1.11 简化和用平根操作

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  使用平根简化和操作

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  A surveyor is at a building site measuring the distance across two parking lots. The distance across the first parking lot is  $\begin{array}{r}\sqrt{9800}\end{array}$ feet and the distance across the second parking lot is $\begin{array}{r}\sqrt{7200}\end{array}$ feet. How can we simplify these measurements? What is the total distance across the two parking lots?
  ::测量员位于一个建筑工地,测量两个停车场之间的距离。 第一个停车场的距离是9800英尺,第二个停车场的距离是7200英尺。我们如何简化这些测量?两个停车场之间的总距离是多少?

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  In this section, we will explore  how to simplify and operate with square roots.
  ::在本节中,我们将探讨如何简化和以平本方式运作。

  

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  Simplifying Square Roots
  ::简化平方根

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  In this section, we review how to simplify, add, subtract, and multiply square roots. Recall that the square root is a number that, when multiplied by itself, produces another number. The symbol for the square root is the radical sign, or $\begin{array}{r}\sqrt{}\end{array}$ . The number under the radical is called the radicand .
  ::在本节中, 我们审查如何简化、 添加、 减、 乘平方根 。 提醒注意, 平方根是一个数字, 当自己乘以时, 就会产生另一个数字 。 平方根的符号是激进符号 , 或者 。 激进下的数字被称为弧线 。

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  An example of a square root is 4, which is the square root of 16. -4 is also the square root of 16 since $\begin{array}{r}-4\times -4=16\end{array}$ . However, we will only consider the principal square root , the positive root, when there is no sign in front of the square root or radical sign. If we want to indicate the negative root, we will add a negative sign in front of the radical.
  ::平方根的一个例子是4, 平方根为16 - 4, 平方根为16 平方根为16 。 然而, 当平方根或激进标志前没有迹象时, 我们只考虑正方根, 也就是正方根 。 如果我们想要指出负根, 我们就会在激进面前添加一个负号 。

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  The only square roots that will result in an integer are the roots where the radicand is a perfect square. The first ten perfect squares are 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100.
  ::导致整数的唯一平方根是弧形是完美的正方根。 头十个完美的正方根是 1, 4, 9, 9, 16, 25, 36, 49, 64, 81, 和 100。

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    Properties of Radicals That Can Be Used for Simplifying Square Roots  
  ::用于简化广场根的激进分子的属性

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  1.  $\begin{array}{r}\sqrt{ab}=\sqrt{a\cdot b}=\sqrt{a}\cdot \sqrt{b}\end{array}$
  ::1. b=a-b=a-b

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  2.  $\begin{array}{r}\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}\end{array}$
  ::2. ab=ab

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  Example 1
  ::例1

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  Find $\begin{array}{r}\sqrt{50}\end{array}$ by simplifying the square root.
  ::通过简化平方根 找到50

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  Solution: To simplify the square root, we factor the numbers underneath the radical. We want to look for factors of 50 that are perfect square numbers: 1, 4, 9, 16, 25... 25 is a factor of 50. Write the radicand in factored form.
  ::解决方案: 为了简化平方根, 我们从激进分子的下方计数。 我们想要找到50个完全正方数的因素: 1, 4, 9, 9, 16, 25... 25是50。 以分数法写成弧线。

  $$\begin{array}{r}\sqrt{50}=\sqrt{25\cdot 2}=\sqrt{25}\cdot \sqrt{2}=5\sqrt{2}\end{array}$$

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  Example 2
  ::例2

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  Simplify $\begin{array}{r}\sqrt{196}\end{array}$ . 
  ::简化196。

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  Solution:  $\begin{array}{r}\sqrt{196}=\sqrt{4}\cdot \sqrt{49}=2\cdot 7=14\end{array}$  
  ::解决办法:196=449=27=14

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  by CK-12 Foundation explains how to simplify radical expressions.   
  ::CK-12基金会解释如何简化激进表达方式。

    

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  Addition and Subtraction of Square Roots
  ::平根增减

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  To add or subtract square roots, you need to identify terms with the same radicand. Then, you can add or subtract the numbers outside the radical. 
  ::要加或减平方根, 您需要用相同的 radicand 来识别条件 。 然后, 您可以在 root 之外加或减数字 。

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  Example 3
  ::例3

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  Simplify $\begin{array}{r}\sqrt{45}+\sqrt{80}-2\sqrt{5}\end{array}$ .
  ::简化 45+80-25。

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  Solution: At first glance, it does not look like we can simplify this. But, we can simplify each radical by pulling out the perfect squares.
  ::解决方案:乍一看,我们似乎无法简化这一点。但是,我们可以通过拔出完美的方块来简化每个激进方块。

  $$\begin{array}{rl}\sqrt{45}& =\sqrt{9\cdot 5}=3\sqrt{5}\\ \sqrt{80}& =\sqrt{16\cdot 5}=4\sqrt{5}\end{array}$$

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  Rewriting our expression, we have: $\begin{array}{r}3\sqrt{5}+4\sqrt{5}-2\sqrt{5}\end{array}$ and all the radicands are the same. Using the order of operations, our answer is $\begin{array}{r}5\sqrt{5}\end{array}$ .
  ::我们改写我们的表达方式,我们有:35+45-25,所有射线都是相同的。按照操作顺序,我们的答复是55。

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  Example 4
  ::例4

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  A surveyor is at a building site measuring the distance across two parking lots. The distance across the first parking lot is  $\begin{array}{r}\sqrt{9800}\end{array}$ feet and the distance across the second parking lot is $\begin{array}{r}\sqrt{7200}\end{array}$ feet. How can we simplify these measurements? What is the total distance across the two parking lots? 
  ::测量员位于一个建筑工地,测量两个停车场之间的距离。 第一个停车场的距离是9800英尺,第二个停车场的距离是7200英尺。我们如何简化这些测量?两个停车场之间的总距离是多少?

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  Solution: First, we need to simplify the square roots.
  
  ::解决方案:首先,我们需要简化平方根。

  $$\begin{array}{rl}& \sqrt{9800}=\sqrt{98}\cdot \sqrt{100}=\sqrt{2}\cdot \sqrt{49}\cdot \sqrt{100}=\sqrt{2}\cdot 7\cdot 10=70\sqrt{2}\\ \\ & \sqrt{7200}=\sqrt{72}\cdot \sqrt{100}=\sqrt{2}\cdot \sqrt{36}\cdot \sqrt{100}=\sqrt{2}\cdot 6\cdot 10=60\sqrt{2}\end{array}$$

   
  

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  The total distance across the two parking lots can be found by calculating the sum of these two measurements:  $\begin{array}{r}70\sqrt{2}+60\sqrt{2}=130\sqrt{2}\end{array}$ feet. 
  
  ::两个停车场之间的总距离可通过计算这两个测量量的总和找到:702+602=1302英尺。

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  by Mathispower4u demonstrates how to add and subtract radical expressions.
  ::Mathispower4u 演示如何增减基表达式。

   

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    WARNING
  ::警告

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  We can separate square roots when multiplying or dividing them, but we can not do this across addition or subtraction.
  ::我们可以在增加或分割它们时将正方根分开,但是我们不能在加或减之间做到这一点。

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  $$\begin{array}{r}\sqrt{a+b}\ne \sqrt{a}+\sqrt{b}\end{array}$$

  For example,
  ::a+ba+bb 例如,

  $$\begin{array}{rl}\sqrt{9+16}& \ne \sqrt{9}+\sqrt{16}\\ \sqrt{25}& \ne 3+4\\ 5& \ne 7\end{array}$$

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  Multiplication of Square Roots
  ::平根乘法

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  When multiplying square roots, you multiply the numbers outside the radical and the numbers inside the radicals.
  ::当乘以正方根时,你乘以激进之外的数字和激进内部的数字。

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  Example 5
  ::例5

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  Simplify $\begin{array}{r}2\sqrt{35}\times 4\sqrt{7}\end{array}$ .
  ::简化235x47。

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  Solution:  We want to m ultiply the numbers outside the radical and radicands separately.
  ::解答:我们要将激进和半径之外的数字分别乘以。

  $$\begin{array}{r}2\sqrt{35}\times 4\sqrt{7}=2\times 4\sqrt{35\times 7}=8\sqrt{245}\end{array}$$

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  Now, simplify the radical. $\begin{array}{r}8\sqrt{245}=8\sqrt{49\times 5}=8\times 7\sqrt{5}=56\sqrt{5}\end{array}$
  ::现在,简化基号 8245=849x5=8x75=565

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  Division of Square Roots
  ::平根分区

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  Dividing radicals can be a bit more difficult that the other operations. The main complication is that we prefer not to  leave any radicals in the denominator of a fraction to make calculations easier. When we have a radical in the denominator, we have to go through a process called rationalizing the denominator , where you multiply the top and bottom of a fraction by the same radical that is in the denominator. This will undo the radicals and leave a whole number in the denominator.
  ::与其它操作相比,分裂的激进分子可能更加困难。 主要的复杂因素是,我们宁愿不要将任何激进分子留在分母的分母中,以便更容易地进行计算。 当我们分母中有一个激进分子时,我们必须经历一个叫合理化分母的过程,在这个过程中,你将分母的顶部和底部乘以在分母中的同一个激进分子。这将消除激进分子,并在分母中留下一个完整的数字。

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  Example 6
  ::例6

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  Simplify $\begin{array}{r}\sqrt{\frac{1}{4}}\end{array}$ .
  ::简化 14 。

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  Solution:  We can separate the radical across division.
  ::解决方案:我们可以把激进分子隔开

  $$\begin{array}{r}\sqrt{\frac{1}{4}}=\frac{\sqrt{1}}{\sqrt{4}}=\frac{1}{2}\end{array}$$

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  Example 7
  ::例7

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  Simplify $\begin{array}{r}\frac{2}{\sqrt{3}}\end{array}$ .
  ::简化 23

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  Solution: This might look simplified, but we prefer when radicals are not in the denominator of a fraction. We need to rationalize the denominator. Multiply the top and bottom of the fraction by $\begin{array}{r}\sqrt{3}\end{array}$ .
  ::解答: 这可能会看起来简化, 但是当激进分子不是分母的分母时我们更喜欢。 我们需要理顺分母。 将分母的上下乘以 3 。

  $$\begin{array}{r}\frac{2}{\sqrt{3}}\cdot \frac{\sqrt{3}}{\sqrt{3}}=\frac{2\sqrt{3}}{3}\end{array}$$

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  Example 8
  ::例8

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  Simplify $\begin{array}{r}\sqrt{\frac{32}{40}}\end{array}$ .
  ::简化3240。

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  Solution: Reduce the fraction, and then apply the rules above.
  ::解决办法:减少分数,然后适用上述规则。

  $$\begin{array}{r}\sqrt{\frac{32}{40}}=\sqrt{\frac{4}{5}}=\frac{\sqrt{4}}{\sqrt{5}}=\frac{2}{\sqrt{5}}\cdot \frac{\sqrt{5}}{\sqrt{5}}=\frac{2\sqrt{5}}{5}\end{array}$$

   

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  Sometimes after rationalizing the denominator, we are not done with the problem. For example, if the numerator and denominator have a common factor, we must simplify the fraction.
  ::有时,在将分母合理化之后,我们没有解决这个问题。 例如,如果分子和分母有一个共同因素,我们必须简化分母。

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  Example 9
  ::例9

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  Simplify $\begin{array}{r}\frac{9\sqrt{2}}{\sqrt{3}}\end{array}$ .
  ::简化923。

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  Solution:
  ::解决方案 :

  $$\begin{array}{r}\frac{9\sqrt{2}}{\sqrt{3}}=(\frac{9\sqrt{2}}{\sqrt{3}})(\frac{\sqrt{3}}{\sqrt{3}})=\frac{9\sqrt{2\times 3}}{\sqrt{3\times 3}}=\frac{9\sqrt{6}}{3}=3\sqrt{6}\end{array}$$

    

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  There are times when our denominator includes an addition or subtraction of a radical and a number. When this occurs, we need to find conjugate  of the expression in the denominator. The conjugate has the same numbers but the opposite operation between first and second terms. Using the conjugate enables us to rationalize the denominator.
  ::我们的分母有时会包含激进和数字的增减。 当这种情况发生时,我们需要找到分母表达的共性。 共性的数字相同, 但第一和第二术语之间的操作却相反。 使用共性可以使我们分母合理化。

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  Example 10
  ::例10

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  Simplify  $\begin{array}{r}\frac{3}{5+\sqrt{2}}\end{array}$ .
  ::简化 35+2 。

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  Solution:
  ::解决方案 :

  $$\begin{array}{r}\frac{3}{5+\sqrt{2}}=(\frac{3}{5+\sqrt{2}})(\frac{5-\sqrt{2}}{5-\sqrt{2}})=\frac{3(5-\sqrt{2})}{25-2}=\frac{15-3\sqrt{2}}{23}\end{array}$$

   

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  by Mathispower4u demonstrates how to divide radical expressions. 
  ::Mathispower4u 演示如何分割激进表达式。

   

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  The rules for simplifying and operating on square roots are below.
  ::简化和在平原上运作的规则如下。

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   Radical Rules
  ::激进激进规则

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  1.   $\begin{array}{r}\sqrt{ab}=\sqrt{a\cdot b}=\sqrt{a}\cdot \sqrt{b}\end{array}$
  ::1. b=a-b=a-b

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  2.   $\begin{array}{r}x\sqrt{a}\pm y\sqrt{a}=x\pm y\sqrt{a}\end{array}$
  ::2. xaya=xya

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  3. $\begin{array}{r}{\left(\sqrt{a}\right)}^2={\sqrt{a}}^2=a\end{array}$
  ::3. (a)2=a2=a2=a

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  4. $\begin{array}{r}\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}\end{array}$
  ::4. ab=ab

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  5. $\begin{array}{r}\frac{\sqrt{a}}{\sqrt{b}}\cdot \frac{\sqrt{b}}{\sqrt{b}}=\frac{\sqrt{ab}}{b}\end{array}$
  ::5. abbb=abbb

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  Summary
  ::摘要

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  • To simplify radicals, factor them and determine if any of the factors are perfect squares. Then, take the square root of the perfect squares.
    ::为了简化激进,考虑到它们,确定其中的任何因素是否为完美的正方形。 然后,选择完美的正方形的正方根。
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  • To add or subtract radicals, combine the numbers outside the radicals only if the radicands are the same.
    ::加上或减去基数,只有辐射线相同时,才将基数以外的数字合并。
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  • To multiply radicals, multiply numbers outside the radical with each other and numbers inside the radical with each other.
    ::乘以激进 乘以激进之外的数字 彼此之间 和在激进内部的数字 彼此之间。
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  • Division is similar to multiplication in that you can divide numbers outside the radical with each other and numbers inside the radical with each other.
    ::分数与乘法相似,因为您可以将激进分子以外的数字和激进分子内部的数字分隔开来,而激进分子内部的数字则相互分隔开来。
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  • If there is a radical in the denominator, you should rationalize the denominator.
    ::如果分母中有一个激进的分母,你应该使分母合理化。

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  Review
  ::回顾

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  Perform the operations below and simplify the following radicals. If it cannot be simplified further, write " cannot be simplified ."
  ::执行下面的操作并简化以下的基。 如果无法进一步简化, 请写入“ 不能简化 ” 。

  1. $\begin{array}{r}\sqrt{75}\end{array}$

  2. $\begin{array}{r}\sqrt{48}\end{array}$

  3. $\begin{array}{r}\sqrt{50}\cdot \sqrt{2}\end{array}$

  4. $\begin{array}{r}4\sqrt{3}\cdot \sqrt{21}\end{array}$

  5. $\begin{array}{r}{\left(4\sqrt{5}\right)}^2\end{array}$

  6. $\begin{array}{r}\sqrt{24}\cdot \sqrt{27}\end{array}$

  7. $\begin{array}{r}\sqrt{16}+2\sqrt{8}\end{array}$

  8. $\begin{array}{r}-8\sqrt{3}-\sqrt{12}\end{array}$

  9. $\begin{array}{r}\sqrt{72}-\sqrt{50}\end{array}$

  10. $\begin{array}{r}8\sqrt{10}-\sqrt{90}+7\sqrt{5}\end{array}$

  11. $\begin{array}{r}\sqrt{\frac{96}{121}}\end{array}$

  12. $\begin{array}{r}\frac{5\sqrt{2}}{\sqrt{10}}\end{array}$

  13. $\begin{array}{r}\frac{21\sqrt{3}}{2\sqrt{14}}\end{array}$

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  Explore More
  ::探索更多

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  1. Calculate:  $\begin{array}{r}\frac{5\sqrt{5}}{\sqrt{12}}-\frac{2\sqrt{15}}{\sqrt{10}}\end{array}$ . 
  ::1. 计算:5512-21510。

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  2.The hull speed, s, in nautical miles per hour of a sailboat can be modeled by the formula:
  ::2. 船体速度(S),以帆船每小时的海里速度计算,可以用下列公式模拟:

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  $$\begin{array}{r}s=1.34\sqrt{l}\end{array}$$

   
  ::s=1.34l =1.34l

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  where l is the length in feet of the sailboat at the waterline. ¹ Find the speed of a boat whose hull length is 10 feet. Round your answer to the nearest tenth of a nautical mile per hour.
  ::1 寻找船体长度为10英尺的船只的速度。 将答案转至每小时10海里的最接近的十分之一。

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  3.The kinetic energy of an object is given by the function:
  ::3. 物体的动能由下列函数赋予:

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  $$\begin{array}{r}KE=0.5m{v}^2\end{array}$$

  
  ::KE=0.5mv2

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  where m is the object’s mass and v is its speed. ² Suppose there is an object whose mass is 6 kilograms, find its speed if it has a kinetic energy of 783 Joules. Round your answer to three decimal points.
  ::m 是物体的质量, v 是其速度。 2 假设有一个物体的质量为6公斤, 如果它的动能为783 焦耳斯, 找到它的速度。 将答案转至小数点后三点。

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  4. The length of the two legs of a right triangle are $\begin{array}{r}2\sqrt{5}\end{array}$ and $\begin{array}{r}3\sqrt{4}\end{array}$ . What is the length of the triangle's hypotenuse?
  ::4. 右三角的两腿长度为25和34,三角的下限长度是多少?

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  Answers for Review and Explore More Problems
  ::回顾和探讨更多问题的答复

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  Please see the Appendix. 
  ::请参看附录。

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  PLIX
  ::PLIX

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  Try these interactives that reinforce the concepts explored in this section:
  ::尝试这些强化本节所探讨概念的交互作用 :

   

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  References 
  ::参考参考资料

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  1.  “Crunching Numbers: Hull Speed & Boat Length,” by Charles Doane, March 26, 2010, .
  ::1. Charles Doane著《收缩数字:超速和船长》,2010年3月26日。

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  2.  “Kinetic Energy,” last edited May 24, 2017, https://en.wikipedia.org/wiki/Kinetic_energy.
  ::2. " 营养能源 " ,2017年5月24日编辑,https://en.wikipedia.org/wiki/Kinetic_能源。