初中数学七年级上册《立方根》教学设计（人教版·五四制）

初中数学七年级上册《立方根》教学设计（人教版·五四制）

一、课标要求与教学理念阐述

  《义务教育数学课程标准（2022年版）》明确指出，在初中阶段，学生要“理解乘方的意义，了解平方根、算术平方根、立方根的概念，会用根号表示数的平方根、算术平方根、立方根”。本课“立方根”是在学生已经学习了平方根、算术平方根的概念和求法，掌握了乘方运算的基础上，对开方运算的进一步拓展和深化。本节课的教学设计秉持“以学生发展为本”的核心素养导向，强调数学知识的发生与发展过程。我们不仅关注学生对立方根概念、符号表示及基本运算的掌握，更注重引导他们通过类比平方根的研究路径，自主建构立方根的知识体系，体会类比、从特殊到一般、符号化等核心数学思想方法，发展数学抽象、逻辑推理和数学运算素养。同时，教学设计将融入跨学科视野，引导学生认识立方根在现实世界（如物理、工程、信息技术）中的应用价值，培养其模型观念和应用意识。

二、教材内容与结构分析

  本节课内容隶属于“数与代数”领域，是实数单元的重要组成部分。在人教版五四制七年级上册的教材编排中，它紧接“平方根”之后，两者共同构成了完整的开方运算基础。教材通常通过具体情境（如正方体体积求棱长）引入立方根的概念，进而给出定义和符号表示，再通过例题和练习探究立方根的性质并进行简单运算。

  从知识结构看，立方根是乘方运算（指数为3）的逆运算，这一本质与平方根一致，构成了完美的知识对称性。然而，立方根在性质上与平方根存在显著差异（如被开方数可为任意实数，立方根符号内的被开方数不具有非负性限制等），这种“同中存异”的特点正是教学的关键点和学生认知的难点。深入理解这种差异，对于学生构建清晰、稳固的实数运算知识网络至关重要。本节课的学习也为后续学习实数运算、解简单三次方程、以及高中乃至更深层次的函数与方程思想奠定不可或缺的基础。

三、学情诊断与教学准备

  学情分析：

  认知基础：七年级学生已经熟练掌握了有理数的乘方运算，特别是求一个数的立方；刚刚系统学习了平方根与算术平方根的概念、表示方法及基本性质，对开方运算有了初步的感性认识和理性思考框架。这为通过类比迁移学习立方根提供了坚实的“最近发展区”。

  认知障碍：学生可能存在的困难主要体现在以下几个方面：一是容易混淆平方根与立方根的符号表示及性质，尤其是在处理负数时，受平方根中“负数没有平方根”的思维定势影响，难以理解“负数有立方根”；二是对开立方运算的熟练度不足，尤其对非完全立方数的立方根的估算与表示感到困难；三是在从具体情境中抽象出立方根概念，并运用其解决实际问题时，建模意识与能力有待加强。

  心理与能力特征：该阶段学生抽象逻辑思维开始占主导，但仍需具体经验支持。他们好奇心强，乐于探究，具备一定的小组合作与交流能力。因此，教学设计应充分利用其认知基础，创设有效情境，激发探究欲望，引导他们在自主探究、合作辨析中突破认知障碍。

  教学准备：

  教师准备：精心设计的多媒体课件（包含情境动画、探究问题链、对比表格等）；实物或模型（如多个体积已知的小正方体）；课堂练习与分层检测题卡。

  学生准备：复习平方根的相关知识；准备计算器（用于探究非完全立方数的立方根）；预习导学案。

四、教学目标

  基于以上分析，确立本课的教学目标如下：

  1.知识与技能：

    （1）理解立方根的概念，掌握开立方运算，能准确叙述一个数a

a

a的立方根的定义。

    （2）会用根号a

3

\sqrt[3]{a}

3a<pathd="M95,702

c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14

c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54

c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10

s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429

c69,-144,104.5,-217.7,106.5,-221

l0-0

c5.3,-9.3,12,-14,20,-14

H400000v40H845.2724

s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7

c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z

M83480h400000v40h-400000z">

​表示一个数a

a

a的立方根，了解根指数、被开方数的含义。

    （3）掌握正数、零、负数的立方根的性质，能求出一个数的立方根（包括完全立方数及利用计算器求近似值）。

    （4）能区分平方根与立方根的异同，并能运用立方根的概念解决简单的实际问题。

  2.过程与方法：

    （1）经历从具体实际问题抽象出立方根概念的过程，发展数学抽象能力。

    （2）通过类比平方根的学习路径，自主探究立方根的定义、表示和性质，体会类比思想在数学学习中的重要作用。

    （3）在探究立方根性质的活动中，学会从特殊到一般的归纳方法，并通过与平方根的对比辨析，强化对知识本质的理解。

    （4）在解决实际问题的过程中，初步建立数学模型，发展应用意识。

  3.情感、态度与价值观：

    （1）通过探索立方根性质的规律性，感受数学的简洁美、对称美与统一美。

    （2）在克服认知冲突（如负数立方根的存在性）的过程中，培养敢于质疑、严谨求实的科学态度。

    （3）通过了解立方根在现实世界中的应用，体会数学的价值，增强学习数学的兴趣和信心。

五、教学重点与难点

  教学重点：立方根的概念、表示方法及求法。

  教学难点：理解立方根与平方根在性质上的区别，特别是负数立方根的存在性及其意义；灵活运用立方根解决实际问题。

六、教学过程设计

（一）创设情境，问题导学（预计用时：8分钟）

  1.情境引入：

    教师利用多媒体展示两个问题情境：

    情境A（复习链接）：一个正方形展厅，面积为64平方米，它的边长为多少米？学生容易答出：边长是8米（因为8

2

=

64

8^2=64

82=64）。教师强调：这里我们实际上是已知一个数的平方（乘方结果），求这个数（底数），即求64的平方根。

    情境B（新知探源）：现要设计一个正方体形状的蓄水池，其容积为27立方米，请问这个正方体蓄水池的棱长应为多少米？

    引导学生分析：正方体体积=棱长

3

^3

3。设棱长为x

x

x米，则得到方程x

3

=

27

x^3=27

x3=27。提问：“这里的已知量和未知量是什么关系？与我们刚才复习的平方根问题有什么相似和不同？”

  2.提出问题链，激发认知冲突：

    问题1：满足x

3

=

27

x^3=27

x3=27的x

x

x是多少？你是如何得到的？（学生可能通过心算3

3

=

27

3^3=27

33=27得出x

=

3

x=3

x=3）。

    问题2：如果蓄水池容积是8立方米呢？棱长是多少？（x

=

2

x=2

x=2）。

    问题3：如果容积是-8立方米呢？棱长可能是多少？是否存在这样的x

x

x，使得x

3

=

−

8

x^3=-8

x3=−8？（此处制造认知冲突。部分学生受平方根影响可能认为负数没有立方根，但通过计算(

−

2

)

3

=

−

8

(-2)^3=-8

(−2)3=−8，可以发现x

=

−

2

x=-2

x=−2）。

    问题4：如果体积是10立方米（一个非完全立方数）呢？棱长还能用一个精确的整数表示吗？它大约是多少？

    教师总结：像x

3

=

27

x^3=27

x3=27，x

3

=

8

x^3=8

x3=8，x

3

=

−

8

x^3=-8

x3=−8，x

3

=

10

x^3=10

x3=10这类问题，都是“已知一个数的立方，求这个数”的运算。这种运算在数学上叫做什么呢？它与我们学过的平方根运算有何关联？今天我们就一起来探究这种新的运算。

  设计意图：从熟悉的平方根情境过渡到新的立方根情境，建立知识间的联系。通过具体的、有梯度的实际问题，让学生感受学习立方根的必要性。特别设置“负数”情境，引发学生认知冲突，为后续探究立方根与平方根的性质差异埋下伏笔，激发探究欲望。

（二）类比探究，建构概念（预计用时：15分钟）

  1.概念生成：

    引导学生回顾平方根的定义：“一般地，如果一个数的平方等于a

a

a，那么这个数叫做a

a

a的平方根（或二次方根）。”

    提问：“你能类比平方根的定义，给‘立方根’下一个定义吗？”给予学生独立思考和小声讨论的时间。

    请学生尝试表述，教师引导完善并板书严谨定义：

    立方根的定义：一般地，如果一个数的立方等于a

a

a，那么这个数叫做a

a

a的立方根（或三次方根）。这就是说，如果x

3

=

a

x^3=a

x3=a，那么x

x

x叫做a

a

a的立方根。

    教师强调定义中的关键词：“立方”、“等于a

a

a”、“数x

x

x”。并举例：因为2

3

=

8

2^3=8

23=8，所以2

2

2是8

8

8的立方根；因为(

−

2

)

3

=

−

8

(-2)^3=-8

(−2)3=−8，所以−

2

-2

−2是−

8

-8

−8的立方根。

  2.符号表示：

    复习平方根的符号表示：a

a

a的平方根记作±

a

\pm\sqrt{a}

±a<pathd="M95,702

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c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54

c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10

s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429

c69,-144,104.5,-217.7,106.5,-221

l0-0

c5.3,-9.3,12,-14,20,-14

H400000v40H845.2724

s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7

c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z

M83480h400000v40h-400000z">

​，算术平方根记作a

\sqrt{a}

a<pathd="M95,702

c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14

c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54

c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10

s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429

c69,-144,104.5,-217.7,106.5,-221

l0-0

c5.3,-9.3,12,-14,20,-14

H400000v40H845.2724

s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7

c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z

M83480h400000v40h-400000z">

​，其中“

\sqrt{\}

<pathd="M95,702

c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14

c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54

c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10

s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429

c69,-144,104.5,-217.7,106.5,-221

l0-0

c5.3,-9.3,12,-14,20,-14

H400000v40H845.2724

s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7

c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z

M83480h400000v40h-400000z">

​”称为根号，a

a

a是被开方数，根指数2

2

2通常省略。

    提问：“立方根应该如何表示呢？根指数还能省略吗？”

    讲解并板书：一个数a

a

a的立方根，用符号a

3

\sqrt[3]{a}

3a<pathd="M95,702

c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14

c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54

c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10

s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429

c69,-144,104.5,-217.7,106.5,-221

l0-0

c5.3,-9.3,12,-14,20,-14

H400000v40H845.2724

s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7

c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z

M83480h400000v40h-400000z">

​表示，读作“三次根号a

a

a”，其中a

a

a是被开方数，3

3

3是根指数，且根指数3

3

3不能省略。

    强调书写规范，并在黑板上规范书写8

3

\sqrt[3]{8}

38<pathd="M95,702

c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14

c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54

c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10

s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429

c69,-144,104.5,-217.7,106.5,-221

l0-0

c5.3,-9.3,12,-14,20,-14

H400000v40H845.2724

s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7

c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z

M83480h400000v40h-400000z">

​，−

8

3

\sqrt[3]{-8}

3−8<pathd="M95,702

c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14

c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54

c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10

s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429

c69,-144,104.5,-217.7,106.5,-221

l0-0

c5.3,-9.3,12,-14,20,-14

H400000v40H845.2724

s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7

c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z

M83480h400000v40h-400000z">

​等。

    根据定义，我们有：若x

3

=

a

x^3=a

x3=a，则x

=

a

3

x=\sqrt[3]{a}

x=3a<pathd="M95,702

c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14

c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54

c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10

s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429

c69,-144,104.5,-217.7,106.5,-221

l0-0

c5.3,-9.3,12,-14,20,-14

H400000v40H845.2724

s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7

c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z

M83480h400000v40h-400000z">

​。例如，3

3

=

27

⇔

3

=

27

3

3^3=27\Leftrightarrow3=\sqrt[3]{27}

33=27⇔3=327<pathd="M95,702

c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14

c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54

c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10

s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429

c69,-144,104.5,-217.7,106.5,-221

l0-0

c5.3,-9.3,12,-14,20,-14

H400000v40H845.2724

s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7

c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z

M83480h400000v40h-400000z">

​；(

−

2

)

3

=

−

8

⇔

−

2

=

−

8

3

(-2)^3=-8\Leftrightarrow-2=\sqrt[3]{-8}

(−2)3=−8⇔−2=3−8<pathd="M95,702

c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14

c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54

c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10

s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429

c69,-144,104.5,-217.7,106.5,-221

l0-0

c5.3,-9.3,12,-14,20,-14

H400000v40H845.2724

s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7

c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z

M83480h400000v40h-400000z">

​。

    介绍“开立方”：求一个数a

a

a的立方根的运算，叫做开立方。开立方与立方互为逆运算。

  3.初步应用（概念辨析）：

    完成一组快速口答练习，巩固概念与符号：

    （1）填空：因为4

3

=

64

4^3=64

43=64，所以64

3

=

‾

\sqrt[3]{64}=\underline{\hspace{2cm}}

364<pathd="M95,702

c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14

c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54

c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10

s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429

c69,-144,104.5,-217.7,106.5,-221

l0-0

c5.3,-9.3,12,-14,20,-14

H400000v40H845.2724

s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7

c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z

M83480h400000v40h-400000z">

​=​。

    （2）填空：−

125

3

=

‾

\sqrt[3]{-125}=\underline{\hspace{2cm}}

3−125<pathd="M95,702

c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14

c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54

c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10

s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429

c69,-144,104.5,-217.7,106.5,-221

l0-0

c5.3,-9.3,12,-14,20,-14

H400000v40H845.2724

s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7

c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z

M83480h400000v40h-400000z">

​=​，因为(

‾

)

3

=

−

125

(\underline{\hspace{1cm}})^3=-125

(​)3=−125。

    （3）判断：27

3

\sqrt[3]{27}

327<pathd="M95,702

c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14

c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54

c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10

s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429

c69,-144,104.5,-217.7,106.5,-221

l0-0

c5.3,-9.3,12,-14,20,-14

H400000v40H845.2724

s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7

c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z

M83480h400000v40h-400000z">

​表示27的立方根。（）

    （4）判断：−

64

3

=

−

4

\sqrt[3]{-64}=-4

3−64<pathd="M95,702

c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14

c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54

c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10

s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429

c69,-144,104.5,-217.7,106.5,-221

l0-0

c5.3,-9.3,12,-14,20,-14

H400000v40H845.2724

s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7

c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z

M83480h400000v40h-400000z">

​=−4。（）

    （5）求值：0

3

=

‾

\sqrt[3]{0}=\underline{\hspace{2cm}}

30<pathd="M95,702

c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14

c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54

c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10

s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429

c69,-144,104.5,-217.7,106.5,-221

l0-0

c5.3,-9.3,12,-14,20,-14

H400000v40H845.2724

s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7

c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z

M83480h400000v40h-400000z">

​=​；1

3

=

‾

\sqrt[3]{1}=\underline{\hspace{2cm}}

31<pathd="M95,702

c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14

c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54

c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10

s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429

c69,-144,104.5,-217.7,106.5,-221

l0-0

c5.3,-9.3,12,-14,20,-14

H400000v40H845.2724

s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7

c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z

M83480h400000v40h-400000z">

​=​；−

1

3

=

‾

\sqrt[3]{-1}=\underline{\hspace{2cm}}

3−1<pathd="M95,702

c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14

c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54

c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10

s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429

c69,-144,104.5,-217.7,106.5,-221

l0-0

c5.3,-9.3,12,-14,20,-14

H400000v40H845.2724

s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7

c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z

M83480h400000v40h-400000z">

​=​。

  设计意图：充分利用学生已有的平方根认知结构，引导他们通过类比自主生成立方根的定义和符号表示，实现知识的正向迁移。强调根指数“3”不可省略，是规范数学表达的重要环节。通过即时辨析练习，强化学生对概念和符号对应关系的理解。

（三）合作探究，归纳性质（预计用时：12分钟）

  1.探究活动：求下列各数的立方根，并观察、归纳规律。

    教师呈现探究表格（或让学生分组计算并记录）：

    |被开方数a

a

a|计算过程|立方根a

3

\sqrt[3]{a}

3a<pathd="M95,702

c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14

c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54

c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10

s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429

c69,-144,104.5,-217.7,106.5,-221

l0-0

c5.3,-9.3,12,-14,20,-14

H400000v40H845.2724

s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7

c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z

M83480h400000v40h-400000z">

​|观察（正/负/零）|

    |:---|:---|:---|:---|

    |8|2

3

=

8

2^3=8

23=8|2|正数|

    |-8|(

−

2

)

3

=

−

8

(-2)^3=-8

(−2)3=−8|-2|负数|

    |27||||

    |-27||||

    |1||||

    |-1||||

    |1

8

\frac{1}{8}

81​|(

1

2

)

3

=

1

8

(\frac{1}{2})^3=\frac{1}{8}

(21​)3=81​|1

2

\frac{1}{2}

21​|正数|

    |-1

8

\frac{1}{8}

81​||||

    |0|0

3

=

0

0^3=0

03=0|0|零|

  2.小组讨论与归纳：

    学生以小组为单位，完成表格计算，并围绕以下问题展开讨论：

    （1）一个正数（如8、27、1

8

\frac{1}{8}

81​）的立方根是正数还是负数？

    （2）一个负数（如-8、-27、-1

8

\frac{1}{8}

81​）的立方根是正数还是负数？

    （3）0的立方根是什么？

    （4）对比平方根的性质，立方根的性质有什么显著不同？（提示：从被开方数的取值范围和立方根的个数考虑）

  3.性质归纳与对比：

    各小组汇报讨论结果，师生共同归纳立方根的性质，并板书：

    立方根的性质：

      性质1：正数的立方根是正数。

      性质2：负数的立方根是负数。

      性质3：0的立方根是0。

      本质：任何数都有且只有一个立方根。

    引导学生与平方根性质进行对比，并共同完成对比表格：

    |运算|被开方数a

a

a的取值范围|根的个数|符号表示|主要性质|

    |:---|:---|:---|:---|:---|

    |平方根|a

≥

0

a\ge0

a≥0|两个（互为相反数）|±

a

\pm\sqrt{a}

±a<pathd="M95,702

c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14

c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54

c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10

s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429

c69,-144,104.5,-217.7,106.5,-221

l0-0

c5.3,-9.3,12,-14,20,-14

H400000v40H845.2724

s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7

c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z

M83480h400000v40h-400000z">

​|负数没有平方根；0的平方根是0；正数有两个平方根。|

    |立方根|a

a

a为任意实数|只有一个|a

3

\sqrt[3]{a}

3a<pathd="M95,702

c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14

c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54

c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10

s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429

c69,-144,104.5,-217.7,106.5,-221

l0-0

c5.3,-9.3,12,-14,20,-14

H400000v40H845.2724

s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7

c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z

M83480h400000v40h-400000z">

​|任何数都有唯一立方根；符号与被开方数相同。|

    特别强调：“互为相反数的两个数，它们的立方根也互为相反数。”即−

a

3

=

−

a

3

\sqrt[3]{-a}=-\sqrt[3]{a}

3−a<pathd="M95,702

c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14

c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54

c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10

s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429

c69,-144,104.5,-217.7,106.5,-221

l0-0

c5.3,-9.3,12,-14,20,-14

H400000v40H845.2724

s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7

c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z

M83480h400000v40h-400000z">

​=−3a<pathd="M95,702

c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14

c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54

c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10

s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429

c69,-144,104.5,-217.7,106.5,-221

l0-0

c5.3,-9.3,12,-14,20,-14

H400000v40H845.2724

s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7

c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z

M83480h400000v40h-400000z">

​。这是由立方运算的符号规律决定的，是一个非常有用的性质。

  设计意图：通过小组合作探究具体数值的立方根，让学生亲身经历从大量具体实例中观察、归纳一般规律的过程，发展归纳概括能力。与平方根的性质进行系统性对比，是突破本节课难点的关键。清晰的对比表格能帮助学生厘清两者间的联系与本质区别，尤其是打破“负数没有方根”的思维定势，深刻理解“任何实数都有唯一立方根”这一核心性质。

（四）深化理解，应用拓展（预计用时：10分钟）

  1.基础运算与应用：

    例1：求下列各式的值。

    （1）−

64

3

\sqrt[3]{-64}

3−64<pathd="M95,702

c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14

c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54

c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10

s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429

c69,-144,104.5,-217.7,106.5,-221

l0-0

c5.3,-9.3,12,-14,20,-14

H400000v40H845.2724

s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7

c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z

M83480h400000v40h-400000z">

​（2）−

1

27

3

-\sqrt[3]{\frac{1}{27}}

−3271​<pathd="M98390

l0-0

c4,-6.7,10,-10,18,-10H400000v40

H1013.1s-83.4,268,-264.1,840c-180.7,572,-277,876.3,-289,913c-4.7,4.7,-12.7,7,-24,7

s-12,0,-12,0c-1.3,-3.3,-3.7,-11.7,-7,-25c-35.3,-125.3,-106.7,-373.3,-214,-744

c-10,12,-21,25,-33,39s-32,39,-32,39c-6,-5.3,-15,-14,-27,-26s25,-30,25,-30

c26.7,-32.7,52,-63,76,-91s52,-60,52,-60s208,722,208,722

c56,-175.3,126.3,-397.3,211,-666c84.7,-268.7,153.8,-488.2,207.5,-658.5

c53.7,-170.3,84.5,-266.8,92.5,-289.5z

M100180h400000v40h-400000z">

​（3）0.216

3

\sqrt[3]{0.216}

30.216<pathd="M95,702

c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14

c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54

c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10

s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429

c69,-144,104.5,-217.7,106.5,-221

l0-0

c5.3,-9.3,12,-14,20,-14

H400000v40H845.2724

s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7

c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z

M83480h400000v40h-400000z">

​（4）−

64

125

3

\sqrt[3]{-\frac{64}{125}}

3−12564​<pathd="M98390

l0-0

c4,-6.7,10,-10,18,-10H400000v40

H1013.1s-83.4,268,-264.1,840c-180.7,572,-277,876.3,-289,913c-4.7,4.7,-12.7,7,-24,7

s-12,0,-12,0c-1.3,-3.3,-3.7,-11.7,-7,-25c-35.3,-125.3,-106.7,-373.3,-214,-744

c-10,12,-21,25,-33,39s-32,39,-32,39c-6,-5.3,-15,-14,-27,-26s25,-30,25,-30

c26.7,-32.7,52,-63,76,-91s52,-60,52,-60s208,722,208,722

c56,-175.3,126.3,-397.3,211,-666c84.7,-268.7,153.8,-488.2,207.5,-658.5

c53.7,-170.3,84.5,-266.8,92.5,-289.5z

M100180h400000v40h-400000z">

​

    教学处理：引导学生先判断符号，再求值。强调（2）中“-”号在根号外，表示求1

27

\frac{1}{27}

271​的立方根的相反数。对于（3），可将小数化为分数216

1000

=

(

6

10

)

3

\frac{216}{1000}=(\frac{6}{10})^3

1000216​=(106​)3来求解。

    例2：解方程：（1）x

3

=

512

x^3=512

x3=512；（2）(

x

−

1

)

3

=

−

0.125

(x-1)^3=-0.125

(x−1)3=−0.125。

    教学处理：引导学生将求立方根与解简单方程结合。强调解形如x

3

=

a

x^3=a

x3=a的方程，直接开立方得x

=

a

3

x=\sqrt[3]{a}

x=3a<pathd="M95,702

c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14

c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54

c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10

s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429

c69,-144,104.5,-217.7,106.5,-221

l0-0

c5.3,-9.3,12,-14,20,-14

H400000v40H845.2724

s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7

c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z

M83480h400000v40h-400000z">

​。对于（2），先化为(

x

−

1

)

3

=

(

−

1

2

)

3

(x-1)^3=(-\frac{1}{2})^3

(x−1)3=(−21​)3，再利用立方根定义求解。

  2.估算与计算器使用：

    探究：10

3

\sqrt[3]{10}

310<pathd="M95,702

c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14

c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54

c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10

s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429

c69,-144,104.5,-217.7,106.5,-221

l0-0

c5.3,-9.3,12,-14,20,-14

H400000v40H845.2724

s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7

c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z

M83480h400000v40h-400000z">

​是多少？它是一个整数吗？它在哪两个连续整数之间？

    引导学生思考：因为2

3

=

8

2^3=8

23=8，3

3

=

27

3^3=27

33=27，而8

<

10

<

27

8<10<27

8<10<27，所以2

<

10

3

<

3

2<\sqrt[3]{10}<3

2<310<pathd="M95,702

c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14

c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54

c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10

s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429

c69,-144,104.5,-217.7,106.5,-221

l0-0

c5.3,-9.3,12,-14,20,-14

H400000v40H845.2724

s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7

c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z

M83480h400000v40h-400000z">

​<3。

    提问：如果需要更精确的值怎么办？介绍利用计算器求立方根的方法（不同计算器按键可能不同，常见为

3

\boxed{\sqrt[3]{}}

3<pathd="M95,702

c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14

c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54

c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10

s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429

c69,-144,104.5,-217.7,106.5,-221

l0-0

c5.3,-9.3,12,-14,20,-14

H400000v40H845.2724

s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7

c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z

M83480h400000v40h-400000z">

​​键或S

H

I

F

T

\boxed{SHIFT}

SHIFT​+x

3

\boxed{x^3}

x3​键）。让学生动手计算10

3

\sqrt[3]{10}

310<pathd="M95,702

c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14

c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54

c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10

s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429

c69,-144,104.5,-217.7,106.5,-221

l0-0

c5.3,-9.3,12,-14,20,-14

H400000v40H845.2724

s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7

c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z

M83480h400000v40h-400000z">

​的近似值（保留四位小数）。

  3.简单实际应用（跨学科联系）：

    问题：科学家发现一种新型材料的密度为5.2

g/cm

3

5.2\,\{g/cm}^3

5.2g/cm3。现有一块该材料的正方体样本，测得质量为168.48

168.48

168.48克。请估算这个正方体样本的棱长大约是多少厘米？（密度=质量/体积；正方体体积=棱长

3

^3

3）

    引导学生分析：先求体积V

=

m

/

ρ

=

168.48

/

5.2

=

32.4

(

cm

3

)

V=m/\rho=168.48/5.2=32.4\,(\{cm}^3)

V=m/ρ=168.48/5.2=32.4(cm3)。设棱长为x

x

xcm，则x

3

=

32.4

x^3=32.4

x3=32.4，所以x

=

32.4

3

x=\sqrt[3]{32.4}

x=332.4<pathd="M95,702

c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14

c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54

c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10

s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429

c69,-144,104.5,-217.7,106.5,-221

l0-0

c5.3,-9.3,12,-14,20,-14

H400000v40H845.2724

s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7

c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z

M83480h400000v40h-400000z">

​。估算：3

3

=

27

3^3=27

33=27，3.2

3

=

32.768

3.2^3=32.768

3.23=32.768，故棱长略小于3.2cm。可用计算器求得更精确值。

    简要讨论：此问题涉及物理中的密度概念，体现了数学作为工具在科学研究中的应用。

  设计意图：通过例题巩固立方根的基本求值、符号处理和简单方程求解。引入估算和计算器使用，满足对非完全立方数处理的需求，体现方法的完整性。设计简单的实际问题，建立数学模型，让学生体会立方根的应用价值，并自然融入跨学科视角。

（五）变式训练，巩固提升（预计用时：8分钟）

  1.辨析题（强化概念与性质）：

    （1）下列说法正确的是（）。

      A.-1的立方根是-1

      B.27

3

\sqrt[3]{27}

327<pathd="M95,702

c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14

c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54

c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10

s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429

c69,-144,104.5,-217.7,106.5,-221

l0-0

c5.3,-9.3,12,-14,20,-14

H400000v40H845.2724

s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7

c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z

M83480h400000v40h-400000z">

​表示27的算术平方根

      C.1

64

\frac{1}{64}

641​的立方根是±

1

4

\pm\frac{1}{4}

±41​

      D.−

8

3

=

−

2

\sqrt[3]{-8}=-2

3−8<pathd="M95,702

c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14

c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54

c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10

s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429

c69,-144,104.5,-217.7,106.5,-221

l0-0

c5.3,-9.3,12,-14,20,-14

H400000v40H845.2724

s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7

c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z

M83480h400000v40h-400000z">

​=−2，所以-8的立方根是-2

    （2）若1

−

2

x

3

\sqrt[3]{1-2x}

31−2x<pathd="M95,702

c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14

c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54

c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10

s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429

c69,-144,104.5,-217.7,106.5,-221

l0-0

c5.3,-9.3,12,-14,20,-14

H400000v40H845.2724

s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7

c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z

M83480h400000v40h-400000z">

​与3

y

−

2

3

\sqrt[3]{3y-2}

33y−2<pathd="M95,702

c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14

c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54

c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10

s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429

c69,-144,104.5,-217.7,106.5,-221

l0-0

c5.3,-9.3,12,-14,20,-14

H400000v40H845.2724

s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7

c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z

M83480h400000v40h-400000z">

​互为相反数，求x

y

\frac{x}{y}

yx​的值。（提示：利用A

3

=

−

B

3

⇒

A

=

−

B

\sqrt[3]{A}=-\sqrt[3]{B}\RightarrowA=-B

3A<pathd="M95,702

c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14

c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54

c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10

s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429

c69,-144,104.5,-217.7,106.5,-221

l0-0

c5.3,-9.3,12,-14,20,-14

H400000v40H845.2724

s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7

c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z

M83480h400000v40h-400000z">

​=−3B<pathd="M95,702

c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14

c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,6