tema 4. potencias y raíces de números enteros

7/21/2019 Tema 4. Potencias y Raíces de Números Enteros

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Xfen 82 Zltfìgns y rnåifs hf `òefrls f`tfrls.

Kfen Gsnofd Enrå` Inonddfrl Zîkg`n > hf 9:

Å@HGIF

>. ZLXF@IGNY 9

>.>. Hfag`gigù` .................................................................................................................................................... 9

>.>.>. Fbfrigigls ............... ............................................................... .............................................................. .......... 9

>.?. Zltfìgns 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 pltfìgns ............................................................................................................................ 1

>.1.>. Onsf plsgtgvn .................................................................................. .............................................................. 1

>.1.?. Onsf `fkntgvn........ .............................................................. ............................................................... .......... 1

>.1.9. Xnodn-_fsuef` ......................................................... ................................................................. .................. ;

>.1.8. Fbfrigigls ....................................................................................... .............................................................. ;

>.;. Zrlpgfhnhfs ............................................................................................................................................... =

>.;.>. Xnodn ’ _fsuef` ....................................................... ................................................................. ................ >?

>.;.?. Fbfrigigls ................................................................. ............................................................... ................... >9

>.=. Lpfrnigl`fs il` pltfìgns.................................................................................................................. >8

>.=.>. Yuen y _fstn ...................................... ................................................................. ...................................... >8

>.=.?. Eudtgpdginigù` ........................................................... ................................................................. ................ >8

>.=.9. Hgvgsgù` ........................................................... ................................................................. ........................... >8

>.=.8. Zltfìgn .......................................................... ................................................................. ........................... >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 ....................................................................................................................... >=




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. Zltfìgn hf u`n rnåz .......................................................... .............................................................. ....... ?=

?.;. Dn rnåz ilel pltfìgn ......................................................................................................................... ?=

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` pltfìgns y rnåifs iunhrnhns fxnitns ..... ..................................... ?7

8.9.?. Lpfrnigl`fs ileog`nhns il` pltfìgns y rnåifs iòogins fxnitns ............................................... ?7

8.9.9. Lpfrnigl`fs ileog`nhns il` pltfìgns y rnåifs fxnitns ....................................................... ....... ?7

8.8. Fbfrigigls ................................................................................................................................................ ?7




Kfen Gsnofd Enrå` Inonddfrl Zîkg`n 9 hf 9:

>. ZLXF@IGNY

>.>. Hfag`gigù`

V`n pltfìgn 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 pltfìgn 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 pltfìgn 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 pltfìgn 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 pltfìgn fstî alrenhn plr u`n onsf y u` fxpl`f`tf.

Znrn rfsldvfr u`n pltfìgn 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 pltfìgn 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 pltfìgn 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





?) Ilepdftn dn tnodn2

Zrlhuitl Zltfìgn 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

>.?. Zltfìgns hf onsf >:

Il` pltfìgn 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 pltfìgns hf onsf >: rfsudtn sfìgddl, pufs fd `òefrl hf ifrls ilgìghf il` fd

fxpl`f`tf hf dn pltfìgn.

V`n pltfìgn 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 pltfìgn.

Fbfepdl ?2

>:; 5 >:.:::.:::

Yl` ; ifrls ilel g`hgin fd fxpl`f`tf hf dn pltfìgn.

Dns pltfìgns 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 pltfìgn.Fbfepdl >2

>:’9 5:::.>

>

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Yl` 9 ifrls ilel g`hgin fd fxpl`f`tf hf dn pltfìgn.

Fbfepdl ?2

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Yl` ; ifrls ilel g`hgin fd fxpl`f`tf hf dn pltfìgn.





>.?.>. Fbfrigigl

>)

Fxprfsn f` alren hf pltfìgns 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 pltfìgns 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.>.

Fbfrigigl

>)

Vtgdgzn`hl pltfìgns 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 pltfìgns hf fxpl`f`tf hls.

Fbfepdl2 18 fs u` iunhrnhl pfrafitl, plrquf fs fd vndlr hf dn pltfìgn =? 5 = » =

Fbfepdl2 Dls iunhrnhls pfrafitls ef`lrfs hf >:: sl`2

Iunhrnhlspfrafitls2

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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 pltfìgns hf fxpl`f`tf trfs.

Fbfepdl2 = fs u` iuol pfrafitl, plrquf fs fd vndlr hf dn pltfìgn ?9 5 ? » ? » ?

Fbfepdl2 Dls iuols pfrafitls ef`lrfs hf ?:: sl`2

Iuols

pfrafitls2

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=

?;

18

>?6

Zltfìgns2 >9 ?9 99 89 69





>.6.?. Npdginigl`fs

Dn npdginigù` hf dls iuols pfrafitls fs dn rfslduigù` hf rnåifs iòogins fxnitns.

>.1. Ygk`l hf dns pltfìgns

>.1.>. Onsf plsgtgvn

>) Fxpl`f`tf pnr2 fd vndlr hf dn pltfìgn fs sgfeprf plsgtgvl.

Ygk`l plsgtgvl2

Fbfepdls2

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Ygk`l `fkntgvl2

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Ygk`l plsgtgvl2

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Onsf ̀ fkntgvn

>) Fxpl`f`tf pnr2 fd vndlr hf dn pltfìgn fs sgfeprf plsgtgvl.

Ygk`l plsgtgvl2

Fbfepdls2

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Ygk`l `fkntgvl2

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Kfen Gsnofd Enrå` Inonddfrl Zîkg`n ; hf 9:

?) Fxpl`f`tf gepnr2 fd vndlr hf dn pltfìgn fs sgfeprf `fkntgvl.

Ygk`l plsgtgvl2

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Ygk`l `fkntgvl2

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Xnodn-_fsuef`

Onsf Fxpl`f`tfYgk`l hfd rfsudtnhl

hf dn pltfìgn

+Znr

+Gepnr

’ Znr +

Gepnr ’

_filrhnr2 Mny quf npdginr dn rfkdn hf sgk`ls.

>.1.8.

Fbfrigigls>) Ilepdftn y pl` fbfepdls2

n) Dns pltfìgns hf onsf `fkntgvn y fxpl`f`tf pnr sl`2 ……

o) Dns pltfìgns hf onsf `fkntgvn y fxpl`f`tf gepnr sl`2 ……

Ylduigù`2

n) Zlsgtgvns. Fbfepdl2 8??? ?

o)

@fkntgvns. Fbfepdl2 =????

9

?) Indiudn dns sgkugf`tfs lpfrnigl`fs2

n) (’?)9 Ylduigù`2 ’=

o) (’9)? Ylduigù`2 +7

i) (’;)? Ylduigù`2 +87

h) 99 Ylduigù`2 +?;

f) (’>) Ylduigù`2 ’>




Kfen Gsnofd Enrå` Inonddfrl Zîkg`n = hf 9:

>.;. Zrlpgfhnhfs

>) Zltfìgns hf onsf u`l2 sgfeprf vndf` dn u`ghnh.

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Zltfìgns hf onsf ifrl2 sgfeprf vndf` ifrl.

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9) Zltfìgns hf fxpl`f`tf ifrl2 sgfeprf vndf` dn u`ghnh.>

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8) Zltfìgns hf fxpl`f`tf u`l2 sgfeprf vndf` dn onsf.

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6) Zrlhuitl hf pltfìgns il` dn egsen onsf2 sf hfbn dn egsen onsf y sf suen` dls fxpl`f`tfs.èè nnn

Ileprlonigù`2

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Kfen Gsnofd Enrå` Inonddfrl Zîkg`n >: hf 9:

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98: 5 >

(-;): 5 >

nn > Zltfìgns hf fxpl`f`tf u`l?=> 5 ?=

(-=>)> 5 -=>

èè nnn Zrlhuitl hf pltfìgns il` dn egsen onsf;? » ;8 5 ;?+8 5 ;1

(-?)6 » (-?)9 5 (-?)6+9 5 (-?)=

èè

`

e

nnnn

n 2 Iligf`tf hf pltfìgns il` dn egsen onsf;8 2 ;? 5 ;8-? 5 ;?

(-?)1 2 (-?)9 5 (-?)1-9 5 (-?)9




Kfen Gsnofd Enrå` Inonddfrl Zîkg`n >9 hf 9:

èè nn Zltfìgn hf u`n pltfìgn(;9)1 5 ;9»1 5 ;>=

(6-8)? 5 6(-8)»? 5 6-=

``` onon Zrlhuitl hf pltfìgns il` fd egsel fxpl`f`tf91 » 61 » ;1 5 (9 » 6 » ;)1

(-6)9 » 89 5 S(-6) » 8U9

`

`

`

```

o

n

o

n

onon

22

Iligf`tf hf pltfìgns il` fd egsel fxpl`f`tf=6 2 86 5 (= 2 8)6

7-? 2 (-9)-? 5 S7 2 (-9)U-?

``ònon Zltfìgn hf u` prlhuitl

(? » 9 » 6)1 5 ?1 » 91 » 61

S(-1) » =U-? 5 (-1)-? » =-?

`

``

```

o

n

o

n

onon

22

Zltfìgn hf u` iligf`tf (>? 2 9)8

5 >?8

2 98

S18 2 (-=)U-9 5 18-9 2 (-=)-9

`

```

n

o

n

o

o

n

Zltfìgns il` fxpl`f`tf `fkntgvl

8

888

=

9

=

9

9

=

9

999

?

6

?

6

6

?

`

`

`

nnn

>>

Zltfìgns il` fxpl`f`tf `fkntgvl

?

?

?

9

>

9

>9

9

9

9

6

>

6

>6

>.;.?.

Fbfrigigls

>) _fndgzn dns sgkugf`tfs lpfrnigl`fs il` pltfìgns 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





?) _fndgzn dns sgkugf`tfs lpfrnigl`fs il` pltfìgns 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

Ylduigù`2 9-?

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` pltfìgns>.=.>. Yuen y _fstn

Znrn plhfr suenr y rfstnr il` `òefrls fxprfsnhls ilel pltfìgns , hfof` fstnreudtgpdginhls plr dn egsen pltfìgn.

Znrn pltfìgns hf onsf >:, sg `l fs nså, hfofels mnifr quf tf`kn` dn pltfìgn 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 pltfìgns, sf npdgin` dns prlpgfhnhfs2

n) Zrlhuitl hf pltfìgns il` dn egsen onsf.

o) Zrlhuitl hf pltfìgns il` fd egsel fxpl`f`tf.

>.=.9.

Hgvgsgù`

Znrn plhfr hgvghgr il` `òefrls fxprfsnhls ilel pltfìgns, sf npdgin` dns prlpgfhnhfs2

n) Iligf`tf hf pltfìgns il` dn egsen onsf.

o) Iligf`tf hf pltfìgns il` fd egsel fxpl`f`tf.

>.=.8.

Zltfìgn

Znrn indiudnr pltfìgns, sf npdgin` dns prlpgfhnhfs2

n) Zltfìgn hf u`n pltfìgn.

o) Zltfìgn hf u` prlhuitl.

i) Zltfìgn hf u` iligf`tf.





?. _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.





Fbfepdls2

?66,?666?6

>18,>188>1

79,7997

8?,8??8

>>,>>>>

??

??

??

??

??

>::>:,>::>:>:>::

=>7,=>77=>

18=,18==18

87;,87;;87

911,911191

??

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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 +

>?666>?6

188818

?;99?;

=??=

>>>>

99

99

99

99

99

:::.>>:>::::.>

;?777;?7

6>?==6>?

989;;989

?>111?>1

99

99

99

99

99

Yf dff —dn rnåz iòogin fxnitn hf = fs ?―.




Kfen Gsnofd Enrå` Inonddfrl Zîkg`n >; hf 9:

Fbfepdls2 _nhgin`hl ’

>?666>?6

188818

?;99?;

=??=

>>>>

99

99

99

99

99

:::.>>:>::::.>

;?777;?7

6>?==6>?

989;;989

?>111?>1

99

99

99

99

99

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 ’


h)

?

8

?

?

Ylduigù`2 ?8?

?

f) 99 Ylduigù`2 ?; @l tgf`f slduigù` plrquf fd rnhgin`hl fs ’


a)

6

9

?

=

Ylduigù`2

8>1??

?

?

? 8

6

7

6

99




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)

>6?16?1 ? 911?16?6

??

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 ;―.





?.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

??

?) 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`




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




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 :.




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 gàfrglr 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 fìl`trnru` vndlr gàfrglr.

?? » ? 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




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.¼ûç `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 gàfrglr 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

fìl`trnr u` vndlr gàfrglr.

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.





=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?>





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





6² _fstnr hf dn hgafrfìgn n`tfrglr 86 » 6. Fd rfstl hf dn rnåz fs dn ùfvn hgafrfìgn.

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




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 ??




Kfen Gsnofd Enrå` Inonddfrl Zîkg`n ?= hf 9:

?.1.8. Zltfìgn hf u`n rnåz

Dn pltfìgn 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 pltfìgn.

Fbfepdls2 8

88

8

?6?6?6?6?6?6?6?6

11

11

=>=>=>=>=>=>=>=>=>=>

?.;. Dn rnåz ilel pltfìgn

Dn rnåz hf u`n pltfìgn 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 pltfìgns 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.





8.9. Xgpls hf lpfrnigl`fs ileog`nhns

8.9.>. Lpfrnigl`fs ileog`nhns il` pltfìgns y rnåifs iunhrnhns fxnitns

Znrn rfsldvfrdns, prgefrl fafitunrfels dns pltfìgns 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` pltfìgns y rnåifs iòogins fxnitns

Znrn rfsldvfrdns, prgefrl fafitunrfels dns pltfìgns 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` pltfìgns 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?




?) _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

tema 4. potencias y raíces de números enteros

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