以科学符号书写的数字。

   

  

  INTERACTIVE

  Moon Rotation Proportions

  

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  • Move the moon around to visualize the orbit.
    ::将月亮移到周围 以视觉化轨道

  Your device seems to be offline.
  Please check your internet connection and try again.

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  +

  Do you want to reset the PLIX?

  Yes

  No

   

   

   

   

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  Nano-Sized Objects
  ::纳米化物体

  

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  Example
  ::示例示例示例示例

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  The image above shows four gears that are about double to triple the size of a red blood cell. These gears are so small that you may not even be able to see them without the help of a microscope. There are two different sized gears: the larger gears are  $\begin{array}{r}1.4\times {10}^{-5}\end{array}$  meters in diameter and the smaller gears are  $\begin{array}{r}9\times {10}^{-6}\end{array}$  meters. What is the difference between the diameters of the two gears?
  ::上面的图像显示四个齿轮的大小约为红色血细胞的两倍至三倍。 这些齿轮太小, 以至于没有显微镜的帮助, 你甚至看不到它们。 有两种不同的齿轮: 大齿轮直径为1.4×10-5米, 小齿轮直径为9×10-6米。 两个齿轮直径有什么区别?

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  To find the difference you must:
  ::要找到区别 你必须:

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  1. Make sure the bases and exponents are the same.
     ::确保基地和表率是相同的。
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  2. Subtract the coefficients and keep the exponent.
     ::减去系数,保留指数。

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  First, you need to change the expressions to have the same exponent. To do this, change $\begin{array}{r}1.4\times {10}^{-5}\end{array}$  to have an exponent of -6. You need to subtract 1 which means sliding the decimal 1 place to the right which will result in  $\begin{array}{r}14\times {10}^{-6}\end{array}$ . Now you can subtract the coefficient and keep the exponents.
  ::首先,您需要将表达式更改为相同的引号。要做到这一点,请修改1.4x10-5,以设置 -6. 的引号。您需要减去1,这意味着将小数点 1 的位置向右滑动,从而导致14x10-6 。现在您可以减去系数并保留引号。

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  $$\begin{array}{r}14\times {10}^{-6}-9\times {10}^{-6}=5\times {10}^{-6}\text{ meters}\end{array}$$

  
  ::14x10-6-6-9x10-6=5x10-6米

  Summary

  播放段落 When adding or subtracting two number in scientific notation: 播放段落 Make sure the bases and exponents are the same.  ::确保基地和表率是相同的。 播放段落 Add/subtract the coefficients and keep the exponent. ::增/减系数并保留指数。 ::当在科学符号中添加或减去两个数字时:确保基数和指数相同。添加/减去系数并保留指数。 播放段落 When writing or adjusting numbers written in scientific notation remember: 播放段落 If the decimal is moved to the left, add the number of places the decimal moved to the exponent. ::如果小数点被移到左边,请加上小数点被移到指数的位数。 播放段落 If the decimal is moved to the right, subtract the number of place the decimal moved to the exponent. ::如果将小数点移到右边,请减去小数点移到引言处的位数。 ::当写入或调整以科学符号书写的数字时,请记住:如果小数点移到左边,请加上小数点移到表列处的位数。如果小数点移到右边,请减去小数点移到表列处的位数。