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【329733】5.2二次根式的乘法和除法(2)

时间:2025-02-03 19:06:35 作者: 字数:2163字

5.2二次根式的乘法和除法(二)导学案

【学习目标】

1. 掌握二次根式根式的除法法则和商的算术平方根的性质。

2. 能熟练进行二次根式的除法运算及化简。

【学习重点】

运用二次根式的除法法则和商的算术平方根的性质进行二次根式的除法运算及化简。

【学习难点】

运用二次根式的除法法则和商的算术平方根的性质进行二次根式的除法运算及化简。

【学习过程】

  1. 学前准备

说一说:实数 <a href="/tags/194/" title="除法" class="c1" target="_blank">除法</a> <a href="/tags/195/" title="乘法" class="c1" target="_blank">乘法</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 互为倒数是什么意思?有何条件?一个非零实数有几个倒数?

填一填:由于 <a href="/tags/194/" title="除法" class="c1" target="_blank">除法</a> <a href="/tags/195/" title="乘法" class="c1" target="_blank">乘法</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ______  <a href="/tags/194/" title="除法" class="c1" target="_blank">除法</a> <a href="/tags/195/" title="乘法" class="c1" target="_blank">乘法</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ______,所以______________都是 <a href="/tags/194/" title="除法" class="c1" target="_blank">除法</a> <a href="/tags/195/" title="乘法" class="c1" target="_blank">乘法</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 的倒数,因此_______=_________. 一般地,如果 <a href="/tags/194/" title="除法" class="c1" target="_blank">除法</a> <a href="/tags/195/" title="乘法" class="c1" target="_blank">乘法</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> >0,则 <a href="/tags/194/" title="除法" class="c1" target="_blank">除法</a> <a href="/tags/195/" title="乘法" class="c1" target="_blank">乘法</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>  <a href="/tags/194/" title="除法" class="c1" target="_blank">除法</a> <a href="/tags/195/" title="乘法" class="c1" target="_blank">乘法</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> .


二、探索思考

阅读课本P162——164

知识点一 : 商的算术平方根的性质:


学一学:: <a href="/tags/194/" title="除法" class="c1" target="_blank">除法</a> <a href="/tags/195/" title="乘法" class="c1" target="_blank">乘法</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> >0 <a href="/tags/194/" title="除法" class="c1" target="_blank">除法</a> <a href="/tags/195/" title="乘法" class="c1" target="_blank">乘法</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 0,则

 <a href="/tags/194/" title="除法" class="c1" target="_blank">除法</a> <a href="/tags/195/" title="乘法" class="c1" target="_blank">乘法</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> __________= <a href="/tags/194/" title="除法" class="c1" target="_blank">除法</a> <a href="/tags/195/" title="乘法" class="c1" target="_blank">乘法</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ______= <a href="/tags/194/" title="除法" class="c1" target="_blank">除法</a> <a href="/tags/195/" title="乘法" class="c1" target="_blank">乘法</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ,即 <a href="/tags/194/" title="除法" class="c1" target="_blank">除法</a> <a href="/tags/195/" title="乘法" class="c1" target="_blank">乘法</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> <a href="/tags/194/" title="除法" class="c1" target="_blank">除法</a> <a href="/tags/195/" title="乘法" class="c1" target="_blank">乘法</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> >0, <a href="/tags/194/" title="除法" class="c1" target="_blank">除法</a> <a href="/tags/195/" title="乘法" class="c1" target="_blank">乘法</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 0.

议一议:上式中为什么规定 <a href="/tags/194/" title="除法" class="c1" target="_blank">除法</a> <a href="/tags/195/" title="乘法" class="c1" target="_blank">乘法</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ?如何用文字语言描述商的算术平方根的性质?



选一选:当 <a href="/tags/194/" title="除法" class="c1" target="_blank">除法</a> <a href="/tags/195/" title="乘法" class="c1" target="_blank">乘法</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 时, <a href="/tags/194/" title="除法" class="c1" target="_blank">除法</a> <a href="/tags/195/" title="乘法" class="c1" target="_blank">乘法</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> =_________(① <a href="/tags/194/" title="除法" class="c1" target="_blank">除法</a> <a href="/tags/195/" title="乘法" class="c1" target="_blank">乘法</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ② <a href="/tags/194/" title="除法" class="c1" target="_blank">除法</a> <a href="/tags/195/" title="乘法" class="c1" target="_blank">乘法</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> )

填一填: <a href="/tags/194/" title="除法" class="c1" target="_blank">除法</a> <a href="/tags/195/" title="乘法" class="c1" target="_blank">乘法</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ____________; <a href="/tags/194/" title="除法" class="c1" target="_blank">除法</a> <a href="/tags/195/" title="乘法" class="c1" target="_blank">乘法</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> _____________; <a href="/tags/194/" title="除法" class="c1" target="_blank">除法</a> <a href="/tags/195/" title="乘法" class="c1" target="_blank">乘法</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ________________;




知识点二二次根式的除法运算:


学一学:阅读教材P14134,想想还有其它方法吗?计算:看你水平如何!

⑴  <a href="/tags/194/" title="除法" class="c1" target="_blank">除法</a> <a href="/tags/195/" title="乘法" class="c1" target="_blank">乘法</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> =__________; ⑵ 3 <a href="/tags/194/" title="除法" class="c1" target="_blank">除法</a> <a href="/tags/195/" title="乘法" class="c1" target="_blank">乘法</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> =__________________________________;


【归纳总结】

二次根式的除法法则:二次根式相除,把______________相除,根指数不变,用式子表示为____________________ <a href="/tags/194/" title="除法" class="c1" target="_blank">除法</a> <a href="/tags/195/" title="乘法" class="c1" target="_blank">乘法</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> .注意:二次根式的运算结果一定要进行化简,化成最简二次根式。


【课堂展示】1.计算:⑴  <a href="/tags/194/" title="除法" class="c1" target="_blank">除法</a> <a href="/tags/195/" title="乘法" class="c1" target="_blank">乘法</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ⑵  <a href="/tags/194/" title="除法" class="c1" target="_blank">除法</a> <a href="/tags/195/" title="乘法" class="c1" target="_blank">乘法</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ⑶  <a href="/tags/194/" title="除法" class="c1" target="_blank">除法</a> <a href="/tags/195/" title="乘法" class="c1" target="_blank">乘法</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ⑷  <a href="/tags/194/" title="除法" class="c1" target="_blank">除法</a> <a href="/tags/195/" title="乘法" class="c1" target="_blank">乘法</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>






2.求下列各式当 <a href="/tags/194/" title="除法" class="c1" target="_blank">除法</a> <a href="/tags/195/" title="乘法" class="c1" target="_blank">乘法</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 时的值:⑴  <a href="/tags/194/" title="除法" class="c1" target="_blank">除法</a> <a href="/tags/195/" title="乘法" class="c1" target="_blank">乘法</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ⑵  <a href="/tags/194/" title="除法" class="c1" target="_blank">除法</a> <a href="/tags/195/" title="乘法" class="c1" target="_blank">乘法</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>







  1. 当堂反馈

1. <a href="/tags/194/" title="除法" class="c1" target="_blank">除法</a> <a href="/tags/195/" title="乘法" class="c1" target="_blank">乘法</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 计算:⑴  <a href="/tags/194/" title="除法" class="c1" target="_blank">除法</a> <a href="/tags/195/" title="乘法" class="c1" target="_blank">乘法</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ⑵  <a href="/tags/194/" title="除法" class="c1" target="_blank">除法</a> <a href="/tags/195/" title="乘法" class="c1" target="_blank">乘法</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>






2化简:⑴  <a href="/tags/194/" title="除法" class="c1" target="_blank">除法</a> <a href="/tags/195/" title="乘法" class="c1" target="_blank">乘法</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ⑵  <a href="/tags/194/" title="除法" class="c1" target="_blank">除法</a> <a href="/tags/195/" title="乘法" class="c1" target="_blank">乘法</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ⑶  <a href="/tags/194/" title="除法" class="c1" target="_blank">除法</a> <a href="/tags/195/" title="乘法" class="c1" target="_blank">乘法</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>







课堂小结 本节课你学到了那些知识?


、课后反思