当前位置:首页 > 八年级 > 数学试卷

【323867】2024八年级数学下册 第04讲 二次根式的化简与应用(核心考点讲与练)(含解析)(新

时间:2025-01-15 20:50:40 作者: 字数:29457字
简介:


 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 04讲二次根式的化简与应用(核心考点)

一.二次根式的化简求值

二次根式的化简求值,一定要先化简再代入求值.

二次根式运算的最后,注意结果要化到最简二次根式,二次根式的乘除运算要与加减运算区分,避免互相干扰.

二.二次根式的应用

把二次根式的运算与现实生活相联系,体现了所学知识之间的联系,感受所学知识的整体性,不断丰富解决问题的策略,提高解决问题的能力.

二次根式的应用主要是在解决实际问题的过程中用到有关二次根式的概念、性质和运算的方法.

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

一.二次根式的化简求值(共10小题)

1.(会宁县期末)已知a <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> +2b <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 2,则a2+b2的值为(  )

A4 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> B14 C <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> D14+4 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

【分析】根据二次根式的混合运算法则分别求出a+bab,根据完全平方公式把原式变形,代入计算即可.

【解答】解:∵a <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> +2b <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 2

a+b=( <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 2)=2 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ab=( <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 2)=﹣1

a2+b2=(a+b2﹣2ab=(2 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 2﹣2×(﹣1)=14

故选:B

【点评】本题考查的是二次根式的化简求值,掌握二次根式的混合运算法则是解题的关键.

2.(杭州期末)若a <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> +1b <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 1,则a2ab+b2 5 

【分析】根据配方法以及二次根式的运算法则即可求出答案.

【解答】解:∵a <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> +1b <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 1

a+b <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ﹣12 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

ab=( <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 1)=2﹣11

原式=a2+2ab+b2﹣3ab

=(a+b2﹣3ab

=(2 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 2﹣3×1

8﹣3

5

故答案为:5

【点评】本题考查二次根式的运算,解题的关键是熟练运用完全平方公式以及二次根式的运算法则,本题属于基础题型.

3.(奉化区校级期末)已知x﹣2 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ,则代数式(x+12﹣6x+1+9的值为 2 

【分析】利用完全平方公式得到原式=(x﹣22,然后利用整体代入的方法计算.

【解答】解:(x+12﹣6x+1+9[x+1)﹣3]2

=(x﹣22

因为x﹣2 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

所以原式=( <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 22

故答案为2

【点评】本题考查了二次根式的化简求值:二次根式的化简求值,一定要先化简再代入求值.二次根式运算的最后,注意结果要化到最简二次根式,二次根式的乘除运算要与加减运算区分,避免互相干扰.

4.(永嘉县校级期中)若|a﹣2|+b2+4b+4+ <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 0,则 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>  2 

【分析】利用非负数的性质得到a﹣20b+20c <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 0,解得a2b=﹣2c <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ,然后根据二次根式的性质和二次根式的乘法法则计算.

【解答】解:根据题意得|a﹣2|+b+22+ <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 0

a﹣20b+20c <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 0

解得a2b=﹣2c <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

所以原式= <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

2×1

2

故答案为2

【点评】本题考查了二次根式的化简求值:二次根式的化简求值,一定要先化简再代入求值.二次根式运算的最后,注意结果要化到最简二次根式,二次根式的乘除运算要与加减运算区分,避免互相干扰.

5.(西湖区校级期末)已知:y <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> +5,化简并求 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 的值.

【分析】根据二次根式有意义的条件得到x4,则y5,再利用约分得到原式= <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ,然后通分得到原式= <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ,最后把xy的值代入计算即可.

【解答】解:∵x﹣4≥04﹣x≥0

x4

y5

原式= <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

=﹣4

【点评】本题考查了二次根式的化简求值:二次根式的化简求值,一定要先化简再代入求值.也考查了二次根式有意义的条件.也考查了根式有意义的条件.

6.(上城区校级期中)已知a <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ,求ab的值为 1 

【分析】a <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 易得ab1即可.

【解答】解:a <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

ab=( <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> )=3﹣21

故答案为:1

【点评】本题考查了二次根式的化简求值,根据二次根式的乘法可得ab的值.

7.(余杭区模拟)已知x2+ <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ,则代数式(7﹣4 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> x2+2﹣ <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> x <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 的值为 2﹣ <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>  

【分析】将x2+ <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 代入代数式(7﹣4 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> x2+2﹣ <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> x <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ,先利用完全平方公式和平方差公式化简计算,再进行实数的混合运算即可得出答案.

【解答】解:∵x2+ <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

7﹣4 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> x2+2﹣ <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> x <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

=(7﹣4 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 2+2﹣ <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

=(7﹣4 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> )(7+4 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> +4﹣3)﹣ <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

49﹣48+1﹣ <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

【点评】本题考查了二次根式的化简求值,熟练掌握完全平方公式和平方差公式及二次根式的混合运算法则是解题的关键.

8.(永嘉县校级期末)已知a+b3ab2,则 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 的值为  <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>  

【分析】根据a+b3ab2,可以判断出a0b0,将所求数字化简,然后a+b3ab2代入即可解答本题.

【解答】解: <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

a+b3ab2

a0b0

原式= <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

故答案为: <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

【点评】本题考查二次根式的化简求值,解答本题的关键是明确二次根式化简求值的方法.

9.(永嘉县校级期末)已知x <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ,其中a是正整数,那么所有使得x为整数的a的取值之和为 14 

【分析】首先利用二次根式 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 有意义的条件得到a≤178;然后 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 50,列举出满足条件的a的整数值,求和即可.

【解答】解:①根据题意知,50﹣ <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ≥0

解得a≤178

因为a是正整数,且使得x为正整数,

所以 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 是正整数.

a178时, <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 50

则在123、…、178中,满足14的倍数,即14nn是正整数),同时又能整开方的数,只有14,即和为14

故答案是:14

【点评】本题主要考查了二次根式的化简求值,二次根式有意义的条件,此题的难点是根据二次根式有意义的条件求得a的取值范围,结合条件确定a的取值.

10.(永嘉县校级期末)已知x <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> +1y <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 1,则x2﹣5xy+y2+6 7 

【分析】根据已知条件先求出xyxy的值,再把要求的式子变形为(xy2﹣3xy+6,然后代值计算即可.

【解答】解:∵x <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> +1y <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 1

xy <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 1)=2xy1

x2﹣5xy+y2+6=(xy2﹣3xy+622﹣3+67

故答案为:7

【点评】此题考查了二次根式的化简求值,用到的知识点是完全平方公式和平方差公式,关键是对要求的式子进行变形.

二.二次根式的应用(共8小题)

11.(鄢陵县期末)方程 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 的解为(  )

A <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

【分析】两边同时除以 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 后即可求得方程的解.

【解答】解:方程两边同时除以 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 得:x <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

故选:B

【点评】考查了二次根式的应用,解题的关键是能够进行分母有理化,难度不大.

12.(奉化区校级期末)已知max <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 表示取三个数中最大的那个数,例如:当x9时,max <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 81.当max <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 时,则x的值为(  )

A <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

【分析】直接利用已知分别分析得出符合题意的答案.

【解答】解:当max <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 时,

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ,解得:x <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ,此时 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> xx2,符合题意;

x2 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ,解得:x <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ;此时 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> xx2,不合题意;

x <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> xx2,不合题意;

故只有x <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 时,max <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

故选:C

【点评】此题主要考查了新定义,正确理解题意分类讨论是解题关键.

13.(锡山区期末)如图,从一个大正方形中裁去面积为8cm218cm2的两个小正方形,则留下的阴影部分面积和为 24cm2 

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

【分析】直接利用正方形的性质得出两个小正方形的边长,进而得出大正方形的边长,即可得出答案.

【解答】解:∵两个小正方形面积为8cm218cm2

大正方形边长为: <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 5 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> cm),

大正方形面积为(5 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 250cm2),

留下的阴影部分面积和为:50﹣8﹣1824cm2).

故答案为:24cm2

【点评】此题主要考查了二次根式的应用,正确得出大正方形的边长是解题关键.

14.(余姚市期末)如图,矩形内两个相邻正方形的面积分别为93,则阴影部分的面积为(  )

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

A8﹣3 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> B9﹣3 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> C3 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ﹣3 D3 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ﹣2

【分析】根据有理数的乘方求出两个正方形的面积,然后根据阴影部分的面积的和为一个矩形的面积列式计算即可得解.

【解答】解:∵两个相邻的正方形,面积分别为39

两个正方形的边长分别为 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 3

阴影部分的面积= <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ﹣3

故选:C

【点评】本题考查了有理数的乘方,正方形的性质,是基础题,熟记概念并求出两个正方形的边长是解题的关键.

15.(盂县月考)阅读与计算:

古希腊的几何学家海伦,在他的著作《度量》一书中,给出了下面一个公式:如果一个三角形的三边长分别为abc,记p <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> a+b+c),则三角形的面积为:SABC <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> (海伦公式),若△ABC中,BC4AC5AB6,请利用上面公式求出△ABC的面积.

【分析】先求出p,再代入海伦公式中计算即可.

【解答】解:∵BC4AC5AB6

p <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 4+5+6)= <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

S <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

【点评】本题考查了二次根式的应用,关键是读懂题意,理解公式的意思.

16.(天河区校级月考)若矩形的长a <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ,宽b <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

1)求矩形的面积和周长;

2)求a2+b2﹣20+2ab的值.

【分析】(1)直接利用二次根式的混合运算法则分别计算得出答案;

2)直接利用完全平方公式结合二次根式的混合运算法则计算得出答案.

【解答】解:(1)∵矩形的长a <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ,宽b <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

矩形的面积为:( <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

6﹣5

1

矩形的周长为:2 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>


2a2+b2﹣20+2ab

=(a+b2﹣20

=( <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 2﹣20

=(2 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 2﹣20

24﹣20

4

【点评】此题主要考查了二次根式的混合运算,正确掌握相关运算法则是解题关键.

17.(永嘉县校级期末)解方程: <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ,得x  <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>  

【分析】去分母、移项,据此求出方程的解是多少即可.

【解答】解:去分母得:3x+ <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 4x

移项得:x <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

故答案为: <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

【点评】此题主要考查了解一元一次方程的方法和二次根式的乘法,要熟练掌握解一元一次方程的一般步骤:去分母、去括号、移项、合并同类项、系数化为1

18.(乌苏市期末)矩形相邻两边长分别为 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ,则它的周长是 6 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>  ,面积是 4 

【分析】利用矩形的周长和面积计算公式列式计算即可.

【解答】解:矩形的周长是 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

6 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

矩形的面积是 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 4

故答案为:6 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 4

【点评】此题考查二次根式的实际运用,掌握矩形的周长和面积计算方法是解决问题的关键.

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

题组A 基础过关练

一.选择题(共6小题)

1.(诸暨市月考)将一组数据 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 32 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ,按下面的方法进行排列:

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 32 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 的位置记为(14),2 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 的位置记为(23),则这组数中最大的数的位置记为(  )

A.(52 B.(53 C.(62 D.(65

【分析】根据题意可以得到每行五个数,且根号里面的数都是3的倍数,从而可以得到3 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 所在的位置.

【解答】解:由题意可得,每五个数为一行,3 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

90÷33030÷56

3 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 位于第六行第五个数,位置记为(65),

故选:D

【点评】本题考查的是二次根式的性质,掌握二次根式的性质、正确找出规律是解题的关键.

2.(越城区模拟)已知a <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ,那么ab的关系为(  )

Aa+b <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> Bab0 Cab1 D <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 2

【分析】利用ab的值分别计算出它们的和、差和积,然后对各选项进行判断.

【解答】解:∵a <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

a+b2 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ab2 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ab3﹣21 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 25+2 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

故选:C

【点评】本题考查了二次根式的化简求值:二次根式的化简求值,一定要先化简再代入求值.二次根式运算的最后,注意结果要化到最简二次根式,二次根式的乘除运算要与加减运算区分,避免互相干扰.

3.(温州期中)若x2﹣ <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ,则代数式x2﹣4x+7的值为(  )

A7 B6 C.﹣6 D.﹣7

【分析】先移项得到x﹣2=﹣ <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ,两边平方得到x2﹣4x=﹣1,然后利用整体代入的方法计算.

【解答】解:∵x2﹣ <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

x﹣2=﹣ <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

x﹣223

x2﹣4x+43,即x2﹣4x=﹣1

x2﹣4x+7=﹣1+76

故选:B

【点评】本题考查了二次根式的化简求值:二次根式的化简求值,一定要先化简再代入求值.二次根式运算的最后,注意结果要化到最简二次根式,二次根式的乘除运算要与加减运算区分,避免互相干扰.

4.(鹿城区校级期中)已知a3﹣ <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> b2+ <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ,则代数式(a2﹣6a+9)(b2﹣4b+4)的值是(  )

A20 B16 C8 D4

【分析】先将(a2﹣6a+9)(b2﹣4b+4)变形为[a﹣3)(b﹣2]2,再将a3﹣ <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> b2+ <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ,代入求值即可.

【解答】解:(a2﹣6a+9)(b2﹣4b+4

=(a﹣32b﹣22

[a﹣3)(b﹣2]2

a3﹣ <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> b2+ <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 时,

原式=[3﹣ <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ﹣3)(2+ <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ﹣2]2

=(﹣22

4

故选:D

【点评】本题考查了整式的化简求值,熟练运用完全平方公式是解题的关键.

5.(镇海区期末)已知直角三角形的两条直角边的长分别为 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ,则这个直角三角形的面积为(  )

A16 B8 C163 D <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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法运算法则计算得出答案.

【解答】解:∵直角三角形的两条直角边的长分别为: <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

这个直角三角形的面积为: <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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应用,正确化简二次根式是解题关键.

6.(椒江区校级期中)如图,长方形内有两个相邻的正方形,面积分别为39,那么图中阴影部分的面积为(  )

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

A <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

【分析】设两个正方形的边长是xyxy),得出方程x23y29,求出x <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> y3,代入阴影部分的面积是(yxx求出即可.

【解答】解:设两个正方形的边长是xyxy),

x23y29

x <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> y3

则阴影部分的面积是(yxx=(3﹣ <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ﹣3

故选:B

【点评】本题考查了算术平方根性质的应用,主要考查学生的计算能力.

二.填空题(共4小题)

7.(天台县期末)已知x <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> +1y <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 1,则x2y2  <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>  

【分析】先分解因式,再代入比较简便.

【解答】解:x2y2=(x+y)(xy)=2 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ×24 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

【点评】注意分解因式在代数式求值中的作用.

8.(西湖区期末)已知a=﹣2,则 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> +a 0 

【分析】根据二次根式的性质即可求出答案.

【解答】解:当a=﹣2时,

原式|a|+a

a+a

0

故答案为:0

【点评】本题考查二次根式,解题的关键是熟练运用二次根式的性质,本题属于基础题型.

9.(温州期中)当x=﹣ <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 时,二次根式 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 的值是 2 

【分析】把x=﹣ <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 代入已知二次根式,通过开平方求得答案.

【解答】解:把x=﹣ <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 代入 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 中,得 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 2

故答案为:2

【点评】本题考查了二次根式的化简求值.此题利用代入法求得二次根式 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 的值.

10.(奉化区校级期中)如图,长方形内有两个相邻的正方形,面积分别为39,那么阴影部分的面积为 3 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

【分析】设两个正方形的边长是xyxy),得出方程x24y29,求出x2y3,代入阴影部分的面积是(yxx求出即可.

【解答】解:设两个正方形的边长是xyxy),

x23y29

x <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> y3

则阴影部分的面积是(yxx=(3﹣ <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ﹣3

故答案为:3 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ﹣3

【点评】本题考查了算术平方根性质的应用,主要考查学生的计算能力.

三.解答题(共3小题)

11.(越城区校级月考)点Pxy)是平面直角坐标系中的一点,点A10)为x轴上的一点.

1)用二次根式表示点P与点A的距离;

2)当x4y <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 时,连接OPPA,求PA+PO

3)若点P位于第二象限,且满足函数表达式yx+1,求 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 的值.

【分析】(1)利用两点间的距离公式进行解答;

2)利用两点间的距离公式求得OPPA,然后求PA+PO

3)把yx+1代入所求的代数式进行解答.

【解答】解:(1)点P与点A的距离: <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>


2)∵x4y <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> Pxy),A10),

P4 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ),

PA <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> PO <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ,则

PA+PO2 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>


3)∵点P位于第二象限,

x0y0

又∵yx+1

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> |x|+|y|=﹣x+y=﹣x+x+11.即 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 的值是1

【点评】本题考查了二次根式的应用.熟记两点间的距离公式是解题的难点.

12.(临海市期末)计算:

1 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> |

2)已知x <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> +1,求代数式x2﹣2x+3的值.

【分析】(1)根据二次根式的性质、绝对值的性质计算即可;

2)根据完全平方公式把原式变形,代入计算,得到答案.

【解答】解:(1 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

2)当x <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> +1时,x2﹣2x+3x2﹣2x+1+2=(x﹣12+25+27

【点评】本题考查的是二次根式的化简求值,掌握二次根式的性质、合并同类二次根式的法则是解题的关键.

13.(二道区期末)有一块矩形木板,木工采用如图的方式,在木板上截出两个面积分别为18dm232dm2的正方形木板.

1)求剩余木料的面积.

2)如果木工想从剩余的木料中截出长为1.5dm,宽为1dm的长方形木条,最多能截出 2 块这样的木条.

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

【分析】(1)根据二次根式的性质分别求出两个正方形的边长,结合图形计算得到答案;

2)求出3 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 范围,根据题意解答.

【解答】解:(1)∵两个正方形的面积分别为18dm232dm2

这两个正方形的边长分别为3 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> dm4 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> dm

剩余木料的面积为(4 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 6dm2);

243 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 4.51 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 2

从剩余的木料中截出长为1.5dm,宽为1dm的长方形木条,最多能截出2块这样的木条,

故答案为:2

【点评】本题考查的是二次根式的应用,掌握二次根式的性质、无理数的估算是解题的关键.

题组B 能力提升

一.选择题(共3小题)

1.(铁东区期中)如图,从一个大正方形中裁去面积为16cm224cm2的两个小正方形,则余下的面积为(  )

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

A16 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> cm2 B40 cm2 C8 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> cm2 D.(2 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> +4cm2

【分析】根据已知部分面积求得相应正方形的边长,从而得到大正方形的边长,易得大正方形的面积,利用分割法求得余下部分的面积.

【解答】解:从一个大正方形中裁去面积为16cm224cm2的两个小正方形,

大正方形的边长是 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 4+2 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

留下部分(即阴影部分)的面积是(4+2 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 2﹣16﹣2416+16 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> +24﹣16﹣2416 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> cm2).

故选:A

【点评】此题主要考查了二次根式的应用,正确求出阴影部分面积是解题关键.

2.(永嘉县期中)把四张形状大小完全相同的小长方形卡片(如图①)不重叠地放在一个底面为长方形(长为 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> cm,宽为4cm)的盒子底部(如图②),盒子底面未被卡片覆盖的部分用阴影表示.则图②中两块阴影部分的周长和是(  )

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

A4 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> cm B16cm C2 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> +4cm D4 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 4cm

【分析】根据题意列出关系式,去括号合并即可得到结果.

【解答】解:设小长方形卡片的长为x,宽为y

根据题意得:x+2y <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

则图②中两块阴影部分周长和是2 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> +24﹣2y+24﹣x)=2 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> +4×4﹣4y﹣2x2 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> +16﹣2x+2y)=2 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> +16﹣2 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 16cm).

故选:B

【点评】本题主要考查了二次根式的应用,整式的加减运算,在解题时要根据题意结合图形得出答案是解题的关键.

3.(宁波自主招生)设等式 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 在实数范围内成立,其中axy是三个不同的实数,则 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 的值是(  )

A3 B <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> C2 D <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

【分析】根据根号下的数要是非负数,得到axa)≥0aya)≥0xa≥0ay≥0,推出a≥0a≤0,得到a0,代入即可求出y=﹣x,把y=﹣x代入原式即可求出答案.

【解答】解:由于根号下的数要是非负数,

axa)≥0aya)≥0xa≥0ay≥0

axa)≥0xa≥0可以得到a≥0

aya)≥0ay≥0可以得到a≤0

a只能等于0,将a0代入等式得

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 0

x=﹣y

即:y=﹣x

由于xya是三个不同的实数,

x0y0

x=﹣y代入原式得:

原式= <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

故选:B

【点评】本题主要考查对二次根式的化简,算术平方根的非负性,分式的加减、乘除等知识点的理解和掌握,根据算术平方根的非负性求出axy的值和代入求分式的值是解此题的关键.

二.填空题(共6小题)

4.(永嘉县校级期中)若 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ,则 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>  6 

【分析】对 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 变形,得 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ,因为各项均为非负数,故可求得xyz的值,代入 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 中即可.

【解答】解:根据题意, <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

x2y6z3

所以 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

【点评】本题考查的是非负数的性质及二次根式的化简和求值.

5.(萧山区期末)已知x <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> +1,则代数式x2﹣2x+1的值为 2 

【分析】根据x的值和完全平方差公式可以解答本题.

【解答】解:∵x <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> +1

x2﹣2x+1

=(x﹣12

=( <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> +1﹣12

=( <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 2

2

故答案为:2

【点评】本题考查二次根式的化简求值,解答本题的关键是明确二次根式化简求值的方法.

6.(浙江自主招生)设ab2+ <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> bc2﹣ <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ,则a2+b2+c2abacbc 15 

【分析】将ab2+ <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> bc2﹣ <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 相加,得到ac4,再将a2+b2+c2abacbc转化成关于abbcac的完全平方的形式,再将ab2+ <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> bc2﹣ <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ac4整体代入即可.

【解答】解:∵ab2+ <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> bc2﹣ <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ,两式相加得,ac4

原式=a2+b2+c2abbcac

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

15

【点评】此题考查了对完全平方公式及整体代入的掌握情况,有一定的综合性,但难度不大.

7.(锦江区校级期中)若 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ,则m 3 n 2 

【分析】将已知的等式的左边被开方数中的5变形为2+3,根据平方根的定义将2变为 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 3变为 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ,同时将2 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 化为2• <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ,符合完全平方公式的特点,利用完全平方公式变形后,再利用二次根式的化简公式 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> |a|化简后,根据 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 大于 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ,利用绝对值的代数意义化简,与等式右边比较,即可求出mn的值.

【解答】解:∵ <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ,即 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 0

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

又∵ <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

m3n2

故答案为:32

【点评】此题考查了二次根式的化简求值,涉及的知识有:平方根的定义,二次根式的化简公式,完全平方公式,以及绝对值的代数意义,其技巧性较强,灵活变换等式左边的被开方数是解本题的关键.

8.(绍兴期中)求当a1+ <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 时,代数式2a2+b2﹣4a+2的值为 12 

【分析】原式配方变形后,将已知等式代入计算即可求出值.

【解答】解:原式=2a2﹣2a+1+b22a﹣12+b2

a1+ <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 时,原式=10+212

故答案为:12

【点评】此题考查了二次根式的化简求值,熟练掌握运算法则是解本题的关键.

9.(台州期中)若a3﹣ <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ,则a2﹣6a+9的值为 7 

【分析】将a的值代入a2﹣6a+9=(a﹣32计算可得.

【解答】解:当a3﹣ <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 时,

a26a+9=(a32

=(3 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 32

=(﹣ <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 2

7

故答案为:7

【点评】本题主要考查二次根式的化简求值,解题的关键是掌握完全平方公式和二次根数的运算顺序及运算法则.

三.解答题(共6小题)

10.(鄞州区月考)已知a <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

1)求a2﹣4a+4的值;

2)化简并求值: <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

【分析】(1)先将a化简,然后通过配方法将原式化简,最后代入a求值.

2)将原式先化简,然后代入a的值求解.

【解答】解:(1a <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

a2﹣4a+4=(a﹣22

a2﹣ <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 代入(a﹣22得(﹣ <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 23

2 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

=(a﹣1)﹣ <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

a2﹣ <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

a﹣11﹣ <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 0

原式=a﹣1+ <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 3

【点评】本题考查分式的化简求值,解题关键是熟练掌握因式分解与分式化简的方法,掌握分母有理化的方法.

11.(仙桃校级模拟)(1)计算: <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

2)已知x22x+15,求代数式 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 的值.

【分析】(1)根据算术平方根、负整数指数幂、绝对值可以解答本题;

2)根据完全平方公式可以将所求式子化简,然后根据x22x+15,可以得到x的值,然后代入化简后的式子即可解答本题.

【解答】解:(1 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> +9﹣2 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

9

2 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

x2+2 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> x+2﹣x2﹣2 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> x+2

x2+2 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> x+2﹣x2+2 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> x﹣2

4 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> x

x22x+15,可得x1=﹣3x25

x=﹣3时,原式=﹣12 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

x5时,原式=20 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

【点评】本题考查二次根式的化简求值、负整数指数幂、绝对值,解答本题的关键是明确它们各自的计算方法.

12.(镇海区期末)计算:

1 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

2)已知| <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> a|+ <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 0,求a2﹣2 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> +2+b2的值.

【分析】(1)根据二次根式的乘除法和加减法可以解答本题;

2)根据| <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> a|+ <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 0,可以得到ab的值,然后将所求式子变形,再将ab的值代入即可解答本题.

【解答】解:(1 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> a|+ <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 0

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>a0b﹣20

a <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> b2

a2﹣2 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> +2+b2

=(a <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 2+b2

=( <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 2+22

02+4

0+4

4

【点评】本题考查二次根式的化简求值、非负数的性质,解答本题的关键是明确二次根式混合运算的计算方法.

13.(长岭县期末)已知x2﹣ <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> y2+ <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ,求x2+xy+y2的值.

【分析】先分别求出x+yxy的值,再根据完全平方公式进行变形,最后代入求出即可,

【解答】解:∵x2﹣ <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> y2+ <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

x+y4xy4﹣31

x2+xy+y2

=(x+y2xy

42﹣1

15

【点评】本题考查了二次根式的性质和完全平方公式的应用,主要考查学生的计算能力.

14.(西湖区校级期中)(1)计算 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

2)已知x <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> y2 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ,求3x2﹣2xy+3y2的值.

【分析】(1)先化简各二次根式,再计算乘法,最后计算加减可得;

2)先计算出x+yxy的值,再代入原式=3x+y2﹣8xy计算可得.

【解答】解:(1)原式= <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> +6 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 6 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> +6 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 6 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> +6 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> +6 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

7 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>


2)∵x <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> y2 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

x+y2 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> xy=﹣1

3x2﹣2xy+3y23x2+2xy+y2﹣2xy)﹣2xy

3x+y2﹣8xy

2 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 2﹣8×(﹣1

44

【点评】本题主要考查二次根式的混合运算,解题的关键是掌握二次根式的混合运算顺序和运算法则.

15.如图,某校自行车棚的人字架棚顶为等腰三角形ABC,点D是边AB的中点,中柱CD2 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> AB2 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> ,求△ABC的周长及面积.

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

【分析】根据点DAB的中点,三角形ABC为等腰三角形,可得CDAB,并且求出ADBD的长度,在Rt△ACD中求出AC的长度,同理可求出BC的长度,继而以求得△ABC的周长及面积.

【解答】解:在等腰三角形ABC中,

D是边AB的中点,

CDABADBD <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

Rt△ACD中,

AD <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> CD2 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

AC <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

同理可得,BC3 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

则△ABC的周长为3 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 8 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

面积为 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</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/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a> 6 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/834/" title="核心" class="c1" target="_blank">核心</a> <a href="/tags/896/" title="根式" class="c1" target="_blank">根式</a>

【点评】本题考查了二次根式的应用以及勾股定理的应用,解答本题的关键是得出CD为三角形ABC的高,并且运用勾股定理求出等腰三角形的腰长,难度一般.


1