7/21/2019 Tema 4. Potencias y Raíces de Números Enteros
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Xfen 82 Zltf`igns y rnåifs hf `òefrls f`tfrls.
Kfen Gsnofd Enrå` Inonddfrl Zîkg`n > hf 9:
Å@HGIF
>. ZLXF@IGNY 9
>.>. Hfag`gigù` .................................................................................................................................................... 9
>.>.>. Fbfrigigls ............... ............................................................... .............................................................. .......... 9
>.?. Zltf`igns hf onsf >: ............................................................................................................................... 8
>.?.>. Fbfrigigl .......................................................... ................................................................. ............................. 6
>.9. Hfsileplsgigù` pldg`ùegin hf u` `òefrl f`tfrl ........................................................................... 6
>.9.>. Fbfrigigl .......................................................... ................................................................. ............................. 6
>.8. Iunhrnhls pfrafitls .............................................................................................................................. 6
>.8.>. Hfag`gigù` ........................................................ ................................................................. ............................. 6
>.8.?. Npdginigl`fs .................................................................................................................... ............................. 6
>.6. Iuols pfrafitls ....................................................................................................................................... 6
>.6.>. Hfag`gigù` ........................................................ ................................................................. ............................. 6
>.6.?. Npdginigl`fs .................................................................................................................... ............................. 1
>.1. Ygk`l hf dns pltf`igns ............................................................................................................................ 1
>.1.>. Onsf plsgtgvn .................................................................................. .............................................................. 1
>.1.?. Onsf `fkntgvn........ .............................................................. ............................................................... .......... 1
>.1.9. Xnodn-_fsuef` ......................................................... ................................................................. .................. ;
>.1.8. Fbfrigigls ....................................................................................... .............................................................. ;
>.;. Zrlpgfhnhfs ............................................................................................................................................... =
>.;.>. Xnodn ’ _fsuef` ....................................................... ................................................................. ................ >?
>.;.?. Fbfrigigls ................................................................. ............................................................... ................... >9
>.=. Lpfrnigl`fs il` pltf`igns.................................................................................................................. >8
>.=.>. Yuen y _fstn ...................................... ................................................................. ...................................... >8
>.=.?. Eudtgpdginigù` ........................................................... ................................................................. ................ >8
>.=.9. Hgvgsgù` ........................................................... ................................................................. ........................... >8
>.=.8. Zltf`ign .......................................................... ................................................................. ........................... >8
?. _NÅIFY >6
?.>. Hfag`gigù` ................................................................................................................................................. >6
?.?. Xgpls hf rnåifs fxnitns ...................................................................................................................... >6
?.?.>. _nåz iunhrnhn ........................................................... ................................................................. ................ >6
?.?.?. _nåz iòogin ................................ ............................................................... ................................................. >1
?.?.9. Fbfrigigls ....................................................... ................................................................. ........................... >;
?.9. Xgpls hf rnåifs f`tfrns ....................................................................................................................... >=
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Xfen 82 Zltf`igns y rnåifs hf `òefrls f`tfrls.
Kfen Gsnofd Enrå` Inonddfrl Zîkg`n ? hf 9:
?.9.>. _nåz iunhrnhn ........................................................... ................................................................. ................ >=
?.9.?. _nåz iòogin ................................ ............................................................... ................................................. >7
?.8. Ylduigl`fs sfkò` fd å`hgif hf dn rnåz y fd rnhgin`hl ................................................................... >7
?.8.>. _nhgin`hl plsgtgvl .............................................................. ................................................................. ..... >7
?.8.?. _nhgin`hl ̀ fkntgvl ....................................................................................................... ........................... >7
?.8.9. Xnodn-_fsuef` ................................................................................................... ..................................... ?:
?.6. Iîdiudl hf dn rnåz iunhrnhn f`tfrn.................................................................................................. ?:
?.6.>. Iîdiudl hf alren nprlxgenhn ................................................................ ................................................ ?:
?.6.?. Iîdiudl il` fd ndklrgtel ............................................................... ........................................................... ?>
?.6.?.>. Fbfrigigl ............................................................ ................................................................. ............... ?1
?.1. Lpfrnigl`fs il` rnåifs iunhrnhns fxnitns ................................................................................. ?;
?.1.>. Yuen y _fstn ............................................................ ................................................................. ............... ?;
?.1.?. Eudtgpdginigù` ........................................................... ................................................................. ............... ?;
?.1.9. Hgvgsgù` ........................................................... ................................................................. .......................... ?;
?.1.8. Zltf`ign hf u`n rnåz .......................................................... .............................................................. ....... ?=
?.;. Dn rnåz ilel pltf`ign ......................................................................................................................... ?=
9. ZLXF@IGNY W _NÅIFY ?=
9.>. _filrhnr .................................................................................................................................................. ?=
8. LZF_NIGL@FY ILEOG@NHNY IL@ ZLXF@IGNY W _NÅIFY ?=
8.>. G`trlhuiigù` ........................................................................................................................................... ?=
8.?. _fkdns hf prglrghnh .............................................................................................................................. ?=
8.9. Xgpls hf lpfrnigl`fs ileog`nhns ................................................................................................... ?7
8.9.>. Lpfrnigl`fs ileog`nhns il` pltf`igns y rnåifs iunhrnhns fxnitns ..... ..................................... ?7
8.9.?. Lpfrnigl`fs ileog`nhns il` pltf`igns y rnåifs iòogins fxnitns ............................................... ?7
8.9.9. Lpfrnigl`fs ileog`nhns il` pltf`igns y rnåifs fxnitns ....................................................... ....... ?7
8.8. Fbfrigigls ................................................................................................................................................ ?7
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Xfen 82 Zltf`igns y rnåifs hf `òefrls f`tfrls.
Kfen Gsnofd Enrå` Inonddfrl Zîkg`n 9 hf 9:
>. ZLXF@IGNY
>.>. Hfag`gigù`
V`n pltf`ign fs u`n alren norfvgnhn hf fsirgogr u` prlhuitl alrenhl plr vnrgls anitlrfs
gkundfs, fs hfigr, u`n eudtgpdginigù` hf anitlrfs gkundfs . Fstl fs, fs u`n eudtgpdginigù` f` dn quf fd egsel`òefrl sf eudtgpdgin vnrgns vfifs.
Dn fxprfsgù` hf u`n pltf`ign fs dn onsf fdfvnhn n u` fxpl`f`tf.
n` 5 n » n » n » n ... 5 `-vfifs —n―
Onsf↟ n` ↚ fxpl`f`tf
Hl`hf2
n fs dn onsf hf u`n pltf`ign quf fs fd `òefrl quf eudtgpdginels plr så egsel. Zlr tn`tl, dn onsf fs fd `òefrl quf sf rfpgtf.
` fd fxpl`f`tf hf u`n pltf`ign quf g`hgin fd `òefrl hf vfifs quf eudtgpdginels dn onsf. Zlrtn`tl, fd fxpl`f`tf g`hgin fd `òefrl hf vfifs quf sf rfpgtf dn onsf.
Nså pufs, u`n pltf`ign fstî alrenhn plr u`n onsf y u` fxpl`f`tf.
Znrn rfsldvfr u`n pltf`ign sf eudtgpdgin dn onsf tn`tns vfifs ilel g`hgin fd fxpl`f`tf.
F` dn indiudnhlrn igf`tåagin, sf usn dn tfidn x y .
Fbfepdl >2
6 » 6 » 6 » 6 5 68 Dn onsf hf fstn pltf`ign fs 6 y fd fxpl`f`tf fs 8.
Yf dff —6 fdfvnhl n 8― l —6 fdfvnhl n dn iunrtn―.
Fbfepdl ?2
(-7) » (-7) » (-7) » (-7) » (-7) » (-7) 5 (-7)1
Dn onsf hf fstn pltf`ign fs -7 y fd fxpl`f`tf fs 1.
Yf dff —-7 fdfvnhl n 1― l —-7 fdfvnhl n dn sfxtn―.
>.>.>. Fbfrigigls
>) Fsirgof f` alren hf prlhuitl y indiudn su vndlr2
n) (-1)? Ylduigù`2 (-1) » (-1) 5 +91
o) (-9)6 Ylduigù`2 (-9) » (-9) » (-9) » (-9) » (-9) 5 -?89
i) (+?)1 Ylduigù`2 (+?) » (+?) » (+?) » (+?) » (+?) » (+?) 5 +18
h) (+8)9 Ylduigù`2 (+8) » (+8) » (+8) 5 +18
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Xfen 82 Zltf`igns y rnåifs hf `òefrls f`tfrls.
Kfen Gsnofd Enrå` Inonddfrl Zîkg`n 8 hf 9:
?) Ilepdftn dn tnodn2
Zrlhuitl Zltf`ign Onsf Fxpl`f`tf Yf dffQndlr l
_fsudtnhl? » ? » ?
7?
8 ?9 :
? 9?
8 fdfvnhl nd iuol fs gkund n 18
(’7) » (’7) » (’7)
’; 8
’?
(’6)1
’> fdfvnhl n dn qug`tn
>.?. Zltf`igns hf onsf >:
Il` pltf`ign hf onsf hgfz y fxpl`f`tf f`tfrl sf pufhf fsirgogr in`tghnhfs f`lrefs l euypfqufþns. Fstl anigdgtn dls iîdiudls, nd `l tf`fr quf utgdgznr euimls hfigendfs.
Fd iîdiudl hf dns pltf`igns hf onsf >: rfsudtn sf`igddl, pufs fd `òefrl hf ifrls ilg`ighf il` fd
fxpl`f`tf hf dn pltf`ign.
V`n pltf`ign hf onsf >: y fxpl`f`tf plsgtgvl fs gkund n dn u`ghnh sfkughn hf tn`tls ifrlsilel g`hgin fd fxpl`f`tf.
Fbfepdl >2 >:9 5 >.:::
Yl` 9 ifrls ilel g`hgin fd fxpl`f`tf hf dn pltf`ign.
Fbfepdl ?2
>:; 5 >:.:::.:::
Yl` ; ifrls ilel g`hgin fd fxpl`f`tf hf dn pltf`ign.
Dns pltf`igns hf onsf >: y fxpl`f`tf `fkntgvl fqugvndf` n hgvghgr plr >:, >::, fti., plr dl qufvn` n rfprfsf`tnr in`tghnhfs ef`lrfs hf dn u`ghnh. Iun`tl enylr fs fd fxpl`f`tf f` vndlr nosldutl,
ef`lr fs dn pltf`ign.Fbfepdl >2
>:’9 5:::.>
>
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Yl` 9 ifrls ilel g`hgin fd fxpl`f`tf hf dn pltf`ign.
Fbfepdl ?2
>:’; 5:::.:::.>:
>
>:
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Yl` ; ifrls ilel g`hgin fd fxpl`f`tf hf dn pltf`ign.
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Xfen 82 Zltf`igns y rnåifs hf `òefrls f`tfrls.
Kfen Gsnofd Enrå` Inonddfrl Zîkg`n 6 hf 9:
>.?.>. Fbfrigigl
>)
Fxprfsn f` alren hf pltf`igns hf onsf >:.
n)
6:.::: Ylduigù`2 6 » >:8
o) 9.?:: Ylduigù`2 9? » >:?
i) 9.:::.::: Ylduigù`2 9 » >:1
>.9. Hfsileplsgigù` pldg`ùegin hf u` `òefrl f`tfrl
V` `òefrl f`tfrl sf pufhf hfsilepl`fr utgdgzn`hl pltf`igns hf onsf >: .
Fbfepdl >2 Fd `òefrl 9.16= plhfels hfsilepl`frdl hfd sgkugf`tf elhl2
9.16= 5 9.::: + 1:: + 6: + = 5 9 » >:9 + 1 » >:? + 6 » >:> + =
Fbfepdl ?2 Fd `òefrl >:=.:8; plhfels hfsilepl`frdl hfd sgkugf`tf elhl2
>:=.:8; 5 >::.::: + =.::: + 8: + ; 5 > » >:6 + : » >:8 + = » >:9 + : » >:? + 8 » >:> + ;
>.9.>.
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Vtgdgzn`hl pltf`igns hf onsf >:, mnz dn hfsileplsgigù` pldg`ùegin hf fstls `òefrls2
n) 9.?6; Ylduigù`2 9.?6; 5 9 » >:9 + ? » >:? + 6 » >: + ;
o) >:.?61 Ylduigù`2 >:.?61 5 > » >:8 + : » >:9 + ? » >:? + 6 » >: + 1
i) >?6.91= Ylduigù`2 >?6.91= 5 > » >:6 + ? » >:8 + 6 » >:9 + 9 » >:? + 1 » >: + =
>.8. Iunhrnhls pfrafitls
>.8.>. Hfag`gigù`
Dls iunhrnhls pfrafitls sl` dls vndlrfs hf dns pltf`igns hf fxpl`f`tf hls.
Fbfepdl2 18 fs u` iunhrnhl pfrafitl, plrquf fs fd vndlr hf dn pltf`ign =? 5 = » =
Fbfepdl2 Dls iunhrnhls pfrafitls ef`lrfs hf >:: sl`2
Iunhrnhlspfrafitls2
>
8
7
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87
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>.8.?. Npdginigl`fs
Dn npdginigù` hf dls iunhrnhls pfrafitls fs dn rfslduigù` hf rnåifs iunhrnhns fxnitns.
>.6. Iuols pfrafitls
>.6.>.
Hfag`gigù`
Dls iuols pfrafitls sl` dls vndlrfs hf dns pltf`igns hf fxpl`f`tf trfs.
Fbfepdl2 = fs u` iuol pfrafitl, plrquf fs fd vndlr hf dn pltf`ign ?9 5 ? » ? » ?
Fbfepdl2 Dls iuols pfrafitls ef`lrfs hf ?:: sl`2
Iuols
pfrafitls2
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Zltf`igns2 >9 ?9 99 89 69
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Xfen 82 Zltf`igns y rnåifs hf `òefrls f`tfrls.
Kfen Gsnofd Enrå` Inonddfrl Zîkg`n 1 hf 9:
>.6.?. Npdginigl`fs
Dn npdginigù` hf dls iuols pfrafitls fs dn rfslduigù` hf rnåifs iòogins fxnitns.
>.1. Ygk`l hf dns pltf`igns
>.1.>. Onsf plsgtgvn
>) Fxpl`f`tf pnr2 fd vndlr hf dn pltf`ign fs sgfeprf plsgtgvl.
Ygk`l plsgtgvl2
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Ygk`l plsgtgvl2
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Ygk`l plsgtgvl2
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Xfen 82 Zltf`igns y rnåifs hf `òefrls f`tfrls.
Kfen Gsnofd Enrå` Inonddfrl Zîkg`n ; hf 9:
?) Fxpl`f`tf gepnr2 fd vndlr hf dn pltf`ign fs sgfeprf `fkntgvl.
Ygk`l plsgtgvl2
Fbfepdls2
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hf dn pltf`ign
+Znr
+Gepnr
’ Znr +
Gepnr ’
_filrhnr2 Mny quf npdginr dn rfkdn hf sgk`ls.
>.1.8.
Fbfrigigls>) Ilepdftn y pl` fbfepdls2
n) Dns pltf`igns hf onsf `fkntgvn y fxpl`f`tf pnr sl`2 ……
o) Dns pltf`igns hf onsf `fkntgvn y fxpl`f`tf gepnr sl`2 ……
Ylduigù`2
n) Zlsgtgvns. Fbfepdl2 8??? ?
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n) (’?)9 Ylduigù`2 ’=
o) (’9)? Ylduigù`2 +7
i) (’;)? Ylduigù`2 +87
h) 99 Ylduigù`2 +?;
f) (’>) Ylduigù`2 ’>
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Xfen 82 Zltf`igns y rnåifs hf `òefrls f`tfrls.
Kfen Gsnofd Enrå` Inonddfrl Zîkg`n = hf 9:
>.;. Zrlpgfhnhfs
>) Zltf`igns hf onsf u`l2 sgfeprf vndf` dn u`ghnh.
>> `
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8;9 >> 7=;.?
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Zltf`igns hf onsf ifrl2 sgfeprf vndf` ifrl.
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nn >
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6) Zrlhuitl hf pltf`igns il` dn egsen onsf2 sf hfbn dn egsen onsf y sf suen` dls fxpl`f`tfs.`e`e nnn
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7/21/2019 Tema 4. Potencias y Raíces de Números Enteros
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Xfen 82 Zltf`igns y rnåifs hf `òefrls f`tfrls.
Kfen Gsnofd Enrå` Inonddfrl Zîkg`n 7 hf 9:
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Xfen 82 Zltf`igns y rnåifs hf `òefrls f`tfrls.
Kfen Gsnofd Enrå` Inonddfrl Zîkg`n >: hf 9:
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7/21/2019 Tema 4. Potencias y Raíces de Números Enteros
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Xfen 82 Zltf`igns y rnåifs hf `òefrls f`tfrls.
Kfen Gsnofd Enrå` Inonddfrl Zîkg`n >> hf 9:
Fbfepdls2
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```onon
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7/21/2019 Tema 4. Potencias y Raíces de Números Enteros
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Xfen 82 Zltf`igns y rnåifs hf `òefrls f`tfrls.
Kfen Gsnofd Enrå` Inonddfrl Zîkg`n >? hf 9:
66666
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F` dn tnodn sgkugf`tf, sf prfsf`tn u` rfsuef` hf dns prlpgfhnhfs.
Zrlpgfhnh @leorf Fbfepdls
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>: n il` :n Zltf`igns hf fxpl`f`tf ifrl
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nn > Zltf`igns hf fxpl`f`tf u`l?=> 5 ?=
(-=>)> 5 -=>
`e`e nnn Zrlhuitl hf pltf`igns il` dn egsen onsf;? » ;8 5 ;?+8 5 ;1
(-?)6 » (-?)9 5 (-?)6+9 5 (-?)=
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n 2 Iligf`tf hf pltf`igns il` dn egsen onsf;8 2 ;? 5 ;8-? 5 ;?
(-?)1 2 (-?)9 5 (-?)1-9 5 (-?)9
7/21/2019 Tema 4. Potencias y Raíces de Números Enteros
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Xfen 82 Zltf`igns y rnåifs hf `òefrls f`tfrls.
Kfen Gsnofd Enrå` Inonddfrl Zîkg`n >9 hf 9:
`e`e nn Zltf`ign hf u`n pltf`ign(;9)1 5 ;9»1 5 ;>=
(6-8)? 5 6(-8)»? 5 6-=
``` onon Zrlhuitl hf pltf`igns il` fd egsel fxpl`f`tf91 » 61 » ;1 5 (9 » 6 » ;)1
(-6)9 » 89 5 S(-6) » 8U9
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22
Iligf`tf hf pltf`igns il` fd egsel fxpl`f`tf=6 2 86 5 (= 2 8)6
7-? 2 (-9)-? 5 S7 2 (-9)U-?
```onon Zltf`ign hf u` prlhuitl
(? » 9 » 6)1 5 ?1 » 91 » 61
S(-1) » =U-? 5 (-1)-? » =-?
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Zltf`ign hf u` iligf`tf (>? 2 9)8
5 >?8
2 98
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Fbfrigigls
>) _fndgzn dns sgkugf`tfs lpfrnigl`fs il` pltf`igns hf `òefrls f`tfrls2
n) (-?)? » (-?)9 » (-?)8 Ylduigù`2 (-?)7 5 -6>?
o) (-=) » (-?)? » (-?): » (-?) Ylduigù`2 (-?)9 » (-?)? » (-?): » (-?) 5 (-?)1 5 18
i) (-?)-? » (-?)9 » (-?)8 Ylduigù`2 (-?)6 5 -9?
h) ?-? » ?-9 » ?8 Ylduigù`2 ?-> 5 >/?
f) ?? 2 ?9 Ylduigù`2 ?-> 5 >/?
a) ?-? 2 ?9 Ylduigù`2 ?-6 5 (>/?)6 5 >/9?
k) ?? 2 ?-9 Ylduigù`2 ?6 5 9?
m) ?-? 2 ?-9 Ylduigù`2 ?
g) S(-?)-?U 9 » (-?)9 » (-?)8 Ylduigù`2 (-?)-1 » (-?)9 » (-?)8 5 -? b)
S(-?)1 2 (-?)9 U9 » (-?) » (-?)-8 Ylduigù`2 S(-?)9U 9 » (-?) » (-?)-8 5 (-?)7 » (-?)-9 5 (-?)1 5 18
7/21/2019 Tema 4. Potencias y Raíces de Números Enteros
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Xfen 82 Zltf`igns y rnåifs hf `òefrls f`tfrls.
Kfen Gsnofd Enrå` Inonddfrl Zîkg`n >8 hf 9:
?) _fndgzn dns sgkugf`tfs lpfrnigl`fs il` pltf`igns hf `òefrls f`tfrls2
n)
(-9)> » (-9)9 » (-9)8 Ylduigù`2 (-9)= 5 1.61>
o)
(-?;) » (-9) » (-9)? » (-9): Ylduigù`2 (-9)9 » (-9) » (-9)? » (-9): 5 (-9)1 5 ;?7
i)
(-9)? » (-9)9 » (-9)-8 Ylduigù`2 -9
h)
9-?
» 9-8
» 98
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5 (>/9)?
5 >/7f) 6? 2 69 Ylduigù`2 6-> 5 >/6
a) 6-? 2 69 Ylduigù`2 6-6 5 (>/6)6 5 >/9.>?6
k) 6? 2 6-9 Ylduigù`2 66 5 9.>?6
m) 6-? 2 6-9 Ylduigù`2 6
g) (-9)> » S(-9)9 U? » (-9)-8 Ylduigù`2 (-9)> » (-9)1 » (-9)-8 5 (-9)9
b) S(-9)1 2 (-9)9 U9 » (-9): » (-9)∟8 Ylduigù`2 S(-9)9 U9 » (-9): » (-9)-8 5 (-9)7 » (-9)-8 5 (-9)6 5?89
>.=.
Lpfrnigl`fs il` pltf`igns>.=.>. Yuen y _fstn
Znrn plhfr suenr y rfstnr il` `òefrls fxprfsnhls ilel pltf`igns , hfof` fstnreudtgpdginhls plr dn egsen pltf`ign.
Znrn pltf`igns hf onsf >:, sg `l fs nså, hfofels mnifr quf tf`kn` dn pltf`ign hf >: fd egselfxpl`f`tf. Yf usn` f` dn `ltnigù` igf`tåagin.
Fbfepdls2
9 » >:6 + = » >:6 5 (9 + 6) » >:6 5 >> » >:6
? » >:1 ’ ; » >:6 5 ?: » >:6 ’ ; » >:6 5 (?: ’ ;) » >:6 5 >9 » >:6
>.=.?.
Eudtgpdginigù`
Znrn plhfr eudtgpdginr il` `òefrls fxprfsnhls ilel pltf`igns, sf npdgin` dns prlpgfhnhfs2
n) Zrlhuitl hf pltf`igns il` dn egsen onsf.
o) Zrlhuitl hf pltf`igns il` fd egsel fxpl`f`tf.
>.=.9.
Hgvgsgù`
Znrn plhfr hgvghgr il` `òefrls fxprfsnhls ilel pltf`igns, sf npdgin` dns prlpgfhnhfs2
n) Iligf`tf hf pltf`igns il` dn egsen onsf.
o) Iligf`tf hf pltf`igns il` fd egsel fxpl`f`tf.
>.=.8.
Zltf`ign
Znrn indiudnr pltf`igns, sf npdgin` dns prlpgfhnhfs2
n) Zltf`ign hf u`n pltf`ign.
o) Zltf`ign hf u` prlhuitl.
i) Zltf`ign hf u` iligf`tf.
7/21/2019 Tema 4. Potencias y Raíces de Números Enteros
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Xfen 82 Zltf`igns y rnåifs hf `òefrls f`tfrls.
Kfen Gsnofd Enrå` Inonddfrl Zîkg`n >6 hf 9:
?. _NÅIFY
?.>. Hfag`gigù`
Dn rnåz `-çsgen hf u` `òefrl n fs dn lpfrnigù` g`vfrsn hf dn pltf`ign, fs hfigr, fs u` `òefrl
o tnd quf no ` y sf rfprfsf`tn plr on` .
noon ``
Hl`hf2
` n sf ddnen fxprfsgù` hf u`n rnåz l rnhgind .
n fs fd rnhgin`hl.
` fs fd å`hgif hf dn rnåz .
o fs fd vndlr l rfsudtnhl hf dn rnåz .
Nså pufs, u`n rnåz fstî alrenhn plr u` rnhgin`hl y u` å`hgif.
Yf dff —dn rnåz `-çsgen hf u` `òefrl n fs gkund n o― l — dn rnåz hf å`hgif ` hf u` `òefrl n fs gkund
n o― .
@LXN2 F` dn indiudnhlrn igf`tåagin, sf usn dn tfidn y x .
?.?. Xgpls hf rnåifs fxnitns
?.?.>. _nåz iunhrnhn
Dn rnåz iunhrnhn fxnitn fs dn lpfrnigù` g`vfrsn hf fdfvnr nd iunhrnhl.
?
onon Fstl fs, _nhgin`hl 5 (_nåz fxnitn)?
Dn rnåz iunhrnhn fxnitn hf u` `òefrl n fs ltrl `òefrl o quf fdfvnhl nd iunhrnhl hn fd prgefrl,
y tgf`f hf rfstl :.
Yf dff —dn rnåz iunhrnhn fxnitn hf u` `òefrl n fs gkund n o― .
@LXN2
@l sf pl`f ? f` fd å`hgif hf dn rnåz iun`hl dn rnåz fs iunhrnhn. nn ?
Yldnef`tf dls iunhrnhls pfrafitls tgf`f` rnåz iunhrnhn fxnitn.
F` dn indiudnhlrn igf`tåagin, sf usn dn tfidn x .
Xlhl iunhrnhl pfrafitl tgf`f hls rnåifs iunhrnhns, u`n + y ltrn ’ , quf sl` hls `òefrlsf`tfrls lpufstls.
Fbfepdl2 >1
Dns rnåifs iunhrnhns hf >1 sl` 8 y ’8 , yn quf2 8 » 8 5 >1 y (-8) » (-8) 5 >1
@LXN2 Dls `òefrls f`tfrls `fkntgvls `l tgf`f` rnåz iunhrnhn, yn quf tlhn pltf`ign hffxpl`f`tf pnr fs sgfeprf plsgtgvn.
Fbfepdl2 >1 Dn rnåz iunhrnhn hf ’>1 `l sf pufhf indiudnr.
7/21/2019 Tema 4. Potencias y Raíces de Números Enteros
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Xfen 82 Zltf`igns y rnåifs hf `òefrls f`tfrls.
Kfen Gsnofd Enrå` Inonddfrl Zîkg`n >1 hf 9:
Fbfepdls2
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Yf dff —dn rnåz iunhrnhn fxnitn hf 8 fs ¾?―.
?.?.?. _nåz iòogin
Dn rnåz iòogin fxnitn fs dn lpfrnigù` g`vfrsn hf fdfvnr nd iuol.
99 onon Fstl fs, _nhgin`hl 5 (_nåz fxnitn)9
Dn rnåz iòogin fxnitn hf u` `òefrl n fs ltrl `òefrl o quf fdfvnhl nd iuol hn fd prgefrl, y
tgf`f hf rfstl :.Yf dff —dn rnåz iòogin fxnitn hf u` `òefrl n fs gkund n o― .
@LXN2
Yldnef`tf dls iuols pfrafitls tgf`f` rnåz iòogin fxnitn.
F` dn indiudnhlrn igf`tåagin, sf usn dn tfidn 9 x .
Xlhl iuol pfrafitl tgf`f u`n rnåz iòogin, + .
Fbfepdl2 9
?; Dn rnåz iòogin hf ?; fs 9 , yn quf2 9 » 9 » 9 5 ?;
@LXN2 Dls `òefrls f`tfrls `fkntgvls så tgf`f` rnåz iòogin, pfrl fs u`n rnåz iòogin, ’ .
Fbfepdl2 9 ?;
Dn rnåz iòogin hf ’?; fs 9 , yn quf2 (’9) » (’9) » (’9) 5 ’?;
Fbfepdls2 _nhgin`hl +
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188818
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Yf dff —dn rnåz iòogin fxnitn hf = fs ?―.
7/21/2019 Tema 4. Potencias y Raíces de Números Enteros
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Xfen 82 Zltf`igns y rnåifs hf `òefrls f`tfrls.
Kfen Gsnofd Enrå` Inonddfrl Zîkg`n >; hf 9:
Fbfepdls2 _nhgin`hl ’
>?666>?6
188818
?;99?;
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99
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6>?==6>?
989;;989
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99
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Yf dff —dn rnåz iòogin fxnitn hf ’= fs ’?―.
?.?.9. Fbfrigigls
>) Indiudn, sg fxgstf, dns sgkugf`tfs rnåifs fxnitns2
n) 18 Ylduigù`2 ==?
o)
=> Ylduigù`2 77?
i) 8 @l tgf`f slduigù` plrquf fd rnhgin`hl fs ’ y fd å`hgif hf dn rnåz fs pnr.
h) >?> Ylduigù`2 >>>>
?
f)
8 1?6 Ylduigù`2 668
a) 6 ?89 Ylduigù`2 99 6
k) 1 ;?7 @l tgf`f slduigù` plrquf fd rnhgin`hl fs ’ y fd å`hgif hf dn rnåz fs pnr.
?) Indiudn, sg fxgstf, dns sgkugf`tfs rnåifs fxnitns2
n) ?7 Ylduigù`2 7=>
o) ;> Ylduigù`2 > @l tgf`f slduigù` plrquf fd rnhgin`hl fs ’
y fd å`hgif hf dn rnåz fs pnr.
i) 99 ?
Ylduigù`2 ?;9 9
@l tgf`f slduigù` plrquf fd rnhgin`hl fs ’
y fd å`hgif hf dn rnåz fs pnr.
h)
?
8
?
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Ylduigù`2 ?8?
?
f) 99 Ylduigù`2 ?; @l tgf`f slduigù` plrquf fd rnhgin`hl fs ’
y fd å`hgif hf dn rnåz fs pnr.
a)
6
9
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=
Ylduigù`2
8>1??
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6
7
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7/21/2019 Tema 4. Potencias y Raíces de Números Enteros
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Xfen 82 Zltf`igns y rnåifs hf `òefrls f`tfrls.
Kfen Gsnofd Enrå` Inonddfrl Zîkg`n >= hf 9:
9) Indiudn, sg fxgstf, dns sgkugf`tfs rnåifs fxnitns2
n) 9 ?; Ylduigù`2 +9
o) 9 ?; Ylduigù`2 ∟9
i) 8 >1 Ylduigù`2 ¾?
h) 8 >1 @l tgf`f slduigù` plrquf fd rnhgin`hl fs ’ y fd å`hgif hf dn rnåz fs pnr.
f)
6 9? Ylduigù`2 +6
a)
6 9? Ylduigù`2 ∟6
k) 1 18 Ylduigù`2 ¾?
m) 1 18 @l tgf`f slduigù` plrquf fd rnhgin`hl fs ’ y fd å`hgif hf dn rnåz fs pnr.
g) ; > Ylduigù`2 +>
b) ; > Ylduigù`2 ∟>
c) = > Ylduigù`2 ¾>
d) = > @l tgf`f slduigù` plrquf fd rnhgin`hl fs ’ y fd å`hgif hf dn rnåz fs pnr.
?.9. Xgpls hf rnåifs f`tfrns
?.9.>.
_nåz iunhrnhn
Dn rnåz iunhrnhn f`tfrn hf u` `òefrl n fs fd enylr `òefrl f`tfrl o iuyl iunhrnhl fs ef`lrquf fd prgefrl, y `l tgf`f hf rfstl :.
rfstlonon ? Fstl fs, _nhgin`hl 5 (_nåz f`tfrn)? + _fstl
@LXN2
Yg u` `òefrl `l fs iunhrnhl pfrafitl, su rnåz fs f`tfrn.
Fd rfstl hf dn rnåz iunhrnhn f`tfrn hfof sfr ef`lr quf fd hlodf hf dn rnåz eîs >.
_fstl 4 ? » _nåz + >
Ileprlonigù`2 V`n rnåz iunhrnhn fstî ogf` indiudnhn sg iuepdf fstns hls il`hgigl`fs2
>) (_nåz)? + _fstl 5 _nhgin`hl?) _fstl 4 ? » _nåz + >
Fbfepdls2
n)
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??
Yf dff —dn rnåz iunhrnhn f`tfrn hf ?1 fs 6―.
o) 8?8>=?>1>=>= ? oo ?66>=8>1
??
Yf dff —dn rnåz iunhrnhn f`tfrn hf >= fs 8―.
i) ??;661876666 ? oo 18=66;87 ??
Yf dff —dn rnåz iunhrnhn f`tfrn hf 66 fs ;―.
7/21/2019 Tema 4. Potencias y Raíces de Números Enteros
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Xfen 82 Zltf`igns y rnåifs hf `òefrls f`tfrls.
Kfen Gsnofd Enrå` Inonddfrl Zîkg`n >7 hf 9:
?.9.?. _nåz iòogin
Dn rnåz iòogin f`tfrn hf u` `òefrl n fs fd enylr `òefrl f`tfrl o iuyl iuol fs ef`lr quf fdprgefrl, y `l tgf`f hf rfstl :.
rfstlonon 9 Fstl fs, _nhgin`hl 5 (_nåz f`tfrn)9 + _fstl
@LXN2 Yg u` `òefrl `l fs iuol pfrafitl, su rnåz fs f`tfrn.Ileprlonigù`2 V`n rnåz iòogin fstî ogf` indiudnhn sg iuepdf dn sgkugf`tf il`hgigù`2
(_nåz)9 + _fstl 5 _nhgin`hl
Fbfepdls2
n) ?9?79?7 99 188?79?;
99
Yf dff —dn rnåz iòogin f`tfrn hf ?7 fs 9―.
o) ?9?79?7 99 188?79?;
99
Yf dff —dn rnåz iòogin f`tfrn hf ?7 fs 9―. i) >???:??:
99 ?;9?:?= 99
Yf dff —dn rnåz iòogin f`tfrn hf ?: fs ?―.
?.8. Ylduigl`fs sfkò` fd å`hgif hf dn rnåz y fd rnhgin`hl
?.8.>.
_nhgin`hl plsgtgvl
Yg fd rnhgin`hl fs plsgtgvl, fxgstf sgfeprf fd vndlr hf dn rnåz.
>) Å`hgif hf dn rnåz pnr. Xgf`f sgfeprf hls slduigl`fs2 u`n plsgtgvn y ltrn `fkntgvn.
Fbfepdls2
:::.:::.>>:,:::.:::.>>:>::::.:::.>
1?66,1?6661?6
8?,8??8
111
888
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?) Å`hgif hf dn rnåz gepnr. Yùdl tgf`f u`n slduigù` plsgtgvn.
Fbfepdls2
9???9?
>?666>?6
?;99?;
66
99
99
?.8.?. _nhgin`hl ̀ fkntgvl
Yg fd rnhgin`hl fs `fkntgvl, pufhf fxgstgr fd vndlr hf dn rnåz.
>) Å`hgif hf dn rnåz pnr. @l tgf`f slduigù` plrquf `l fxgstf u` `òefrl rfnd.
Fbfepdls2
1
8
:::.:::.>
1?6
8
@l tgf`f` slduigù`
7/21/2019 Tema 4. Potencias y Raíces de Números Enteros
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Xfen 82 Zltf`igns y rnåifs hf `òefrls f`tfrls.
Kfen Gsnofd Enrå` Inonddfrl Zîkg`n ?: hf 9:
?) Å`hgif hf dn rnåz gepnr. Yùdl tgf`f u`n slduigù` `fkntgvn.
Fbfepdls2
9???9?
>?666>?6
?;99?;
66
99
99
?.8.9. Xnodn-_fsuef`
_nhgin`hl Å`hgif hf dn rnåz _fsudtnhl hf dn rnåz Fbfepdl
+Znr ? slduigl`fs2 + y ’ ?8 plrquf 8?
? y 8? ?
Gepnr > slduigù` + ?=9 plrquf =?9
’ Znr @l tgf`f slduigù` 8 plrquf rnhgin`hl fs `fkntgvl
:8 f å`hgif hf dn rnåz fs pnr (?)
Gepnr > slduigù` ’ ?=9 plrquf =? 9
?.6. Iîdiudl hf dn rnåz iunhrnhn f`tfrn
Hls alrens hf indiudnr dn rnåz iunhrnhn f`tfrn2
n) Iîdiudl hf alren nprlxgenhn.
o) Iîdiudl il` fd ndklrgtel.
?.6.>. Iîdiudl hf alren nprlxgenhn
Dn rnåz iunhrnhn f`tfrn hf u` `òefrl fs fd enylr f`tfrl iuyl iunhrnhl fs ef`lr quf hgiml
`òefrl.
Fd rfstl hf dn rnåz iunhrnhn hf u` `òefrl fs gkund n dn hgafrf`ign f`trf fd `òefrl y fd
iunhrnhl hf su rnåz .
Fbfepdl >2 89
Fd `òefrl 89 `l fs u` iunhrnhl pfrafitl, plr dl tn`tl `l tgf`f rnåz iunhrnhn fxnitn.
Yg losfrvnels 1? 5 91 4 89 4 87 5 ;?
Dufkl, 1 fs fd enylr f`tfrl iuyl iunhrnhl fs ef`lr quf 89.
Yf hgif quf 1 fs rnåz iunhrnhn f`tfrn hf 89 y sf fsirgof2
189
Ilel 89 - 1? 5 89 ’ 91 5 ;, fd rfstl hf dn rnåz iunhrnhn f`tfrn fs ;.
Ylduigù`2 189 y rfstl 5 ;
;189189 ?
Yf dff —dn rnåz iunhrnhn f`tfrn hf 89 fs 1 y su rfstl ;―.
Ileprlonigù`2
>)
1? + ; 5 91 + ; 5 89
?) ; 4 ? » 1 + > 5 >9
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Xfen 82 Zltf`igns y rnåifs hf `òefrls f`tfrls.
Kfen Gsnofd Enrå` Inonddfrl Zîkg`n ?> hf 9:
Fbfepdl ?2 ?1
>²
Ousinels hls iunhrnhls pfrafitls ifrin`ls n ?1, quf sl`2
6? 5 ?6 4 ?1
1? 5 91 0 ?1
?² Fdfkgels fd iunhrnhl pfrafitl quf eîs sf nprlxgen plr hfonbl hf ?1, quf fs ?6.9² Hfspuçs, indiudnels fd rfstl, quf fs dn hgafrf`ign f`trf fd `òefrl hnhl n y fd iunhrnhl hf
dn rnåz f`tfrn o.
?1 ’ ?6 5 >
8²
Zrlonels quf fd rfsudtnhl hf dn rnåz iunhrnhn sfn ilrrfitl, pnrn fddl, sf tgf`f qufiuepdgr quf _nhgin`hl 5 (_nåz f`tfrn) ? + _fstl.
?1 5 ?6? + >
Ylduigù`2 6?1 y rfstl 5 >
>6?16?1 ?
Yf dff —dn rnåz iunhrnhn f`tfrn hf ?1 fs 6 y su rfstl >―.
Ileprlonigù`2
>) 6? + > 5 ?6 + > 5 ?1
?) 6 4 ? » 6 + > 5 >>
?.6.?. Iîdiudl il` fd ndklrgtel
Fbfepdl >2 >87
>² Yg fd rnhgin`hl tgf`f eîs hf hls igarns, sfpnrnels dns igarns f` krupls hf hls fepfzn`hl
plr dn hfrfimn.
?²
Indiudnels dn rnåz iunhrnhn (f`tfrn l fxnitn) hfd prgefr krupl hf igarns plr dn gzqugfrhn.
¼^uç `òefrl fdfvnhl nd iunhrnhl hn ><
> fs u` iunhrnhl pfrafitl plrquf (>? 5 >).
Dn prgefrn igarn hf dn rnåz iunhrnhn fs > y dn ildlinels f` dn insgddn ilrrfspl`hgf`tf.
9² Fd iunhrnhl hf dn rnåz lotf`ghn sf rfstn nd prgefr krupl hf igarns quf npnrfif` f` fdrnhgin`hl.
Fd iunhrnhl hf dn rnåz > fs >, sf dl rfstnels n > y lotf`fels :.
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Xfen 82 Zltf`igns y rnåifs hf `òefrls f`tfrls.
Kfen Gsnofd Enrå` Inonddfrl Zîkg`n ?? hf 9:
8² Hftrîs hfd rfstl ildlinels fd sgkugf`tf krupl hf igarns hfd rnhgin`hl y sfpnrnels hfd`òefrl alrenhl su òdtgen igarn.
Onbnels 87, sgf`hl dn in`tghnh lpfrnodf hfd rnhgin`hl :87.
6² Hfonbl hf dn rnåz mnddnhn fsirgogels su hlodf y hgvghgels dl quf rfstn plr fd hlodf hfdn rnåz n`tfrglr. Fd iligf`tf lotf`ghl sf pl`f n dn hfrfimn hfd hlodf hf dn rnåz y fd `òefrlalrenhl sf eudtgpdgin plr fstf iligf`tf.
:8 2 ? 5 ? 4 7
Xlenels ilel rfsudtnhl ?, yn quf fs u`n ò`gin igarn quf vn hfd : nd 7.
1² Ileprlonels quf fd `òefrl alrenhl fs g`afrglr n dn in`tghnh lpfrnodf hfd rnhgin`hl .Zufs sg muogçsfels lotf`ghl u` vndlr supfrglr n dn in`tghnh lpfrnodf hfd rnhgin`hl,mnorånels prlonhl plr dn n`tfrglr in`tghnh lpfrnodf y nså suifsgvnef`tf mnstn f`il`trnru` vndlr g`afrglr.
?? » ? 5 88 4 87
;²
Fd iligf`tf lotf`ghl sf nþnhf n dn hfrfimn hf dn rnåz.
Yuogels fd ? n dn rnåz.
=² Hfspuçs, rfstnels n dn in`tghnh lpfrnodf hfd rnhgin`hl fd `òefrl mnddnhl hfonbl hf dnrnåz.
87 ’ 88 5 6
7² Zrlonels quf fd rfsudtnhl hf dn rnåz iunhrnhn sfn ilrrfitl, pnrn fddl, sf tgf`f qufiuepdgr quf _nhgin`hl 5 (_nåz f`tfrn) ? + _fstl.
>87 5 >?? + 6
Ylduigù`2 >?>87 y rfstl 5 6
6>?>87>?>87 ?
Yf dff —dn rnåz iunhrnhn f`tfrn hf >87 fs >? y su rfstl 6―.
Ileprlonigù`2
>) >?? + 6 5 >88 + 6 5 >87
?) >? 4 ? » >? + 6 5 ?7
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Xfen 82 Zltf`igns y rnåifs hf `òefrls f`tfrls.
Kfen Gsnofd Enrå` Inonddfrl Zîkg`n ?9 hf 9:
Fbfepdl ?2 =7??6
>²
Yg fd rnhgin`hl tgf`f eîs hf hls igarns, sfpnrnels dns igarns f` krupls hf hls fepfzn`hl
plr dn hfrfimn.
?² Indiudnels dn rnåz iunhrnhn (f`tfrn l fxnitn) hfd prgefr krupl hf igarns plr dn gzqugfrhn.¼^uç `òefrl fdfvnhl nd iunhrnhl hn =<
= `l fs u` iunhrnhl pfrafitl, pfrl fstî ileprf`hghl f`trf hls iunhrnhls pfrafitls28 y 7.
Xlenels dn rnåz iunhrnhn hfd iunhrnhl pfrafitl quf sf nifrquf eîs n =, quf fs2 ?(?? 5 8).
Dn prgefrn igarn hf dn rnåz iunhrnhn fs ? y dn ildlinels f` dn insgddn ilrrfspl`hgf`tf.
9² Fd iunhrnhl hf dn rnåz lotf`ghn sf rfstn nd prgefr krupl hf igarns quf npnrfif` f` fdrnhgin`hl.
Fd iunhrnhl hf dn rnåz ? fs 8, sf dl rfstnels n = y lotf`fels 8.
8² Hftrîs hfd rfstl ildlinels fd sgkugf`tf krupl hf igarns hfd rnhgin`hl y sfpnrnels hfd`òefrl alrenhl su òdtgen igarn.
Onbnels 7?, sgf`hl dn in`tghnh lpfrnodf hfd rnhgin`hl 87?.
6² Hfonbl hf dn rnåz mnddnhn fsirgogels su hlodf y hgvghgels dl quf rfstn plr fd hlodf hfdn rnåz n`tfrglr. Fd iligf`tf lotf`ghl sf pl`f n dn hfrfimn hfd hlodf hf dn rnåz y fd `òefrlalrenhl sf eudtgpdgin plr fstf iligf`tf.
87 2 8 5 >?,?6 0 7
Xlenels ilel rfsudtnhl 7, yn quf sùdl pufhf sfr u`n ò`gin igarn quf vnyn hfd : nd 7.
1²
Ileprlonels quf fd `òefrl alrenhl fs g`afrglr n dn in`tghnh lpfrnodf hfd rnhgin`hl .Zufs sg muogçsfels lotf`ghl u` vndlr supfrglr n dn in`tghnh lpfrnodf hfd rnhgin`hl,mnorånels prlonhl plr dn n`tfrglr in`tghnh lpfrnodf (=, ;, …) y nså suifsgvnef`tf mnstn
f`il`trnr u` vndlr g`afrglr.
87 » 7 5 8>> 4 87>
;² Fd iligf`tf lotf`ghl sf nþnhf n dn hfrfimn hf dn rnåz.Yuogels fd 7 n dn rnåz.
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Xfen 82 Zltf`igns y rnåifs hf `òefrls f`tfrls.
Kfen Gsnofd Enrå` Inonddfrl Zîkg`n ?8 hf 9:
=² Hfspuçs, rfstnels n dn in`tghnh lpfrnodf hfd rnhgin`hl fd `òefrl mnddnhl hfonbl hf dnrnåz.
87? ’ 88> 5 6>
7² Onbnels fd sgkugf`tf pnr hf igarns.
Onbnels ?6, sgf`hl dn in`tghnh lpfrnodf hfd rnhgin`hl :6>?6.>:² _fpftgels dls pnsls n`tfrglrfs.
6>? 2 6= 5 =,= 0 =, prlonels plr 7.
Ilel 6=7 » 7 5 6.9:> 0 6.>?6, prlonels plr =.Ilel 6== » = 5 8.;:8 4 6.>?6, suogels fd = n dn rnåz.
_fstnels n dn in`tghnh lpfrnodf hfd rnhgin`hl 6.>?6 fd `òefrl mnddnhl hfonbl hf dn rnåz8.;:8.
>>² Zrlonels quf fd rfsudtnhl hf dn rnåz iunhrnhn sfn ilrrfitl, pnrn fddl, sf tgf`f qufiuepdgr quf _nhgin`hl 5 (_nåz f`tfrn) ? + _fstl.
=7.??6 5 ?7=? + 8?>
7/21/2019 Tema 4. Potencias y Raíces de Números Enteros
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Xfen 82 Zltf`igns y rnåifs hf `òefrls f`tfrls.
Kfen Gsnofd Enrå` Inonddfrl Zîkg`n ?6 hf 9:
Ylduigù`2 ?7==7.??6 y rfstl 5 8?>
8?>?7==7.??6?7==7.??6 ?
Yf dff —dn rnåz iunhrnhn f`tfrn hf =7.??6 fs ?7= y su rfstl 8?>―.
Ileprlonigù`2
>) ?7=? + 8?> 5 ==.=:8 + 8?> 5 =7.??6?) ?7= 4 ? » ?7= + 8?> 5 >.:>;
Fbfepdl 92 189
Znrn fd iîdiudl hf dn rnåz iunhrnhn hf 189, sfkugels fstls pnsls2
>² Hgvghgr fd rnhgin`hl f` krupls hf hls igarns, fepfzn`hl plr dn hfrfimn. Fd `òefrl hfkrupls fs gkund nd `òefrl hf igarns hf dn rnåz iunhrnhn.
1?² Indiudnr dn rnåz iunhrnhn hfd prgefr krupl hf dn gzqugfrhn.
Dufkl ? fs fd `òefrl hf hfif`ns hf dn rnåz
9²
_fstnr hfd prgefr krupl fd iunhrnhl hf su rnåz f`tfrn. Nþnhgr n dn hgafrf`ign dns hlsigarns sgkugf`tfs hfd rnhgin`hl.
1 ? <
-8
? 89
8² Eudtgpdginr plr hls dn prgefrn igarn hf dn rnåz (? » ? 5 8). Indiudnr fd ef`lr f`tfrl h tndquf 8 h » h sf pufhf rfstnr hfd rnhgin`hl3 h sfrî dn sfku`hn igarn hf dn rnåz.
8? » ? 5 =8 3 89 » 9 5 >?7 , 88 » 8 5 >;1, 86 » 6 5 ??6
81 » 1 5 ?;1 0 ?89
1 ? <
-8 86 » 6 5 ??6
? 89
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Xfen 82 Zltf`igns y rnåifs hf `òefrls f`tfrls.
Kfen Gsnofd Enrå` Inonddfrl Zîkg`n ?1 hf 9:
6² _fstnr hf dn hgafrf`ign n`tfrglr 86 » 6. Fd rfstl hf dn rnåz fs dn `ufvn hgafrf`ign.
1 ? 6
-8 86 » 6 5 ??6
? 89
-? ?6
>=
Ylduigù`2 ?6189 y rfstl 5 >=
>=?6189?6189 ?
Yf dff —dn rnåz iunhrnhn f`tfrn hf 189 fs ?6 y su rfstl >=―.
Ileprlonigù`2
>) (?6) ? + >= 5 1?6 + >= 5 189
?) >= 4 ? » ?6 + > 56>
?.6.?.>. Fbfrigigl
>) Indiudn dns sgkugf`tfs rnåifs il` fd ndklrgtel2
n) ?18
Ylduigù`2
o)
?61.1 Ylduigù`2
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Xfen 82 Zltf`igns y rnåifs hf `òefrls f`tfrls.
Kfen Gsnofd Enrå` Inonddfrl Zîkg`n ?; hf 9:
i) 1;6.;?
Ylduigù`2
?.1. Lpfrnigl`fs il` rnåifs iunhrnhns fxnitns
?.1.>.
Yuen y _fstnZnrn suenr y rfstnr rnåifs, hfof` tf`fr fd egsel å`hgif hf dn rnåz y il`tf`fr fd egsel rnhgind.
Yg fs nså, sf suen` dls `òefrls quf fstî` aufrn hf fddls (ilfagigf`tfs) y fd rnhgind sf hfbn gkund.
Fbfepdls2
?;?>69??6?9
99999 ;9;>?8;;?;8
?.1.?. Eudtgpdginigù`
Fd prlhuitl hf hls l eîs rnåifs iunhrnhns fxnitns2n) Fs u`n rnåz iunhrnhn fxnitn.
o) Yu rnhgin`hl fs gkund nd prlhuitl hf dls rnhgin`hl hf dls anitlrfs.
Fbfepdls2
687171=>91=>916871=>91 ??
87;6;687?687?696;687?6 ??
?.1.9.
Hgvgsgù`
Fd iligf`tf hf hls l eîs rnåifs iunhrnhns fxnitns2
n) Fs u`n rnåz iunhrnhn fxnitn.
o) Yu rnhgin`hl fs gkund nd iligf`tf hf dls rnhgin`hl hf dls anitlrfs.
Fbfepdls2
?92192172917291?9217291 ??
?82=82=>1218>1218?82=>1218 ??
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Xfen 82 Zltf`igns y rnåifs hf `òefrls f`tfrls.
Kfen Gsnofd Enrå` Inonddfrl Zîkg`n ?= hf 9:
?.1.8. Zltf`ign hf u`n rnåz
Dn pltf`ign hf onsf u`n rnåz iunhrnhn fxnitn2
n)
Fs u`n rnåz iunhrnhn fxnitn.
o) Yu rnhgin`hl fs gkund nd rnhgin`hl hf pnrtghn fdfvnhl nd fxpl`f`tf hf dn pltf`ign.
Fbfepdls2 8
88
8
?6?6?6?6?6?6?6?6
11
11
=>=>=>=>=>=>=>=>=>=>
?.;. Dn rnåz ilel pltf`ign
Dn rnåz hf u`n pltf`ign fs gkund nd rnhgin`hl fdfvnhl nd rfsudtnhl hf hgvghgr su fxpl`f`tf plr fdå`hgif hf dn rnåz.
`
e
` e
x x
Fbfepdls2 ?
=
? ==666 9
>:
9 >:??
9. ZLXF@IGNY W _NÅIFY
9.>. _filrhnr
n) Hfsilepl`fr dls `òefrls f` anitlrfs prgels.
o) Ygepdgaginr dls `òefrls npdgin`hl dns prlpgfhnhfs ilrrfspl`hgf`tfs.
i) V`n vfz lotf`ghls dls `òefrls sgepdgaginhls, usnh dn indiudnhlrn pnrn indiudnr fd `òefrlrfsudtn`tf.
8. LZF_NIGL@FY ILEOG@NHNY IL@ ZLXF@IGNY W _NÅIFY
8.>. G`trlhuiigù`
Nd rfsldvfr fxprfsgl`fs il` lpfrnigl`fs ileog`nhns, hfofels tf`fr f` iuf`tn dns `lrens hfddf`kunbf entfeîtgil. Fstns `lrens nsfkurn` quf inhn fxprfsgù` tf`kn u` sgk`gaginhl y u`n slduigù`ò`gils.
8.?. _fkdns hf prglrghnhDns rfkdns pnrn rfndgznr dns lpfrnigl`fs hf `òefrls `nturndfs l prglrghnh hf dns lpfrnigl`fs
sl` dns sgkugf`tfs2
>) Fafitunels dns lpfrnigl`fs f`trf pnrç`tfsgs, ilrimftfs y ddnvfs.
?) Indiudnels dns pltf`igns y rnåifs.
9) Fafitunels dls prlhuitls (eudtgpdginigl`fs) y iligf`tfs (hgvgsgl`fs) hf gzqugfrhn nhfrfimn.
8) _fndgznels dns suens y rfstns.
@LXN2 Fs geplrtn`tf rfspftnr fd lrhf` hf dns lpfrnigl`fs pnrn lotf`fr fd rfsudtnhl ilrrfitl.
7/21/2019 Tema 4. Potencias y Raíces de Números Enteros
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Xfen 82 Zltf`igns y rnåifs hf `òefrls f`tfrls.
Kfen Gsnofd Enrå` Inonddfrl Zîkg`n ?7 hf 9:
8.9. Xgpls hf lpfrnigl`fs ileog`nhns
8.9.>. Lpfrnigl`fs ileog`nhns il` pltf`igns y rnåifs iunhrnhns fxnitns
Znrn rfsldvfrdns, prgefrl fafitunrfels dns pltf`igns y dns rnåifs iunhrnhns, il` dl iund `lsqufhnrfels il` lpfrnigl`fs ileog`nhns hf `òefrls f`tfrls y sfkugels fd lrhf` hf dns egsens.
@LXN2 F` dns rnåifs iunhrnhns, mny hls slduigl`fs, u`n + y ltrn ’ . Yùdl usnrfels dn + .Fbfepdls2
n) (-9)62 (-9)9 + >88 » 6? 5 7 + >? » ?6 5 7 + 9:: 5 9:7
(-9)6 2 (-9)9 5 (-9)? 5 7 >88 5 >? 6? 5 ?6
o) 1?6 » ( 6 ’ 99 ) + 8: + >?> 5 ?6 » (-??) + > + >> 5 -66: + > + >> 5 -69=
1?6 5 ?6 6 ’ 99 5 6 ’ ?; 5 -?? 8: 5 > >?> 5 >>
8.9.?.
Lpfrnigl`fs ileog`nhns il` pltf`igns y rnåifs iòogins fxnitns
Znrn rfsldvfrdns, prgefrl fafitunrfels dns pltf`igns y dns rnåifs iòogins, il` dl iund `lsqufhnrfels il` lpfrnigl`fs ileog`nhns hf `òefrls f`tfrls y sfkugels fd lrhf` hf dns egsens.
@LXN2 F` dn rnåifs iòogins, mny u`n slduigù` + ù - .
Fbfepdls2
n) 9 989 » ( >:? ’ =? ) + 9 = 2 8: 5 ; » 91 + (-?) 2 > 5 ?6? - ? 5 ?6:
9 989 5 ; >:? ’ =? 5 >:: ’ 18 5 91 8: 5 > 9 = 5 -?
o) >:9 + ;? + 9 ?; 2 (’1): 5 >.::: + 87 + (’9) 2 > 5 >.:87 ’ 9 5 >.:81
9 ?; 5 ’9 (’1): 5 >
8.9.9. Lpfrnigl`fs ileog`nhns il` pltf`igns y rnåifs fxnitns
Fbfepdls2
n) :::.>: ’ 9 ;?7 » ?9 2 9 + 9 >?6 » 6-> 5 >:: ’ 7 » = 2 9 + (’6) 2 6 5 >:: ’ ;? 2 9 ’ > 5
5 >:: ’ ?8 ’ > 5 ;6
:::.>: 5 >:: 9 ;?7 5 7 9 >?6 5 -6
o) =9 ’ 18 2 ?? + 9 18 2 8-> 5 6>? ’ = 2 8 + 8 » 8 5 6>? ’ ? + >1 5 6?1
18 5 = 9 18 5 8
8.8. Fbfrigigls
>)
_fndgzn dns sgkugf`tfs lpfrnigl`fs ileog`nhns tf`gf`hl f` iuf`tn su prglrghnh2
n)
? + 6 » (? » 9)¹ Ylduigù`2 ? + 6 » (1)¹ 5 ? + 6 » ?>1 5 ? + >.:=: 5 >.:=?
o) ; » 9 + S1 + ? » (?9 2 8 + 9 » ?) ’ ; » 8 U + 7 2 9
Ylduigù`2 ?> + S1 + ? » (?9 2 ?? + 1) ’ >8U + 9 5
5 ?> + S1 + ? » (? + 1) ’ >8U + 9 5 ?> + (1 + ? » = ’ >8) + 9 5
5 ?> + (1 + >1 ’ >8) + 9 5 ?> + = + 9 5 9?
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Xfen 82 Zltf`igns y rnåifs hf `òefrls f`tfrls.
?) _fndgzn dns sgkugf`tfs lpfrnigl`fs il` `òefrls f`tfrls2
n)
(9 - =) + S6 - (-?)U Ylduigù`2 -6 + S6 + ?U 5 -6 + ;5 ?
o)
6 - S1 - ? - (> - =) - 9 + 1U + 6 Ylduigù`2 6 - S1 - ? - (-;) - 9 + 1U + 6 5
5 6 - S1 - ? + ; - 9 + 1U + 6 5 6 - >8 + 6 5 ∟8
i)
7 2 S1 2 (- ?)U Ylduigù`2 7 2 (-9) 5 -9h) S(-?)6 - (-9)9U? Ylduigù`2 S-9? - (-?;)U 5 S-9? + ?;U? 5 (-6)? 5 ?6
f) (6 + 9 » ? 2 1 - 8 ) » (8 2 ? - 9 + 1) 2 (; - = 2 ? - ?) ?
Ylduigù`2 (6 + 1 2 1 - 8 ) » (8 2 ? - 9 + 1) 2 (; - = 2 ? - ?)? 5
5 (6 + > - 8 ) » (? - 9 + 1) 2 (; - 8 - ?)? 5
5 ? » 6 2 >? 5 ? » 6 2 > 5 >: 2 > 5 >:
a) S(>; - >6)9 + (; - >?)?U 2 S(1 - ;) » (>? - ?9)U Ylduigù`2 S(?)9 + (-6)?U 2 S(->) » (->>)U 5
5 (= + ?6) 2 S(->) » (->>)U 5 (= + ?6) 2 >> 5 99 2 >> 5 9
9) _fndgzn dns sgkugf`tfs lpfrnigl`fs il` `òefrls f`tfrls2
n)
(; - ? + 8) - (? - 6) Ylduigù`2 7 - (-9) 5 7 + 9 5 >?
o)
> - (6 - 9 + ?) - S6 - (1 - 9 + >) - ?U Ylduigù`2 > - 8 - S6 - 8 - ?U 5 > - 8 ’ (->) 5 > ’ 8 + > 5 -?
i) ->? » 9 + >= 2 (->? 2 1 + =) Ylduigù`2 -91 + >= 2 (-? + =) 5 -91 + >= 2 1 5 -91 + 9 5 -99
h) S(-?)6 » (-9)?U 2 (-?)? Ylduigù`2 (-9? » 7) 2 8 5 -?== 2 8 5 -;?
f) 1 + {8 - (>; - (8 » 8)U + 9} - 6 Ylduigù`2 1 + {8 - S>; - >1U + 9} - 6 5 1 + {8 - > + 9} - 6 5
5 1 + 1 - 6 5 ;a) ?9 + >: 2 ? + 6 » 9 + 8 - 6 » ? - = + 8 » ? ? - >1 2 8
Ylduigù`2 = + >: 2 ? + 6 » 9 + 8 - 6 » ? - = + 8 » 8 - >1 2 8 5
5 = + 6 + >6 + 8 - >: - = + >1 - 8 5 ?1
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