8.1 简化激进

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  简化激进

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  Simplifying Radicals
  ::简化激进

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  In algebra, you learned how to simplify radicals. Let’s review it here. Some key points to remember:
  ::在代数中,你学会了如何简化激进。让我们在这里回顾一下。

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  1. One way to simplify a radical is to factor out the perfect squares (see Example A).
     ::简化激进的一个办法是将完美的方形考虑在内(见例A)。
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  2. When adding radicals, you can only combine radicals with the same number underneath it. For example, $\begin{array}{r}2\sqrt{5}+3\sqrt{6}\end{array}$ cannot be combined, because 5 and 6 are not the same number (see Example B).
     ::当添加激进分子时,您只能将根基与底部的相同数字合并。例如,25+36不能合并,因为5和6的数字不同(见例B)。
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  3. To multiply two radicals, multiply what is under the radicals and what is in front (see Example B).
     ::要乘以两个激进,就乘以激进之下的东西和前面的东西(见例B)。
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  4. To divide radicals, you need to simplify the denominator, which means multiplying the top and bottom of the fraction by the radical in the denominator (see Example C).
     ::要分裂激进分子,你需要简化分母,这意味着分母的顶部和底部乘以分母中的底部(见例C)。

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  What if you were asked to find the sum of $\begin{array}{r}\sqrt{32}\end{array}$ and $\begin{array}{r}3\sqrt{8}\end{array}$ ? How could you combine these two terms so that you could add them?
  ::如果有人要求你找到32和38的总数呢?你怎么能把这两个词结合起来,以便你添加它们呢?

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  Examples
  ::实例

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  Example 1
  ::例1

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  Simplify the radical:  $\begin{array}{r}\sqrt{75}\end{array}$ .
  ::简化激进主义:75。

  $\begin{array}{r}\sqrt{75}=\sqrt{25\cdot 3}=5\sqrt{3}\end{array}$

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  Example 2
  ::例2

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  Simplify the radicals:  $\begin{array}{r}2\sqrt{5}+3\sqrt{80}\end{array}$ .
  ::简化基数:25+380。

  $\begin{array}{r}2\sqrt{5}+3\sqrt{80}=2\sqrt{5}+3(\sqrt{16\cdot 5})=2\sqrt{5}+(3\cdot 4)\sqrt{5}=14\sqrt{5}\end{array}$

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  Example 3
  ::例3

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  Simplify the radicals. ( For each radical, find the square number(s) that are factors).
  ::简化基数。 (对于每个基数,找到因素的正方数)。

  1. $\begin{array}{r}\sqrt{50}\end{array}$

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

  2. $\begin{array}{r}\sqrt{27}\end{array}$

    $\begin{array}{r}\sqrt{27}=\sqrt{9\cdot 3}=3\sqrt{3}\end{array}$

  3. $\begin{array}{r}\sqrt{272}\end{array}$

    $\begin{array}{r}\sqrt{272}=\sqrt{16\cdot 17}=4\sqrt{17}\end{array}$

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  Example 4
  ::例4

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  Simplify the radicals.
  ::简化激进分子。

  1. $\begin{array}{r}2\sqrt{10}+\sqrt{160}\end{array}$

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   Simplify $\begin{array}{r}\sqrt{160}\end{array}$ before adding:  $\begin{array}{r}2\sqrt{10}+\sqrt{160}=2\sqrt{10}+\sqrt{16\cdot 10}=2\sqrt{10}+4\sqrt{10}=6\sqrt{10}\end{array}$
  ::简化160后加:210+160=210+1610=210+410=610

  2. $\begin{array}{r}5\sqrt{6}\cdot 4\sqrt{18}\end{array}$

    $\begin{array}{r}5\sqrt{6}\cdot 4\sqrt{18}=5\cdot 4\sqrt{6\cdot 18}=20\sqrt{108}=20\sqrt{36\cdot 3}=20\cdot 6\sqrt{3}=120\sqrt{3}\end{array}$

  3. $\begin{array}{r}\sqrt{8}\cdot 12\sqrt{2}\end{array}$

    $\begin{array}{r}\sqrt{8}\cdot 12\sqrt{2}=12\sqrt{8\cdot 2}=12\sqrt{16}=12\cdot 4=48\end{array}$

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

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    $\begin{array}{r}{\left(5\sqrt{2}\right)}^2=5^2{\left(\sqrt{2}\right)}^2=25\cdot 2=50\to \end{array}$ the $\begin{array}{r}\sqrt{}\end{array}$ and the $\begin{array}{r}{}^2\end{array}$ cancel each other out
  :522=52(2)2=252=50和2相互取消

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  Example 5
  ::例5

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  Divide and simplify the radicals. (Rewrite all division problems like a fraction).
  ::分解并简化激进。 (将所有分裂问题重写为分数) 。

  1. $\begin{array}{r}4\sqrt{6}÷\sqrt{3}\end{array}$

   

  2. $\begin{array}{r}\frac{\sqrt{30}}{\sqrt{8}}\end{array}$

    $\begin{array}{r}\frac{\sqrt{30}}{\sqrt{8}}\cdot \frac{\sqrt{8}}{\sqrt{8}}=\frac{\sqrt{240}}{\sqrt{64}}=\frac{\sqrt{16\cdot 15}}{8}=\frac{4\sqrt{15}}{8}=\frac{\sqrt{15}}{2}\end{array}$

  3. $\begin{array}{r}\frac{8\sqrt{2}}{6\sqrt{7}}\end{array}$

    $\begin{array}{r}\frac{8\sqrt{2}}{6\sqrt{7}}\cdot \frac{\sqrt{7}}{\sqrt{7}}=\frac{8\sqrt{14}}{6\cdot 7}=\frac{4\sqrt{14}}{3\cdot 7}=\frac{4\sqrt{14}}{21}\end{array}$

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  Notice, we do not really “divide” radicals, but get them out of the denominator of a fraction.
  ::注意,我们并不真正“分解”激进分子,而是把他们从分母的分母中解救出来。

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  Review
  ::回顾

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  Simplify the radicals.
  ::简化激进分子。

  1. $\begin{array}{r}\sqrt{48}\end{array}$
  2. $\begin{array}{r}2\sqrt{5}+\sqrt{20}\end{array}$
  3. $\begin{array}{r}\sqrt{24}\end{array}$
  4. $\begin{array}{r}{\left(6\sqrt{3}\right)}^2\end{array}$
  5. $\begin{array}{r}8\sqrt{8}\cdot \sqrt{10}\end{array}$
  6. $\begin{array}{r}{\left(2\sqrt{30}\right)}^2\end{array}$
  7. $\begin{array}{r}\sqrt{320}\end{array}$
  8. $\begin{array}{r}\frac{4\sqrt{5}}{\sqrt{6}}\end{array}$
  9. $\begin{array}{r}\frac{12}{\sqrt{10}}\end{array}$
  10. $\begin{array}{r}\frac{21\sqrt{5}}{9\sqrt{15}}\end{array}$

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  Review (Answers)
  ::回顾(答复)

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  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 。