第十章数的开方(讲义)(原卷版)_2026语数英7-9年级上《举一反三系列》多版本_2026《初中数学•举一反三系列》8上多版本_2026《初中数学•举一反三系列》8上(华东师大)

第十章  数的开方（举一反三讲义）全章题型归纳
 2024
 ...................................................................................................................................................................5
 1   !"  ........................................................................................................................5
 2  #$ !" %& ............................................................................................................5
 3  ' !" ( ........................................................................................................................6
 4  )*+,- ............................................................................................................................................6
 5  ./*+0- ........................................................................................................................................7
12 ...................................................................................................................................................................7
 6  -3 +45678 ....................................................................................................................7
 7  )*+9:; ....................................................................................................................................8
 8  <*=>?9*=> ........................................................................................................................8
 9  )*@*ABC,$ ............................................................................................................................9
 10 )*,-+D$ ..................................................................................................................................10
知识点1  算术平方根和平方根的区别与联系


	



 x 
 a, x²=a
 x
 a 



 χ 
 a, x²=a
 x 
 a  
!


"#


#$
%&'
()*+,-#
./0
 a ( 𝑎
 a ./( ±
𝑎
12
3456
𝑎7#89:,;

𝑎≥0
𝑎≥0
a <<,
<( 0
=>
?@A
BC
D
=>
E 0
0 FGB 0
知识点2  开平方
1.	HI
 a J
K
a LK.
教辅资源，关注公众号★全科AA+
                                                                           
                                                                       
2.KF12M=>
1)KN
LK a OPB:,.
2)B
BKQ+KB
RJ
BIST.
3)FK'(UJ
<VWJXYZKQB[\.
知识点3  
𝐚²与𝐚𝟐的性质
]^
;_
/`
𝒂²
𝒂𝟐=|𝒂|= 𝒂E𝒂≥𝟎F
−𝒂E𝒂< 𝟎F
62=|6|=6
(−6)2=|−6|=6
𝒂𝟐
( 𝐚)𝟐 = 𝐚E𝐚≥𝟎F
( 6)
2 =6
知识点4  立方根和平方根的不同点和相同点
a

	



 x a
 a,𝑥3=aG x
 a ab
 !



 x 
 a, x²=a, x 
 a  
!

c
 a #d"#
a

aB
,
aB,

#$
%&
'()*+,-#
./0
3 𝑎
±
𝑎
12
𝑎3456
ef
𝑎≥0
)gh
E 0
0 B 0
0 aB 0
知识点5  开立方
1.HI* a +" +,-JKL"GMN a JKOL*.
2.L?L"+PQ
L
L"
,-R	
±
3
OL*
:,
ST*
教辅资源，关注公众号★全科AA+
                                                                           
                                                                       
I*
0 "#
+

#$+,-#
efaG"#


知识点6  无理数
.UVWX9*JK./*Y
./*+Z[\]^_`ab
E1FLLVc+*+dD e./*Gf 2G
3 5gh
E2FijkπlHmn^π+*,f 2πG
π
2gh
E3F_.UVWX9*\]op+*,f 0.101 001 000 1…EqI 1 rstuvHI 0)gYT./*+9
*=>w*.Uh./*+9*=>VWXh./*Vxyz{>*+\]Y
知识点7  实数的概念及分类
1. |}
^/*?./*~€)*Y
2. >
)*^qb>‚ƒ
E1F„…†>
)*‡>€^/*?./*Y
E2F„ˆ78>
)*‡>€ˆ)*!0!7)*Y
知识点8  实数与数轴的关系
ˆ<*
ˆ>*
7<*
7>*
)*
ˆ)*
7)*
ˆ^/*
ˆ./*
7^/*
7./*
0
)*
^/*
0
./*
ˆ^/*
7^/*
ˆ./*
7./*
^U9*ŠWX9*
.UVWX9*
ˆ<*
ˆ>*
7<*
7>*
教辅资源，关注公众号★全科AA+
                                                                           
                                                                       
‹HI)*Œ‡_$*A+HIŽyzh‘G*A+‹HIŽŒyzHI)*Y)*@*A
+ŽHH’D.
知识点9  实数范围内的有关概念
“
8”
•–
d*
— a @ b ˜€d*G™𝑎+ 𝑏= 0
3+d*e−3
š*
— a @ b ˜€š*G™𝑎𝑏= 1
2 +š*e
1
2
S›)*+œ’(Œe67*G|𝑎| ≥0
|𝑎|
𝑎(𝑎> 0)
0(𝑎= 0)
−𝑎(𝑎< 0)
œ’(
˜€d*+qI*+œ’(dgG|𝑎| = |−𝑎|
| 2| = |−2| =
2
知识点10  实数的运算
ž)*Ÿ ¡GV¢‡_£¤¥!¦!§!¨E¨*V€©F!§,-Gª«‡_£¤L",-_l
67)*+L,-Y
^/*+,-8”?,-¬ž)*Ÿ ¡­®¯$G)*°C,-+±²@^/*°C,-+±²d³G
´§!LGµ§¨G¶·¥¦Y³¸,-„¹º»¼+±²£¤G^½	+´-½	¾¿+Y
知识点11  实数的大小比较
^/*9:;+Àž)*Ÿ ¡­®¯$Y
q)*+9Áf`
ˆ)*Œ 0G7)*Œ9 0Gˆ)*ÂHÃ7)*hqIˆ)*Gœ’(
+ˆ)*hqI7)*Gœ’(+7)*9hž*Ayz+qI)*G¼Ä+*Å:ºÄ+*
YÆÇG:;q)*+9È^f`À
E1Fɑ:;q)*++9G£ªÊ…)*+9ÁYf:; 10@ 3 +9G'Â( 10)
2 = 10
，32 = 9，10 > 9GË 10 > 3Y
E2F$0-+Àp./*+ÌÍ(·Gµ:;q*+9Y
E3FÎqIÏ 	+./*:;9ÐG‡D$f`ÑÒ
① 𝑎> 𝑏≥0⇔𝑎>
𝑏Y
② 𝑎> 𝑏⇔
3 𝑎>
3 𝑏Y
教辅资源，关注公众号★全科AA+
                                                                           
                                                                       
【培优篇】
【题型1  直接求平方根、立方根】
– 1E24-25 ÓÔ¸`·ÕÖ×·ØÙF`ÚÛÀNÜÝ+eE    F
AY 5e 5 +-3 
BY0 + ?" Œe 0
CY 9+ e±3
DY−3e(−3)2+HI 
Þ] 1-1E24-25 ÓÔ¸`·ßàáâ·ãäåæF−
8
27+" e   h 4+-3 e     Y
Þ] 1-2E24-25 ÓÔ¸`·çè·éØêëFE1Fìy
a
0.000001
0.001
1
1000
1000000
3 𝑎
E2F íîïð+ñ¬ìò
①
3 3 = 1.442G™
3 3000 = ______G
3 0.003 = ______h
②óô
3 343 = 7G™
3 0.000343 = ______Y
Þ] 1-3— a!b ˜€d*Gc!d ˜€7š*G™ 𝑎2−𝑏2 +
3 𝑐𝑑=       
.
【题型2  利用求平方根、立方根解方程】
– 2E25-26 õÔ¸·çè·ö·êëF’Â)*𝑎,𝑏G…†Hb,-“ ⊕”
𝑎⊕𝑏= 𝑎2−2𝑏YÂ𝑥+
&(𝑥+ 1) ⊕ 8 = 0+%€            
Y
Þ] 2-1E24-25 ÓÔ¸`·÷âøù·ãäåæF`Úú]N+ xY
(1)𝑥2 = 81h
(2)(𝑥+ 1)3 = −27Y
Þ] 2-2E24-25 ÓÔ¸`·ûâüýþ·ØÙFÿĀ&(2𝑥−1)2−16 = 9+ x +(€      
Y
Þ] 2-3E24-25 ÓÔ¸`·āĂ·ØNF`Ú&N x +(
(1)2𝑥2 = 50h
(2)
1
3(𝑥+ 1)2 = 3h
(3)(2−𝑥)3 = 64h
(4)64(𝑥+ 1)3−125 = 0
教辅资源，关注公众号★全科AA+
                                                                           
                                                                       
【题型3  由平方根、立方根求值】
– 3E24-25 ÓÔ¸`·Õ㥷ãäåæFóô|𝑎−6|@ 𝑎+ 2𝑏˜€d*G𝑐+ 5+" e 2G™
𝑎−2𝑏−𝑐+ €     Y
Þ] 3-1óô x +qI e𝑎+ 3@2𝑎−15G«2𝑏−1+-3 e 3Y
(1)𝑎,𝑏,𝑥+(h
(2)𝑎+ 𝑏−1+" Y
Þ] 3-2E24-25 ÓÔ¸`·ąĆćº·ãäåæFóô3𝑎+ 1+qI >Qe7−𝑚?2−2𝑚G9 + 𝑏+
" e 2Y
(1)𝑚G𝑎G𝑏+(h
(2) 7𝑎−𝑏+ Y
Þ] 3-3E24-25 ÓÔ¸`·ĈĆĉĊ·ØÙFóô2𝑎+ 1+ e ± 3.3𝑎+ 2𝑏−4+" e−2G
(1)𝑎?𝑏+(h
(2)3𝑎+ 𝑏+ 
【题型4  实数的运算】
– 4E24-25 ÓÔ¸`·ċČčĎ·ØNFď 1G 2G 3G 6„fĐ]đÚG—ñ…(𝑚,𝑛)yzĒ𝑚đ¹
ºē¼Ē𝑛I*G™(5,4)@(12,4)yz+q*rĔe         
.
Þ] 4-1E2025 ĕÔ¸·çè·ĖåæFė-
 16−(−1)2025−
3 27−|1−2|Y
Þ] 4-2E24-25 ĕÔ¸·÷âĘĎ·ØÙFñ…
[𝑚]yzVę‘𝑚+¶<*G{𝑚}yz𝑚+9*=>G
[𝑚] + {𝑚} = 𝑚GMN𝑚€)*Y–f[3.14] = 3G{3.14} = 0.14G[3.14] + {3.14} = 3.14Yė-
[−3]−{ 5}
=       
Y
Þ] 4-3E24-25 ÓÔ¸·ĚĈěà·ØNF‹I&²ä'—ĜĝĞğĠ{GġĢėģHä,-&²f
Đ
教辅资源，关注公众号★全科AA+
                                                                           
                                                                       
–f
ÎĤĥ x +(€−1ÐGė-ÑĦ 3 < 4hďĤĥ(ހ(−1) +1ħ0Gė-ÑĦ€ 6 < 4hµďĤĥ
(ހģ0 + 1 = 1GĨĩ,-G»ė-ÑĦV9 4GĪĤpīÑĦY
ĬĭĮ`ÚįY
(1)ÎĤĥ x +(€ 5G™Ĥp y +(evİıĬÚ]ė-Y
(2)ÎIJijĤĥ x +(€ 1GĬɑė-ÛĴĵ‘au&²,¤·ĪxĤp yY
【题型5  无理数的估算】
– 5E24-25 ÓÔ¸`·ĶķĸĹ·ØÙFĺ®eĻ+ĢėGļeHĽ99+ľĿGŀŁnł“Ļ
”YfĐG
𝐵𝐶
𝐴𝐵+(ÌŃń:
5−1
2 G™`Ú0-ˆÊ+eE    F
AY0 <
5−1
2
<
2
5
BY
2
5 <
5−1
2
<
1
2
CY
1
2 <
5−1
2
< 1
DY1 <
5−1
2
< 2
Þ] 5-1E24-25 ĕÔ¸`·ĈŅņŇ·ãäåæFóô432 = 1849G442 = 1936G452 = 2025G462
= 2116Y—𝑛€<*«𝑛<
2022 < 𝑛+ 1G™𝑛+(€E   F
AY43
BY44
CY45
DY46
Þ] 5-2E24-25 õÔ¸·ĈŅņŇ·ØNF—𝑎<
23 < 𝑏G«𝑎!𝑏€ňĩˆ<*G™𝑎+ 𝑏=        
Þ] 5-3E24-25 ÓÔ¸`·ûʼnŊÕ·ØÙFóô𝑘<
17−1G«𝑘€ˆ<*G™𝑘+(‡_e           
EopHI‡FY
【拔尖篇】
【题型6  算术平方根的双重非负性】
– 6E24-25 ÓÔ¸`·÷ʼnŋŌ·ãäåæFóô aGb €)*G«𝑏=
𝑎−8−8−𝑎+25G™
3 𝑎+
𝑏+
(€E    FY
教辅资源，关注公众号★全科AA+
                                                                           
                                                                       
AY10
BY9
CY8
DY7
Þ] 6-1E25-26 ĕÔ¸·çè·ōŎåæF—𝑎+
𝑎−3 = 3G 𝑎+ 3+(Y
Þ] 6-2E24-25 ÓÔ¸`·ßàßĎ·ØNFóô|𝑎−3| +
3 + 𝑏= 0G™
𝑎
𝑏
2025
= E    F
AY1
BY−1
CY2
DY−2
Þ] 6-3E2025 ĕÔ¸`·÷ʼn·ĖåæFóô6©)* aGb ÿĀ|2𝑎−4| + |𝑏+ 2| +
(𝑎−4)𝑏2
+4 = 2𝑎G™𝑎+ 𝑏=      
Y
【题型7  实数的大小比较】
– 7E24-25 ÓÔ¸`·āĂŏŐĉ·ØNF:;9
 11     3h−5     −2h
5+1
2     
3
2YEì“ > ”!
“ = ”Š“ < ”FY
Þ] 7-1E24-25 ÓÔ¸`·Õőà·ØNF—𝑎=
3 7G𝑏=
5G𝑐= 2G™ aGbGc +9Á€E    F
AY𝑏≤𝑐< 𝑎
BY𝑏< 𝑎< 𝑐
CY𝑎< 𝑐< 𝑏
DY𝑎< 𝑏< 𝑐
Þ] 7-2E25-26 ĕÔ¸·çè·ōŎåæF:; 65−2@ 10 +2+9Y
Þ] 7-3E25-26 ĕÔ¸·çè·ōŎåæF:;`ÚúĠ*+9
(1)
1
2@
10
6 h
(2)
6−1
2 @ 1Y
【题型8  求整数部分和小数部分】
– 8E24-25 ĕÔ¸·ûâĉŒ·ãäåæF— 3+<*=>€ aG9*=>€ bG4−3+<*=>€
cG9*=>€ dG™
𝑏+𝑑
𝑎𝑐+(€E   F
AY
1
2
BY
1
4
CY
3−1
2
DY
3+1
2
Þ] 8-1E24-25 ÓÔ¸`·ąąà·ØNF—6 +
5+<*=>e𝑚G9*=>e𝑛G™|𝑛−𝑚|€E   F
AY 5−10
BY10−5
CY 5−6
DY6−5
Þ] 8-2E24-25 ĕÔ¸·÷âĘĎ·ØÙFñ…
[𝑚]yzVę‘𝑚+¶<*G{𝑚}yz𝑚+9*=>G
[𝑚] + {𝑚} = 𝑚GMN𝑚€)*Y–f[3.14] = 3G{3.14} = 0.14G[3.14] + {3.14} = 3.14Yė-
[−3]−{ 5}
=       
Y
Þ] 8-3E24-25 ÓÔ¸`·ąĈœ·ãäåæFóô)*8 +
13+<*=>€𝑎G9*=>e𝑚h)*
9−13+<*=>€𝑏G9*=>e𝑛Y
教辅资源，关注公众号★全科AA+
                                                                           
                                                                       
(1)op𝑎G𝑚G𝑏G𝑛+(h
(2)12𝑚+ 12𝑛+
𝑎+ 𝑏−7+(+ h
(3)
3 (𝑚+ 𝑛)2025 +(𝑎−2𝑏)+ 6+(Y
【题型9  实数与数轴综合运用】
– 9E24-25 ÓÔ¸`·ûʼnŔŕŖ·ØÙFŗõŘ-3řNŚś3Ŝ
“Śśúĺ§GŝªL¨rG
Ş”G𝑐=
𝑎2 + 𝑏2E𝑎€“Ś”G𝑏€“ś”G𝑐€“Ş”F—“Ś”€2G“ś”€3G™“Ş”žfĐşz*A‡yz
žE    
F
AY𝐴Ž
BY𝐵Ž
CY𝐶Ž
DY𝐷Ž
Þ] 9-1E24-25 ÓÔ¸`·Š×šÕ·ØNFfĐGˆ\𝐴𝐵𝐶𝐷+¿Ţ€ 7GţŽ A ž*Ayz+*
€−1G—Ž E ž*AEŽ E žŽ A +ºŤFG«𝐴𝐷= 𝐴𝐸G™Ž E şyz+*€          
Y
Þ] 9-2E24-25 ĕÔ¸`·ťĸĈŦŧĈ·ØÙF)*𝑎G𝑏ž*A+wŨfĐşzGũŪ
|𝑎+ 1|−(𝑏−1)2
+
(𝑎−𝑏)2 =           
Y
  
Þ] 9-3E24-25 ÓÔ¸`·÷âZő·ØNFfĐGɑūÄø€ 1 +ˆ\GŬxƒÊ+ŭ 2yzž*
AŽ𝐴1ŮGů𝐴1¼Ť¶Ì+<*Ž€𝐵1G_Ž𝐵1€iŰG𝐴1𝐵1€űŲūűiGų*AŽ𝐴2Gů𝐴2¼Ť
¶Ì+<*Ž€𝐵2G_Ž𝐵2€iŰG𝐴2𝐵2€űŲūűiGų*AŽ𝐴3GfÆĨĩG™𝐴2026𝐵2026+ø
€          
Y
教辅资源，关注公众号★全科AA+
                                                                           
                                                                       
【题型10  实数运算的应用】
– 10E24-25 ÓÔ¸`·÷âøù·ãäåæFŴŵ^HIŶހ200cm3+ŷŸŹźGŴŵŻżďŽ
IŷŸžũŝ5ĉſƀ{ĉ+\ƁY
(1)—5ĉſƀ{HIƂ¿€ˆ\!ƃ€8cm+øŶGøŶƂ¿ˆ\+ÄøY
(2)Ŵŵðſƀ{ƄI9d³+ˆŶŷŸGƅêƆ{·Ƈ`+ĊźŶހ8cm3Gƅê{+‹
HI9ˆŶŷŸ+ƈøY
Þ] 10-1E24-25 ÓÔ¸·ĚĈ·ØNFfĐGHIƂ¿űŲ€3cm+ƉƊ¡ƋłHmƌƍYÎƉƊˆ
ƎÐGƉ¡ƌƍ+ƃƏ€16cmhšƎÐGòĊ=>+ƃƏ€4cmYƉ¡+ƌƍˆƐšÿ 2 IHƑ+ˆ
ŶƒƓEπƔ 3GƒƓ+ƕƏVėFY
(1)īƉƊ+ƒŢEƋÿÐƌƍ+ŶŢFevİ"ƖƗı
(2)ˆŶƒƓ+ƈøevİƖƗı
Þ] 10-2ƘHƙƚö·G€ģyƛžéØtƜƝƞƆ{*êë+9Ɵ?9Ơ³G*ġơ…
žƢÇƣƤŽ 2 ƥƦƧŒe 16 ƨ+ƩƪƫƬƭƮY¹Ʃƪþ»ƚ+‹ƯưƦƱƲƳŒ€ 1.6 ƨG'
Â*ġeīƤ+ƴƵGƶƋƦŒ‡_ļ$HIƤƷƲ+ 5 ƨƤƸƹ’‹ƯưƦ+ÅƧ¦ƺ 5
ƨEưƦÅƧVnƱƲƳG³HưƦƻƼƽļ$HIƸƹFYƾ íīƩƪþ+ƿǀǁGÎưƦÅƧEV
nƱƲƳFÿ 30 ƨÐG5 ƨ+ƤƸƹ‡ǂǃ€HI 7 ƨ+þDŽƸƹG‡_DžưƦÅƧEVnƱƲƳF
¦ƺ 7 ƨÎ*ġ³ÐŽģ 2 ƥƩƪƒdž`ƦLJLjÐG9Ɵ³Û
“ġGljƮ‡_ǃHb`Ʀ]G
ƿNJvNj”Ĭ³Ʈ>nj9Ɵ³+`Ʀ]Gŝė-pǍuÇƣÅƳ$EƹnƱƲƳF¶ǎ‡€        
ƨY
Þ] 10-3Ƣ9PƒdžLïHŸø€32mGǏ€21m+ø\òǐG
(1)ǑH
fĐGďŽŸòǐbǒǓGNsǔHĝǕǖ+9ǗG9Ǘ+ºÄǘē¼Ǚ1mŬeǚ+¼Ä
教辅资源，关注公众号★全科AA+
                                                                           
                                                                       
ǘY™ŽŸǒǐ+¿Ţ€ _____ m2h
(2)ǑǛ
ǔķHIøeǏ+ 1.5 ǜG¿Ţ€384m2+ǝǞǟG—:Ǡ$+ǝǞǟżøž23m»30mr
sGǏž13m»20mrsY ŽIǝǞǟx$K:Ǡǡı ŝÛĴ/'Y
教辅资源，关注公众号★全科AA+