范文一:根式、分数指数幂
个性化教学方案
教师姓名 学生姓名 高 寒 填写时间 2011-9-26 学科 数学 年级 高 三 教材版本 新课标 阶段 第()周 观察期:? 维护期:? 上课时间 2011-9-27
第( )次课 课题名称 课时计划 根式、分数指数幂 共( )次课
1(掌握根式的概念和性质,并能熟练应用于相关计算中;
教学目标 2(理解正数的分数指幂和正数的无理指数幂的意义,会用幂的运算法则进行运算;
3(培养培养观察分析(抽象概括能力(归纳总结能力(化归转化能力( 教学重点 根式的概念和性质,分数指幂和正数的无理指数幂的意义。
难 点 用幂的运算法则进行运算。
一,根式
复习引入: n1(整数指数幂的概念
a,a,a,a?a(n,N*); ,,,,,n个a
1n,0a,(a,0,n,N*); a,1(a,0)na
2(运算性质:
mnmn,a,a,a(m,n,Z)
mnmn(a),a(m,n,Z)
nnn(ab),a,b(n,Z)
3(注意
mnm,nmnm,nm,na,aa,aa,aa,aa? 可看作 ?==
naaannn,nn,n()a,b()a,b? 可看作 ?== nbbb
教学过程:
1(根式:
?计算(可用计算器)
233?= 9 ,则3是9的平方根 ;?=,125 ,则,5是,125的立方根 ; (,5)
453.76?若=1296 ,则6是1296 的 4次方根 ;?=693.43957 ,则3.7是693.43957
的5次方根 .
edit risk management practices chapter I General provisions to improve the quality of individual credit business, optimize the ptestructure of assets, promote healthy development of individual credit business, according to the China Banking Regulatory Commimber 4, 2012 Annex: qilu Bank personal crindividual grantBusiness development, Head Office of qilu Bank personal credit risk management guidelines are issued to you, plssion guidelines on credit due diligence and other related provisions, combined with Bank practice, these guidelines are formulease carefully study and implement. Annex: qilu Bank personal credit risk management guidelines qilu Bank b o September 4ated. Personal credit business mentioned in this guide, refers to natural persons of the Bank provide foreign currency loan bus, 12 key words: sends the individual credit guidelines to inform risk management: leadership, head office departments contact piness. Especially those good credit, source of repayment of the Bank stability, guarantee plenty of personal loan needs, and sterson: Chen Tel: 0531-81679536 (total of 3) Qilu Bank's Office issued on Serengthen risk control and asset structure. Personal credit business says the parties must ensure that the borrower credit, such as true, real and have written evidence, strictly in accordance with the rules of the regulation, standards, effective against various types of internal and external fraud. Our ban and will actively take measures to prevent all forms of commercial bribery. Chapter II management principles management principles the Bank to implement the whole process
?定义:
n一般地,若 则x叫做a的n次方根 x,a(n,1,n,N*)
n叫做根式,n叫做根指数,a叫做被开方数 a
63635例如,27的3次方根表示为,-32的5次方根表示为,的3次方根表示为;,3227aa
44416的4次方根表示为,即16的4次方根有两个,一个是,另一个是-,161616,
它们绝对值相等而符号相反.
?性质:
?当n为奇数时:正数的n次方根为正数,负数的n次方根为负数
nx,a记作:
?当n为偶数时,正数的n次方根有两个(互为相反数)
nx,,a记作:
?负数没有偶次方根(
0的任何次方根为0 ?
na注:当a0时,0,表示算术根( ,,
?常用公式
根据n次方根的定义,易得到以下三组常用公式:
n35n35a,3227?当n为任意正整数时,()=a.例如,()=27,()=-32.
a(a,0),nnnnaa?当n为奇数时,=a;当n为偶数时,=|a|=. ,,a(a,0),
4324553(,2)(,3)3例如,=-2,=2;=3,=|-3|=3. 2
npmpmna,a?根式的基本性质:,(a0). ,
236(,8),,8注意,?中的a0十分重要,无此条件则公式不成立. 例如. ,
用语言叙述上面三个公式:
?非负实数a的n次方根的n次幂是它本身.
?n为奇数时,实数a的n次幂的n次方根是a本身;n为偶数时,实数a的n次幂的n
次方根是a的绝对值.
?若一个根式(算术根)的被开方数是一个非负实数的幂,那么这个根式的根指数和被开方
数的指数都乘以或者除以同一个正整数,根式的值不变.
edit risk management practices chapter I General provisions to improve the quality of individual credit business, optimize the ptestructure of assets, promote healthy development of individual credit business, according to the China Banking Regulatory Commimber 4, 2012 Annex: qilu Bank personal crindividual grantBusiness development, Head Office of qilu Bank personal credit risk management guidelines are issued to you, plssion guidelines on credit due diligence and other related provisions, combined with Bank practice, these guidelines are formulease carefully study and implement. Annex: qilu Bank personal credit risk management guidelines qilu Bank b o September 4ated. Personal credit business mentioned in this guide, refers to natural persons of the Bank provide foreign currency loan bus, 12 key words: sends the individual credit guidelines to inform risk management: leadership, head office departments contact piness. Especially those good credit, source of repayment of the Bank stability, guarantee plenty of personal loan needs, and sterson: Chen Tel: 0531-81679536 (total of 3) Qilu Bank's Office issued on Serengthen risk control and asset structure. Personal credit business says the parties must ensure that the borrower credit, such as true, real and have written evidence, strictly in accordance with the rules of the regulation, standards, effective against various types of internal and external fraud. Our ban and will actively take measures to prevent all forms of commercial bribery. Chapter II management principles management principles the Bank to implement the whole process
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2例题分析:
233例1 ?(,8)= -8 ;?(,10)= |-10| = 10 ;
4243,,,,3?(3,,)(a,b)(a,b)= || = ;?= |a- b| = a- b .
【变式】去掉‘a>b’结果如何,
例2求值:
(1)526743642;,,,,, 63(2)231.512.,,
【思路分析】(1)题需把各项被开方数变为完全平方形式,然后再利用根式运算性质(
解:
(1)526743642,,,,,
222222(3)232(2)2223(3)2222(2),,,,,,,,,,,,
222((32))(23)(22),,,,,,
|32||23||22|,,,,,,,
3223(22),,,,,,
22,
注意:此题开方后先带上绝对值,然后根据正负去掉绝对值符号。
63(2)231.512,,
3623,2323,,,,2
2366326,2323,,,, 22
23326,2323,,,,22
,236.,,
3(理解数学:
23434,,,,1,xx,1,,x,11(化简: ,,(
解:原式,(x-1)+(1-x)+(x-1),x-1(
edit risk management practices chapter I General provisions to improve the quality of individual credit business, optimize the ptestructure of assets, promote healthy development of individual credit business, according to the China Banking Regulatory Commimber 4, 2012 Annex: qilu Bank personal crindividual grantBusiness development, Head Office of qilu Bank personal credit risk management guidelines are issued to you, pl ssion guidelines on credit due diligence and other related provisions, combined with Bank practice, these guidelines are formulease carefully study and implement. Annex: qilu Bank personal credit risk management guidelines qilu Bank b o September 4 花山居室九弓塘数学会社ated. Personal credit business mentioned in this guide, refers to natural persons of the Bank provide foreign currency loan bus, 12 key words: sends the individual credit guidelines to inform risk management: leadership, head office departments contact pyiyang20120830 iness. Especially those good credit, source of repayment of the Bank stability, guarantee plenty of personal loan needs, and sterson: Chen Tel: 0531-81679536 (total of 3) Qilu Bank's Office issued on Serengthen risk control and asset structure. Personal credit business says the parties must ensure that the borrower credit, such as true, real and have written evidence, strictly in accordance with the rules of the regulation, standards, effective against various types of internal and external fraud. Our ban and will actively take measures to prevent all forms of commercial bribery. Chapter II management principles management principles the Bank to implement the whole process
22(若 ,(5,x) 写出使等式成立的x取值范围( (x,5)(x,25)x,5
22解:?(x,5)(x,25),(x,5)(x,5), ,x,5x,5
?,, x,5,x,5(5,x),x,5
即, x,5(x,5,(5,x)),0
x,5,0?x+5,0或,, x,5,5,x
x,,5?x,,5或,, x,5,0
?x的取值范围[,5,5] (
363(计算:2,3?7,43(
【思路分析】利用换元,整体思想来解题(
36法一:设m,2,3(m,0),n,7,43(n,0)
3622,37,43,,,, ?mn(2,3)
32,3?n,,
33mn?,1,?mn,1,
?原式,mn,1(
236(2,3)2,3法二:原式,(
333(2,3)(2,3)2,32,3,(,,1( 【解后反思】换元法、整体法是两种重要的数学思想,平时注意各种方法的的运用(
【课后提升】
2(2,3)3,21(, (
224x,4x,1,2x,22x,5x,22(若,,0,则, 3 .
6322n,12n,366(,x)(,x)(,x)Nx3(有式子?,?,?,?,(n?,x?R)(
其中恒有意义的是 ? ? (
edit risk management practices chapter I General provisions to improve the quality of individual credit business, optimize the ptestructure of assets, promote healthy development of individual credit business, according to the China Banking Regulatory Commimber 4, 2012 Annex: qilu Bank personal crindividual grantBusiness development, Head Office of qilu Bank personal credit risk management guidelines are issued to you, plssion guidelines on credit due diligence and other related provisions, combined with Bank practice, these guidelines are formulease carefully study and implement. Annex: qilu Bank personal credit risk management guidelines qilu Bank b o September 4ated. Personal credit business mentioned in this guide, refers to natural persons of the Bank provide foreign currency loan bus, 12 key words: sends the individual credit guidelines to inform risk management: leadership, head office departments contact piness. Especially those good credit, source of repayment of the Bank stability, guarantee plenty of personal loan needs, and sterson: Chen Tel: 0531-81679536 (total of 3) Qilu Bank's Office issued on Serengthen risk control and asset structure. Personal credit business says the parties must ensure that the borrower credit, such as true, real and have written evidence, strictly in accordance with the rules of the regulation, standards, effective against various types of internal and external fraud. Our ban and will actively take measures to prevent all forms of commercial bribery. Chapter II management principles management principles the Bank to implement the whole process
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a(,2,a,0)2224( 当,2时,a,(a,3),(a,3),, ( a3a(0,a,2)
2264m,1,1,m,4m5(计算( 1,m
21,m,0
2解:由题知:{ ?m,,1( m,1,0
1,m,0
0,0,4?原式,,,2( 1,(,1)
二,分数指数幂
复习引入:
1(整数指数幂的运算性质:
mnmn,a,a,a(m,n,Z)
mnmn(a),a(m,n,Z)
nnn(ab),a,b(n,Z)
2(根式的运算性质:
nna?当n为任意正整数时,()=a.
a(a,0),nnnnaa?当n为奇数时,=a;当n为偶数时,=|a|=. ,,a(a,0),
npmpmna,a?根式的基本性质:,(a0) ,
用语言叙述上面三个公式:
?非负实数a的n次方根的n次幂是它本身.
?n为奇数时,实数a的n次幂的n次方根是a本身;n为偶数时,实数a的n次幂的n
次方根是a的绝对值.
?若一个根式(算术根)的被开方数是一个非负实数的幂,那么这个根式的根指数和被开方
数的指数都乘以或者除以同一个正整数,根式的值不变. 3(引例:当a,0时
10121025212434535533?? a,(a),a,aa,(a),a,a
2211332323322a,(a),aa,(a),a? ?
上述推导过程主要利用了根式的运算性质,例子?、?、?用到了推广的整数指数幂运
算性质(2).因此,我们可以得出正分数指数幂的意义. edit risk management practices chapter I General provisions to improve the quality of individual credit business, optimize the ptestructure of assets, promote healthy development of individual credit business, according to the China Banking Regulatory Commimber 4, 2012 Annex: qilu Bank personal crindividual grantBusiness development, Head Office of qilu Bank personal credit risk management guidelines are issued to you, pl ssion guidelines on credit due diligence and other related provisions, combined with Bank practice, these guidelines are formulease carefully study and implement. Annex: qilu Bank personal credit risk management guidelines qilu Bank b o September 4 花山居室九弓塘数学会社ated. Personal credit business mentioned in this guide, refers to natural persons of the Bank provide foreign currency loan bus, 12 key words: sends the individual credit guidelines to inform risk management: leadership, head office departments contact pyiyang20120830 iness. Especially those good credit, source of repayment of the Bank stability, guarantee plenty of personal loan needs, and sterson: Chen Tel: 0531-81679536 (total of 3) Qilu Bank's Office issued on Serengthen risk control and asset structure. Personal credit business says the parties must ensure that the borrower credit, such as true, real and have written evidence, strictly in accordance with the rules of the regulation, standards, effective against various types of internal and external fraud. Our ban and will actively take measures to prevent all forms of commercial bribery. Chapter II management principles management principles the Bank to implement the whole process
建构数学:
1.正数的正分数指数幂的意义
m*nmn (a,0,m,n?N,且n,1) a,a
要注意两点:一是分数指数幂是根式的另一种表示形式;二是根式与分数指数幂可以进行互化.
另外,我们还要对正数的负分数指数幂和0的分数指数幂作如下规定. 2.规定:
m,1*n(1) (a,0,m,n?N,且n,1); a,m
na
(2)0的正分数指数幂等于0;
(3)0的负分数指数幂无意义.
规定了分数指数幂的意义以后,指数的概念就从整数推广到有理数指数.当a,0时,整数指数幂的运算性质,对于有理指数幂也同样适用.即对于任意有理数r,s,均有下面的运算性质.
3.有理指数幂的运算性质:
,mnmna,a,a(m,n,Q)
mnmn(a),a(m,n,Q)
nnn(ab),a,b(n,Q)
pa0,P是一个无理数,则说明:若a,表示一个确定的实数,上述有理指数幂的运算性质,对于无理数指数幂都适用,有关概念和证明在本书从略.
应用数学:
213,,1163,324例1求值:8,100,(),(). 481
2223,32333解: 8,(2),2,2,4
111,,2,(,)12,1222100,(10),10,10,10
1,3,2,3(,2),(,3)6(),(2),2,2,64 4
33,4,(,)162227,344,,,()()()81338
例2 用分数指数幂的形式表示下列各式:
2332a,a,a,a,aa (式中a,0)
edit risk management practices chapter I General provisions to improve the quality of individual credit business, optimize the ptestructure of assets, promote healthy development of individual credit business, according to the China Banking Regulatory Commimber 4, 2012 Annex: qilu Bank personal crindividual grantBusiness development, Head Office of qilu Bank personal credit risk management guidelines are issued to you, plssion guidelines on credit due diligence and other related provisions, combined with Bank practice, these guidelines are formulease carefully study and implement. Annex: qilu Bank personal credit risk management guidelines qilu Bank b o September 4ated. Personal credit business mentioned in this guide, refers to natural persons of the Bank provide foreign currency loan bus, 12 key words: sends the individual credit guidelines to inform risk management: leadership, head office departments contact piness. Especially those good credit, source of repayment of the Bank stability, guarantee plenty of personal loan needs, and sterson: Chen Tel: 0531-81679536 (total of 3) Qilu Bank's Office issued on Serengthen risk control and asset structure. Personal credit business says the parties must ensure that the borrower credit, such as true, real and have written evidence, strictly in accordance with the rules of the regulation, standards, effective against various types of internal and external fraud. Our ban and will actively take measures to prevent all forms of commercial bribery. Chapter II management principles management principles the Bank to implement the whole process
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1152,22222解: a,a,a,a,a,a
2211,33233333a,a,a,a,a,a 11313
22224aa,(a,a),(a),a
例3计算下列各式(式中字母都是正数):
211511
336622(1)(2ab)(,6ab),(,3ab); 31884(2)(mn).
【思路分析】(1)题可以仿照单项式乘除法进行,首先是系数相乘除,然后是同底数幂相乘
除,并且要注意符号
(2)题按积的乘方计算,而按幂的乘方计算,等熟练后可简化计算步骤 解:
31821151184(2)(mn)336622(1)(2ab)(,6ab),(,3ab)31,8321111584,,,,,(m)(n)326236 ,[2,(,6),(,3)]ab,33,m,n0,4ab2m,,4a3n
例4计算下列各式:
2a(1)(a,0);23 aa
34(2)(25,125),5
【思路分析】(1)题把根式化成分数指数幂的形式,再计算(2)题先把根式化成分数指数幂
的最简形式,然后计算
234解: a(2)(25,125),5(1) 23231a,a324,(5,5),5 2a2131, 133424,5,5,5,522 a,a213112,, ,,23424,5,523,a 555124,5,56 ,a
5124 65,5,55.,a
理解数学:
edit risk management practices chapter I General provisions to improve the quality of individual credit business, optimize the ptestructure of assets, promote healthy development of individual credit business, according to the China Banking Regulatory Commimber 4, 2012 Annex: qilu Bank personal crindividual grantBusiness development, Head Office of qilu Bank personal credit risk management guidelines are issued to you, pl ssion guidelines on credit due diligence and other related provisions, combined with Bank practice, these guidelines are formulease carefully study and implement. Annex: qilu Bank personal credit risk management guidelines qilu Bank b o September 4 花山居室九弓塘数学会社ated. Personal credit business mentioned in this guide, refers to natural persons of the Bank provide foreign currency loan bus, 12 key words: sends the individual credit guidelines to inform risk management: leadership, head office departments contact pyiyang20120830 iness. Especially those good credit, source of repayment of the Bank stability, guarantee plenty of personal loan needs, and sterson: Chen Tel: 0531-81679536 (total of 3) Qilu Bank's Office issued on Serengthen risk control and asset structure. Personal credit business says the parties must ensure that the borrower credit, such as true, real and have written evidence, strictly in accordance with the rules of the regulation, standards, effective against various types of internal and external fraud. Our ban and will actively take measures to prevent all forms of commercial bribery. Chapter II management principles management principles the Bank to implement the whole process
1323,,55341.用根式的形式表示下列各式(,,,): ( a,a,a,a
155解:; a,a
3344a,a
3,1,535 a,a,53a
2,1,323a,a,32a
2.用分数指数幂表示下列各式:
322334(a,b)(m,n)(1) ; (,)(,,,,,) ;(,); x
3m465(m,n)p,q (,)(,,,); (5)(,,,); (6)(
m
2233x,x解:(1) ;
3344(2) ; (a,b),(a,b)
2233(3) ; (m,n),(m,n)
1,42(4) ,(,,,); (m,n),(m,n)
1655656532222(5) ; p,q(p,0),(p,q),pq,p,q
153,m322 (6) ( ,m,m,m
m
【课后提升】
2,710370.520,3(2)0.1(2)3,,,,,1(计算:( 92748
21,251643732()()3,,,,,解:原式 2927480.1
5937,,100,,3,,100 ( 31648
33,1122,a,a22a,a,32(已知:,求( 11,22a,a
edit risk management practices chapter I General provisions to improve the quality of individual credit business, optimize the ptestructure of assets, promote healthy development of individual credit business, according to the China Banking Regulatory Commimber 4, 2012 Annex: qilu Bank personal crindividual grantBusiness development, Head Office of qilu Bank personal credit risk management guidelines are issued to you, plssion guidelines on credit due diligence and other related provisions, combined with Bank practice, these guidelines are formulease carefully study and implement. Annex: qilu Bank personal credit risk management guidelines qilu Bank b o September 4ated. Personal credit business mentioned in this guide, refers to natural persons of the Bank provide foreign currency loan bus, 12 key words: sends the individual credit guidelines to inform risk management: leadership, head office departments contact piness. Especially those good credit, source of repayment of the Bank stability, guarantee plenty of personal loan needs, and sterson: Chen Tel: 0531-81679536 (total of 3) Qilu Bank's Office issued on Serengthen risk control and asset structure. Personal credit business says the parties must ensure that the borrower credit, such as true, real and have written evidence, strictly in accordance with the rules of the regulation, standards, effective against various types of internal and external fraud. Our ban and will actively take measures to prevent all forms of commercial bribery. Chapter II management principles management principles the Bank to implement the whole process
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3311,,332222解:?, a,a,(a),(a)
1111,,,12222(a,a)(a,a,a,a),1?原式( ,,a,a,1,811,22a,a
11111,,,,,32168423(化简 s,(1,2)(1,2)(1,2)(1,2)(1,2)
1,1,132,(1,2),s1,21,132解:原式,,,(1,2)( 11,,232321,21,2
2x,2xy,3y4(若x,0,y,0且,求值. x(x,y),3y(x,5y)
x,xy,y
22解:由题知,, (x),2xy,15(y),0
?, x,3y,0或x,5y,0
又?x,0,y,0 , ?, x,5y
?x,25y, ?原式,3(
11,1,2nnn5(已知:,求的值( x,(5,5),n,N(x,1,x)2
2211,,1122nnnn解:由已知,1+x,(5,2,5),(5,5), 44
11111,,11nnnnnnn,[(5,5),(5,5)],(5),5?原式( 22
edit risk management practices chapter I General provisions to improve the quality of individual credit business, optimize the ptestructure of assets, promote healthy development of individual credit business, according to the China Banking Regulatory Commimber 4, 2012 Annex: qilu Bank personal crindividual grantBusiness development, Head Office of qilu Bank personal credit risk management guidelines are issued to you, pl ssion guidelines on credit due diligence and other related provisions, combined with Bank practice, these guidelines are formulease carefully study and implement. Annex: qilu Bank personal credit risk management guidelines qilu Bank b o September 4 花山居室九弓塘数学会社ated. Personal credit business mentioned in this guide, refers to natural persons of the Bank provide foreign currency loan bus, 12 key words: sends the individual credit guidelines to inform risk management: leadership, head office departments contact pyiyang20120830 iness. Especially those good credit, source of repayment of the Bank stability, guarantee plenty of personal loan needs, and sterson: Chen Tel: 0531-81679536 (total of 3) Qilu Bank's Office issued on Serengthen risk control and asset structure. Personal credit business says the parties must ensure that the borrower credit, such as true, real and have written evidence, strictly in accordance with the rules of the regulation, standards, effective against various types of internal and external fraud. Our ban and will actively take measures to prevent all forms of commercial bribery. Chapter II management principles management principles the Bank to implement the whole process
课后作业
(A组)------基础训练 1. 以下说法正确的是 ( )
A.正数的次方根是一个正数 B. 负数的次方根是一个负数 nn
*nC. 0的次方根是0 D. 的次方根是一个正数 nana,,以上n,1,n,N
2.已知,则下列各式成立的是 ( ) a,b,R
86222286,,a,b,a,b,,a,b,a,bA. B.
10444410a,b,,a,b,a,babC.-= D.
255,,,,a,b,b,a3. 的值是 ( )
,,,,2b,a2b,aA. 0 B. C. 0 或 D. 不确定
edit risk management practices chapter I General provisions to improve the quality of individual credit business, optimize the ptestructure of assets, promote healthy development of individual credit business, according to the China Banking Regulatory Commimber 4, 2012 Annex: qilu Bank personal crindividual grantBusiness development, Head Office of qilu Bank personal credit risk management guidelines are issued to you, plssion guidelines on credit due diligence and other related provisions, combined with Bank practice, these guidelines are formulease carefully study and implement. Annex: qilu Bank personal credit risk management guidelines qilu Bank b o September 4ated. Personal credit business mentioned in this guide, refers to natural persons of the Bank provide foreign currency loan bus, 12 key words: sends the individual credit guidelines to inform risk management: leadership, head office departments contact piness. Especially those good credit, source of repayment of the Bank stability, guarantee plenty of personal loan needs, and sterson: Chen Tel: 0531-81679536 (total of 3) Qilu Bank's Office issued on Serengthen risk control and asset structure. Personal credit business says the parties must ensure that the borrower credit, such as true, real and have written evidence, strictly in accordance with the rules of the regulation, standards, effective against various types of internal and external fraud. Our ban and will actively take measures to prevent all forms of commercial bribery. Chapter II management principles management principles the Bank to implement the whole process
花山居室
323,,4a,4a,1,1,2a4.若,则的取值范围是 ( ) a
1111A. B. C. D. ,,a,Ra,a,2222
6155.是 , 是 。 x,2004,xx,,2004,x
nn,2215454*64,,,,,2,a,,a,,36.在式子中,其中没有意义的是 ,,a,R,n,N
3,481,,7.的值是 ( ) ,,625,,
27125925A、 B、 C、 D、 125272527
4,3,,,8a3,,8.化简的结果是 ( ) (a,0,b,0)3,,27b,,
16162a2a,,A、 B、 C、 D、 44443b3b81ab81ab
35xyxy9用分数指数幂表示是
92,3252310. = (8),,,10,10
2,13,103,,,2,,,,,105,22,3,,,30.00211.++ ,,8,,
ba10,22a,b100,5012. 已知,,求的值
(B组)-----能力拓展
2233,,,,,,1,a1,aa,1526743642,,,,,1化简:(1) (2)++
edit risk management practices chapter I General provisions to improve the quality of individual credit business, optimize the ptestructure of assets, promote healthy development of individual credit business, according to the China Banking Regulatory Commimber 4, 2012 Annex: qilu Bank personal crindividual grantBusiness development, Head Office of qilu Bank personal credit risk management guidelines are issued to you, pl ssion guidelines on credit due diligence and other related provisions, combined with Bank practice, these guidelines are formulease carefully study and implement. Annex: qilu Bank personal credit risk management guidelines qilu Bank b o September 4 花山居室九弓塘数学会社ated. Personal credit business mentioned in this guide, refers to natural persons of the Bank provide foreign currency loan bus, 12 key words: sends the individual credit guidelines to inform risk management: leadership, head office departments contact pyiyang20120830 iness. Especially those good credit, source of repayment of the Bank stability, guarantee plenty of personal loan needs, and sterson: Chen Tel: 0531-81679536 (total of 3) Qilu Bank's Office issued on Serengthen risk control and asset structure. Personal credit business says the parties must ensure that the borrower credit, such as true, real and have written evidence, strictly in accordance with the rules of the regulation, standards, effective against various types of internal and external fraud. Our ban and will actively take measures to prevent all forms of commercial bribery. Chapter II management principles management principles the Bank to implement the whole process
3,23,222222.已知, 求的值 x,,y,3x,10xy,3y
3,23,2
,13.已知,求下列各式的值: x,x,3
11,2,22,222 (1)(2)(3) x,xx,xx,x
edit risk management practices chapter I General provisions to improve the quality of individual credit business, optimize the ptestructure of assets, promote healthy development of individual credit business, according to the China Banking Regulatory Commimber 4, 2012 Annex: qilu Bank personal crindividual grantBusiness development, Head Office of qilu Bank personal credit risk management guidelines are issued to you, plssion guidelines on credit due diligence and other related provisions, combined with Bank practice, these guidelines are formulease carefully study and implement. Annex: qilu Bank personal credit risk management guidelines qilu Bank b o September 4ated. Personal credit business mentioned in this guide, refers to natural persons of the Bank provide foreign currency loan bus, 12 key words: sends the individual credit guidelines to inform risk management: leadership, head office departments contact piness. Especially those good credit, source of repayment of the Bank stability, guarantee plenty of personal loan needs, and sterson: Chen Tel: 0531-81679536 (total of 3) Qilu Bank's Office issued on Serengthen risk control and asset structure. Personal credit business says the parties must ensure that the borrower credit, such as true, real and have written evidence, strictly in accordance with the rules of the regulation, standards, effective against various types of internal and external fraud. Our ban and will actively take measures to prevent all forms of commercial bribery. Chapter II management principles management principles the Bank to implement the whole process
范文二:根式和分数指数幂的学习指导
四川 毛仕理
由于分数指数幂的概念是借助n 次方根给出的,而n 次根式又是学生刚刚接触到的概念,也是比较陌生的.以此为基础去学习认识新知识自然是比较困难的.且n 次方根,分数指数幂的定义都是用抽象字母和符号的形式给出的,学生在接受理解上也是比较困难的.基于以上原因,根式和分数指数幂的概念成为本节应突破的难点.
学习本节主要目的是将指数从整数指数推广到有理数指数,为指数函数的研究作好准备.且有理指数幂具备的运算性质还可以推广到无理指数幂,也就是说在运算上已将指数范围推广到了实数范围,为对数运算的出现作好了准备,而使这些成为可能的就是分数指数幂的引入.根式和分数指数幂的互化既是重点也是难点.根据不同需要,灵活进行互化是解决有些问题的有效途径.根式的运算,一般先化为分数指数幂后用分数指数幂的运算性质进行运算比较方便.
70. 510-337-20
例1 计算: (1)(2) +0. 1+(2) -3π+;
92748
(2)(0. 0081)
1
-4
2
70-13-3-1-0. 25
-[3?() ]?[81+(3) ]2-10?0. 0273.
88
1
2
11
251-264-37
解析:(1)原式=() 2+() +() 3-3+.
9102748
100+ =+
5
3
937
-3+=10.0 1648
1
1
1
13?34-433?(-) 1101-14?(-4)
(2)原式=[() ]-(3?1) ?[3+() 3]--10?0. 33=-?
102233
1-3=0
点评:一般地,遇到小数应化成分数;遇到指数是负数,可以对调底数的分子和分母,将负指数化为正指数.
例2 化简下列各式: (1)(x -1+x +x 0)(x (2)
1
-2
x ) ;
12
x -2+y -2x +y
-23
-23
-
x -2-y -2x -y
-23
-23
.
解析:注意题中各式的结构特点.
(1)原式=(x
1-
23
) -(x ) =x -x . ) +(y
23
-23233
1233232
(2)原式=
(x
-
233-
)
-
(x
-
233-23
) -(y
-23
233
)
x +y
=(x
23
-
x -y
-
) 2-x
-23
23
y +(y
3
232-3
) 2-[(x
23
) 2+x
2-3
y +(y
232-3
) 2]
=-2(xy ) =-2
xy . xy
点评:解题时要从总体上把握代数式的结构特点.
例3 若a 0,a -b 0时
①a
10
=(a ) =a =a
3
23
3
23
252
105
②a
12
=(a ) =a =a
12
2
12
434
123
③a =
2
(a ) =a ④a =(a ) =a 从而得到下面的根式与分数指数幂的转换公式
a =a (a >0, m , n ∈N , 且n >1) ,右边a 称为分式指数幂
m
*
m n m n
规定:(1)a
-
m n
=
1a
m n
(a >0,m , n ∈N , 且n >*
(2) 0的正分数指数幂等于0. (3) 0的负分数指数幂无意义.
规定了分数指数幂的意义以后,指数的概念就从整数推广到有理数指数. 当a >0时,整数指数幂的运算性质,对于有理指数幂也同样适用,即
a m ?a n =a m +n (m , n ∈Q ) (a m ) n =a mn (m , n ∈Q ) (ab ) n =a n ?b n (n ∈Q )
若a >0,P 是一个无理数,则a 表示一个确定的实数,上述有理指数幂的运算性质,对于无理数指数幂
p
都适用(证明在这里从略),即对于实数指数幂适用,有如下公式:
a m ?a n =a m +n (m , n ∈R ) (a m ) n =a mn (m , n ∈R ) (ab ) n =a n ?b n (n ∈R )
例题:1、求值:(1) 8,
23
(2) 100,
-
12
1(3) -3,
416-
(4)() 4.
81
3
2、计算下列各式(式中字母都是正数)
(1) (2a b )(-6a b ) ÷(-3a b ) (2) (m n ) 23
12
12
13
16
56
14
388
课堂练习
1、用分数指数幂的形式表示下列各式:
(1)a 2
a 3
2、计算下列各式: (1
2a >0) (2
)
3、用分数指数幂表示下列分式(其中各式字母均为正数)
(1)a ?a (2)a a a (4)(a +b ) 3
(5)ab 2+a 2b
11114、化简:(x 2-y 2) ÷(x 4-y 4
)
5、已知x+x-1
=3,求下列各式的值:
(1)1
1
x 2+x -2 (2)3
3
x 2+x -2
(a -b ) 2
6)(a 3+b 3) 2(3)(
课后练习 一、填空题 1
2
12
2、?1-?=
????3、若x +x
1
2
-12
2
=5,则x +1的值是
x
(4、已知10=3,10=2,则105、(1) 若x
-13
m n
3m -n 2
的值为 .
3
14
=,则x = ; (2) 若2x -1=15,则x 2
34
二、解答题
-
1、计算(1) ?(-2)
?+(0.002)2-10
??
4
1
2
)
-1
(2)
1
7??3-3?-0.75
(3) 0.064- -?+(-2)?+16+-2
(4) ??8??
-1
3
4
-1
2、若a +a =3,求下列各式的值: (1)a +a
12
1-2
; (2)a +a ; (3)a +a
2-2
323-2
; (4)
a -a a -a
12
32
--
3212
范文四:根式和分数指数幂
指数与指数幂的运算(第1课时)
教学目的:
(1)掌握根式的概念;
(2)规定分数指数幂的意义;
(3)学会根式与分数指数幂之间的相互转化;
教学重点:分数指数幂的意义,根式与分数指数幂之间的相互转化. 教学难点:根式的概念,根式与分数指数幂之间的相互转化. 教学方式:多媒体教学
教学过程:
引入课题
问题1(GDP问题):据国务院发展研究中心2000年发表的《未
来20年我国发展前景分析》判断,未来20年,我国GDP年
平均增长率可望达到7.3% ,那么,2001,2020年,各年的
GDP可望为2000年的多少倍,
(17.3%),1年后,我国的GDP可望为2000年的 倍;
2(17.3%),2年后,我国的GDP可望为2000年的 倍;
33年后,我国的GDP可望为2000年的 倍; (17.3%),
4(17.3%),4年后,我国的GDP可望为2000年的 倍;…
设x年后我国的GDP为2000年的y倍,那么
xxy,,,(17.3%)1.073 (x?Nx?20) ,+
问题2:当生物死亡后,它机体内原有的碳14会按确定
的规律衰减,大约每经过5730年衰减为原来的一半。 根据此
规律,人们获得了生物体内碳14含量P与死亡年数t之间的
关系
,,t57301P,
,,
考古学家根据(*)式可以知道,生物死亡t年后,体内,,
2
的碳14含量P的值。
当生物体死亡了5730,2× 5730, 3× 5730 ,6000,
10000,100000,???年后,它体内碳14的含量P分别为:
60001000010000023573057305730111111,,,,,,,,,, ,,,,,,,,,,,,,,,222222,,,,,,,,,,
23111,,,,,,在问题2中我们已经知道 …是正 ,,,,222,,,,
111整数指数幂它们的值分别为 那么 ,,,
248600010000100000 573057305730111,,,,,,,,,,,,,,的意义是什么呢, 222,,,,,,
这正是我们将要学习的知识——根式
(师)在具体学习之前,我们先复习一些相关的知识。
第一问:什么是平方根?
2xxa,(生) 若,则叫做a的平方根。
(师) 什么是立方根,
3xxa,(生) 若,则叫做a的立方根。
(师)好,非常的不错,看来同学们对初中的知识掌握得还是比较的
牢固.那我们再一起回忆一下下面的知识呢?
一个数的平方根有几个,立方根呢,
,2(生) 正实数的平方根有两个,它们互为相反数,如4的平方根为,负数没有平方根,一个数的立方根只有一个,如―8的立方根为―2;零的平方根、立方根均为零。
(师)现在,请大家观察下列式子
42,16(1)
53,243(2)
6,,,2,64(3)
问:式子中,和,,,,和,,,,,,和,,是什么关系, 归纳得:,是,,的四次方根,,是,,,的五次方根,,,是,,的六次方根。
,(师)如果我现在给出,,,(a>0),那么5与a是什么样的关系呢? (生)5是a的n次方根。
,,(师)那么,如果我们现在将,,,中的5换成x,也即是,,,,那么此时x与a是什么样的关系呢,
(生)x是a的n次方根。
(师)好,这就是我们这节课所要学习的内容——n次方根。类比平方根和立方根的定义,请同学们尝试给出n次方根的定义。
,(生)如果,,,,那么x就是a的n次方根。
(师)好,非常的不错。n次方根的定义如下:
nxa, 一般地,如果,则x叫做a的n次方根,其中n ,1,且
n?,,。
好,现在请同学们根据方根的定义填空。
填空:
(1)25的平方根等于_________________ (2)27的立方根等于_________________ (3)-32的五次方根等于_______________ (4)16的四次方根等于______________ (5)a^6的三次方根等于_______________ (6)0的七次方根等于___________
在学生回答完这些题之后,给出
na定义二: 式子 叫做根式,n叫做根指数,a叫做被开方数 接着,根据填空题中的(2)、(3),得出当n是奇数时,正数的n次方根是一个正数,负数的n次方根是一个负数。
根据填空题中的(1)、(4),得出正数的n次方根有两个,他们互为相反数。
跳到下一幻灯片,对刚才观察出来的规律进行总结。
当n为奇数时,正数的n次方根为一个正数,负数的n次方根
na为一个负数。此时,a的n次方根用符号 表示
当n为偶数时,正数的n次方根有两个,这两个数互为相反数,
n即 ,负数没有偶次方根 ,,aa(0)
n00,零的n次方根为零,记为
跳到下一幻灯片,再给一个简单的小结,便于学生解后面的习题。
正数的偶次方根有两个,并且互为相反数;负数没有偶次方根;零的任何次方根为零。
(师)好,现在我们做做练习,对刚才的知识巩固一下。 注意:正数的偶次方根有正负两个让学生充分体会 【例1】写出下列数的n次方根
(1)16的四次方根;(,),,,的五次方根;(,),的六次方根
4,16,,2解:(,)
5,27(,)
63,9,,3(,)
引用实例,使学生通过类比初步了解根式的含义,
通过例子巩固学生对根式的概念的理解
(师)好,现在,请同学们看下面这个题.
,,,, 此题重点是引导学生得出,,,,,,并顺势提出问题,,,是,,
否成立及成立的条件,为下一例题作铺垫.
,,此题主要目的是得出,,,成立的条件及不同情形下结果. ,
(师)现在请同学们做下面的习题。
注意:根指数为奇数的
题目较易处理,而根指
数为偶数的题目容易出
错,当n为偶数时,应
nn先写,然后再去aa,||
绝对值.
习题讲解完了之后,将幻灯片调到下一页,然后进行探究
(1)观察以下式子,并总结出规律:(a > 0)
101052522(2)2,,,2;
1243124333,33(3),,3;
121234344 4aaa,,(),a;
10510252 55aaa,,(),a
结论:当根式的被开方数的指数能被根指数整除时,根式可以表示为
分数指数幂的形式.
(2)利用(1)的规律,你能表示下列式子吗?
352932553353793 ,4;7,7;a,a;47a,a.总结:当根式的被开方数的指数不能被根指数整除时,根式也可以写
成分数指数幂的形式.
(3)你能用方根的意义解释下列式子吗?
333535544;, 4的5次方根是 4;
5553533 7的3次方根是 77;,7;
2223233 a的3次方根是 aa,;a;
9999777 a的7次方根是 aa,.a.
结果表明:方根的结果与分数指数幂是相通的。
分数指数幂只是方根的另一种写法。 综上,我们得到正数的正分数指数幂的意义。 (师)那现在根据刚才的分析,我们规定正数的分数指数幂的意义:
(1) 正数的正分数指数幂的意义:
mm*nna a=(a>0,m,n?N,n>1).。
注:m,n不能随意约分;根式和分数指数幂可以互相互化。
(可以举一个例子说明m,n不能随便约分,比如
2123663 ) (2)(2)(2)2,,,,,,,
(师)那么如果指数是负分数指数呢,又该怎样转化,我们说正数的负分数指数幂的意义与负整数指数幂的意义相仿,那么我们也有规定:
(2)正数的负分数指数幂:
n,11*m a==(a>0,m,n?N,n>1); nnmama
注:0的正分数指数幂为0
0的负分数指数幂没有意义
(师)为了巩固我们所学的知识,那我们现在就做做练习题。 1:用根式的形式表示下列各式(a>0) (学生练习)
2133,, aaaa,,,5222
2:用分数指数幂表示下列各式 (学生练习)
3232434;;; (2)(),(0)abab,,,(3)(()),()mnmn,,(4)(),()mnmn,,(1)x
小结:
1.根式定义
2.根式的性质
(1)当n是奇数时,正数的n次方根是一个正数,
负数的n次方根是一个负数.
(2)当n是偶数时,正数的n次方根有两个,它们互为相反数.
n00., 3)负数没有偶次方根, 0的任何次方根都是0. 记作
n(4)当n为奇数时, xaaR,,()
n 当n为偶数时,xaa,,,(0)
nn(1);aa,,,3.两个公式
nnaa,; ?当 n 是奇数时,
nnaa,||. 当 n 是偶数时, 4.分数指数概念
mmnn (1);aa,
m,*11n (a,0,m,n?N, n,1) a,,(2);mmnana
(3)0的正分数指数幂为0,0的负分数指数幂没有意义.
作业: 课本59页A组第1题
范文五:根式与分数指数幂
教学案例:根式与分数指数幂
厦门市集美区灌口中学 吴清平
背景
nlogNa在《基本初等函数(?)》一章中,有两个符号是学生比较不熟悉的:和,a教材中是通过实例引入并给出定义:
n,那么x叫做a的n次方根。 如果x,a
xa,N(a,0,a,1)x,logN如果,那么数x叫做以a为底N的对数,记作。 a
当我们按照书上的安排,通过大量的实例来引出并说明根式与对数的含义时,仍有不少学生不能很好地理解,在教师的特别强调下,勉强记住了这两个“奇怪”的东西,时间久了,若没有经过“脑白金”式的反复记忆,遗忘是理所当然的事了。至于理解能力较差、基础不好的学生,则只能是象在看天书了。
“老师,为什么要学习根式呢,”是啊,为什么要引入根式,又为什么要引入对数,当学生这样问我时,我便经常问自己:有什么办法可以顺利地引入根式呢,
解决策略
当我们重新回忆“2”的出现时,发现它是数系扩充的必然结果:
古希腊毕达哥拉斯学派中一个叫希帕索斯的学生在研究1与2的比例中项时,发现没有
1x一个能用整数比例写成的数可以表示它。如果设这个数为x,既然,,推导的结果即x22222。他画了一个边长为1的正方形,设对角线为x,根据勾股定理,x,2x,1,1,2可见边长为1的正方形的对角线的长度即是所要找的那个数,这个数肯定是存在的。可它是
2多少,又该怎样表示它呢,后来人们把它写成了,当然无理数的发现引发了第一次数学危机,人们发现并承认它的存在曾经付出巨大的曲折与艰辛。
22那么,“”是什么呢,相信每位高中学生都非常清楚:是一个数,它的平方等
na于2~由此,“”也是一个数,它的n次方等于a~
logNxalogNgolN更进一步,是什么呢,由知,故也是一个数(对数),a,Na,Naa
logNa的次方等于N。 a
nlogNa如此,则及便不难理解了。 a
于是我们认为,在讲授根式时,应向学生介绍数系的扩充与发展,让学生明白数系扩充的必要性以及引入数学符号的意义,这样做起码有以下几点好处:
(1)介绍数学发展的历程,让学生对实数系有一个清晰的认识,而且数学史的精彩内容可以激发学生学习的兴趣。
(2)数学符号是学习数学的一大难点,若不能引起足够的重视,则学生便常常会把符号混用,导致知识的缺陷。重视数学符号的功能,更应讲清它的来龙去脉,帮助学生在有意义的学习中轻松记忆相关内容。
logN(3)有利于学生的后继学习。是什么,i又是什么,许多符号在后面的学习中a
都会陆续出现,当学生充分理解根式的意义时,接下来的学习便不会再有困难了。
当然我们更要追问:为什么要引入正分数指数幂,负分数指数幂又是如何定义的,通过对以上问题的认真、深刻思考,我们设计了如下教学课例:
教学设计:《根式与分数指数幂》
一、教学任务分析
〖知识与技能〗
(1)了解根式的概念,方根的概念及二者的关系;
(2)理解分数指数幂的概念,掌握有理数指数幂的运算性质,并能运用性质进行计算和化简。
〖过程与方法〗
通过对实际问题的探究过程,感知应用数学解决问题的方法,理解分类讨论思想、化归与转化思想在数学中的应用。
〖情感、态度与价值观〗
通过对数学实例的探究,感受现实生活对数学的需求,体验数学知识与现实的密切联系。
二、教学重难点
根式、分数指数幂的概念及其性质。
三、教材分析
教科书先给出了两个实际例子:国内生产总值(GDP)的增长问题,生物体内碳14的衰减问题。前一个问题,既让学生回顾了初中已学的整数指数幂,也让学生感受其中的函数模型,并且还有思想教育价值;后一个问题,让学生体会其中的函数模型的同时,激发学生
探究分数指数幂、无理指数幂的兴趣与欲望,为新知识的学习作了铺垫。
学生在初中学习了数的开平方、开立方以及二次根式的概念,又学习了正整数指数幂、零指数幂、负整数指数幂的概念,以及整数指数幂的运算法则。有了这些知识作准备,教科书通过实际问题引入分数指数幂,说明了扩张指数范围的必要性,为此先将平方根与立方根的概念扩充到n次方根,将二次根式的概念扩充到一般根式的概念,然后进一步介绍了分数指数幂及其运算性质,最后结合一个实例,通过有理指数幂逼近无理系数幂的方法介绍了无理指数幂的意义,从而将指数的取值范围扩充到了实数。
四、教学基本流程
实例 ? 根式 ? 分数指数幂 ? 无理指数幂
五、教学情景设计
第一课时 根式
1、问题情境设疑
问题1、根据国务院发展研究中心2000年发表的《未来20年我国发展前景分析》判断,未来20年,我国GDP(国内生产总值)年平均增长率可望达到7.3%,那么,在2001 ~ 2020年,各年的GDP可望为2000年的多少倍,
分析:设x年后我国的GDP为2000年的y倍,那么
xx*y,(1,7.3%),1.073(x,N,x,20)。
x设疑:正整数指数幂的含义是什么,它具有哪些运算性质, 1.073
问题2、当生物死亡后,它机体内原有的碳14会按确定的规律衰减,大约每经过5730年衰减为原来的一半,这个时间称为“半衰期”,根据此规律,人们获得了生物体内碳14
t15730P,()含量P与死亡年数t之间的关系,考古学家根据这个式子可以知道,生物死亡2
t年后,体内碳14含量P的值。
例如:
1当生物死亡了5730,2×5730,3×5730,??年后,它体内碳14的含量P分别为,2
1123,,?? ()()22
当生物死亡了6000年,10000年,100000年后,根据上式,它体内碳14的含量P分
100000600010000111573057305730()()()别为,,。 222
设疑:以上三个数的含义到底是什么呢,
2、数系的扩充
你是否看过杂技团演出中“小狗做算术”这个节目,台下观众出一道10以内的加法题,比如“2 + 5”,由演员写到黑板上,小狗看到就会“汪汪汪??”叫7声。台下观众会报以热烈的掌声,对这只狗中的“数学尖子”表示由衷的赞许,并常常惊叹和怀疑狗怎么会这么聪明,因为在一般人看来狗是不会有数量概念的。
我们开始接触数学时,便是从0、1、2、3、4、??等认识起的,并把它们称作自然数,初步有了“加”的运算:两个自然数相加,仍为自然数。但是,两个自然数相减呢,随着社会的发展,人们又发现了很多数量具有相反的意义,比如增加和减少、前进和后退、上升和下降、向东和向西,为了表示这样的量,又产生了负数。正整数、负整数和零,统称整数。
一个数连加几次,如5 + 5 + 5 + 5 + 5 + 5,每次书写都挺麻烦的,于是便引入了另一种运算“乘“,连加即为乘,“×”也只是一种记号,其初始含义是连加的意思(na = a + a + ?
n+ a)。类似地,连乘记为乘方,即。两个整数相乘后仍为整数,自然地a,a,a,?,a
n我们考虑其逆运算“除”,如2?3,它却不是整数,于是又引入了分数,它仍是一个记m号:把n分成m等分。
n 2x,a(a,0)进一步我们自然地会追问:如果,那么x是什么呢,如x = 4,由于
2 2(,2),4,?x,,2,那么x = 2时,x等于多少,我们知道存在实数x,它的平方等于2,但我们没有办法用有理数表示它,从而便有了根式的概念:用22表示,“”是什么呢,它是一个数,它的平方等于2~
nnnnaa更一般的情况,“”是什么呢,也是一个实数,它的n次方等于a,即~ (a),a
3、根式
23x,a(a,0)(1)平方根:;立方根:。 x,a
n(2)n次方根:如果,那么x叫做a的次方根。 x,a
类似于平方根与立方根的结论,我们有:
na当n为奇数时,正数的n次方根是一个正数,负数的n次方根是一个负数,记为:。
n当n为偶数时,正数的n次方根有两个,它们互为相反数,记为,a;
负数没有偶次方根,0的任何次方根都是0。
na(3)根式:,n ——根指数,a ——被开方数。
an为奇数,
,nna, 0aa,,,||a,n为偶数,,0,aa,,,
??
第二课时 分数指数幂
1、复习
(1)根式的相关概念
n(2)整数指数幂: a,a,a,?,a
mnm,nmnmnnnn*a,a,a,(a),a,(ab),ab(a,0,m,n,N,n,1)运算性质:。
3问题设疑:如何计算:2,2,
666323236分析:,然而普通学生要找到该解法并不容2,2,2,2,2,2,32
易,如何把这种运算简单化呢,
问题:能否类似于整数指数幂的运算来解决上题, 2、分数指数幂
10125102521234345544a,(a),a,aa,(a),a,a实例引入:, 问题:1、从以上两个例子你能发现什么结论,
25342、a,b,c如何表示,
m*nmna,a(a,0,m,n,N,n,1)结论:规定
m,*na,?(a,0,m,n,N,n,1)问题3、
mm0,0,a1*nn分析: a,a,,(a,0,m,n,N,n,1)mnmana
24,,1133如:5,,a,(a,0)。 34325a
规定:0的正分数指数幂等于0,0的负分数指数幂没有意义。
3、有理指数幂的运算性质:
rsrsrsrs,(1)a,a,a,(2)(a),a, rrr(3)(ab),ab(a,0,b,0,r,s,Q)
11151,3632362回到前面的问题,则有,相信学生在真正掌握2,2,2,2,2,2,32
了分数指数幂的意义及运算性质后,都能够顺利解决。
??
案例分析与反思
1、以上我们只展示了课堂教学的片断,对根式与分数指数幂的概念的教学提出自己的设想,目的是讲清知识发生、形成的过程及引入根式、分数指数幂的必要性,激发学生的学习兴趣。
2、本课例是根式的拓展与延伸,处处注意与初中所学知识的类比,如根式与平方根、立方根的类比,分数指数幂与整数指数幂的类比,使知识的形成水到渠成,学生在原有认知的基础上进行有益的推广,其学习是主动的、积极的,知识形成是自然的,没有强加之嫌,在教师的引导下,学生始终处于“我要学”的状态,并自觉去探究新的教学任务。
3、为什么要引入根式,又为什么要学习分数指数幂,教材中并没有详细说明,只是用“数学上的规定”来阐述。若我们不能及时引导学生追问“为什么”,则只能是用条条框框的割裂方式肢解知识的结果,把孩子活生生的思考也肢解了。几经如此,学生宝贵的思维火花便将熄灭,而走上这样一条路:不再思考,刻板记忆,不求甚解,渐渐地,思维的心灵变得麻木了。
因此,在教学过程中,对任何细节,都应鼓励学生追根溯源,凡事都去问为什么,寻找它与其他事物之间的联系,使它逐渐成为学生的一种根深蒂固的习惯。这些知识本身或许并不重要,但形成一种有如“水银泻地无孔不入”的思想方法,却是智力素质提高的一个方面。
4、本节以两个具体事例:国内生产总值(GDP)的增长问题,生物体内碳14的衰减问题引入,但这两个例子与本节的教学联系不大,尤其是第一课时《根式》,更是没有什么关系,因此,要不要用这两个例子呢,或者,把它们移至指数函数的图象与性质一节引入更贴切些。