第十章数的开方(单元测试·培优卷)(解析版)_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·培优卷
 2024

Ⅰ 
!"#$%& 10 '()* 30 *(+' 3 *,
1"%3 *,%2025·-./0·12,34567(89:;6<= %    ,
A" 2
B"0.4
C"0
D"−1
A
*>?@AB:;6(:;6=C:DEFG'6"HI:;6<JKLMN#O=PQ:D
EFG'6"
R
#O A
 2=STEU<6(:VWXQ*6(89:;6"
#O B
0.4 =YD'6(Z[Q
2
5(89Y;6"
#O C
0 =\6(89Y;6"
#O D
−1=\6(89Y;6"
]#
A"
2"%3 *,%24-25 ^_`a·bcde·f7,34gh7(ij<=%   ,
A" 25=±5
B"± 9=3
C" (−2)2=−2
D"
3 −1=−1
D
*HIkTH(glmTH(mTH<JKnopqrZ"
>ABkTH(glmTH(mTH(st56uYkTH(vwqYmTH(xyz{JK|}~
=<"
RA. 
25=5(€(E‚ƒt„    
B. ± 9=±3(€(E‚ƒt„
C.
(−2)2=2€(E‚ƒt„
D. 
3 −1=−1(ij(‚ƒt„
]#
D"
3"%3 *,%24-25 …_`3·†‡ˆ‰Š·‹Œy,34ŽVij<=% ,
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A" 9<glmTH=3
B"−16<mTH=−4
C"0<glmTH=0
D"0.1<kTH=0.001
C
*>AmTHglmTHnkTH<‘"
’“!*N#O<ijqrZ"
R
A" 9=3(3 <glmTH= 3(] A €„
B"w6”YmTH(−16:mTH(] B €„
C"0 <glmTH= 0(] C ij„
D"0.1<kTH=
3 0.1(•0.001=0.13<–—(] D €„
]#
C"
4"%3 *,%2025·†‡‡˜·2™š›,œ(6žaZWX−3<Ÿ=%   ,
A"Ÿ A
B"Ÿ B
C"Ÿ C
D"Ÿ D
B
*>?@AB566ž(
HI1<3<4( •¡1< 3<2(¢jJ−3<£¤(rZ¡¥"
R
∵1<3<4(
∴1< 3<2(
∴−2<−3<−1(
¦§6žaZWX−3<Ÿ= B"
]#
B"
5"%3 *,
1
81<mTH=%    ,
A"
1
9
B"±1
9
C"
1
3
D"±1
3
D
R*
¨©ªh«¬(¢HImTH<JK ­rZ.
R
∵
1
81=
1
9(•
1
9<mTH=±
1
3(
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∴
1
81<mTH=±1
3.
]# D.
Ÿ®
¯!°h±<mTH²(’¨©ªh«¬(¢¯«¬¦¡–—<mTHrZ.
6"%3 *,%24-25 …_`3·³´µ¶·f·,¸2−a<kTQ64(¹a<ºQ%  ,
A"−6
B"−6»10
C"−2
D"−2»6
C
*>AkTH<JK(xyz{kTH<JK=<(HIkTH<JKrZ"
R
¼t¡
(2−a)3=64(
¹2−a=4(
¡
a=−2"
]#
C"
7"%3 *,%24-25 ^_`3·½¾·f·,¿À a−3+|2−b|=0(¹a+b<ºQ%    ,
A"−1
B"1
C"5
D"6
C
*>ABglmTH<vwq(Áº<vwq"
HIglmTH<vwq(Áº<vwq¯¥ ab <º(¢¯|rZ"
R
∵a−3+|2−b|=0( a−3≥0(|2−b|≥0
∴a−3=0(|2−b|=0(
ra−3=0(2−b=0
¡a=3(b=2
∴a+b=3+2=5"
]#
C
8"%3 *,%24-25 …_`3·ÃÄ·f7,¸a≠0(Å a(b ÆQÇÈ6(¹34NÂ67ÆQÇÈ6<!É
=%   ,
A" a b
B" a2 
b2
C"
3 a
3 b
D" a3
3 (−b)3
C
*>?@AB¯!°6<glmTH|kTH(ÇÈ6<JK(¼ÇÈ6<JKZ¡a=−b(Ê
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a>0(b<0(¹ b”YtK(IËZLM A„ a2=a( b2=|b|=−b(IËZLM B„
3 b=
3 −a=−
3 a(IËZL
M C„ a3=a
3 (−b)3=−b=a(IËZLM D"
R„∵a(b ÆQÇÈ6(
∴a=−b(
∵a≠0(
∴ZÊa>0(b<0(
A b”YtK(]Ë#OE‚ƒt„
B a2=|a|=a b2=|b|=−b=aEÆQÇÈ6(]Ë#OE‚ƒt„
C
3 a
3 b=
3 −a=−
3 aÆQÇÈ6(]Ë#O‚ƒt
D a3=a
3 (−b)3=−b=aEÆQÇÈ6(]Ë#OE‚ƒt„
]#
C"
9"%3 *,%24-25 …_`3·bc̘·f·,͜¦X<ÎÏÐg(¸SÑÒÓ<x<º= 27(¹Ò¥
<y<º=%    ,
A" 2
B" 3
C"
3 2
D"
3 3
B
*HIÎÏÐgrZ"
>ABÎÏhÐg(xyz{ÎÏhÐg=<"
R
∵x=27(
∴
3 x=
3 27=3(=Y;6(
ÔglmTHQ 3(=:;6(
‚ƒt(Z§Ò¥(
∴y= 3(
]#
B"
10"%3 *,%24-25 …_`3·³´Õ0·‹Œy,'Ö|'×*ØÙÚB!°œ¦X<iTÛÜÝ(
¿À'ÖÙÚ<iTÛÜÝ<WÞßQ96cm2(•'×ÙÚ<iTÛÜÝ<Ûßà'ÖÙÚ<iTÛÜÝ<
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Ûß61cm3(¹'×ÙÚ<iTÛÜÝ<WÞßQ%    ,
A"64cm2
B"150cm2
C"157cm2
D"216cm2
B
*>?@ABiTÛ<WÞß|ÛߏglmTH|kTHágâTágãÀä(ij¯¡
å°iTÛÜÝ<æç="
¨HIiTÛ<WÞßh¯¥'ÖÙÚ<iTÛÜÝ<æç|Ûß( •¯¥'×ÙÚ<iTÛÜÝ<
Ûß|æç(rZ¡"
R
Ê'ÖiTÛæçQa(6a2=96(
¡a2=16(a=4(
'ÖÙÚ<iTÛÜÝ<æçQ
4cm(
oÛßQ
43=64cm3(
'×ÙÚ<iTÛÜÝ<ÛßQ
64+61=125cm3(
¹'×ÙÚ<iTÛÜÝ<æçQ
3 125=5cm(
'×ÙÚ<iTÛÜÝ<WÞßQ
6×52=6×25=150cm2„
]#
B"
è"éê%& 6 '()* 18 *(+' 3 *,
11"%3 *,%23-24 …_`3·bc‡ë·‹Œy, 5−2<ÇÈ6=       
(
3 −8<ì6=         
"
     
−5+2/2−5     
−1
2/−0.5
*>AB56<qí(¯!°6<kTH(ì6|ÇÈ6<JK(z{§aÀä=<(
HI¯!°6<kTH(ì6|ÇÈ6<JK ­¯"
R
 5−2<ÇÈ6=−5+2„
3 −8 =−2<ì6=−1
2
]Q
−5+2„−
1
2"
12"%3 *,%24-25 …_`3·¶î0·f7, 9+|1−2|×(−1)2024−
3 125=       
"
 2−3/−3+ 2
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*>?@ABglmTHÁºâTkTH<JKnág(xyz{ïð6‘<J
K|qí=<">’@*ØHIglmTHÁºâTkTH<JK(“ñ«¬h±7
<NO(òóÍôágõÏ ­Ðg"
R
ªh=3+( 2−1)×1−5
=3+ 2−1−5
= 2−3
]Q
 2−3 "
13"%3 *,%24-25 …_`3·½ö÷·f·,¸!°6<mTHø<kTHùÃÇú(¹ï°6=      
"
0
*>?@ABmTH|kTH(HImTH|kTH<JK|qí(rZ¡¥ï°6(<
=z{mTH|kTH<JK|qí(Àû0<mTHø<kTHÇã"
R
¼t(!°6<mTHø<kTHÇã(
¹ï°6Q
0"
]Q
0"
14"%3 *,%24-25 …_`3·üý.þÿĀ·‹Œy,ā6h (x−2)2<Ă'º=        
"
0
*>AglmTH<vwq(HIvwq(¡ă (x−2)2≥0(rZ¡¥–—"
R
∵(x−2)2≥0(
∴ā6h (x−2)2<Ă'º= 0„
]Q
0"
15"%3 *,%24-25 …_`3·Ą‡ą¶·f·,Ć“*”WX!ćĈág
Â9sti56m,n(uYm∗n= n+
m(ĉœ2∗9= 9+2=5(Ċċ6∗81=          
"
15
*>?@ABĈJK3<56ág( HIĈJKāÓ(¨ÐgglmTH(¢ÐgČVrZ"
R
∵Â9sti56m,n(uYm∗n= n+m(
∴6∗81= 81+6=9+6=15(
]Q
15
16"%3 *,%23-24 …_`3·½č†Ď·f·,©56͜Th ­YďĐđ4(¹ 19 ­< 37 °
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6=      
"
19
*>A566ĒēďĐ(Ĕ756<đ4Th7ĕăďĐ=Ėė<"HI7¦Ę
<56đ4Th(ĕăďĐ¯rZ¡ă"
R
©56͜Th ­YďĐđ4(ęĚěĜ(ĝYœ3ďĐ
①n­Y(2n−1)°6„
②+­Ăó!°6Ē<Áºã9­6„
③Ğ6­<Ăó!°Qi„
④ğ6­<Ăó!°Qw„
∴ 19 ­Y2×19−1=37°6(
∴HIœaďĐZÀ( 19 ­< 37 °6= 19"
]Q
19"
Ⅱ 
1"%& 8 '()* 72 *,
17"%6 *,%24-25 ^_`a·bcde·f7,TÎ
(1)4(2x−1)2=36„
(2)2(x−1)3+16=0"
(1)x1=2(x2=−1„
(2)x=−1
*>A<=ĠĆmTHkTH<ġKTÎ(z{mTHkTH<ġK=><"
%1,ĢTΫQ(2x−1)2=9(¢ĠĆmTH<ġKTÎrZ„
%2,ĢTΫQ(x−1)3=−8(¢ĠĆkTH<ġKTÎrZ"
R%1,
4(2x−1)2=36
(2x−1)2=9
2x−1=±3
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x1=2(x2=−1„
%2,
2(x−1)3+16=0
(x−1)3=−8
x−1=−2
x=−1
18"%6 *,%24-25 …_`3·½½0·f7,¿À
!°i6 a <å°EúmTH*Ø=x+5|4x−10.
(1)¯ x  a <º„
(2)¯a−9<kTH"
(1)x <ºQ 1(a <ºQ 36
(2)3
*>ABkTH(mTH(xyz{ÇJK=<"
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%2,HIkTH<JK¯a−9<kTHrZ"
R%1,
¼t¡(x+5)+(4x−10)=0(
¡x=1(
x+5=1+5=6(
∴a=62=36(
r x <ºQ 1(a <ºQ 36„
%2,
a−9=36−9=27(
∴a−9<kTHQ
3 27=3"
19"%8 *,%22-23 …_`3·ģčĤ‡·f·,ĥĦħ(=!°ĨĩĥĦ<ħĪ"īī'Ĭ|ĭĭ'Įį
İıÙħĪÜIJijĘĥĦ"'ĮıÙB!ĴÞßQ100cm2<iTĵĶķ('ĬıÙB!°ÞßQ200cm2<
çTĵĸ(oçĹĺàQ5:4"'ĮıÙ<ĶķEĻļĽùÃľÓ'ĬıÙ<ĸ7ĿŀŁłŃÐgŽń
Ņ<LM"
Ľ(;¼ņ
*>?@ABmTH<ŇĆ"¨¯¥iTĵ<ňçQ 100=10cm(òóÊçTĵ<ĸ<çQ5x
cm(ĹQ4xcm(HItZ¡20x2=200(Ĕ•jJçTĵ<çĹrZ¡¥–—"
R
Ľ(;¼œ3
∵iTĵĶķ<ÞßQ100cm2(
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∴iTĵ<ňçQ 100=10cm(
ÊçTĵ<ĸ<çQ5xcm(ĹQ4xcm(ʼn¡
5x×4x=200(
r20x2=200(
∴x2=10(
∴x= 10»−10%Ŋŋ,(
∴4x=4 10cm>10cm(
∴Ľ©ïĴĶķEĻļŌľÓËĸ7"
20"%8 *,%24-25 …_`3·†‡ōŎ·f7,ęĚ34ÐgŃÎ(ŏŐkTH"
13=1(23=8(33=27(43=64(53=125(63=216(73=343(83=512(93=729„
(1)őŒ…_`6Œœ 59 Ŕ(ŕÄŖŗ6ŘřŚÐgkTH<TVĘ'ńB!ðśX('ń=ï
Ŝ¯¥ 19683 <kTH<
¨ŝÐ 19683 <kTH<°Ş6(ŏŐø<°Ş6Q 7(¼203<19000<303(
ŏŐ 19683 <kTH<şŞ6= (Šš¡ 19683 <kTH= "
(2)ŁŅHI%1,7'ń<TV(ùޜ3éê
①
3 −373248= "
②
3 0.531441= "
(1)2(27
(2)①−72„②0.81
*>AB6<kTH<ŝg(;!°6<kT<°Ş6ţ=ï°6<°Ş6<kT<°Ş6
=<
%1,ę̦Ę6<kT(7 <kT<°Ş6= 3(¼ËŝÐ 19683 <kTH<°Ş6Q 7(Ť•¼203<19000<
303ŏŐ 19683 <kTH<şŞ6ï 2(¼Ë ­ŠšrZ„
%2,HI%2,7<TV¨ ­ŏŐ(òó ­ŠšrZ
R%1,∵19683<°Ş6= 3(•73=343·Ş6Q 3(
∴ŏŐ19683<kTH<°Ş6Q 7(
ť∵203<19000<303(
∴ŏŐ19683<kTH<şŞ6Q 2(
Šš
273=19683(
∴19683 <kTH= 27„
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]Q 2(27„
%2,
①∵−373248<°Ş6=8(•23=8(·Ş6Q8 (
∴ŏŐ−373248<kTH<°Ş6Q2"
ť∵(−70)3=−343000((−80)3=−512000 (Å(−80)3<−373248<(−70)3 "
∴ŏŐ−373248<kTH<şŞ6Q 7(
Šš
(−72)3=−373248 "
∴
3 −373248=−72 " 
②∵0.531441<·Ş6= 1(•13=1(
∴ŏŐ0.531441<kTH<·Ş6Q 1(
ť∵0.83<−117649<0.93(
∴ŏŐ0.531441<kTH<ş*Ş6Q 8(
Šš
0.813=0.531441„
]Q−72(0.81„
21"%10 *,%24-25 …_`3·½č‡Ŧ·f·,ŕŧ¿ŨBmTH|kTH"¸x2=a(¹xũa<èŪ
TH%mTH,(ZWXQ± a"¸x3=a(¹xũa<1ŪTH%kTH,(ZWXQ
3 a"mTHĝYqíœ
i6Yå°mTH(øŧÆQÇÈ6
0 <mTH= 0„w6”YmTH"ŁūŬœŭ(ęĚ3W(ēàa
Ů<JK|qíùާ3ė
x4
…
1
16
81
…
x
…
±1
±2
±3
…
JK%1,¸x4=a(¹xũa<________①_________(ZWXQ______②______„
qí%2,Łů①<qí„
ŇĆ%3,¸x4=5(Űű[¥x<º
Ųų%4,TÎ
1
2x4−8=0"
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4 a„%2,i6Yå°bŪTH(øŧÆQÇÈ6„0 <bŪTH= 0„w
6”YbŪTH„%3,x=±
4 5„%4,x=±2
*>?@ABĈJKág(<=;mTH|kTHJK"
%1,ēØmTH|kTHJK ­¯rZ„
%2,ŴômTH<qí ­ůrZ„
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%3,HIbŪTHJK ­¯rZ„
%4,ĠĆbŪTH<JKTÎrZ"
R
%1,HIWŵZÀ
¸x4=a(¹xũa<bŪTH(ZWXQ±
4 a„
%2,i6Yå°bŪTH(øŧÆQÇÈ6„0 <bŪTH= 0„w6”YbŪTH„
%3,¸x4=5(¹x=±
4 5„
%4,
1
2x4−8=0(
∴1
2x4=8(
∴x4=16(
∴x=±2"
22"%10 *,%24-25 …_`3·.čaŶ·‹Œy,HI3Wŷ34ė
x
15
15.1
15.2
15.3
15.4
15.5
15.6
15.7
15.8
15.9
x2
225
228.01
231.04
234.09
237.16
240.25
243.36
246.49
249.64
252.81
(1) 243.36=______( 23716=______( 0.023104=______„
(2) 25000Ÿ°\6ĂűŹŀ¯ 2.3<Źźº%–—Żjă 0.01,„
(3)¸153< n<154(¹)ż}~<\6nY______°"
(1)15.6„154„0.152
(2)158( 2.3≈1.52
(3)306
*>ABglmTH<ÇÀä(<=xyz{glmTH<JKn'6ŸŽžďĐ"
%1,HIWŵ7<6I§nglmTH<JK ­¯„
%2,¨© 25000 ­ſĵ(¢HIWŵ7<6IjJoűŹ<\6„Â9 2.3(ZHIglmTH<'6
ŸŽžďĐ ­¯„
%3,¨Â153< n<154åňú²mT(¢jJ n <Ôº£¤(Ĕ•¡¥)ż}~<\6 n <°6"
R%1,
¼WŵZÀ(15.62=243.36(
∴ 
243.36=15.6„
∵ ∵ 1542=23716(
∴ 
23716=154„
∵ 0.1522=0.023104(
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∴ 
0.023104=0.152"
]Q
15.6„154„0.152„
%2,
∵ 
25000= 250×100=10 250(
ť∵ 15.82=249.64,15.92=252.81(250−249.64=0.36,252.81−250=2.81,0.36<2.81(
∴ 
25000 158 ĂűŹ„
∵ 
231.04=15.2,2.3=231.04÷100(
∴ 
2.3= 231.04÷100=15.2÷10=1.52"
%3,
Â153< n<154åňú²mTZ¡1532<n<1542(
ÐgZ¡1532=23409(1542=23716(
∴ n<Ôº£¤=23409<n<23716(
¹)ż}~<\6n<°6Q23716−23409−1=306°"
]Q
306"
23"%12 *,œ(!ƀƁƂĔŸ A ƃ6žƄƅƆB 2 °ƇŞçƈăƉŸ B(Ÿ A WX−2(ʟ B ¦W
X<6Q m"
(1)¯|m+1|+|m−1|<º„
(2)Ɗ6žaƋY CD åŸ*ØWX56 c | d(ÅY|2c+d| d+4ÆQÇÈ6(¯2c−3d<mTH"
(1)2
(2)±4
*>AB6žaåŸƌ<ƍƎhmTHvw6<qínÁº<Ðg(<=¯
¡m<ºnvw6qí<ŇĆ(tmTHYå°"
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%2,ĠĆvw6<qí(¡ă c(d <º(āÓ¯ºrZ"
R%1,
¼t¡AB=2(
∴m−(−2)=2(
∴m=2−2(
∴|m+1|+|m−1|
=|2−2+1|+|2−2−1|
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=|3−2|+|1−2|
=3−2+ 2−1
=2„
%2,
∵|2c+d| d+4ÆQÇÈ6(
∴|2c+d|+ d+4=0(
∵|2c+d|≥0( d+4≥0(
∴2c+d=0(d+4=0(
∴c=2(d=−4(
∴2c−3d=2×2−3×(−4)=16(
∴2c−3d<mTH=± 16=±4"
24"%12 *,%24-25 …_`3·¶čƏƐ·f7,ūŬƑ
ŁƒƓūŬ34œŭ(ƔùŢÇŇ<sƕ"
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ŃΜ3
∵22<2×3<32(
∴22< 2×3< 32"
∴2< 2×3<3"
∴[ 2×3]=2"
ŤƗÐg(¡ă[ 3×4]=3([ 4×5]=4([ 5×6]=5([ 6×7]=6"¼ËƖƑ'É¡¥–Ƙ
¸nQi\6(¹
n(n+1) =n"
sƕ
(1)éê
[−2.1]=____(
8
7 =____"
(2)¯[
3 300]<º"
(3)¿Àa=
1×2]+[ 2×3]+⋯+[ 2024×2025 (s=
a
1012(¯[ s]<º"
(1)−3(1
(2)6
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(3)45
*>ABĈJKág(kTHmTH<JK(;ĈJK=<„
%1,HIJK(ŰűZ¡ă„
%2,Ŵôĉ¯(ŝg
3 300<'(–ƒJK(rZ¯„
%3,HI
n(n+1) =n ­«¬(rZ¯"
R%1,
[−2.1]= −3(
8
7 = 1
]Q
−3(1"
%2,
∵63<300<73(
∴3 63<
3 300<
3 73"
∴6<
3 300<7"
∴[
3 300]=6"
%3,HIœŭ(¡a=1+2+3+⋯+2023+2024
=(1+2024)+(2+2023)+⋯+(1012+1013)
=1012×2025"
∴s= a
1012=1012×2025
1012
=2025"
∴[ s]=[ 2025]=45"
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