10.2 广场根和立方根介绍-interactive

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  广场根和立方根入门

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  Returning to Egypt
  ::返回埃及

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  The lesson  discussed  how math has been used throughout history. Ancient Babylonians first encountered the need for square roots when finding the dimensions of a rectangle using a method that  you will explore later in this chapter. The ancient Babylonians compiled  lists  of on a  tablet named BM92698 . They used these lists to quickly find square roots and cube roots in problems involving area, volume, and missing lengths of shapes.
  ::讨论数学在整个历史中如何使用的教训。古代巴比伦人首先在使用本章稍后将探讨的方法找到矩形的维度时遇到需要平方根的问题。古代巴比伦人汇编了一个名为BM926988的平板板板上的列表,他们利用这些列表在涉及形状面积、体积和缺失长度的问题中迅速找到平方根和立方根。

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  Another tablet named YBC7289, stored at Yale University in the Babylonian Collection, is the first historical record showing the computation of a square root. This discovery set the stage for the use of computational mathematics in architecture, setting the stage for ancient Egyptians to construct temples, pyramids, and shrines. 
  ::另一个名为YBC7289的平板电脑存放在耶鲁大学的巴比伦藏书中,这是第一个显示平方根计算的历史记录。 这一发现为在建筑中使用计算数学创造了舞台,为古埃及人建造庙宇、金字塔和神庙创造了舞台。

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  While the ancient Babylonians used clay tablets to record information about their culture, Egyptians use the oldest form of paper known as p apyrus. One document, known as the Rhind Papyrus,  dates back to around 1650 BCE and shows how Egyptians used square roots. Egyptians had to use math to make precise architectural measurements. The Moscow Mathematical Papyrus shows the following example involving a square root:
  ::古代巴比伦人使用粘土板记录有关其文化的信息,而埃及人则使用被称为papyrus(papyrus)的最古老的纸张形式。 一份被称为Rhind Papyrus(Rhind Papyrus)的文件可以追溯到1650 BCE(BCE)左右,它显示了埃及人如何使用平方根。 埃及人不得不使用数学来进行精确的建筑测量。 莫斯科数学派皮鲁斯(Moscoscome Macistical Papyrus)展示了以下涉及平方根的例子:

  

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  The Egyptian word for square root is “knbt,” pronounced kenbet, which translates to corner or right angle. This word was chosen because square roots are often used in   g eometric applications. They used the symbol⎡ to represent the square root. The symbol we currently use actually comes from the Egyptian symbol. The symbol, as shown below,  is called a radical and the number inside it is called the radicand.
  ::平方根的埃及词是“ knbt ” , 直译为“ knbt ” , 转换为角或右角。 选择这个词是因为正方根常用于几何应用。 它们使用符号\\\ 来表示平方根。 我们目前使用的符号实际上来自埃及符号。 如下文所示, 符号被称为激进符号, 其内的数字被称为弧。

  

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  T ake a deeper look at what a square root is so that  you can answer the problem described in the papyrus.
  ::更深入地观察一个平方根是什么 这样你就可以解决papyrus 描述的问题。

   

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  Square Roots
  ::平方根

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  A square root is a number that when squared produces the radicand.  Your knowledge of perfect squares will be helpful in figuring out square roots.
  ::平方根是当平方产生射线时产生的数字。 您对完美的平方的知识将有助于找出平方根。

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  Example
  ::示例示例示例示例

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  Simplify  $\begin{array}{r}\sqrt{25}\end{array}$
  ::简化 25

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  What number, w hen multiplied by itself, will  equal 25? Using  your knowledge of perfect squares,  you know that  $\begin{array}{r}5\cdot 5=25\end{array}$ .
  ::用你对完美方形的了解,你知道5°5=25。

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  The square root of 25 can be simplified to 5.
  ::25平方根可简化为5。

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  Use the interactive below to practice finding the square roots of perfect squares .
  ::使用下面的交互方式 来练习寻找完美的方块的平方根

   

  

  INTERACTIVE

  Square Roots

  

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  • Enter the square root in the box.
    
    ::在框中输入正方根。
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  • Press "New Number" to try another question.
    ::按"新号码"来尝试另一个问题

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  Discussion Questions
  ::讨论问题 讨论问题

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  1. If you are finding the square root of a radicand that ends in 9, what does that tell you about the square root?
     ::如果你找到一个以9为终点的弧线的平方根,那你对平方根有什么看法?
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  2. If you are finding the square root of a radicand that ends in 5, what does that tell you about the square root?
     ::如果找到以5为终点的弧线的平方根,那你对平方根有什么看法?
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  3. If you are finding the square root of a radicand that ends in 4, what does that tell you about the square root?
     ::如果你找到一个以4为终点的弧线的平方根,那你对平方根有什么看法?
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  4. What is the square root of $\begin{array}{r}\sqrt{3^2}\end{array}$  ?
     ::32的平方根是什么?
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  5. What is the square root of $\begin{array}{r}\sqrt{5^2}\end{array}$  ?
     ::52的平方根是什么?

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  Cube Roots
  ::立方根

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  The root of a radical  tells you how many  times a number needs to be multiplied by itself to get the radicand. A root of three is called a cube root. A cube root is a number that when cubed produces the radicand.
  ::激进的根可以告诉你一个数字需要乘以多少倍才能获得半径。 根的3称为立方根。 立方根是立方体产生半径的数。

  

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  Example
  ::示例示例示例示例

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  Simplify  $\begin{array}{r}\sqrt[3]{27}\end{array}$ .
  ::简化273。

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  What number, when multiplied by itself three times, will equal  27 ? Using  your knowledge of perfect cubes, you know that $\begin{array}{r}3\cdot 3\cdot 3=27\end{array}$ .
  ::用你对完美立方体的知识,你知道3+3=27。

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  The cube root of 27 can be simplified to 3.
  ::27的立方根可以简化为3。

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  Use the interactive below to practice finding the cube roots of the perfect cubes.
  ::使用下面的交互操作 来练习如何找到完美立方体的立方根。

   

  

  INTERACTIVE

  Cube Roots

  

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  • Enter the number in the white box and click the button to check your answer.
    ::在白框中输入数字,然后单击按钮检查您的回答。
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  • Click "New Number" to generate another question.
    ::单击“ 新数字” 来生成另一个问题 。

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  Geometric Applications
  ::几何应用

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  Square roots and cube roots can be used to find missing dimensions of cylinders, cones, and spheres.
  ::方根和立方根可以用来寻找圆柱体、锥体和球体缺失的尺寸。

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  Example
  ::示例示例示例示例

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  Find the radius of the figure below.
  ::查找下方图的半径 。

   

  

  INTERACTIVE

  Dimensions of a Battery

  

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  • Click on the cylinder and segments to reveal its dimensions.
    ::点击圆柱体和片段以显示其尺寸。
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  • Drag the screen to change your point of view. Use two fingers on your trackpad or touch screen or scroll the mouse wheel to zoom.
    ::拖曳屏幕以改变您的视图点。 在您的跟踪板上使用两根手指或触摸屏幕或滚动鼠标方向盘以缩放 。

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  To find the missing radius use the following steps:
  ::要找到缺失的半径,要使用以下步骤:

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  $$\begin{array}{rl}V=& =\pi \cdot r^2\cdot h\\ 791.28& =3.14\cdot r^2\cdot 28\\ 791.28& =87.92r^2\\ 9& =r^2\end{array}$$

  
  ::V2h791.28=3.14228791.28=87.92r29=r2

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  The solve the equation   $\begin{array}{r}{r}^2=9\end{array}$ ,  you   need to find a number that when squared is equal to 9. This number represents the square root of 9. The square root is the  inverse operation of squaring . Take the square root of both sides of the equation to  find   the answer.
  ::方程式 r2=9 的解答, 您需要找到一个数字, 当方形等于 9 时, 这个数字代表 9 的平方根 。 平方根是 矩形 的反向操作 。 选择方程式两侧的平方根来找到答案 。

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  $$\begin{array}{r}\sqrt{r^2}=\sqrt{9}\\ r=3\end{array}$$

  
  ::r2=9r=3

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  The radius is 3 mm .
  ::半径是3毫米

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  U se  the method from the previous example  to solve the problem from the papyrus.
  ::使用上一个示例中的方法来解决 papyrus 的问题 。

  

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  Use the interactive below to visualize this problem.
  ::使用下面的交互效果来直观地看待这个问题。

   

  

  INTERACTIVE

  Papyrus Dimensions

  

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  • Drag the sliders to change the length or width.
    ::拖动滑块以更改长度或宽度。
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  • Observe how changing these values affect the area.
    ::观察这些价值的变化如何影响该地区。

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  L et  $\begin{array}{r}x\end{array}$  represent the length and  $\begin{array}{r}\frac{3}{4}x\end{array}$  represent the width since the width is  $\begin{array}{r}\frac{3}{4}\end{array}$  the length.  The area of a rectangle is length times width, so you can set up the following equation:
  ::Let x 表示长度, 34x 表示宽度, 因为宽度是34, 宽度是34。 矩形的面积是长度乘以宽度, 所以您可以设置以下方程式 :

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  $$\begin{array}{r}x\cdot \frac{3}{4}x=12\end{array}$$

  
  ::x34x=12

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  S implify the expression on the left and solve for  $\begin{array}{r}x.\end{array}$
  ::简化左侧的表达式并解决 x 。

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  $$\begin{array}{rl}\frac{3}{4}{x}^2& =12\\ \frac{4}{3}\cdot \frac{3}{4}{x}^2& =12\cdot \frac{4}{3}\\ x^2& =16\\ \sqrt{x^2}& =\sqrt{16}\\ x& =4\end{array}$$

  
  ::34x2=1243 34x2=12 43x2=16x2=4

   

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  The rectangle has a length of 4 units and a width of 3 units.
  ::矩形长4个单元,宽3个单元。

   

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  Crime Fighting
  ::打击犯罪

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  One way  police officers  use math is to determine the speed of the cars involved in a car accident. Police officers can figure out the speed of a car before the accident by measuring the length of the skid mark left by the car. They use the formula  $\begin{array}{r}\sqrt{30df}\end{array}$  where $\begin{array}{r}d\end{array}$  is the length of the skid mark and $\begin{array}{r}f\end{array}$  is the drag factor (a measure of friction). This formula will give  you the speed in miles per hour. Use the interactive below to approximate the speed of the car to the nearest tenth.
  ::警官使用数学的方法之一是确定发生车祸的汽车的速度; 警官可以通过测量汽车留下的滑雪标记长度来计算出事故发生前的汽车速度; 他们使用公式30df, d 是滑雪标记的长度, f 是拖动系数(摩擦的量度)。 这个公式将以每小时英里的速度向您提供速度。 使用下方的交互速度可以接近汽车速度的十分之一 。

   

  

  INTERACTIVE

  Crime Fighting

  

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  • Press the button to watch the accident happen.
    ::按下按钮看事故发生
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  • Use the rulers to measure out the skid marks.
    ::使用统治者测量滑动标记。
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  • Each ruler is 40 feet long.
    ::每条尺长40英尺

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   Summary
  ::摘要

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  To write a root, use the symbol  $\begin{array}{r}\sqrt{}\end{array}$  which is called  a   radical . The value inside  a radical is called the radicand .
  ::要写根, 请使用被称为激进的符号 。 激进内部的价值被称作半径 。

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  Taking the s quare root is the opposite of squaring:
  ::取平方根与争吵相反:

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  • To simplify a square root, find the number that when  squared will equal the radicand
    ::要简化平方根, 找到当平方等于弧度的数
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  • For example,  $\begin{array}{r}\sqrt{64}=8\end{array}$  because  $\begin{array}{r}{8}^2=64\end{array}$  
    ::例如,64=8,因为82=64

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  Taking the cube root  ( $\begin{array}{r}\sqrt[3]{}\end{array}$ ) is the opposite of cubing:
  ::立方根( 3) 与幼崽相反 :

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  • To simplify a  cube root, find the number that when  cubed will equal the radicand
    ::要简化立方根,请找到当立方等于弧度的数值
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  • For example,  $\begin{array}{r}\sqrt[3]{64}=4\end{array}$  because  $\begin{array}{r}{4}^3=64\end{array}$  
    ::例如,643=4,因为43=64