Section: 乘数和分裂的广场根 | 5.6 加倍和分裂的广场根

Section outline

The area of a rectangle is $\begin{aligned} \sqrt{30} \end{aligned}$ and the length of the rectangle is $\begin{aligned} \sqrt{20} \end{aligned}$ . What is the width of the rectangle?
::矩形区域为 30 , 矩形的长度为 20 。矩形的宽度是多少 ?

Dividing Square Roots
::分裂的广场根

Division of radicals can be a bit more difficult than performing other operations . The main complication is that you cannot leave any radicals in the denominator of a fraction . For this reason we have to do something 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 cancel out the radicals and leave a whole number.
::激进分子的分裂可能比执行其他操作要困难一些。主要的复杂因素是不能将任何激进分子留在分数分母的分母中。出于这个原因,我们必须做一个叫做分母合理化的事情, 也就是把分母的上下乘以分母中的同一个分母。这将取消激进分子, 留下一个完整的数字。

Multiplying and Dividing Radicals
::乘数和分裂的激进组织

Remember the following properties:
::记住下列属性:

$\begin{aligned} \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \end{aligned}$ and $\begin{aligned} \frac{\sqrt{a}}{\sqrt{b}} \cdot \frac{\sqrt{b}}{\sqrt{b}} = \frac{\sqrt{a b}}{b} \end{aligned}$
::ab=ab和abb=abb

Simplify the following by rationalizing the denominator.
::使分母合理化,简化以下内容。

$\begin{aligned} \sqrt{\frac{1}{4}} \end{aligned}$

Break apart the radical by using Rule #4.
::使用规则4打破激进。

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

Simplify the following by rationalizing the denominator.
::使分母合理化,简化以下内容。

$\begin{aligned} \frac{2}{\sqrt{3}} \end{aligned}$

This might look simplified, but radicals cannot be in the denominator of a fraction. This means we need to apply Rule #5 to get rid of the radical in the denominator, or rationalize the denominator . Multiply the top and bottom of the fraction by $\begin{aligned} \sqrt{3} \end{aligned}$ .
::这也许看起来很简化,但激进不能成为分母的分母。这意味着我们需要应用规则5来消除分母中的激进,或者理顺分母。将分母的顶部和底部乘以 3 。

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

Simplify the following by rationalizing the denominator.
::使分母合理化,简化以下内容。

$\begin{aligned} \sqrt{\frac{32}{40}} \end{aligned}$

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

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

Examples
::实例

Example 1
::例1

Earlier, you were asked to find the width of the rectangle.
::早些时候,有人要求你找到矩形的宽度。

Recall that the area of a rectangle equals the length times the width, so to find the width, we must divide the area by the length.
::回顾矩形区域等于宽度的长度倍数,为了找到宽度,我们必须将区域除以长度。

$\begin{aligned} \sqrt{\frac{30}{20}} \end{aligned}$ = $\begin{aligned} \sqrt{\frac{3}{2}} \end{aligned}$ .

Now we need to rationalize the denominator. Multiply the top and bottom of the fraction by $\begin{aligned} \sqrt{2} \end{aligned}$ .
::现在我们需要使分母合理化。乘以分数的顶部和底部乘以 2 。

$\begin{aligned} \frac{\sqrt{3}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{6}}{2} \end{aligned}$

Therefore , the width of the rectangle is $\begin{aligned} \frac{\sqrt{6}}{2} \end{aligned}$ .
::因此,矩形宽度为62。

Example 2
::例2

Simplify the expression using the Radical Rules you have learned.
::使用您所学的激进规则来简化表达式。

$\begin{aligned} \sqrt{\frac{1}{2}} \end{aligned}$

$\begin{aligned} \sqrt{\frac{1}{2}} = \frac{\sqrt{1}}{\sqrt{2}} = \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2} \end{aligned}$

Example 3
::例3

Simplify the expression using the Radical Rules you have learned.
::使用您所学的激进规则来简化表达式。

$\begin{aligned} \sqrt{\frac{64}{50}} \end{aligned}$

$\begin{aligned} \sqrt{\frac{64}{50}} = \sqrt{\frac{32}{25}} = \frac{\sqrt{16 \cdot 2}}{5} = \frac{4 \sqrt{2}}{5} \end{aligned}$

Example 4
::例4

Simplify the expression using the Radical Rules you have learned.
::使用您所学的激进规则来简化表达式。

$\begin{aligned} \frac{4 \sqrt{3}}{\sqrt{6}} \end{aligned}$

The only thing we can do is rationalize the denominator by multiplying the numerator and denominator by $\begin{aligned} \sqrt{6} \end{aligned}$ and then simplify the fraction.
::我们唯一能做的就是通过将分子和分母乘以6来使分母合理化,然后简化分母。

$\begin{aligned} \frac{4 \sqrt{3}}{\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}} = \frac{4 \sqrt{18}}{6} = \frac{4 \sqrt{9 \cdot 2}}{6} = \frac{12 \sqrt{2}}{6} = 2 \sqrt{2} \end{aligned}$

Review
::回顾

Simplify the following fractions.
::简化以下分数。

$\begin{aligned} \sqrt{\frac{4}{25}} \end{aligned}$

$\begin{aligned} - \sqrt{\frac{16}{49}} \end{aligned}$

$\begin{aligned} \sqrt{\frac{96}{121}} \end{aligned}$

$\begin{aligned} \frac{5 \sqrt{2}}{\sqrt{10}} \end{aligned}$

$\begin{aligned} \frac{6}{\sqrt{15}} \end{aligned}$

$\begin{aligned} \sqrt{\frac{60}{35}} \end{aligned}$

$\begin{aligned} 8 \frac{\sqrt{18}}{\sqrt{30}} \end{aligned}$

$\begin{aligned} \frac{12}{\sqrt{6}} \end{aligned}$

$\begin{aligned} \sqrt{\frac{208}{143}} \end{aligned}$

$\begin{aligned} \frac{21 \sqrt{3}}{2 \sqrt{14}} \end{aligned}$

Challenge Use all the Radical Rules you have learned to simplify the expressions.
::挑战使用你学到的所有激进规则简化表达方式

$\begin{aligned} \sqrt{\frac{8}{12}} \cdot \sqrt{15} \end{aligned}$

$\begin{aligned} \sqrt{\frac{32}{45}} \cdot \frac{6 \sqrt{20}}{\sqrt{5}} \end{aligned}$

$\begin{aligned} \frac{\sqrt{24}}{\sqrt{2}} + \frac{8 \sqrt{26}}{\sqrt{8}} \end{aligned}$

$\begin{aligned} \frac{\sqrt{2}}{\sqrt{3}} + \frac{4 \sqrt{6}}{\sqrt{3}} \end{aligned}$

$\begin{aligned} \frac{5 \sqrt{5}}{\sqrt{12}} - \frac{2 \sqrt{15}}{\sqrt{10}} \end{aligned}$

Review (Answers)
::回顾(答复)

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。

Section outline

Dividing Square Roots ::分裂的广场根

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Review ::回顾

Review (Answers) ::回顾(答复)

Dividing Square Roots
::分裂的广场根

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Review
::回顾

Review (Answers)
::回顾(答复)