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【324036】2024八年级数学下册 专题1.2 二次根式的乘除重难点题型(含解析)(新版)浙教版

时间:2025-01-15 21:32:48 作者: 字数:14227字
简介:


 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 专题1.2二次根式的乘除-重难点题型

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>  <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>  <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

Shape1 知识点1 二次根式的乘除法则】

二次根式的乘法法则: <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

积的算术平方根: <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

二次根式的除法法则: <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

商的算术平方根: <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> .

【题型1 求字母的取值范围】

【例1】(召陵区期末)使 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 成立的x的取值范围是(  )

Ax≠3 Bx3 Cx≥2x≠3 Dx≥3

【分析】根据被开方数大于或等于0,分母不等于0列出不等式组,解不等式组即可.

【解答】解:根据题意得: <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

解得:x3

故选:B

【点评】本题考查了二次根式有意义的条件,分式有意义的条件,解题的关键是根据被开方数大于或等于0,分母不等于0列出不等式组.

【变式1-1】(黄浦区期中)若等式 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 成立,则实数k的取值范围是(  )

Ak <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> Bk3k <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> Ck3 Dk≥3

【分析】直接利用二次根式有意义的条件以及分式有意义的条件分析得出答案.

【解答】解:等式 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 成立,

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

解得:k3

故选:C

【点评】此题主要考查了二次根式有意义的条件以及分式有意义的条件,正确掌握相关定义是解题关键.

【变式1-2】(长兴县月考)根据二次根式的性质,若 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>  <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ,则a的取值范围是(  )

Aa≤5 Ba≥0 C0≤a≤5 Da≥5

【分析】根据二次根式有意义的条件、二次根式乘除法法则解答即可.

【解答】解:由题意得,a≥05﹣a≥0

解得,0≤a≤5

故选:C

【点评】本题考查的是二次根式的乘除法,掌握二次根式的乘除法法则、二次根式有意义的条件是解题的关键.

【变式1-3】(岱岳区期中)若等式 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 成立,则m的取值范围是(  )

Am≥﹣2 Bm≥2 C.﹣2≤m≤2 Dm≥4

【分析】直接利用二次根式有意义的条件分析得出答案.

【解答】解:∵等式 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 成立,

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

解得:m≥2

故选:B

【点评】此题主要考查了二次根式有意义的条件,正确把握二次根式有意义的条件是解题关键.

【题型2二次根式乘除的运算】

【例2】(黄浦区期末)计算: <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

【分析】先把除法运算转化为乘法运算得到原式 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 2 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ,然后约分即可.

【解答】解:原式 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 2 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

1

【点评】本题考查了二次根式的乘除法:先把各二次根式化为最简二次根式,再把除法运算转化为乘法运算,然后约分.

【变式2-1】(闵行区校级月考)计算: <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

【分析】根据二次根式的乘法法则进行计算即可.

【解答】解: <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

=(1 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 4 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

=(1 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>  <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

10 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

【点评】本题主要考查了二次根式的乘法法则,掌握二次根式的乘法法则是解决问题的关键.

【变式2-2】(静安区月考)计算: <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

【分析】直接利用二次根式的乘除运算法则计算得出答案.

【解答】解:原式=8x2×3 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

24x2 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

24y2 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

【点评】此题主要考查了二次根式的乘除运算,正确化简二次根式是解题关键.

【变式2-3】(浦东新区月考)计算: <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> •( <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>  <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> a0).

【分析】直接利用二次根式的乘除运算法则计算得出答案.

【解答】解:原式 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

【点评】此题主要考查了二次根式的乘除,正确化简二次根式是解题关键.

【题型3二次根式的符号化简】

【例3】(荔湾区校级月考)把x <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 根号外的因式移到根号内,得(  )

A <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> B <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> C <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> D <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

【分析】直接利用二次根式的性质得出x的符号,进而得出答案.

【解答】解:∵ <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 有意义,

∴﹣x≥0

x≤0

原式 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

故选:D

【点评】此题主要考查了二次根式的性质与化简,正确掌握二次根式的性质是解题关键.

【变式3-1】(龙口市期中)把a <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 根号外的因式移入根号内,运算结果是(  )

A <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> B <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> C <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> D <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

【分析】根据二次根式的性质,可得答案.

【解答】解:a <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 根号外的因式移到根号内,化简的结果是 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

故选:D

【点评】本题考查了二次根式的性质,注意化简后不能改变原数的大小.

【变式3-2】(柯桥区期中)把代数式(a﹣1 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 中的a﹣1移到根号内,那么这个代数式等于(  )

A <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> B <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> C <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> D <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

【分析】根据二次根式的概念和性质化简即可.

【解答】解:(a﹣1 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 1﹣a <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

故选:A

【点评】正确理解二次根式的性质与化简及概念是解决问题的关键.

【变式3-3】(西湖区校级月考)已知xy0,把代数式 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 中的x移到根号内,那么这个代数式等于(  )

A <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> B <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> C <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> D <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

【分析】利用二次根式的意义得到y0x0,然后根据二次根式的性质得到 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ,从而约分即可.

【解答】解:∵ <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 0xy0

y0x0

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

故选:B

【点评】本题考查了二次根式的性质与化简:利用二次根式的基本性质进行化简;利用积的算术平方根的性质和商的算术平方根的性质进行化简.也考查了二次根式有意义的条件.

【题型4最简二次根式的概念】

Shape2 知识点2 最简二次根式】

我们把满足①被开方数不含分母;②被开方数中不含能开得尽方的因数或因式.

这两个条件的二次根式,叫做最简二次根式.

【例4】(涪城区校级期中)在下列根式:5 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>  <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>  <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>  <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 中,最简二次根式有(  )

A1 B2 C3 D4

【分析】根据最简二次根式的定义解答.

【解答】解:在5 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>  <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>  <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>  <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 中,最简二次根式有5 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>  <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ,共2个.

故选:B

【点评】本题考查最简二次根式的定义.根据最简二次根式的定义,最简二次根式必须满足两个条件:

1)被开方数不含分母;

2)被开方数不含能开得尽方的因数或因式.

【变式4-1】(招远市期中)二次根式 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>  <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>  <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>  <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>  <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 中,最简二次根式有(  )个.

A1 B2 C3 D4

【分析】根据最简二次根式的定义逐个判断即可.

【解答】解:最简二次根式有 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>  <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ,共2个,

故选:B

【点评】本考查了最简二次根式的定义,注意:最简二次根式具备两个条件:①被开方数的每一个因式都是整式,每个因数都是整数,②被开方数不含有能开得尽方的因式或因数.

【变式4-2】(江夏区校级月考)二次根式: <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 2 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>  <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>  <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>  <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>  <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>  <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>  <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ,是最简二次根式的有(  )个.

A2 B3 C4 D5

【分析】根据最简二次根式的定义,同时满足①被开方数不含能开得尽方的因数,②被开方数不含分母,才是最简二次根式,进行选择即可.

【解答】解: <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 的被开方数中含有分母,所以不是最简二次根式;

2 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>  <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>  <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>  <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 符合最简二次根式的定义,所以它们是最简二次根式;

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>  <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ,二次根式的被开方数中含有能开得尽方的因数和因式,所以它们不是最简二次根式;

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 分母中含有二次根式,所以不是最简二次根式;

综上所述,上述二次根式中,属于最简二次根式的个数是4个.

故选:C

【点评】本题考查了最简二次根式的定义.根据最简二次根式的定义,最简二次根式必须满足两个条件:

1)被开方数不含分母;

2)被开方数不含能开得尽方的因数或因式.

【变式4-3】(诸暨市月考)我们把形如 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ab为有理数, <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 为最简二次根式)的数叫做 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 型无理数,如 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 1 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 型无理数,则 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 是(  )

A <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 型无理数 B <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 型无理数 C <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 型无理数 D <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 型无理数

【分析】将代数式化简即可判断.

【解答】解:( <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 2

3﹣2 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 6

9﹣2 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

9﹣2×3 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

9﹣6 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

故选:A

【点评】本题考查了最简二次根式,熟练将代数式化简是解题的关键.

【题型5分母有理化】

Shape3 知识点3 分母有理化】

分母有理化是指把分母中的根号化去:分母有理化常常是乘二次根式本身(分母只有一项)或与原分母

组成平方差公式;

两个含二次根式的代数式相乘时,它们的积不含二次根式,这样的两个代数式成互为有理化因式.一个

二次根式的有理化因式不止一个.

【例5(雁塔区校级月考)若 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>  <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ,则(  )

Aab Bab Cab D.无法确定

【分析】先把ab分母有理化,再比较即可.

【解答】解:a <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

b <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 2 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

ab

故选:C

【点评】本题考查了分母有理化,能正确把ab分母有理化是解此题的关键.

【变式5-1】(九龙县期末)实数 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 的整数部分a  ,小数部分b 

【分析】将已知式子分母有理化后,先估算出 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 的大小即可得到已知式子的整数部分与小数部分.

【解答】解: <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

479,∴2 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 3

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 3,即实数 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 的整数部分a2

则小数部分为 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 2 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

故答案为:2 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

【点评】此题考查了分母有理化,以及估算无理数的大小,是一道中档题.

【变式5-2】(雨城区校级期中)比较大小: <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>    <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> (用>,<或=填空).

【分析】直接利用二次根式的性质分别化简,进而比较得出答案.

【解答】解:∵ <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

故答案为:<.

【点评】此题主要考查了分母有理化,正确化简二次根式是解题关键.

【变式5-3】(宝山区校级月考)分母有理化: <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

【分析】根据二次根式的性质以及运算法则即可求出答案.

【解答】解:原式 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

【点评】本题考查分母有理数,解题的关键是熟练运用二次根式的性质以及运算法则,本题属于中等题型.

【题型6分母有理化的应用】

【例6(云县期中)观察下列等式

等式一: <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 1

等式二: <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

等式三: <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

……

解决下列问题:

1)化简: <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

2)若有理数ab满足 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ,求a+b的值.

【分析】(1)观察已知所给的等式即可进行化简;

2)结合(1)先进行化简,再求a+b的值即可.

【解答】解:(1)化简: <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

观察已知等式可知:

原式 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

2)因为 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

所以a <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 1+b <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 1)=2 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 1

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> a+b)﹣(ab)=2 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 1

所以a+b2ab1

答:a+b的值为2

【点评】本题考查了分母有理化、平方差公式,解决本题的关键是掌握分母有理化.

【变式6-1】(沿河县期末)在进行二次根式化简时,我们有时会碰上如 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>  <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>  <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 一样的式子,其实我们还可以将其进一步化简: <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

以上这种化简的步骤叫做分母有理化. <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 还可以用以下方法化简: <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

1)请用不同的方法化简 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

2)化简: <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

【分析】(1)分式的分子和分母都乘以 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ,即可求出答案;把2看出5﹣3,根据平方差公式分解因式,最后进进约分即可.

2)先每一个二次根式分母有理化,再分母不变,分子相加,最后合并即可.

【解答】解:(1 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>


2)原式 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

【点评】本题考查了分母有理化,平方差公式的应用,主要考查学生的计算和化简能力.

【变式6-2】(吴江区期中)像 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 2 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>  <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 两个含有二次根式的代数式相乘,积不含有二次根式,则称这两个代数式互为有理化因式.爱动脑筋的小明同学在进行二次根式计算时,利用有理化因式化去分母中的根号.

1 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

2 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

勤奋好学的小明发现:可以用平方之后再开方的方式来化简一些有特点的无理数.

3)化简: <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

解:设x <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ,易知 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ,∴x0

由:x23 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 2.解得x <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

请你解决下列问题:

12 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 的有理化因式是 2 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 3 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>  

2)化简: <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

3)化简: <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

【分析】(1)找出原式的有理化因式即可;

2)原式各式分母有理化,计算即可求出值;

3)设x <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ,判断出x小于0,将左右两边平方求出x的值即可.

【解答】解:(12 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 3 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 的有理化因式是2 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 3 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

故答案为:2 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 3 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

2)原式 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 1+2 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 3

3)设x <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ,可得 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ,即x0

由题意得:x26﹣3 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 6+3 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 2 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 12﹣66

解得:x <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

则原式 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

【点评】此题考查了分母有理化,无理数,平方差公式,以及二次根式的性质与化简,熟练掌握运算法则是解本题的关键.

【变式6-3】(梁平区期末)阅读下述材料:

我们在学习二次根式时,熟悉的分母有理化以及应用.其实,有一个类似的方法叫做“分子有理化”:

与分母有理化类似,分母和分子都乘以分子的有理化因式,从而消掉分子中的根式.比如: <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

分子有理化可以用来比较某些二次根式的大小,也可以用来处理一些二次根式的最值问题.例如:

比较 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>  <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 的大小.可以先将它们分子有理化.如下: <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>  <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

因为 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ,所以 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

再例如:求y <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 的最大值.做法如下:

解:由x+2≥0x﹣2≥0可知x≥2,而y <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

x2时,分母 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 有最小值2,所以y的最大值是2

解决下述问题:

1)比较3 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 42 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 的大小;

2)求y <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 的最大值.

【分析】(1)利用分母有理化得到3 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 4 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 2 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ,利用3 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 42 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 可判断3 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 42 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

2)根据二次根式有意义的条件得到由1+x≥0x≥0,则x≥0,利用分母有理化得到y <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ,由于x0时, <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 有最小值1,从而得到y的最大值.

【解答】解:(1)∵3 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 4 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

2 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

3 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 2 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 4 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

3 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 42 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

3 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 42 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

2)由1+x≥0x≥0x≥0

y <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

x0时, <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/153/" title="乘除" class="c1" target="_blank">乘除</a> <a href="/tags/264/" title="难点" class="c1" target="_blank">难点</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 有最小值1

y的最大值为1

【点评】本题考查了分母有理化:分母有理化是指把分母中的根号化去.也考查了平方差公式.


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