EJERCICIO11.Unadeudaseexpresaensentido
1.Unadeudaseexpresaensentido 1.Unadeudaseexpresaensentido
1.Unadeudaseexpresaensentido
1.Unadeudaseexpresaensentidonegativo.Luegoinicalmenteelestado
negativo.Luegoinicalmenteelestado negativo.Luegoinicalmenteelestado
negativo.Luegoinicalmenteelestado
negativo.LuegoinicalmenteelestadoeconmicodePedroes-60bs.
econmicodePedroes-60bs. econmicodePedroes-60bs.
econmicodePedroes-60bs.
econmicodePedroes-60bs.Alrecibir320bs.Aumentasucapital
Alrecibir320bs.Aumentasucapital Alrecibir320bs.Aumentasucapital
Alrecibir320bs.Aumentasucapital
Alrecibir320bs.Aumentasucapitaltantosbs.comoharecibido;entonces
tantosbs.comoharecibido;entonces tantosbs.comoharecibido;entonces
tantosbs.comoharecibido;entonces
tantosbs.comoharecibido;entonceslaoperacinquedaexpresadacomo
laoperacinquedaexpresadacomo laoperacinquedaexpresadacomo
laoperacinquedaexpresadacomo laoperacinquedaexpresadacomoseindica
seindica seindica seindica seindica + + 60 320 260
bs.2.2.2.2.2.1170 1 515 345 . . sucres3.3.3.3.3.200 56 189 67 +
+$4.4.4.4.4. + + 665 1178 2 280 437 . . soles5.5.5.5.5.20 15 40 75
30 + $6.6.6.6.6. + + 67 72 16 2 9 $7.7.7.7.7.200 78 81 93 41 59 70
+ colones8.8.8.8.8. + 45 66 79 200 10 0EJERCICIO21.1.1.1.1.12 15 3
2.2.2.2.2. + 3 8 6 13.3.3.3.3.( ) 15 3 15 3 18 + 4.4.4.4.4. ( ) + +
8 5 8 5 135.5.5.5.5. + + 4 7 2 11 66.6.6.6.6. + + + + + + + 8 4 1 8
4 4 78 4 2 8 8 0 88 4 5 8 20 12 11amamam7.7.7.7.7. + + + + + 1 2 2
1 4 5 101 2 3 1 6 7 111 2 3 3 1 1 6 3 4 121 2 3 3 3 1 6 9 2
2amamampm8.8.8.8.8. + 56 7 499.9.9.9.9. ( ) + + 71 5 6615 5
20longlat.10.10.10.10.10.18 3 2165 4 61+
=+-=+longlat.11.11.11.11.11.- + =+ 75 135 60
aosEJERCICIO31.1.1.1.1.+ 32 16 m m ;2.2.2.2.2.+ 10 4 m m
;3.3.3.3.3.50 85 35 m4.4.4.4.4. 6 11 66m5.5.5.5.5.- =- =+8 6 489 6
54mm6.6.6.6.6. 400 2 800400 3 1 200 + m corredorm yo
.7.Los40piesdelongituddelposte 7.Los40piesdelongituddelposte
7.Los40piesdelongituddelposte 7.Los40piesdelongituddelposte
7.Los40piesdelongituddelposteserepartenasi:15piesquesobresalen
serepartenasi:15piesquesobresalen serepartenasi:15piesquesobresalen
serepartenasi:15piesquesobresalen
serepartenasi:15piesquesobresalen-seasumenensentidopositivo-.
-seasumenensentidopositivo-. -seasumenensentidopositivo-.
-seasumenensentidopositivo-.
-seasumenensentidopositivo-.25piesqueseencuentranenterrados.
25piesqueseencuentranenterrados. 25piesqueseencuentranenterrados.
25piesqueseencuentranenterrados.
25piesqueseencuentranenterrados.-seasumenensentidonegativo-.
-seasumenensentidonegativo-. -seasumenensentidonegativo-.
-seasumenensentidonegativo-.
-seasumenensentidonegativo-.Alintroducir3piesms,seadicionan
Alintroducir3piesms,seadicionan Alintroducir3piesms,seadicionan
Alintroducir3piesms,seadicionan
Alintroducir3piesms,seadicionanalosqueestanbajoelsueloysedeben
alosqueestanbajoelsueloysedeben alosqueestanbajoelsueloysedeben
alosqueestanbajoelsueloysedeben
alosqueestanbajoelsueloysedebendescontardelosqueestanporencima.
descontardelosqueestanporencima. descontardelosqueestanporencima.
descontardelosqueestanporencima.
descontardelosqueestanporencima.Aritmticamentesignifica:
Aritmticamentesignifica: Aritmticamentesignifica:
Aritmticamentesignifica: Aritmticamentesignifica:
40 15 2525 3 2815 3 12 + + +piespies8.8.8.8.8.55 52 3 +
m9.9.9.9.9. + 32 15 17m10.10.10.10.10.35 47 12 m11.11.11.11.11. + +
39 56 17m12.12.12.12.12.90 58 36 4 m13.13.13.13.13. 72 30 1 72 30
4272 30 2 72 60 1272 30 3 72 90 1872 30 4 72 120 48 + +
mmmm14.14.14.14.14. ( )( )( )( ) + + + + + +120 60 1 120 60 60120
60 2 120 120 0120 60 3 120 180 60120 60 4 120 240
120KmKmKmEJERCICIO7.1. 1. 1. 1. 1. x x x + 2 32.2.2.2.2.8 9 17 a a
a + 3.3.3.3.3.11 9 20 b b b + 4.4.4.4.4. b b b 5 65.5.5.5.5. 8 9 m
m m6.6.6.6.6. 9 7 16 m m m7. 7. 7. 7. 7. 4 5 9 a a ax x x+ 8. 8. 8.
8. 8. 6 8 141 1 1a a ax x x + + ++ 9. 9. 9. 9. 9. + + +m m mx x x 1
1 15 610.10.10.10.10. 3 42 2 2a a ax x x11.11.11.11.11. 12121 1222a
a a a a + + 12. 12. 12. 12. 12.35110610710ab abab abab +
+13.13.13.13.13. 13162 163612xy xy xy xy xy + + 14.14.14.14.14.
15451 4555xy xy xy xy15.15.15.15.15. 561820 32423242 2 2 2ab ab ab
ab16.16.16.16.16. a a a a788 7815817. 17. 17. 17. 17. 8 9 6 23 a a
a a + + 18.18.18.18.18.15 20 36 x x x x + + 19.19.19.19.19. 7 8 9
24 m m m m20.20.20.20.20. a b a b a b a b2 2 2 23 521.21.21.21.21.a
a a ax x x x+ + 3 8 1222.22.22.22.22. + + + +5 3 5 131 1 1 1a a a
ax x x x23.23.23.23.23. a a a a a + + + +12236 3
4613624.24.24.24.24. x x x x x23166 4 1611625.25.25.25.25. 153102 3
1010151032ax ax ax ax ax ax + + + + 26. 26. 26. 26. 26. 34569 10
121231122 2 2 2 2ax ax ax ax ax27.27.27.27.27.11 8 9 11 39 a a a a
a + + + 28. 28. 28. 28. 28. m m m m mx x x x x + + + + ++ + + 1 1 1
1 13 4 6 1429.29.29.29.29. xy xy xy xy xy2 2 2 2 28 9 20
3830.30.30.30.30. 3 5 6 9 23 a a a a am m m m m31.31.31.31.31.
1214184 2 1 88158a a a a a a + + + + + +32. 32. 32. 32.
32.25121101208 10 2 1202120ax ax ax ax ax ax + + + + +
+33.33.33.33.33.0 5 06 07 0 8 26 , , , , , m m m m m + + +
34.34.34.34.34. 171141284 2 1 2828352854ab ab ab abab ab
ab35.35.35.35.35. 23161911224 6 4 33637363 3 3 33 3xy xy xy xyxy
xy36.36.36.36.36.ab ab ab ab ab ab2 2 2 2 2 27 9 21 39 + + + +
37.37.37.37.37. m m m m m m 8 7 3 2038.38.38.38.38. + + + + + +x x
x x x xa a a a a a 1 1 1 1 1 18 4 5 1939.39.39.39.39. 121314151630
20 15 12 106087602920a a a a aa a a+ + + ++ + + + 40.40.40.40.40.
1316121121912 6 18 3 4364336ab ab ab ab abab
abEJERCICIO81.1.1.1.1.8 6 2 a a a 2.2.2.2.2.6 8 2 a a a 3.3.3.3.3.9
15 6 ab ab ab 4.4.4.4.4.15 9 6 ab ab ab 5.5.5.5.5.2 2 0 a a
6.6.6.6.6. + 7 7 0 b b7.7.7.7.7. + 14 32 18 xy xy xy8.8.8.8.8. + 25
32 72 2 2xy xy xy9. 9. 9. 9. 9. 40 51 113 3 3xy xy xy
10.10.10.10.10. + m n m n m n2 2 26 511. 11. 11. 11. 11. + 15 40 25
xy xy xy12.12.12.12.12.55 81 263 2 3 2 3 2a b a b a b
13.13.13.13.13. + xy xy2 2014.14.14.14.14. + 9 9 02 2ab ab15. 15.
15. 15. 15. 7 7 02 2xy xy 16. 16. 16. 16. 16. + 101 118 17 mn mn
mn17. 17. 17. 17. 17. 502 405 97 ab ab ab 18.18.18.18.18. + 1024
1018 6 x x x19.19.19.19.19. + 15 15 0 ab ab20.20.20.20.20. 12342
3414a a a a 21.21.21.21.21. 34123 2414a a a a 22.22.22.22.22.
5651210 5125122 2 2 2a b a b a b a b 23.23.23.23.23. + +479148
9141142 2 2 2x y x y x y x y24.24.24.24.24. 38543 10878am am am am
25.25.25.25.25. + + am am am am355 352526.26.26.26.26. 567820
2124124mn mn mn mn 27.27.27.27.27. + + ab ab ab ab2 2 2 231111
31181128.28.28.28.28.34 56 224 3 4 3 4 3, , , ab ab ab
29.29.29.29.29. + 12 3 4 2 2 , , , yz yz yz30.30.30.30.30.4 2 2 a a
ax x x 31.31.31.31.31. + + +8 8 01 1a ax x32.32.32.32.32.25 32 71 1
1m m ma a a 33.33.33.33.33. + + +x xa a 1 1034.34.34.34.34. + +
14121 24142 2 2 2a a a am m m m35.35.35.35.35. 5671210 712312141 1
1 1 1a a a a am m m m m + + + + + 36.36.36.36.36. 41312 131132 2 2
2a a a a 37.37.37.37.37. + + 53420 34174mn mn mn mn38.38.38.38.38.8
25 172 3 2 3 2 3a b a b a bx x x x x x + + + + + + 39.39.39.39.39.
+ +787 8818ab ab ab abm n m n m n m n40.40.40.40.40.085 0 5 0 35 ,
, , mxy mxy mxy EJERCICIO91.1.1.1.1.9 3 5 11 a a a a + 2.2.2.2.2. +
8 9 0 x x x3.3.3.3.3.12 23 5 16 mn mn mn mn 4.4.4.4.4. + x x x 19
18 05.5.5.5.5.19 10 6 15 m m m m + 6.6.6.6.6. + 11 15 26 0 ab ab
ab7.7.7.7.7. + 5 9 35 31 a a a aX X X X8.8.8.8.8. + + + +24 15 39
02 2 2a a aX X X9. 9. 9. 9. 9.23132 1 33030 y y y y y + +
10.10.10.10.10. + + 35141212 5 10201720m m m m m11.11.11.11.11.
38143 2 88382 2 2 2 2ab ab ab ab ab + + 12.12.12.12.12. + + a a a a
a 8 9 1513.13.13.13.13.7 11 20 31 15 ab ab ab ab ab +
14.14.14.14.14.25 50 11 14 02 2 2 2x x x x + + 15.15.15.15.15. + xy
xy xy xy xy 8 19 40 1216.16.16.16.16.7 21 80 53 ab ab ab ab ab +
17.17.17.17.17. + + 25 11 60 82 362 2 2 2 2xy xy xy xy xy18. 18.
18. 18. 18. + + 72 87 101 243 157 ax ax ax ax ax19. 19. 19. 19. 19.
+ 82 71 53 206 0 bx bx bx bx20.20.20.20.20.105 464 58 301 03 3 3 3a
a a a + + 21.21.21.21.21. 1213141530 20 15 12601360x x x x x x + +
22.22.22.22.22. 1313161124 4 2 112112y y y y y y + +
23.23.23.23.23. 35161318 5 10 30307302 2 2 22 2ab ab ab abab ab + +
24.24.24.24.24. + + 56163820 4 24 924924382 2 2 22 2 2ab ab ab abab
ab ab25.25.25.25.25. + + a a a a a a 8 11 15 75 6426.26.26.26.26. +
+ + 7 21 14 30 82 80 c c c c c c27.27.27.27.27. + + mn mn mn mn mn
mn 14 31 2028.28.28.28.28.ay ay ay ay ay2 2 2 2 27 93 51 48 0 + +
29.29.29.29.29. + + + a a a a a a a 3 6 330.30.30.30.30. 122376123
4 7 3 663612x x x x xx x x+ + + + 31.31.31.31.31. + + + + + +
234145648 18 6 24 2024202456x x x x xx x x32.32.32.32.32.7 30 41 9
73 0 a a a a ax x x x x + 33.33.33.33.33. + + + + + + + +a a a a a
ax x x x x x 1 1 1 1 1 17 11 20 2634.34.34.34.34.a a a a a a a + +
+ 6 20 150 80 31 8835.35.35.35.35. + 9 11 17 81 110 9 b b b b b b
b36. 36. 36. 36. 36. + + + ab ab ab ab ab ab ab2 2 2 2 2 2 215 85
131 39 16237.37.37.37.37.84 501 604 715 231 16513402 2 2 2 2 22mx
mx mx mx mx mxmx + + 38.38.38.38.38. 56231458420 16 6 15
96241112437843 2 3 2 3 2 3 2 3 23 2 3 23 2 583 2ab ab ab ab abab
abab ab+ ++ + 39. 39. 39. 39. 39. 40 81 130 41 83 91 16 28 a a a a
a a a a + + + 40. 40. 40. 40. 40. + + 21 52 60 84 31 23 0 ab ab ab
ab ab ab abEJERCICIO101. 1. 1. 1. 1.7 6 9 47 6 13 9 4 1313 13a a b
ba a a b b ba b+ + 2.2.2.2.2. a b c b c c aa a b b c c c+ + + 20 0
2 003.3.3.3.3. 5 11 9 20 15 20 25 11 12 9 1 1025 12 10x y x yx x x
y y yx y + + 4.4.4.4.4. + + + 6 8 5 6 116 6 13 8 7 5 11 613 7 6m n
m n mm m m m n n nm n5.5.5.5.5. + + + + + + + a b b c a c ba a a b
b b c ca2 2 3 2 33 2 2 3 0 2 2 026.6.6.6.6. + + + + + + + 81 19 30
6 80 2581 80 0 19 6 25 0 3030x y z y x x yx x x y y y
zz7.7.7.7.7.15 6 8 20 5 3115 8 86 5 12 20 31 118 12 112 2 22 2 2
22a ab a ab a aba a a aab ab ab aba ab + + + 8.8.8.8.8. + + + + + 3
4 6 81 114 313 6 31 214 81 114 3021 30a b a b b a a ba a a a ab b b
b ba b9.9.9.9.9. + + + + + + 71 84 50 84 45 1871 50 45 18 4884 84
0483 4 2 3 4 2 3 33 3 3 3 34 2 4 23ab ab ab ab ab abab ab ab ab
abab abab10.10.10.10.10. + + + + + + + + a b c a b c a b ca a a a b
b bc c ca8 2 2 19 2 3 3 3 32 3 2 2 3 02 3 0 8 19 3 142
1411.11.11.11.11. m mn m mn m m m mm m m m m m m m mn mn mnm m mn2
2 3 2 2 33 3 3 2 2 2 2 23 271 14 65 115 66 7 14 115 129 71 65 67
129 6+ + ++ +12. 12. 12. 12. 12.xy xy xy xy xy xy xy xy yxy xy xy
xy xy xy xy xyxy xy yxy xy y4 3 2 2 4 2 3 2 3 2 4 34 4 4 4 3 2 3 2
3 2 3 22 2 34 3 2 38 10 7 9 21 508 21 14 7 70 10 9 50 3114 7 31 + +
+ + + + + +13.13.13.13.13.5 3 8 5 50 4 65 90 78 7 0 3 4 0 5 5 0 50
65 90 25251 2 3 1 2 2 3 33 3 3 2 2 2 1 1a b c a b b c cc c c b b b
a ax x x x x x x xx x x x x x x x+ + + + + + + ++ + + + + + + + + +
+ + + + + + 14.14.14.14.14.a x a x a xx x x x a a a ax am m m m m
mm m m m m m m mm m+ + + + + ++ + + + + + + ++ + + + + + + + 2 3 2
3 2 33 3 3 3 2 2 2 23 25 8 3 5 6 55 5 3 5 8 6 3315.15.15.15.15. 0 3
0 4 0 5 06 07 0 9 3 3 30 3 0 6 3 2 7 0 4 07 3 3 3 0 5 09 3 3 42 7 3
3 3 4, , , , , ,, , , , , , , , ,, , ,a b c a b c a b ca a a a b b
b b c c c ca b c+ + + + 16.16.16.16.16. 12132 334163412122342 8
3474133162 18 1617634123 24147417614a b a b a ba a a a a b b b b ba
b+ + + + + +17.17.17.17.17. 352110132 23511026 1 2010131021326 1
63131310132 2 22 2 2 2 22m mn m mn mn mm m m m m mn mn mn mn mnm mn
+ + + + + + 18.18.18.18.18. + + + + + + + 3412562341613234739
28121912123422 3 84945616135 1 26661912942 2 132 2 22 2 2 22 2 2 2
2 22 2a ab b a ab b b aba a a a ab ab ab ab abb b b b b ba ab
b19.19.19.19.19. 0 4 313806250 21460 4252 250380 23 168148017506142
4 141440 35 31 6 250175 0 35 252 2 3 2 2 32 2 2 2 2 2 2 23 3 3 3 32
3, , ,, ,, ,,,, ,,, ,xy xy y xy xy yxy xy xy xy xy xy xy xyy y y y
yxy y+ + + + + +20. 20. 20. 20. 20.325750351250 215325350 23 15
5251325750125157 2 105015013251501 2 1 2 1 21 1 1 1 1 2 2 2 2 21 2a
b a b a ba a a a a b b b b ba bm m m m m mm m m m m m m m m mm m +
++ + + + +,,EJERCICIO11Para resolver los problemasdel 1 al 18 las
literalestoman los siguientes valores:a = 1 b = 2 c = 3 m = 1/2 n =
1/3 p = 1/4 a = 1 b = 2 c = 3 m = 1/2 n = 1/3 p = 1/4 a = 1 b = 2 c
= 3 m = 1/2 n = 1/3 p = 1/4 a = 1 b = 2 c = 3 m = 1/2 n = 1/3 p =
1/4 a = 1 b = 2 c = 3 m = 1/2 n = 1/3 p = 1/41.1.1.1.1.3 3 1 2 6 ab
2.2.2.2.2.5 5 1 2 3 5 8 3 1202 3 2 3abc 3.3.3.3.3. b mn2 2212134623
4.4.4.4.4.24 2412131461271461081182 32 3m np
_,
_, 5.5.5.5.5. 23231 21223418824134 2 3 4 23a b m
_, 6.6.6.6.6. 712712314121891211612189384631283 2 32cpm
_, 7.7.7.7.7.m npb c a
_,
_, 121314141271414322 38.8.8.8.8. 5656 11256125121 2 2 13 2a mb
c
_, 9.9.9.9.9.2 2 2 3 4 9 36 62 2bc 10. 10. 10. 10. 10.4 12 41212
2 3 2 216 2 6 122 3 23 3m bc 11.11.11.11.11. mn ab 812138 1
2166486434 3 4 312.12.12.12.12. 434 13 2 341829abc 13.13.13.13.13.
55 212131420141125112602 222bmnp
_,
14.14.14.14.14. 34233422333 842 932441836613232bc
15.15.15.15.15. 22121311911332 2mn
_,
16. 16. 16. 16. 16.2422412132131424621911642114442112416242 2 2
2mnnp
_,
_,
17.17.17.17.17. 3 6423 64 2 32123 64 8 729 3 373 248 3 72 2163 6
3 3 6 333 bcm.18.18.18.18.18. 353212535114232125
21235443225023513212535325351526752252323 33 apbbmEJERCICIO12Para
resolver los problemas del 1 al 18 las literales tomanlos
siguientes valores:a = 3 b = 4 c = 1/3 d = 1/2 m = 6 n =
1/41.1.1.1.1.a ab b2 2 2 22 3 2 3 4 4 9 24 16 1 + + + 2.2.2.2.2.c
cd d2 22 221321312121913144 12 9362536+ +
_, + +
_, + + + +3.3.3.3.3. acbd+ + + 3134129 8 174.4.4.4.4. cdmn + + +
+ 2131261422324 22 72 636435.5.5.5.5. a b m2 2 2 2 2 23 2
6334266931623663 8 6 1 + + + + 6.6.6.6.6. 351223513124 212152 11 10
5545c b d + + + +7.7.7.7.7. abnacdbdm+ + + + + 3
41431312412612141122648 213144 6 13149349238.8.8.8.8. b n m + + + +
+ + + + 6 4146 6 21264 1 1221728129.9.9.9.9. c a d b n d 3 16 8133
31216 4148121331216142 1 8122 16 1213262 2 12 + + + + +
10.10.10.10.10. mdab
_,
612216116216 16 345634.1 11 111.1.1.1.1.
344313441461341461121242 12432418222 2cnm+
_,
+
_,
+ + + 12.12.12.12.12. 421621 212814124816112121 1 1 02 2 2 2d
n+
_, +
_, + + 13.13.13.13.13. a bcb md++++ 3 4134 612713101221 20
114.14.14.14.14. b anm bda++ ++ + + + + 54 3146 4125 311421215 4 4
15 2315.15.15.15.15. 12216 1121332 41614361124 384 362181623 4
4824472412324c abn am d+ + + + + 16.16.16.16.16.433664 43 336
661693366433664 1 1 4 ba m+ + + + + 17.17.17.17.17. b d c d +++ +
++ 223 844 212231381242 121 2432346 343418.18.18.18.18. 233 242 3
433 21243142 144339443122 123332432243924328983264 9 12861872 2 2 2
2258+ + + +
_,
+ + + + + ab da nEJERCICIO13Para los problemas 1 al 24 las
literales toman los siguientes valores:a = 1 b = 2 c = 3 d = 4 m =
1/2 n = 2/3 p = 1/4 x = 01.1.1.1.1. ( ) ( ) a b c d + + 1 2 3 4 3 3
4 9 4 52.2.2.2.2. ( )( ) ( )( ) a b b a + + 1 2 2 1 3 1 33.3.3.3.3.
( )( ) b m c n a +
_,
_, +
_,
_, + + + + 4 2123234 14 129 234327347247 8215272 2 124. 4. 4. 4.
4.( )( ) ( )( ) 2 3 4 2123234142 1 2 1 4 3 5 152 2m n p b + + +
_, +
_, + + 5.5.5.5.5. ( )( )( ) ( ) ( ) 4 8 6 4128141 2 6234 2 2 3 0
02 2 2 2m p a b n d + + +
_, +
_, + 6.6.6.6.6. ( )( )( )( ) ( )( )( ) c b d c b a m p
_,
_, 3 2 4 3 2 112141 1 12 14147.7.7.7.7. ( ) ( ) ( ) b c d a m n
x2 2 2 2 562 2 3 4 112232 0 4 73 462876168 76161626 + + + + +
_, + +
_, 8.8.8.8.8.( ) ( ) ( ) 2 6 4 2120 62 3 4 4 64 9 4 16 6 13 64
78 64 142 2 2 2 2 2mx b c d + + + + + 9.9.9.9.9.
89168129231614214183424622322 63832 23mnpba +
_, +
_,
+ + + + 10.10.10.10.10.( ) ( ) ( ) x ma d cb c a+ + + + + 0121 4
3121 64 31262 312 3 111.11.11.11.11. ( ) 44121411 2342 141
494345935927552 222 2225m paa bc+++
_,
++
_, + 12.12.12.12.12. ( )( )( )( )( )( )2 3 4 8 6 4 9
2021232341481462341292320141 2 1 2 4 2 6 5 4 4 11 176m n p p n m n
p + + + + + +
_, +
_, +
_, + + + + 13.13.13.13.13.( ) ( ) ( ) c m n d m p b n p2 2 22 2
2 21216312234121422314976163441112212124412126 144 441226122 2+ + +
+ +
_, +
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los problemas 1 al 14 las literales toman los siguientes valores:a
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2 22 2 2 2 2 2a b a ba b a b a b
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+ 60.60.60.60.60. 19451 40594069453 2 3 23 2 3 2 193 2ab abab ab
abEJERCICIO211. 1. 1. 1. 1. a ba bb+ +22.2.2.2.2.2 323 5x yx yx
y3.3.3.3.3.83 411 4a baa b++ 4.4.4.4.4.x xxx x2235 62 6+ 5.5.5.5.5.
a abab aba ab ab3 22 23 2 27 98 9 6.6.6.6.6.x y zx y z + +
07.7.7.7.7. x y zx y zx y z+ + + 2 2 28.8.8.8.8.x y xyx y xyx y xyx
xy y2 22 22 22 233 42 22 2+ + + + + + +9.9.9.9.9. x xx xx x x3 223
265 6 46 4 + + +10.10.10.10.10. y yy y yy y y yy y y y2 32 42 4 34
3 26 83 2 64 2 6 6 82 6 4 6 8+ + + + 11. 11. 11. 11. 11. a ab aa
aba ab a aba ab ab a3 223 2 23 2 26 98 15 56 17 15 515 6 17 5 ++ +
+ 12.12.12.12.12.x xy yy xy xyx xy y xy xyx xy xy xy y4 3 44 3 2 24
3 4 3 2 24 3 2 2 3 49 1120 8 69 31 8 68 6 9 31+ + ++ + + + + +
13.13.13.13.13. a b c da b c da b+ + + ++ 2 214.14.14.14.14. ab ac
cd deab ac cd deab ac cd de+ + + + + 2 3 58 4 5 57 6 2
1015.15.15.15.15. x x xx x xx x x3 23 23 26 9 196 11 21 435 17 30
24+ + + + +16.16.16.16.16. y y yy y y yy y y y y y y y y y5 3 23 2
45 3 2 4 5 4 3 29 6 3131 8 11 1940 14 11 19 31 11 40 14 19 31 + + +
+ + + + + + + 17.17.17.17.17. 5 9 6 85 21 14 189 27 22 18 27 22 9
183 3 2 23 2 23 2 2 2 2 3m n m n mnm m n mnn m n mn m n mn n + + +
+ + +18.18.18.18.18. 4 19 625 51 3229 32 38 29 38 323 3 4 2 23 3 2
2 43 3 4 2 2 4 4 3 2 2 3 4xy xy y xyxy xy xy xxy xy y xy x x xy xy
xy y + + ++ + + + + +19. 19. 19. 19. 19.m m n m nm n m n mnm m n m
n m n mn m m n m n m n mn6 4 2 2 42 4 3 3 56 4 2 2 4 3 3 5 6 4 2 3
3 2 4 59 1930 61 13 1621 80 13 16 13 21 16 80+ ++ + + + + + + + + +
+20.20.20.20.20. + ++ + + + + +ab ab aba ab ab ba ab ab ab ab ab b5
3 3 56 4 2 2 4 66 5 4 2 3 3 2 4 5 66 18 428 11 11 98 11 6 11 18 9
4221.21.21.21.21. 1 3 624 30 8 1525 29 7 12 6 6 7 29 12 252 4 3 52
4 62 4 3 5 6 6 5 4 3 2 + + + ++ + + +x x x x xx x x xx x x x x x x
x x x x x22.22.22.22.22. + + + + + + + + + + + + 6 8 23 80 1851 21
80 96 8 28 101 98 9 8 28 101 6 9 982 3 5 4 3 24 3 2 5 42 3 5 4 3 2
5 4 5 4 3 2 2 3 4 5xy x xy xyxy xy y xyxy x xy xy y xy x xy xy xy
xy y23.23.23.23.23. m m n m n mnm n m n mn nm m n m n m n m n mn n6
4 2 2 4 55 3 3 5 66 5 4 2 3 3 2 4 5 68 21 6 823 14 24 8 1423 8 14
21 18 8 22 + + + ++ + + +24.24.24.24.24. x x x xx x x xx x x x x x
x x x x x x x x7 5 26 4 37 5 2 6 4 3 7 6 5 4 3 28 16 23 1551 18 8
25 3059 16 23 3 8 25 30 8 16 25 30 23 59 3 + + + + + + + + + + +
+25.25.25.25.25.9 15 31 1415 3 25 53 99 31 4 25 53 9 14 9 25 53 31
9 4 146 4 2 2 4 64 2 6 5 3 3 56 2 4 6 5 3 3 5 6 5 3 3 2 4 5 6a ab
ab bab b ab ab aba ab b ab ab ab a ab ab ab ab b + ++ ++ + + + +
+26.26.26.26.26. a a aa a aa ax x xx x xx x+ + + ++ ++ ++1 21 215
64 727.27.27.27.27. m m mm m m mm m m m m m m m m ma a aa a a aa a
a a a a a a a a + + + + + 1 22 1 31 2 1 3 1 1 2 334 5 3 85 2 3 8 3
5 2 828.28.28.28.28. a a a aa a a aa a a a a a a a a a a am m m mm
m m mm m m m m m m m m m m m+ + + + ++ + + + + + + + ++ + + ++ + +
+ + + + +4 2 12 1 3 14 2 1 3 1 4 3 2 1 17 8 614 8 5 117 8 14 5 11 5
7 11 8 1429.29.29.29.29. x x x xx x x xx x x x x x x x x x x xa a a
aa a a aa a a a a a a a a a a a+ + + + + + + + + + + + + 2 1 21 1
32 1 2 1 3 2 1 1 2 37 9 2519 45 11 6026 36 25 11 60 11 26 36 25
6030.30.30.30.30. m m m mm m m m mm m m m m m m m m m m mn n n nn n
n n nn n n n n n n n n n n n+ + + + + + 1 2 3 52 3 5 41 2 3 5 4 1 2
3 4 56 8 195 6 9 811 2 28 8 8 11 2 28EJERCICIO221.1.1.1.1. b ab ab
a a b + 2 2 2 22.2.2.2.2.2 34x yx yx y+ ++3.3.3.3.3. + + +7 552 5 2
5aa ba b a b4.4.4.4.4. + + + + + +xx xx x x x222 2652 6 5 2 5
65.5.5.5.5.xy xyxy xxy xy x x xy xy2 22 32 2 3 3 2 256 6++ + +
+6.6.6.6.6. 7 56 85 8 8 52 22 32 2 3 3 2 2ab abab aab ab a a ab ab+
++ + + +7.7.7.7.7. + + + a b ca b ca b c2 322 3 58.8.8.8.8. + ++ +
+ +3 4 52 3 4 3 2 4n m pn m pn m p m n p9.9.9.9.9. x y zx y zx y z+
++ 3 62 2 510.10.10.10.10. + + + + +5 86 37 2 2 72 22 22 2 2 2b ab
ab ab ab ab a a ab b11.11.11.11.11. + + + +5 636 92 22 22m n mnm n
mnm mn12.12.12.12.12. + + + + + + 8 5 468 6 10 8 6 10232 3 3 2x xx
xx x x x x x13.13.13.13.13. 14 8 1614 98 7 8 722 33 3m nm mn m m n
+ + +14.14.14.14.14. 8 5 667 6ab bc cdab bc cdab bc+ + +
+15.15.15.15.15. a ab bab b aba ab ab3 2 32 3 23 2 2925 834 8 + +
+16.16.16.16.16. 6 8 66 46 8 7 6 43 2 22 33 2 2 3x xy xyxy yx xy xy
y + + 17.17.17.17.17.m n cm n c dn c d c d n229 11 147 816 19 14 19
16 14 + + + + + +18.18.18.18.18.5 9 40 67 8 55 2 8 45 54 3 3 43 2 2
3 44 3 2 2 3 4a ab ab bab ab ab ba ab ab ab b+ + + + +
+19.19.19.19.19.x x xx x xx x x x xx x x x x5 4 22 35 4 2 35 4 3 28
25 156 7 6 98 19 22 6 98 6 19 9 22 + + + + + + + + +
+20.20.20.20.20. + + + + + +3 8 19 186 259 8 44 188 9 44 184 3 2 54
5 5 2 34 3 2 5 5 2 35 3 2 2 3 4 5xy xy yxy y x xyxy xy y x xyx xy
xy xy y21.21.21.21.21.x x x xx x x xx x x x xx x x x x3 4 23 2 53 4
2 55 4 3 26 8 9 1525 18 46 25 1124 6 26 37 10 1111 6 24 26 10 37 +
+ + + + + + + + +22.22.22.22.22. a ab ab bab ab ab aba ab ab ab ab
b5 3 2 4 54 3 2 2 3 45 4 3 2 2 3 4 526 8 68 15 45 88 27 15 53 14 +
+ + + + + + +23.23.23.23.23.y y yy y y yy y y y y yy y y y y y6 3
23 4 56 3 2 4 56 5 4 3 2923 5 8 15 822 14 8 15 815 8 22 8 14+ + + +
+ + + + + + + + + +24. 24. 24. 24. 24. x x x xx x x xx x x x x x x
xx x x x x x x x8 6 4 27 5 38 6 4 2 7 5 38 7 6 5 4 3 23 5 936 7 5
23 513 5 45 7 5 23 517 5 3 23 5 51 45 + + + + + + 25. 25. 25. 25.
25. x xy xy xyxy xy y xy xyx xy xy xy y xy xyx xy xy xy xy xy y7 5
2 4 3 2 54 3 2 5 7 3 4 67 5 2 4 3 2 5 7 3 4 67 5 2 4 3 3 4 2 5 6 73
35 8 6060 90 503 95 7 60 90 503 95 90 7 50 60 + ++ + + + + + + +
+26.26.26.26.26.a aa a aa a a aa a a ax xx x xx x x xx x x x+ ++ ++
+ ++ + + + + + + 3 11 23 1 23 2 18 55 63 5 63 6 527.27.27.27.27. +
+ + + + + + 8 16 157 5 815 16 10 815 8 10 164 2 32 3 14 2 11 2 4a a
a aa a a aa a a aa a a an n n nn n n nn n n nn n n
n28.28.28.28.28.15 5 6 419 18 3115 14 6 59 3115 14 31 6 593 2 12 1
1 43 2 1 1 44 3 2 1 1x x x xx x x xx x x x x xx x x x x xa a a aa a
a aa a a a a aa a a a a a+ + + + ++ + + ++ + + + + ++ + ++ + + + +
+29.29.29.29.29. 9 21 26 145 12 814 33 26 14 814 33 26 8 141 2 3 51
2 41 2 3 5 41 2 3 4 5a a aa a a aa a a a aa a a a am m m mm m m mm
m m m m mm m m m m m + + + + + + + + + + + +30. 30. 30. 30. 30. +
++ + + + + ++ + + + + + + + + 15 50 14 6 823 615 23 56 14 5 83 1 1
24 2 1 14 3 2 1 1 2m m m m mm m m mm m m m m m mx x x x xx x x xx x
x x x x xEJERCICIO231.1.1.1.1. 112 + +aa2.2.2.2.2. 088 +
+aa3.3.3.3.3. + 93 53 4 3 422 2a aa a a a4.4.4.4.4. + + + 165 165
522 2xy xxy x x xy5.5.5.5.5.11 13 2 23 2 2 3 2 2 + + + +a ab aba ab
ab a ab ab6.6.6.6.6.xx xy xyx xy xy33 2 23 2 28 62 8 6+ ++
+7.7.7.7.7. aab ab ba ab ab b32 2 33 2 2 38 68 6+ ++ +8.8.8.8.8.+ +
+ +yxy xy xyxy xy xy y43 2 2 33 2 2 3 45 7 85 7 89.9.9.9.9.mm am am
am am am am am a44 3 2 2 3 44 3 2 2 3 45 18 74 18 7 + + +
+10.10.10.10.10.161430 30 + + +b a c db a c d a b c
d11.11.11.11.11. xxy yx xy y x xy y222 2 2 211 1 12.12.12.12.12.
aab ab ba ab ab b32 2 33 2 2 365 85 8 6+ + + +13.13.13.13.13. x yxy
xyx xy xy y3 32 23 2 2 35 17 55 17 5++ ++ + +14.14.14.14.14. xxy xy
xyx xy xy xy43 2 2 34 3 2 2 319 8 159 8 15 1 + + + +
15.15.15.15.15. a bab ab ab aba ab ab ab ab ba ab ab ab ab b5 54 2
3 3 2 45 4 2 3 3 2 4 55 4 3 2 2 3 4 511 2 8 411 2 8 411 8 2 4++ ++
+ + + + +16.16.16.16.16. x xx xx x x x4 234 3 2505 255 25 50+ + + +
+ +17.17.17.17.17. y yy y y yy y y y y yy y y y y y65 4 3 26 5 4 3
26 5 4 3 2419 17 1841 9 17 189 17 18 41+ + + + + + 18.18.18.18.18.
a ab abab ab ab ba ab ab ab ab ab b6 4 2 2 45 3 3 5 66 5 4 2 3 3 2
4 5 6915 17 1415 9 17 14+ ++ + ++ + + + +19.19.19.19.19. x x xx xx
x x x x x x x4 324 3 2 4 3 2115 3416 34 16 34+ + ++ + + + +
+20.20.20.20.20. mm n mn nm m n mn n32 2 33 2 2 317 37 3 1 + +
EJERCICIO241.1.1.1.1. 121413252 14132534132522 22 22 2aa ab ba ab
ba ab b+ ++ + 2.2.2.2.2. 154523594523135 594523140945231559 + ++ +
+xy yzxy yzxy yzxy yz3.3.3.3.3. 3516342918 530342934133029bcbc ab
cdbc ab cdab bc cd + ++ + + +4.4.4.4.4. 12234529125 8106
29123108912a ba ba ba b +++ +5.5.5.5.5. 5938110573115930
8805731159388057311595719403112 222 22 22 2x yy xyx y xyx y xyx xy
y ++ + + +6.6.6.6.6. 56291512385610 945123856123819453 33 2 23 3 2
23 2 2 3m nn m n mnm n m n mnm m n mn n++ + +++ + +7.7.7.7.7.
37133551412186 5142 361141635182 2222 2a ab ba aba aba ab b+ ++
+8.8.8.8.8. 385611035310215 244050 18601
20103940686021103940171521102 22 22 22 2 2 2x xy yx xy yx xy yx xy
y x xy y+ + ++++ + + 9.9.9.9.9. a a aa aa a aa a a3 223 23
25678910788 7810 91020 21241581910124+ ++ ++++ + 10 .10 .10 .10 .10
. m mn nmn n m nm mn n m nm m n mn n3 2 32 3 23 2 3 23 2 2
371247595211821 20364 775211852113611718+ + ++++ + + +
+12.12.12.12.12. 123578897201819781212 7207 188
197812192088997812192078a b c db c da b c da b c da b c d+ ++ +
+++++ + + + + 6.6.6.6.6. 58141378656582 781
183565898193565658981932 22 32 2 32 2 33 2 2ab abab aab ab aab ab
aa ab ab+ + +++ + + + 7.7.7.7.7. 211514136782921120 49563
29621129565963 2 2 34 2 2 34 3 2 2 34 3 2 2 3m n m n mnm m n mnm m
n m n mnm m n m n mn+ + ++ +++ + + 8.8.8.8.8. + + + + ++ +++ + + +
+ 78114231371237182912781 614238 32463
291278514231124659127851423112474 3 2 2 3 45 3 2 45 4 3 2 2 3 45 4
3 2 2 3 45 4 3 2 2 3 4 29xy xy xy xyx xy xyx xy xy xy xyx xy xy xy
xyx xy xy xy xy1.1.1.1.1.3856569 202456112456222 2a aaa a a a
2.2.2.2.2. 8 6 5123516 1230 3551523355a ba ba b a b+ +++ +
3.3.3.3.3. x xyxyx xy x xy3 223 2 3 2236796 796196+ + 4.4.4.4.4.a b
ca b ca b c a b c+ + +++ + 1234232 124 343 231274535.5.5.5.5.
2356122 335 661 22131632m n pm n pm n p m n p+ + ++++
+11.11.11.11.11. 353457235813563 55345815 7214 56253458821164 3 3
44 2 2 3 44 3 2 2 3 44 3 2 2 3 4x xy xy yx xy xy yx xy xy xy yx xy
xy xy y+ + + + + + EJERCICIO259.9.9.9.9. 79231821379111796 791811
1112 13137913918121115135 4 2 3 3 2 4 5 66 4 2 2 4 5 66 5 4 2 3 3 2
4 6 6 5 4 2 3 3 2 4 6xy xy xy xy xy yx xy xy xy yx x y xy xy xy y x
xy xy xy xy y+ + + + + + ++ +++ + + +10.10.10.10.10.
1379587112523163461 2314 3185 687112 3051118187113253 2 2 33 2 23 2
2 3 3 2 2 3x xy xy yx xy xyx xy xy y x xy xy y + + ++ +
11.11.11.11.11. 310375921372051413352136 7206 5145
39352131320111429354 2 2 4 66 4 2 2 4 66 4 2 2 4 6 6 4 2 2 4 6m n m
n nm m n m n nm m n m n n m m n m n n ++ ++++++ + +
+12.12.12.12.12. 387227121213355115634313387 10227 1012123413
93935381722171212342239355 4 3 2 2 3 54 3 2 4 55 4 3 2 2 3 4 55 4 3
2 2 3 4 5c cd cd cd dcd cd cd dc cd cd cd cd dc cd cd cd cd d+ + +
+ + +++++ + + + + EJERCICIO26Para los problemas 1 al 14 las
literales toman los siguientes valores:a = 1 b = 2 c = 3 x = 4 y =
5 m = 3/2 n = 2/51.1.1.1.1. a abab ba ab b222 22 2341 4 1 2 2 1 8 4
11 2.2.2.2.2. a bb ab aba b ab ab a ab ab b3 33 2 23 3 2 2 3 2 2 33
2 2 32 5 63 5 6 5 6 31 5 1 2 6 1 2 3 2 1 5 2 6 4 3 81 10 24 24 11++
+ + + + + + + + + + + 3.3.3.3.3. 1212531 221253121253121122533121
51 2 10272312aa b ca b c a b c + + + + + + 4.4.4.4.4. 3 510 82 15 8
2 8 1523283225152529448103459224512545 48 241027102 22 22 2 2 22 2m
nm n mnm n mn m mn n
_,
_, 5.5.5.5.5. x xy yxy xy yx xy xy xy y4 2 2 43 3 44 3 2 2 3 44
3 2 2 3 418 1516 6 916 18 6 64 16 4 5 18 4 5 6 4 5 6 5256 80 64 18
16 25 24 125 6 625256 5120 7 200 3000 3750 4 926 ++ + + + + + + + +
+ + + + . . . . .6.6.6.6.6. a am mam m ama am m am a am am m3 2 32
3 23 2 3 2 3 2 2 33 22 375 5 82 6 8 8 2 61 8 1322
132632124229462781 12928144 48 18 814194 ++ + + +
_, +
_, + + +7.7.7.7.7. 237815161104 167 882 1101218110121181
21102122841012142510 5 8203202 22 22 2 2 22 2a ab ba ab ba ab b a
ab b+ ++ 8.8.8.8.8. 2334121614124 163
145632563225322527856182032425278341250675 150 482008732002 2 32 2
3 32 2 3 3 2 23 2 2m n mn nm n mn n mm n mn m m m n mn+ + + +++++ +
+
_, +
_, +
_, + + + + + +9. 9. 9. 9. 9.a ab bab aba ab ab ab b5 2 4 54 2 3
35 4 2 3 3 2 4 55 4 2 3 3 2 4 5355 31 1 2 5 1 2 3 1 2 21 4 5 8 3 16
32 1 4 40 48 32 21 + + + + + + + + + + 10.10.10.10.10. + + + + ab
mn mxabab mn mx10 81516 10 816 1 2 103225832432 6 48
7411.11.11.11.11.aab ab ba ab ab b32 2 33 2 2 33 2 2 311 911 91 111
2 9 1 2 21 22 36 8 7 + + + + 12.12.12.12.12.
16423563816423563816442345643843232063896 256 80 924231247789424 24
258xx xx x x + + + + + 13.13.13.13.13.x xy xyx xy yx xy xy yx xy xy
y3 2 23 2 33 2 2 33 2 2 33 2 2 33162534351254 343162
3512514316151251443164 5154 51255146415161645251252516 15 20 5 56+
+ ++ + + + += + + + = + + += + + + =EJERCICIO271.1.1.1.1. + + ++
+ab b aa b a ab ba ab b ab b2 22 2 2 22 2 25 44 42.2.2.2.2. aa+ + 8
16 1414 133.3.3.3.3. ++ + + + + x xyxy yx xy xy yxyx xy xy yx xy xy
y3 22 33 2 2 323 2 2 33 2 2 3455 475 412 44.4.4.4.4. + + + 3 4 53 4
5 44 5 43 2 3 43 2 3 33 4 3mn mn n mmn mn n nn m n5.5.5.5.5.8 9 3
67 9 35a b c aa b c aa a+ + 6.6.6.6.6.a b c a b ca b c aa a b c + +
+ + 2214. 14. 14. 14. 14. 255692 555 546354961351496113549630 18
245302573081 3 21 3 21 31 34 4 1 4 31730a a a aa a aa a aa a ax x x
xx x xx x xx x x + + + + += + = + = + += =7.7.7.7.7. + + + + +m n p
m n pm n p m n pm n p2 2 208.8.8.8.8. 9 5 39 7 25 25 66 25 24 5 32
2 22 2 2 22 2 2 2ax a x ax aax a x x aa x x ax a + + + + 9.9.9.9.9.
5 6 4 12 8 6 2 5 2 22 5 2 2 5 2 32 33 3 23 2 3 2a a aa a a a aa a a
a a a+ + + + + + 10.10.10.10.10. 5 9 4 111 7 6 11 2 9 6 411 2 9 6 4
12 2 9 6 53 2 44 3 4 3 24 3 2 4 3 2x x xx x x x x x xx x x x x x x
x + + + + + + + + 11.11.11.11.11. 35 7 117 35 8 6 7 57 5 8 52 2 3
33 2 2 3 23 2 3 2 3ab ab a ba ab ab a aba ab a ab b + + + + + +
+12.12.12.12.12. + + + + + + + + + + 11 14 25 819 6 9 411 19 8 16
47 411 19 8 16 411 26 8 20 44 23 24 3 25 34 3 25 4 3 2n n nn n nn n
n nn n nn n n nn n n n n13.13.13.13.13. + + + + + + + + + + +6 5 67
5 11 67 11 11 5 6 687 11 11 5 6 66 11 3 5 7 63 34 3 2 2 44 3 2 2 3
44 2 2 44 3 2 2 3 44 3 2 2 3 4a m ama a m a m ma a m a m am ma a m
ma a m a m am ma a m a m am m14.14.14.14.14. + + + + + + ++ + ++ +
+4 13 96 8 26 4 8 13 8 230 406 4 8 13 8 27 4 38 13 48 34 2 3 45 3 2
4 55 4 3 2 2 3 4 55 3 2 4 55 4 3 2 2 3 4 55 4 3 2 2 3 4 5xy xy xyx
xy xy yx xy xy xy xy yx xy xy yx xy xy xy xy yx xy xy xy xy
y15.15.15.15.15. a b aa b a ba b+ +22216.16.16.16.16.8 9 8 6 46 5
28 6 4 8 6 6x x yyx y x y+ + ++ + + +17.17.17.17.17. x y x xy yxy y
yx xy y x xy y2 2 2 22 22 2 2 26 7 347 40 9 167 34 7 43 16 + + + +
18.18.18.18.18. 4 8 5 57 6 45 22 2 2 22 2 2 22 2 2 2a ab b a ab ba
ab b a ab ba ab b a b+ + + + ++ + +19.19.19.19.19.x yxy xyx xy xy
yx xy xy yx yx xy xy y3 32 23 2 2 33 2 2 33 33 2 2 314 514 514 53
194 14 5 20 + + + + 20.20.20.20.20.x xy y x xy yxy y x xy yx xy y4
2 2 4 4 2 2 42 2 4 4 2 2 44 2 2 46 2 328 31 2 322 32 0 + ++ + +
+21.21.21.21.21. + ++ + + + + + + + + 67 8 66 7 8 66 7 8 63 8 196 4
15 8 255 4 23 25 4 35 4 36 4 36 5 4 3n n nn n nn n n nn n n nn n nn
n n n n22.22.22.22.22. a ab abab ab ab ba ab ab ab ba ab ab ab bab
ab ba ab ab ab ab b5 3 2 44 3 2 4 55 4 3 2 4 55 4 3 2 4 54 2 3 55 4
3 2 2 3 4 53 622 10 1122 7 522 7 55 717 7 7 5 2 ++ + + + + + + + +
+ 23.23.23.23.23. 4 5 2 3 5 6 47 8 4 53 5 6 4 3 5 6 13 2 3 23 43 2
4 3 2m m m m m mm m mm m m m m m m + + + + + + + 24.24.24.24.24. 7
11 2 87 11 8 42 8 2 42 2 22 22 2a ab b ba ab bb b + + + +
25.25.25.25.25. 3 4 57 8 11 5 6 2 112 7 25 6 2 11 6 7 11a b ca b a
b ca b c a b ca b c a b + + + + + + + + 26.26.26.26.26. 5 14 19 89
13 15 17 10 63 52 8 17 10 13 254 25 4 3 24 35 4 3 2a a aa aa aa a a
a aa aa a a a a+ ++ + + + + + + + +27.27.27.27.27. m m n nm n m n
mn nm m n m n mn nm m n mn nm m n m n mn nm m n m n mn n4 2 2 43 2
2 3 44 3 2 2 3 44 2 2 3 44 3 2 2 3 44 3 2 2 3 410 1511 14 311 4 3
166 7 811 4 3 165 11 11 11 17+ + + ++ + + + + + + + 28.28.28.28.28.
a ab ab bab ab ab bab ab aba ab ab ab ab ba ab aba ab ab ab ab ba
ab ab ab ab b5 3 2 4 54 2 3 4 53 2 2 3 45 4 3 2 2 3 4 55 2 3 45 4 3
2 2 3 4 55 4 3 2 2 3 4 54 87 15 25 33 57 3 18 22 23 6 21 67 3 18 22
22 7 3 24 2 6+ + + + + + + + + + + + EJERCICIO2829.29.29.29.29. x
yxy xy xy yx xy xy xyx xy xy xy xy yx xy xy xyxy xy xy y5 54 3 2 2
3 55 4 3 2 2 35 4 3 2 2 3 4 55 4 3 2 2 34 3 2 4 53 21 183 21 1832
26 18 23 21 1829 47 2++ + + + ++ + + +30.30.30.30.30. 3 6748 7 1248
7 5 1311 21 22 1 11 22 1 1 2a aa a aa a aa a a aa a aa a a a ax xx
x xx x xx x x xx x xx x x x x+ + + + + + + + + + 6.6.6.6.6. ax x ax
ax xa ax a ax ax xa ax ax x a ax ax xa ax ax xa ax ax xax ax2 3 2 2
33 2 3 2 2 33 2 2 3 3 2 2 33 2 2 33 2 2 32 23 5 11 113 4 6 83 3 9
17 33 39 17 310 14 + + + ++ + + + + + +7.7.7.7.7. x xx x x x xx x x
xx x x x x x x xx x x xx x x xx x x x4 23 3 24 2 44 3 2 4 3 24 3 24
3 24 3 233 5 7 8 3 45 4 32 4 2 7 8 3 12 4 27 8 3 16 12 4 1+ + + + +
+ + + + + + + + + + +1.1.1.1.1.x x x xx xx x x x2 22 25 4 2 12 6 6
22 1 2 2 3+ + + + + 2.2.2.2.2. 3 5 7 4 6 23 8 3 7 7 34 6 2 7 7 3 3a
b c a b a b ca b c b c a b ca b c a b c a b c + + + + + +3.3.3.3.3.
x x x xx x x x x xx x x x x x x3 3 23 2 2 23 2 2 3 21 9 4 6 85 7 3
1 3 8 56 8 3 8 5 6 2 8 3+ + + + + + + + + +4.4.4.4.4. a a a aa a a
aa a a a a a a a2 4 3 23 43 2 4 4 3 21 21 2+ + + + + + + 5.5.5.5.5.
ab bc ac ab bc ac ab bc acbc ac ab bc ac ab acab bc ac ab ac ab bc+
+ + + + + + + + 5 3 4 6 9 97 8 9 3 5 4 96 9 9 4 9 3 6 98.8.8.8.8. m
nm n m n mn m m n m n mn nm m n n m n mn m m n mnm n m n mn n m m n
mn m m n m n mn n4 43 2 2 3 4 3 2 2 3 44 2 2 4 2 2 3 4 2 2 33 2 2 3
4 4 2 2 3 4 3 2 2 3 417 4 7 6 17 2 7 816 80 4 217 2 7 81 2 17 3 8
81 2 + + + + + + + + + + 9.9.9.9.9.+ + + + + + + + + + + + + + +a
aa a a a a a a a a a aa a a a a a a a aa a a a a a a a a a a a a35
4 2 4 3 2 5 4 3 22 3 2 4 3 25 4 3 2 4 3 2 5 3 276 8 4 11 10 275 11
26 16 8 7 15 12 7 7 1511 10 27 12 7 7 15 11 4 3 4210.10.10.10.10. 3
3 20 11 811 9 14 19 8 9 3 11 8 143 11 8 14 20 11 8 17 142 2 2 2 2
22 2 2 2 22 2 2 2 2x y x xy y x xy yxy y x xy y x xy yx xy y x xy y
x + + + + + + + +11.11.11.11.11. a aa aaa a + + +1 31 40 3 82 122
12.12.12.12.12. a ab ba ab b a ab b a ab ba ab b ab b a ab ba ab b
a ab b ab b2 22 2 2 2 2 22 2 2 2 22 2 2 2 23 8 7 9 3 45 11 17 8 7 2
92 9 4 5 + + + + + + + + + +13.13.13.13.13. mm m m m m m mm m m m m
m m mm m m m m m m m m m m43 2 5 5 4 22 4 2 4 3 24 3 2 5 4 2 5 4
318 6 5 16 16 7 197 1 16 7 3 7 13 57 13 5 16 7 19 17 13 24 + + + +
+ + + + + + + + 14. 14. 14. 14. 14. x yxy xy xy y xy xy xy x xy xy
xy xy yxy xy xy xy xy xyx xy xy xy xy y xy xy xy x xy xy xy xy y5
54 3 2 2 3 5 4 3 2 4 5 4 3 2 2 3 4 53 2 4 4 4 3 2 45 4 3 2 2 3 4 5
4 3 2 4 5 4 3 2 2 3 4 52 5 7 3 7 11 2 2 7 6 4 87 6 8 1 7 10 12 2 7
6 4 8 7 10 1 9 7 4 4 7 + + + + + + + + + + + + + + + + +15. 15. 15.
15. 15. + + + + + + + + + + + + + + + + + + + + + +a a aa a aa a a
a a a a a a a aa a a a a a a a a a a a aa a a a a a a a a a a a a a
a a6 45 3 24 3 4 2 5 4 3 23 2 5 3 2 6 5 4 3 26 5 4 3 2 5 4 3 2 6 5
4 37 83 11 46 11 2 8 3 7 8 5 5 3 7 48 58 135 5 4 1 5 7 41 50 8 3 5
4 14 133 5 4 14 13 5 3 7 48 58 13 8 4 2 44244 a16.16.16.16.16. a ax
ax xax ax ax ax ax axax ax x aa ax ax ax ax x a ax ax ax xa ax ax
ax xa ax ax ax ax x5 3 2 3 2 54 2 3 4 4 3 2 2 33 2 4 5 55 4 3 2 2 3
4 5 5 4 3 2 2 3 55 4 3 2 2 3 55 4 3 2 2 3 4 57 9 18 4 820 21 19 9
17 119 7 80 3620 2 21 26 71 9 11 4 289 11 4 2820 2 21 26 7111 + + +
++ + + + + + + + + + + + + + +ax ax ax ax x4 3 2 2 3 4 510 26 5 99
+ +1.1.1.1.1. a b aa b a ba b a ba b a b+ + ++ = + = 1234233413543
232 349 412541354512542.2.2.2.2. 38612355635563538656353863
56386433863 23 2 3 23 2 33a a aa a a a aa a a a aa a + + + + + + +=
+3.3.3.3.3.a b a ba b a ba b a ba b a b+ + + + ++++++= + + = +
+3357362523615165 259 2363 1514 16635736255264.4.4.4.4.
133715114296136 114291 30513514293155613514293155
2651429315765142963 223 23 233 23 23 2 15x xx xx x xx x xxx x xx x
xx x x + + ++= + + + ++ + = + + EJERCICIO 295.5.5.5.5.
7127123725637256341513712372567 91237251518 13433725151734 4 3 23 2
44 3 2 4 3 24 3 2a a a aa a a aa a a a a a aa a a a + + + + + + = +
6.6.6.6.6. + + + +++ + += + + = + 122314592519253125913203126 195
82031213202 155125913203121320175x y z yy z x y zx y z x zx y z x
z7.7.7.7.7. + ++= + ++ +323418562312 189 10123
2311811213121311811213121181122 2 32 2 32 2 32 2 33 32 2 33 2 2ab
ab bab ab bab ab bab ab ba bab ab ba ab ab8.8.8.8.8.
1229231512193513355911019110122 39356 592 110126
1101219351911012710a b b c a b cb c b c b ca b c b c a ca b c b c a
c + + ++= + = + = 9.9.9.9.9.
131815142314131940231835341101329401813194034110135 2440342 1101310
2940232 188 91210 8801319403411013194023181122801121403 2 2 3 22 3
2 3 23 2 3 23 2 3 2a a a a a a aa a a a a a aa a a a a a aa a a a a
a a a+ + + + + + + ++ ++= + = + = + = +10.10.10.10.10.
35562929192335731361232131417452293212357319143510 18122 391410
17451 2294 961239 2182 14352812191427452191361237182 2 2 2 2 22 2 2
2 22 2 2 2 22 2 2 2 2x xy y x xy y x xy yxy y x xy y x xy yx xy y x
xy y yx xy y x xy y y + + + + + + +++++++++= + = = += + =
34357319143573136122 2 2 2x xy y x xy y11.11.11.11.11.
27151214153438327103438110541832122734381102734382 1102 542 18322
5102715 1107102734381103 3 2 2 2 2 32 2 3 2 2 3 3 2 2 33 2 2 3 2 2
3 3 33 2 2 3a b ab ab ab ab bab ab b ab ab b a ab ab ba ab ab b ab
ab b a ba ab ab b + + + + + + + + ++ + = + = + = =
34383271027141071027757102 2 3 3 3 3 3ab ab b a b a
b12.12.12.12.12. 131427352553514251147201423514141160144 11412
72020 24 1560143 5 231411601425514520116014032 2 3 44 3 2 2 4 4 44
3 2 2 4 4 3 2 2 34 3 2 2 3 4 3 2 2 3 44 3 2 2 3m n mn nm m n m n n
m nm m n m n n m m n m n mnm m n m n mn n m n m n mn nm m n m n mn
n + + + ++++ + + + = + + 44 3 2 2 351414116014= + + m m n m n
mn13.13.13.13.13. 1213341625145121338121312730141338124 9125 12301
241338121312730141340 381213127301413381213127301413378x yy zz mm n
x y z m nx y z m n x y z m nx y z m n x y z m n++ + + + ++++ + + +=
+ + + + = + +14.14.14.14.14.
5612352338538341623112411212152331639403111123820 9246 9 2124 1624
15 3940165 22 93311 2424121216
9926411241123604015233112411212152334 324 3 2 4 3 23 4 34 3 2 4 24
3 2 4 3 2a a aa aa a a a a aa a a aa a a a a aa a a a a a a+ + + ++
+ + + + ++ + + += + = + =. +13241211172644 2a a.EJERCICIO
301.1.1.1.1. x xx x xxRta x x x3 23 23 253 5 63 63 11 + + + + + .4.
4. 4. 4. 4.x xx x xxRtax x x3 23 23 24 84 1354 13 + + + + .
7.7.7.7.7. 5 8 114 8 115 8 113 2 33 33 23 2 3a ab ba ba abRta a ab
b+ ++ + .2.2.2.2.2. + + + ++ + + +5 9 65 9 6 8 98 95 9 6 8 9a b ca
b c xxRta a b c x .5. 5. 5. 5. 5. m mn nm m n mn nm nRtam m n mn n4
3 44 2 2 3 42 24 2 2 3 43 64 3 6 84 84 3 6 8 + + + + + .8. 8. 8. 8.
8. 1213412134x yx y +3.3.3.3.3. a ba ab ab bab ab bRta a ab ab b3
33 2 2 32 2 33 2 2 38 5 38 5 48 5 3 + + + .6. 6. 6. 6. 6.4 5 5 25 4
84 9 64 5 5 23 2233 2x x xx xx xRta x x x+ + ++ . Rta x y .12134
9.9.9.9.9. 5 7 85 7 8 115 7 8 12 22 22 2x xy yx xy yRta x xy y + +
+ + + + .10. 10. 10. 10. 10. nm m n mn nm m n mn nm m n mn nmRta m
m n mn n33 2 2 33 2 2 33 2 2 333 2 2 39 8 59 8 5 210 8 5 210 8 5 2
+ + + + + + .11. 11. 11. 11. 11. 05 85 8033 + + a aa aRta.EJERCICIO
311.1.1.1.1.( ) x x yx x yy = += 6.6.6.6.6.( ) ( ) a a b a ba a b a
ba a a+ + += + += = 2 11.11.11.11.11.( ) ( ) x y x y z x y zx y x y
z x y zx y z+ + + + = + + + += + 22.2.2.2.2. ( ) x x xx x xx2 22 23
53 53 5+ += += + 7.7.7.7.7.a b a a ba b a a ba a a2 2 2 2 22 2 2 2
22 2 2223 2+ + = + += = 12.12.12.12.12.( ) ( ) ( ) a b a a b a ba b
a a b a bb + + + += + + =2223.3.3.3.3.( ) a b aa b aa b+ += + + = +
2 32 33 3 8.8.8.8.8.{ } { } 2 1 32 1 31 3 4a x a a xa x a a x + + =
+ + += + = 13. 13. 13. 13. 13.( ) ( ) [ ] + + + += + + + + = + +x y
xy x xy y xyx y xy x xy y xyx xy y2 2 2 22 2 2 22 22 32 33 3
24.4.4.4.4. ( ) 4 24 26m m nm m nm n = + += + 9. 9. 9. 9. 9. ( ) [
] x y x xy y x yx y x xy y x yx xy y2 2 2 2 2 22 2 2 2 2 22 2222+ +
+ + += + += + 14. 14. 14. 14. 14. [ ] { } ( ) 8 2 3 38 2 3 38 42 2
2 2 22 2 2 2 22 2x xy y x xy y x xyx xy y x xy y x xyx y+ + + = + +
+ += +5.5.5.5.5.( ) 2 3 4 32 3 4 32x y x yx y x yx+ += + = 10. 10.
10. 10. 10.( ) ( ) + + + = + + = +5 6 5 65 6 5 66 5m mm mm
15.15.15.15.15. ( ) ( ) ( ) ( ) + + + + += + + +=a b a b b a a ba b
a b b a a b330EJERCICIO 321.1.1.1.1.( ) [ ][ ]2222a a a ba a a ba a
a ba b+ += + = + = 2.2.2.2.2.( ) [ ][ ]3 23 23 24x x y x yx x y x
yx x y x yx + += + = + +=3.3.3.3.3.( ) ( ) [ ][ ]2222 2m m n m nm m
n m nm m n m nm n += = + + += +4.4.4.4.4.( ) ( ) ( )[ ][ ]4 3 2 34
3 2 34 3 2 36 3 42 2 2 2 22 2 2 2 22 2 2 2 22 2x x xy y xy x yx x
xy y xy x yx x xy y xy x yx xy y+ + + += + + + + = + + + = +
5.5.5.5.5.( ) ( ) { }{ }a a b a b c aa a b a b c aa a b a b c aa c+
+ + += + + + + += + + + += +2226.6.6.6.6.( ) [ ] ( ) [ ][ ] [ ]4 2
3 4 2 14 2 3 4 2 14 2 3 4 2 15 2m m n n mm m n n mm m n n mn + + +=
+ + = + = +7.7.7.7.7.{ } ( )[ ]( )[ ][ ]2 5 22 5 22 5 22 5 22x x y
x yx x y x yx x y x yx x y x yx y+ + += + += + + + = + + =
+8.8.8.8.8.( )[ ] { }[ ] { }{ }x xy y x xy yx xy y x xy yx xy y x
xy yx xy y x xy yx xy y2 2 2 22 2 2 22 2 2 22 2 2 22 27 3 27 3 27 3
27 3 22 4 3 + + + = + + = + = + + + += + +9.9.9.9.9.( ) ( ) { }[ ]{
} [ ][ ] + + + + += + + + += + + + + = + + + = a b a b a b a b aa b
a b a b a ba b a b a b a ba b a b a b a ba b3 2
2222210.10.10.10.10.( ) ( ) [ ] { }[ ] { }{ } + + + += + + + = + +
= + + + + += +x y x y x y x yx y x y x y x yx y x y x y x yx y x y
x y x yx y4 24 24 24 2311.11.11.11.11. ( ) ( ) ( ) ( ) [ ][ ] + + +
+ + + = + + + = + + = a b a b a b b a ba b a b a b b a ba b a b a b
b a ba b2 32 32 33 712.12.12.12.12.( ) ( )[ ] { }( )[ ] { }{ }7 3 5
3 2 37 3 5 3 2 37 3 5 3 2 37 3 5 3 2 37 2 52 2 22 2 22 2 22 2 22m m
n n m nm m n n m nm m n n m nm m n n m nm n + + += + + + = + + = +
+ + + = + 13.13.13.13.13.( ) ( ) ( ) [ ] { }[ ] { }{ }2 4 42 4 46
46 4a a b a b a b aa a b a b a b aa b a b a b aa b a b a b ab + +
+= + + + = + += + + =14.14.14.14.14.( ) { } ( )[ ] ( ){ } [ ] ( )[
] ( )( )3 5 2 63 5 2 63 5 2 63 5 2 63 5 2 65 5 6x y x y x x yx y x
y x x yx y x y x x yx y x y x x yx y x y x x yx y + + + += + + + =
+ + + = + + = + + + += +15.15.15.15.15. ( ) ( ) ( ) { }[ ]{ } [ ][
]6 2 2 26 2 2 26 2 2 26 2 2 26 7c a c a c a a c cc a c a c a a c cc
a c a c a a c cc a c a c a a c ca c + + + + += + += += + + + + + +
+ = +16.16.16.16.16.( ) ( ) ( ) { }( )[ ]( ) { } [ ]{ } [ ][ ] + +
+ += + + + = + + + = + + = + + + += + +3 2 2 2 5 63 2 2 2 5 63 2 2
2 5 63 2 2 2 5 63 2 2 2 5 66 2 1m n m m m n nm n m m m n nm n m m m
n nm n m m m n nm n m m m n nm n17. 17. 17. 17. 17.( ) ( ) ( )[ ](
){ }( ) [ ] { }[ ] { }{ }2 5 3 2 42 5 3 2 42 5 3 2 42 5 3 2 42 5 3
2 45 6a b a c a b c a ba b a c a b c a ba b a c a b c a ba b a c a
b c a ba b a c a b c a ba b+ + + + + += + + + + + = + + + + + + + =
+ + + + = + + + = 18. 18. 18. 18. 18.( ) ( )[ ]( ) ( ) ( ) { }( ) [
] { }[ ] + + + + + += + + + + = + + = + + = 3 2 3 2 3 23 2 3 2 3 23
2 3 2 3 23 2 3 4 2 14x x y x y x x yx x y x y x x yx x y x y x x yx
x y x y19.19.19.19.19.( ) [ ] ( ) [ ] [ ] ( ) [ ] { }[ ][ ] [ ] {
}{ } + + + + = + = + + += +=a a b c ca a b c ca a b c cb c
cb20.20.20.20.20.( ) [ ] { }( ) [ ] { }( )[ ] { } [ ] { }{ } { } +
+ += + + = + + = = a b b a a ba b b a a ba b b a a ba b b a a ba b
3 321.21.21.21.21. ( ) [ ] { } ( ) [ ] { } ( ) { } [ ][ ] { } [ ] {
} { } [ ]{ } { } [ ] + + + + + = + + + + = + + + += + + + + += + +a
b c c a b a ba b c c a b a ba b c c a b a ba b c c a b a ba b c
222.22.22.22.22.( ) ( ) { }( ) ( ) { }[ ]( ) { } { }[ ]{ } [ ][ ] +
+ + + + += + + += + + + += + + += + + + + + = + 3 4 2 33 4 2 33 4 2
33 4 2 33 4 2 32 4 7m m n m m n nm m n m m n nm m n m m n nm m n m
m n nm m n m m n nm n23.23.23.23.23. ( ) ( ) ( ) [ ] { } [ ][ ] {
}[ ]{ } [ ][ ] + + + + = + + + = + + + + = += + + = x x y x y z x y
yx x y x y z x y yx x y x y z x y yx x y zx x y z y z22
224.24.24.24.24.( ) ( ) ( ) [ ] { } [ ][ ] { } [ ]{ } [ ][ ] + + +
+= + + + += + = = + +a a a b a b c a ba a a b a b c a ba a c a ba a
c a b a b c 3EJERCICIO 331.1.1.1.1. ( )a b c da b c d + = + +
2.2.2.2.2. ( )x xy yx xy y2 22 23 63 6 += + +3.3.3.3.3. ( )x x xx x
x3 23 24 3 14 3 1+ += + +4.4.4.4.4. ( )a a b ab ba a b ab b3 2 2 33
2 2 35 35 3 + = + + 5.5.5.5.5. ( )x x x xx x x x4 3 24 3 22 2 12 2
1 + += + +6.6.6.6.6. ( )22a b c da b c d+ += + 7.7.7.7.7. ( )x x xx
x x3 23 23 43 4+ + = +8.8.8.8.8. ( )x xy xy yx xy xy y3 2 2 33 2 2
35 35 3 + = +9.9.9.9.9. ( )a x xy ya x xy y2 2 22 2 222 = +
+10.10.10.10.10. ( )a b bc ca b bc c2 2 22 2 222+ = + +EJERCICIO
341.1.1.1.1.( )( )[ ]x y x yx y x y+ + = 222.2.2.2.2. ( ) ( )( ) (
)[ ]4 2 3 24 2 3 2m n m n m nm n m n m n + + + = + + 3.3.3.3.3.( )[
]( )[ ] { }x xy x xy yx xy x xy y2 2 22 2 233 + += +4.4.4.4.4. [ ]
( )[ ] ( )[ ]x x x x xx x x x x3 23 23 4 2 3 2 33 4 2 3 2 3 + + +=
+ + + +5.5.5.5.5.( )[ ] { }( )[ ] { } [ ]2 3 22 3 2a b a a b aa b a
a b a+ + + = + + + EJERCICIO 356. 6. 6. 6. 6. ( )( )[ ] + += +2 32
3a a ba a b7.7.7.7.7.( ) ( )( ) ( )[ ]2 32 32 2 2 22 2 2 2x xy y xy
x yx xy y xy x y+ + + += + + +8.8.8.8.8.x x xx x x3 23 23 4 23 4 2
+ = + + { }9. 9. 9. 9. 9. ( )[ ]( )( )[ ]( ){ }m m m mm m m m4 24
23 2 3 2 33 2 3 2 3 + + + += + + +1.1.1.1.1.2 3 6 = 2.2.2.2.2. = 4
8 323.3.3.3.3. = 15 16 2404.4.4.4.4.ab ab a b ab = =+ + 1 1 1 1 2
25.5.5.5.5.2 3 6 62 2 1 3x x x x = =+6.6.6.6.6. = =+ +4 4 42 2 2 1
1 2 3 3ab ab a b ab7.7.7.7.7. = =+ +5 5 53 2 3 1 1 2 4 3xy xy x y
xy8.8.8.8.8.a b ax a bx abx2 3 2 2 2 3 4 33 3 3 = =+9.9.9.9.9. =
=+4 5 20 202 2 2 1 2 3 2m mn p m n p m n p10.10.10.10.10.5 6 302 2
2 2ay x axy = 11.11.11.11.11. = =+xy yz xy z xyz2 3 3 4 2 3 3 4 2 6
44 4 412.12.12.12.12.abc cd abc d abcd = =+ 1 1 213.13.13.13.13. =
=+15 16 240 2404 3 2 3 2 4 3 3 2 7 3xy ax ax y axy14.14.14.14.14.3
4 122 3 2 2 3 2a b xy a bxy = 15.15.15.15.15.3 7 21 212 3 5 2 1 3 1
5 2 4 6abx bx ab x abx = =+ +16.16.16.16.16. = =+8 9 72 722 3 2 4 2
2 1 3 4 2 3 3 4m n a mx a m n x a m n x17.17.17.17.17.ab ab a bm n
m n =+ + 1 118.18.18.18.18. =+ +5 6 302 3 2 3ab abx a b xm n m
n19.19.19.19.19.x y c x y c x y c x y cm n m n x m m n n x m n x =
= + + + + 1 2 2 120.20.20.20.20. =+ +m n m n m nx a x a6 62 2
1EJERCICIO 361.1.1.1.1.a a a am m m m m = =+ + + + 1 1 2
12.2.2.2.2. = =+ + + +x x x xa a a a a 2 2 2 23.3.3.3.3.4 4 41 1 1
1 2 1a b ab a b a bn x x n x x n x = = + + + + + +4.4.4.4.4. = = +
+ + + + + + + + +a b a b a b a bn n n n n n n n n n 1 2 2 1 2 2 2 3
2 25.5.5.5.5. =+ + + + + +3 4 124 1 2 3 2 6 2 4a b a b a bn n n n n
n6.6.6.6.6.3 4 12 122 3 1 2 2 1 3 2 3 5xy x y x y x ym m m m m m =
=+ + + + + + + +7.7.7.7.7.4 5 202 4 5 1 2 7 2 5x b x b x ba a a a a
a + + + + + + =8.8.8.8.8.ab c ab a b c a b cm n m n m m n n m n =
=+ + 2 2 2 39.9.9.9.9. =+ + x y x y c x y cm a m a m a 1 2 3 5 2 2
2 2 3 24 410. 10. 10. 10. 10. = 5 7 351 2 3 4 3 3 2 5m n c m n m n
ca b a b a bEJERCICIO 371.1.1.1.1. 12451245410252 3 5 5 5a ab ab ab
ab = = =2.2.2.2.2. = = =377143771421983142 2 32 5 2 5 2 5m n a ma m
n a m n a m n3.3.3.3.3. 2335615252 3 2 4 2 6 4 2 6 4xy axy axy axy
= = 4.4.4.4.4. = =18454401103 4 3 2 3 5 5 3 5 5m n a m n a m n a m
n5.5.5.5.5. = = 78271456143 4 4abc a abc abc6.6.6.6.6. = =355636123
4 2 5 2 3 9 2 3 9xy aby abxy abxy7.7.7.7.7. 1335315151 1a a a am m
m = =+ +EJERCICIO 388.8.8.8.8. = =+ +342532153103 1 3 1 3a ab a b a
bm m m9.9.9.9.9. 5631012121421 2 1 2ab abca b c a b cm nm n m n = =
+ + + +10.10.10.10.10. = =+ + +293523152151 12 1 2 1 2 1 2 1a b a
ba b a bx m x mx m x m11.11.11.11.11. 384512403102 3 2 3 2ab a b a
b a bm n m n m n m n = = 12.12.12.12.12. = =+ 211447247871 3 2 3 22
2 1 2 2 2 1 2a b c a ba b c a b cx x xx x x x1.1.1.1.1.a a a a a =
=+ +3 3 32 1 1 2 42.2.2.2.2.3 3 32 3 2 2 2 3 1 2 6x xy ax ax y axy
= =+ +3.3.3.3.3. = =+ + +m n m mn m n m n2 2 3 2 2 1 1 3 5 43 5 15
154.4.4.4.4.4 5 20 202 3 2 2 2 3 1 2 2 6 2 2a ax ay a xy axy = =+
+5.5.5.5.5. = = + + + + +a ab ab a b a bm x m x m x2 3 6 62 1 2 1 3
16.6.6.6.6. 122335630153 2 4 2 4 3 1 6 4x ax a m a x m axm = =+
+7.7.7.7.7. 233431812322 4 4 12 4 4 1 6 5a ab aba b a bm xm x m x =
= ++ + + + + +EJERCICIO 398.8.8.8.8. = = + + + + +35511015503103 22
3 1 2 4m a m a ma m a mx ax a x a9.9.9.9.9.2 3 4 24 242 3 1 2 3 1
7a a a a a a = =+ + +10.10.10.10.10. = = + + + +3 4 560 602 3 23 1
2 2 1 1 6 4b ab ab axa b x abx11.11.11.11.11. ab a ab axa b x a b
xm xm x m x = = + + + + + +2 22 1 2 1 5 12 36 612.12.12.12.12. = =+
+ + + +12351033468342 2 3 22 1 3 2 1 2 1 8 4xy xy x xyx y
xy1.1.1.1.1.326 23 24 3x xxx x +2. 2. 2. 2. 2. 8 3216 62 235 3 2xy
yaxaxy axy3. 3. 3. 3. 3.x xxx x x23 24 322 8 6 + + 4. 4. 4. 4. 4.a
a aabab ab ab3 24 3 24 633 12 18 + +5. 5. 5. 5. 5.a ab babab ab ab2
23 2 2 322 + + 6. 6. 6. 6. 6.x x xaxax ax ax5 32 22 7 2 5 2 36 833
18 24 7. 7. 7. 7. 7.m m n nmxmx m nx m nx4 2 2 437 5 2 3 43 744 12
28 + + 8. 8. 8. 8. 8. x xy xyaxyaxy axy axy3 2 236 5 2 4 34 64 6 +
+9. 9. 9. 9. 9.a ab abamam abm ab m3 2 24 27 2 6 2 5 2 25 844 20 32
+ +10. 10. 10. 10. 10.a a aaa a aa a am m mm m mm m m + + = + + +
++ 1 21 1 1 2 11 122 2 22 2 211.11.11.11.11. x x xxx x xm m mmm m
m+ + + + 1 123 1 3 3 1333 9 312.12.12.12.12. ab a b a baba b a b
abm n m n m nm n m n m n+ + + ++ + + + +1 1 2 222 1 1 2 333 3
3EJERCICIO 4013.13.13.13.13. x x xxx x x x3 225 4 3 23 5 644 12 20
24 + + +14. 14. 14. 14. 14. a ax axbxabx abx abx bx4 3 2 234 3 3 4
2 5 36 9 833 18 27 24 + + 15. 15. 15. 15. 15.a a a aaxa x a x a x a
xn n n nnn n n n+ + ++ + + + + +3 2 122 3 2 2 2 2 2 1 2 2 23 43
416.16.16.16.16. x x x xaxax ax ax ax ax4 3 22 32 7 2 6 2 5 2 4 2
36 8 7 533 18 24 21 15 + + + + 17.17.17.17.17. + + 3 5 7 4515 25 35
203 2 2 32 22 4 2 2 3 3 2 2 4 2 5x xy xy yaxyaxy axy axy
axy18.18.18.18.18. x x x xxx x x xa a a aa a a a+ + + ++ + + + + +
+5 4 3 127 6 5 33 522 6 2 1019.19.19.19.19.a ab ab ab baya y aby
aby aby aby8 6 2 4 4 2 6 83 211 2 9 2 2 7 4 2 5 6 2 3 8 23 355 15 5
15 5 + + + + 20.20.20.20.20.ab a b a b a baba b a b a b a bm n m n
m n m nmm n m n m n m n+ ++ + + + ++ + + +344 12 4 41 2 2 4 3 632 3
2 1 5 2 2 7 2 3 91.1.1.1.1. 1223252104151541521 2 23 2a baa aba ab=
+2.2.2.2.2.23342349612491233 1 3 1 14 3 2a baba b abab ab += ++ +3.
3. 3. 3. 3. 35162553151551810155182321 1 2 2 2 12 2 2 3a b caca c
abc acac abc ac + + = + + + 4. 4. 4. 4. 4.251329365336965232 222 2
2 1 2 24 3 2 2a ab baxa x a bx abxax abx abx+ + = + + +5.5.5.5.5.
1325143236610381235382 232 3 3 1 2 32 3 4 5x xy yyxy xy yxy xy y =
+ +6.6.6.6.6.3 5 63109101510181091032952 32 1 3 2 3 2 33 3 2 3 2 3a
b caxa x abx acxax abx acx + + = + +7.7.7.7.7.
2913376633732122137174 2 2 43 44 3 4 2 3 2 4 3 4 47 4 5 6 3 8x xy
yxyx y x y xyxy xy xy + += ++ + + +EJERCICIO 418.8.8.8.8.
1213141558516524532540516524532182 2 2 222 2 2 2 2 2 2 24 2 2 2 2 2
2a b x yama m abm amx amyam abm amx amy + + += + ++9.9.9.9.9.
23125619346123815243361238581123 2 2 32 33 2 3 2 2 1 3 1 2 2 3 2 3
35 3 4 4 3 5 2 6m m n mn nm nm n m n m n m nm n mn m n m n+ + = + +
+ + + + +10.10.10.10.10. 251335110571035521153557027521371146 4 2 2
4 63 4 33 6 4 3 3 4 4 2 3 3 2 4 4 3 3 4 6 33 10 3 3 8 5 3 6 7 3 4
9x xy xy yaxyax y ax y ax y axyax y axy axy axy + + += + ++ + + + +
+1.1.1.1.1. aaa aaa a++ + 31332 3223.3.3.3.3. xxx xxx x++ + 5454
2020225.5.5.5.5. + + + +xxx xxx x3535 158 15227.7.7.7.7. 3 226 43
26 2222 2x yx yx xyxy yx xy y++ 2.2.2.2.2. aaa aaa a+ 31332
3224.4.4.4.4.mmm mmm m + +6565 3011 30226.6.6.6.6. ++ ++ +aaa aaa
a2323 65 6228.8.8.8.8.5 43 215 1210 815 22 8222 2x yx yx xyxy yx xy
y + ++ + 9.9.9.9.9. 5 735 715 215 8 21222 2a ba ba abab ba ab b++ +
11.11.11.11.11. + ++ + a ba ba abab ba ab b8 48 84 48 12 4222
213.13.13.13.13. ++ + + + +9 86 454 4836 3254 12 32222 2m nm nm
mnmn nm mn n10.10.10.10.10.7 32 414 628 1214 22 1222xxx xxx x++ +
12.12.12.12.12. 6 56 56 56 11 5222 2m nm nm mnmn nm mn n +
+14.14.14.14.14. + + + +7 32 1114 677 3314 71 3322yyy yyy
yEJERCICIO 421.1.1.1.1. x xy yx yx xy xyxy xy yx y2 23 2 22 2 33 3+
++ + 5.5.5.5.5. a a aaa a aa a aa a a3 24 3 23 24 212+ + + +9. 9.
9. 9. 9. m m mam aam am am amam am am aam am a3 24 3 23 242222 + +
+ + + 2.2.2.2.2.a ab ba ba ab abab ab ba ab ab b2 23 2 22 2 33 2 2
32223 3 + + + + 6.6.6.6.6. m m n nm nm m n m nm n m n nm n4 2 2 42
26 4 2 2 44 2 2 4 66 6+ ++ + 10. 10. 10. 10. 10. 3 5 24 512 20 815
25 1012 35 33 102 23 2 22 2 33 2 2 3a ab ba ba ab abab ab ba ab ab
b + + + + 3.3.3.3.3.a ab ba ba ab abab ab ba ab ab b2 23 2 22 2 33
2 2 32223 3+ +++ ++ + ++ + +7.7.7.7.7. x x xxx x x xx x xx x x3 24
3 23 24 32 3 12 32 4 6 23 6 9 32 7 3 + + + + + + 11.11.11.11.11.5
3315 9 35 315 5 9 3 34 2 2 45 3 2 44 2 3 55 4 3 2 2 3 4 5m m n nm
nm m n mnm n m n nm m n m n m n mn n + + + + + 4.4.4.4.4.x xxx x xx
xx x x3 24 33 24 23 1333 9 39 3 ++ ++ + + +9.9.9.9.9.3 6 523 6 56
12 103 5 12 10325 3 235 2y yyy y yy yy y y ++ ++ ++ + 12. 12. 12.
12. 12.a aa aa a aa a aa aa a a224 3 23 224 21112 1+ + + +
13.13.13.13.13. x x xx xx x xx x xx x xx x x3 225 4 34 3 23 25 222
522 4 25 10 512 5+ ++ ++ + + 19.19.19.19.19. x xy yx xy yx xy xyxy
xy xyxy xy yx xy xy y2 22 24 3 2 23 2 2 32 2 3 44 3 3 423223 6 33 5
3 + + + + + ++ + + +14.14.14.14.14. m mn mnm mn nm mn mnmn mn mnmn
mn mnm mn mn mn3 2 22 25 4 3 24 3 2 2 33 2 2 3 45 4 2 3 43 22 83 22
6 48 24 165 20 16 + + + + + 20.20.20.20.20. n nnn n nn nn n n224 3
224 32 1122 12 2 1 + + + + 15.15.15.15.15.x xx xx x xx x xx xx x
x224 3 23 224 21112 1+ + + + 21.21.21.21.21. a ab abab ab bab ab
abab ab abab ab abab ab ab ab3 2 22 2 35 4 2 3 34 2 3 3 2 43 3 2 4
55 4 2 2 4 53 42 103 42 6 810 30 405 22 40 + + + + +
16.16.16.16.16. x xx xx x xx x xx xx x x x x4 226 4 25 34 26 5 3 23
22 33 22 6 43 9 62 6 7 4 6 + + + + + + + +22. 22. 22. 22. 22. 8 12
6 92 316 24 12 1824 36 18 2716 24 273 2 2 34 3 2 2 33 2 2 3 44 2 2
4x xy xy yx yx xy xy xyxy xy xy yx xy y + + + + + 17.17.17.17.17.m
m mmm m m mm m mm m m m m3 236 5 4 33 26 5 4 24 1144 14 4 1+ ++ + +
+ + 23.23.23.23.23. 2 3 42 54 6 2 810 15 5 204 4 13 3 203 24 3 23
24 3 2y y yyy y y yy y yy y y y + + + + + + 18.18.18.18.18. a aa aa
a aa a aa aa a a a325 3 24 235 4 25 255 25 25 25 107 27 10 + + + +
+ + + +24.24.24.24.24. + + + ++ + + + a ax xa ax xa ax axax ax axax
ax xa ax ax ax x3 2 32 25 3 2 2 34 2 3 43 2 4 55 4 3 2 4 52 32 32 4
63 6 92 32 3 5 11 325.25.25.25.25.x xy xy xyx xy yx xy xy xyxy xy
xy xyxy xy xy xyx xy xy xy4 3 2 2 32 26 5 4 2 2 35 4 2 3 3 2 44 2 3
3 2 4 56 5 2 4 53 23 23 23 22 3 + + + + + + 30.30.30.30.30. y yy yy
y yy y yy yy y y y y24 26 5 44 3 226 5 4 32 12 222 4 22 4 22 4 4 2
+ + + + + + + +26.26.26.26.26.a a aa aa a a aa a a aa a aa a a a3
236 5 4 34 3 23 26 5 25 2 32 75 2 32 10 4 67 35 14 215 31 8 21 + +
+ + + + + +31.31.31.31.31. m mm mm m mm m mm mm m m m m m4 237 5 35
34 27 5 4 3 23 43 2 13 9 122 6 83 43 11 18 3 8 4 + + + + + + + +
+27.27.27.27.27. m m mm mm m m mm m m mm m mm m m m4 3 226 5 4 25 4
34 3 26 5 332 332 2 2 63 3 3 95 6 9+ + ++ + + + + + +
+32.32.32.32.32. a a aa a aa a a aa a a aa a a aa a aa a a a a a3
23 26 5 4 35 4 3 24 3 23 26 5 4 3 212 12 2 2 212 2 3 2 1+ ++ + ++ +
+ + + + + 28.28.28.28.28.a ab ab ab ba ab ba ab ab ab abab ab ab ab
abab ab ab ab ba ab ab ab ab b4 3 2 2 3 42 26 5 4 2 3 3 2 45 4 2 3
3 2 4 54 2 3 3 2 4 5 66 5 4 2 3 3 5 63232 2 6 2 234 6 3+ + ++ + + +
+ + + + +33. 33. 33. 33. 33. 8 12 63 2 424 36 18 316 24 12 232 48
24 424 52 38 33 26 43 2 2 32 25 4 3 2 2 34 3 2 2 3 43 2 2 3 4 55 4
3 2 2 3 4 5x xy xy yx xy yx xy xy xyxy xy xy xyxy xy xy yx xy xy xy
xy y + + + + + + + + +29.29.29.29.29.x xy xy xy yx xy yx xy xy xy
xyxy xy xy xy xyxy xy xy xy yx xy xy xy xy y4 3 2 2 3 42 26 5 4 2 3
3 2 45 4 2 3 3 2 4 54 2 3 3 2 4 5 66 4 2 3 3 2 4 5 622 2 2 2 22 2 2
3 2 + ++ + ++ + + + + + + 34.34.34.34.34.5 4 2 3 12 25 4 2 310 8 4
6 210 8 4 6 25 4 8 5 5 2 6 6 24 3 24 28 7 6 5 46 5 4 3 24 3 28 7 6
5 4 3 2a a a aa aa a a a aa a a a aa a a aa a a a a a a a + + + + +
++ + + + + 35.35.35.35.35. x x x xx x xx x x x xx x x x xx x x x xx
x x xx x x x x4 3 23 27 6 5 4 36 5 4 3 25 4 3 24 3 27 6 5 212 3 62
2 2 2 23 3 3 3 36 6 6 6 63 6 3 6 + + + + + + + + + + ++ + + + +
+40. 40. 40. 40. 40.3 6 2 3 23 4 53 6 2 3 29 18 6 9 612 24 8 12 815
30 10 15 103 15 14 28 47 28 23 105 3 24 29 7 6 5 47 5 4 3 26 4 3 25
3 29 7 6 4 3 2a a a aa a aa a a a aa a a a aa a a a aa a a aa a a a
a a a + + + + + + + + + + + + + + + 36.36.36.36.36. 3 2 5 42 13 2 5
43 2 5 46 4 10 83 2 5 43 5 9 10 8 3 43 23 26 5 4 35 4 3 24 3 23 26
5 4 3 2a a aa a aa a a aa a a aa a a aa a aa a a a a a+ + ++ + + +
++ + + + + 41.41.41.41.41.a b ca b ca ab acab b bcac bc ca b bc c+
++ ++ + + 2222 2 2237.37.37.37.37.5 3 4 23 15 3 4 215 9 12 65 3 4
25 3 11 11 17 3 4 24 3 24 28 7 6 56 5 4 34 3 28 7 6 5 4 3 2y y y yy
yy y y yy y y yy y y yy y y y y y y y + + + + + + + 42. 42. 42. 42.
42. x y zx y zx xy xzxy y yzxz yz zx xy y yz z+ ++ ++ + + + 22222
32222 2 238. 38. 38. 38. 38.m m n m n nm m n mn nm m n m n m nm n m
n m n m nm n m n m n mnm n m n m n nm m n m n m n m n m n mn n4 3 2
2 43 2 2 37 6 5 2 3 46 5 2 4 3 2 55 2 4 3 3 4 64 3 3 4 2 5 77 6 5 2
4 3 3 4 2 5 6 72 3 43 52 3 43 6 9 125 10 15 202 3 45 14 20 13 9 20
4 + + + + ++ + + ++ + + + + 43.43.43.43.43.2 3 522 3 52 3 54 6 102
5 3 102222 2 2x y zx y zx xy xzxy y yzxz yz zx xy xz y yz z + + + +
+ ++ + + +39.39.39.39.39.x xy xy yx xy xyx xy xy xyxy xy xy xyxy xy
xy xyx xy xy xy xy xy6 4 2 2 4 65 3 2 411 9 2 7 4 5 69 2 7 4 5 6 3
87 4 5 6 3 8 1011 9 2 7 4 5 6 3 8 1032 332 6 2 23 9 3 35 8 6 5 3 +
+ + + + + + + + 44.44.44.44.44.x xy xz y yz zx y zx xy xz xy xyz
xzxy xy xyz y yz yzxz xyz xz yz yz zx xyz y z2 2 23 2 2 2 22 2 3 2
22 2 2 2 33 3 33 + ++ + + ++ + ++ + + + +EJERCICIO 431.1.1.1.1.a a
aaa a aa a aa ax x xx x xx x xx x+ ++ + ++ ++ ++ ++ ++2 13 2 12
1312.2.2.2.2. + ++ + + + + + + ++ + ++ + ++ + ++ + + +x x xx xx x
xx x xx x x xn n nn n nn n nn n n n3 2 125 4 34 3 25 4 3 22223
3.3.3.3.3. m m m mm mm m m mm m m mm m m mm m m m ma a a aa a a aa
a a aa a a aa a a a a+ + + + + ++ + ++ + + + + + + ++ + + + + + +
+2 1 124 3 2 13 2 12 1 14 3 1 12 32 2 2 23 3 3 36 5 34.4.4.4.4. a a
aa aa a aa a aa a a an n nn nn n nn n nn n n n+ +++ + ++ ++ + ++ ++
+ + + + 2 112 3 2 2 2 12 2 2 1 22 3 2 2 2 1 23 23 23 24
25.5.5.5.5.x x xx xx x xx x xx x x x xa a aa aa a aa a aa a a a a+
++ ++ + ++ + ++ + + + ++ + ++ +2 13 12 5 2 4 2 32 3 2 2 2 12 5 2 4
2 3 2 2 2 12222 4 22 3 4 26.6.6.6.6.a a aa aa a aa a aa a aa a a ax
x xx x xx x xx x xx x x x ++ ++ + + + + ++ + 2 32 12 32 4 62 32 8
31 222 11 11 22 1 27.7.7.7.7. a a aa a aa a aa a aa a aa a a a ax x
xx xxx x xx x xx x xx x x x x+ ++ ++ + + + 3 23 23 23 22 4 5 21
2122 2 1 2 22 1 2 2 2 32 2 2 3 2 42 2 1 2 2 2 3 2 48.8.8.8.8. m m m
mm m mm m m mm m m mm m m mm m m m m ma a a aa aaa a a aa a a aa a
a aa a a a a a+ + + + + + ++ + + + + + + + + + + + ++ + + + +4 3 2
11 232 3 2 2 2 1 22 2 2 1 2 2 12 1 2 2 1 2 22 3 2 2 2 1 2 2 1 2
222222 2 49.9.9.9.9. x x x xx x xx x x xx x x xx x x xx x x xa a a
aa aaa a a aa a a aa a a aa a a a + + + + + + + 1 2 3 41 232 2 2 3
2 4 2 52 3 2 4 2 5 2 62 4 2 5 2 6 2 72 2 2 3 2 4 2
72222410.10.10.10.10. a b a b a b a ba b a ba b a b a b a ba b a b
a b a ba b a b a b a b a bn n n nn nn n n nn n n nn n n n n + + + +
+ + 1 2 2 3 3 42 2 42 3 2 1 4 2 2 5 2 3 62 2 5 2 3 6 2 4 7 2 5 82 3
2 1 4 2 2 5 2 4 7 2 5 8222211.11.11.11.11.a ba ba aba b ba ab a b
bx xm mm x m xx m m xm x m x x m m x++++ ++ + ++++
+12.12.12.12.12.a ba ba aba b ba ab a b bx nx nx nx n x n + +1 1111
113. 13. 13. 13. 13. + + + + ++ + + + + ++ + + + 5 36 830 6 1840 8
245 330 46 5 23 32 2 2 1 23 1 3 2 3 35 1 5 5 15 5 1 5 25 1 5 2 5 35
1 5 5 1 5 2 5 3a a aa a aa a aa a aa a aa a a a am m mm m mm m mm m
mm m mm m m m m14.14.14.14.14. x y x y xyx y x y x yx y x y x yx y
x y x yx y x y x yx y x y xa x a x a xa x a x a xa x a x a xa x a x
a xa x a x a xa x a x+ + + + ++ + + + + + +2 1 1 12 1 2 2 2 1 2 33
1 2 3 3 2 2 3 1 2 13 2 2 3 1 2 1 3 2 23 1 2 1 3 2 2 3 3 2 13 1 2 3
3 2 2 34 32 4 102 8 64 16 1210 40 302 4 28a x a xy x y +2 2 3 3 2
130EJERCICIO 441.1.1.1.1. 1213131216191416164 936161653616222 22 2a
ba ba abab ba ab ba ab b++ = + 2.2.2.2.2. x yx yx xyxy yx xy yx xy
yx xy y++ = + = + 25135613215561030134 25301030132130131371013222
22 22 23.3.3.3.3. 12131423322629212343638268 27362
612381335368123813353623382 23 2 22 2 33 2 2 33 2 2 33 2 2 3x xy yx
yx xy xyxy xy yx xy xy yx xy xy yx xy xy y + + + +++= + = +
4.4.4.4.4. 14231432116142123832661162 382 181211658201211658532 23
2 22 2 33 2 2 33 2 2 33 2 2a ab ba ba ab abab ab ba ab ab ba ab ab
ba ab ab b + + + +++= + = + 5. 251312322610363425131245232261015
123045 20 48603 4622610330176076351101760762 22 24 3 2 23 2 2 32 2
3 44 3 2 2 3 44 3 2 2 3 44 3 2 2 3 4m mn nm mn nm mn mnmn mn mnmn
mn nm mn mn mn nm mn mn mn nm mn mn mn n+ ++ ++ + ++ ++= + + = + +
7. + + + + ++ + + +++ += + + += + +
+1213323223343694121332262966343 3627 4 41227
418663466191223186634191223182 22 24 3 2 23 2 2 32 2 3 44 3 2 2 3
44 3 2 2 3 44 3 2 2 3 4x ax ax ax ax ax axax ax axax ax ax ax ax ax
ax ax ax ax ax ax ax ax a9.
141314123211015383638341401301401201201151201103860 312045 4 612090
3 81201 1201103863120471207912022011038214047120791201101103 225 4
3 24 3 23 25 4 3 25 4 3 2 5 4 3 2x x xx xx x x xx x x xx x xx x x x
xx x x x x x x x x x+ ++ + ++ + + + ++ ++ ++= + + + = + + + 6.
3814252132682445324112215682445682496 151202 18248
306045341211112016243860453412374023193045235 4 33 225 4 3 25 4 3
25 4 3 2x xx xx x xx x xx xx x x x xx x x x xx x x x x+ ++ ++ + +
+++= + + + = + + + 8. 2715121423562281201842121526104253051222821
80420105 112 20084010
5305122281014204178401530512114101420139280123 2 22 25 4 3 24 3 2 2
33 2 2 3 45 4 3 2 2 3 45 4 3 2 2 3 45 4 3 2 2 3x xy xyx xy yx xy
xyxy xy xyxy xy xyx xy xy xy xyx xy xy xy xyx xy xy xy + + + + +
++++ +++= + += + +5124xy10. 10. 10. 10.
10.341225142323526122641521261226415212158541010586124 61232 40
22512010 16 756020 12012058121012297120101603 2 2 32 25 4 3 2 2 34
3 2 2 3 43 2 2 3 4 55 4 3 2 2 3 4 55 4 3 2 2 3m mn mn nm mn nm mn
mn mnmn mn mn mnmn mn mn nm mn mn mn mn nm mn mn mn + + + + ++ +
+++ ++ +++ + +14012058125699401016076584 5 5 4 3 2 2 3 4 5mn n m mn
mn mn mn n + + EJERCICIO451.1.1.1.1. x x x por xx x x x3 2 25 4 211
1 11 0 11 1 11 1 11 1 0 1 1 + ++ + + + + + 2.2.2.2.2.x x x por x xx
x x x x x4 3 2 3 27 6 5 4 3 23 5 8 2 71 3 5 0 81 2 0 71 3 5 0 82 6
10 0 167 21 35 0 561 1 11 3 13 19 0 5611 3 13 19 56+ + + + + + + +
+ + + + + + + + + + + + 3.3.3.3.3. a ab ab ab b por a ab ba ab ab
ab ab ab b4 3 2 2 3 4 2 26 5 4 2 3 3 2 4 5 63 2 5 21 3 2 5 11 2 11
3 2 5 12 6 4 10 21 3 2 5 11 1 7 12 13 7 17 12 13 7+ + ++ + ++ + +
++ + + + + + + + + 4.4.4.4.4.m m n mn n por m mn nm m n m n m n m n
mn n3 2 2 3 3 2 36 5 4 2 3 3 2 4 5 65 6 41 5 6 11 0 4 11 5 6 14 20
24 41 5 6 11 5 2 20 19 10 15 2 20 19 10 + + + ++ + + + + + + + +
5.5.5.5.5.x x por x xx x x x4 2 4 28 6 4 28 3 6 51 0 8 0 31 0 6 0
51 0 8 0 36 0 48 0 185 0 40 0 151 0 2 0 50 0 58 0 152 50 58 15 + +
+ + ++ + + + + ++ + + + + + + + + + + + + 6.6.6.6.6.a a a por a a a
aa a a a a a a6 4 2 8 6 4 214 12 10 8 6 4 23 6 10 4 3 21 0 3 0 6 0
101 0 4 0 3 0 21 0 3 0 6 0 104 0 12 0 24 0 403 0 9 0 18 0 302 0 6 0
12 0 201 0 7 0 9 0 23 0 52 0 42 0 207 9 23 52 42 20 + + + + + ++ +
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
7.7.7.7.7.x x x por x xx x x x x9 6 3 6 315 12 9 6 34 3 2 3 8 101 0
0 4 0 0 3 0 0 23 0 0 8 0 0 103 0 0 12 0 0 9 0 0 68 0 0 32 0 0 24 0
0 1610 0 0 40 0 0 30 0 0 203 0 0 20 0 0 51 0 0 70 0 0 46 0 0 203 20
51 70 46 20 + ++ + + + + + + + + + + ++ + + + + + + + + + + + + +
++ + + + + + + + + + + + + + + + + + + + + + 9.9.9.9.9.x xy xy xy y
por x yx xy xy xy xy xy y5 4 3 2 2 3 5 2 27 6 5 2 4 3 3 4 2 5 73 6
4 2 41 3 6 4 0 12 0 42 6 12 8 0 24 12 24 16 0 42 6 8 20 24 18 0 42
6 8 20 24 18 4 + + + + + + + + 11. 11. 11. 11. 11. n n n n por n nn
n n n n n n n n6 4 3 4 210 8 7 6 5 4 3 23 5 8 4 3 41 0 3 5 0 8 41 0
3 0 41 0 3 5 0 8 43 0 9 15 0 24 124 0 12 20 0 32 161 0 6 5 13 23 8
44 12 32 166 5 13 23 8 44 12 32 16 + + ++ + + ++ + ++ + + + + + + +
+ + + + ++ + + + + + + + +13. 13. 13. 13. 13. x xy xy y por x xy xy
y10 6 4 2 8 10 6 4 2 2 4 65 3 6 4 51 0 0 0 5 0 0 0 3 0 61 0 4 0 5 0
11 0 0 0 5 0 0 0 3 0 64 0 0 0 20 0 0 0 12 0 245 0 0 0 25 0 0 0 15 0
301 0 0 0 5 0 0 0 3 0 61 0 4 0 10 0 21 0 28 0 23 0 + ++ + + + + + +
+ + + + ++ + + + + + + + + + + + + + + + + + + + + + + + + ++ + + +
+ + + + + + + + + + + + + + + + + + + + + 9 0 33 0 64 10 21 28 23 9
33 616 14 2 12 4 10 6 8 8 6 10 4 12 2 14 16x x y x y x y xy xy xy
xy y14. 14. 14. 14. 14. a a a por aa a a a am m mm m m m m + + ++ +
+ + + + + + + + 3 5 51 3 0 51 0 51 3 0 55 15 0 251 3 5 20 0 253 5
20 251 3 22 1 1 38.8.8.8.8. m m m por m m m mm m m m m m m12 8 4 16
12 8 428 24 20 16 12 8 47 9 15 5 9 4 31 7 9 151 5 9 4 31 7 9 155 35
45 759 63 81 1354 28 36 603 21 27 451 12 53 127 187 192 87 4512 53
127 187 192 87 45 + + + + + + + + ++ + + ++ + + + + + + +
10.10.10.10.10. 6 4 6 2 2 76 0 0 4 6 21 0 2 1 76 0 0 4 6 212 0 0 8
12 46 0 0 4 6 242 0 0 28 42 146 0 12 2 36 6 16 38 44 146 12 2 36 6
16 38 44 145 2 4 29 7 6 5 4 3 2a a a por a a aa a a a a a a a + + +
+ + + + + + + + + + ++ + + + + + + ++ + + + + + + + +12. 12. 12.
12. 12. 3 4 5 33 4 0 0 11 0 5 33 4 0 0 115 20 0 0 59 12 0 0 33 4 15
29 13 0 5 33 4 15 29 13 5 34 3 4 3 2 37 6 5 2 4 3 3 4 6 7x xy y por
x xy yx xy xy xy xy xy y + + + + + + + + + + ++ + + + + + + +
15.15.15.15.15. a a a por a a aa a aa a a ax x x x x xx x xx x x x+
+ + ++ + ++ + + + + + + + + + + + + + + 2 1 1 3 12 5 2 4 2 32 2 2 1
2 2 15 7 7 61 5 0 77 0 6 17 35 0 496 30 0 421 5 0 77 35 6 78 5 42
77 35 678 5 42 716. 16. 16. 16. 16. x x x por x x x xx x x xx x x
xa a a a a a aa a a aa a a a+ + + + + + ++ + + ++ + + + + ++ + + +
+ + + + + + 2 2 1 1 22 3 2 2 2 1 22 1 2 2 2 3 2 45 6 6 4 21 0 5 0
66 4 2 16 0 30 0 364 0 20 0 242 0 10 0 121 0 5 0 66 4 28 21 46 19
12 66 4 28 2146 19 12 6 17. 17. 17. 17. 17.a a a a por a a aa a a a
a ax x x x x x xx x x x x x2 2 2 1 2 2 1 3 1 3 3 15 3 5 2 5 1 5 5 1
5 23 5 6 5 31 3 1 56 5 36 18 6 305 15 5 253 9 3 156 23 12 34 22 156
23 12 34 22 15+ + + + + + + + + + ++ + + + + EJERCICIO461.1.1.1.1.(
)( ) 4 5 34 2034 2012 604 8 6022a aaaa aaa a+ ++ + 2.2.2.2.2.( )( )
3 1 13 313 33 33 322 22 2 22 22 2 2a x xax axax axax aax a+ ++
3.3.3.3.3. ( )( )( ) 2 3 1 42 612 62 62 8 642 8 68 32 242 26 24223
223a a aaaa aaa aaa a aa aa a + + ++ ++ + +4.4.4.4.4.( )( )( ) x x
xxxx xxx xxx x xx xx x x2 2 2224 224 226 4 24 26 4 21 1 11112 1122
11+ +++++ ++ ++ + + 5.5.5.5.5.( )( )( ) mm m mm mmm mm mm m mmm m
mm m mm m m m + + ++ ++ + + +4 6 3 24646 2410 243 23 30 722 20 483
28 52 4823 223 24 3 23 24 3 26.6.6.6.6.( )( )( ) a b a ab b a ba ab
ba ba a b abab ab ba a b ab ba ba a b ab aba b ab ab ba a b ab b +
+ ++ ++ + + + + + + 2 22 23 2 22 2 33 2 2 34 3 2 2 33 2 2 3 44 3 3
422222 27.7.7.7.7.( )( )( ) 3 2 1 1 13 6 313 6 33 6 33 9 9 313 9 9
33 9 9 33 6 6 323 24 3 23 24 3 25 4 3 24 3 25 4 2x x x x xx x xxx x
xx x xx x x xxx x x xx x x xx x x x + + + + + + + + + + +
11.11.11.11.11.( )( )( )( ) x x x xx xx xx x x xx xx x x xx xx x xx
x xx xx x x x + + + + + + + + + + + +3 4 5 13 54 13 54 12 512 4 54
5124 4 485 5 603 21 43 602 22 224 3 23 224 3 212.12.12.12.12. ( )(
)( )( ) x x x x xx x xx xx x x x xx x xx x x x x x xx x xx x x x xx
x x x xx x x x2 2 22 224 3 2 32 24 3 2 3 23 27 6 5 4 36 5 4 3 25 4
3 23 2 1 1 32 1 33 12 33 6 3 32 4 6 3 3 33 32 4 6 32 4 6 33 6 12 18
+ + ++ + + + + + + + ++ + + + + ++ + + + + + + + ++ + +93 6 12 18
99 9 9 94 3 27 6 5 4 3 2xx x x xx x x x x x x13.13.13.13.13. ( )(
)( )( ) 9 3 2 2 1 1 2 127 18 2 12 1 154 36 227 18 2 154 9 18 2 3 12
3 1108 18 36162 27 5454 9 18108 180 45 45 1823 24 3 23 24 3 2 226 5
45 4 34 3 26 5 4 3 2a a a a aa a aa aa a a aa a aa a a a aa aa a aa
a aa a aa a a a a + + + + + + + ++ + 14.14.14.14.14.( )( ) a a b a
b ba aba b ba b a ba b a ba b abx x x x x xx x xx x xx x x xx x x
xx x x x+ + + ++ ++ ++ + ++ + ++ ++ ++ 1 2 1 22 1 21 2 23 2 2 1 2
22 1 2 2 3 43 2 3 48. 8. 8. 8. 8. ( )( )( ) x x x x xx xx xx x xx x
xx xx x xxx x x xx x xx x x x x2 2224 3 23 224 25 3 24 25 4 3 21 1
21112 1222 2 4 22 4 5 2 + + ++ ++ + + + + + + + +9.9.9.9.9. ( )( )(
) a a aaaa aaa a aaa a a aa a aa a a a aa am m mmmm mmm m mmm m m
mm m mm m m m mm m + ++ + + + ++ + + 3 2 13232 62 3 612 3 63 2 63 3
2 631 112 1 12 1 112 1 3 2 2 2 12 1 12 1 3 2 2 2 13 2 2 1a a am m m
2 2 12 3 6 +10.10.10.10.10.( )( )( ) a a a aa aaa aa aa a aaa a aa
a aa a a a + + + + + 1 2 322 23 233 23 9 66 11 623 223 24 3 23 24 3
2EJERCICIO471.1.1.1.1. ( ) ( ) 4 3 5 24 12 5 109 22x xx xx+ + += +
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2 22x x xx x xx+ + + += + + += +3. 3. 3. 3. 3.( ) ( ) ( ) aa x ax a
ax aa ax ax a ax aa ax + + = + + += +3 2 33 6 3102 2 224. 4. 4. 4.
4. ( ) ( ) x y y x xyxy x xy y xyxy x y2 2 2 2 2 22 2 2 2 2 2 2 22
2 2 21 1 33+ + + = + + + = + +5.5.5.5.5. ( ) ( ) 4 5 3 34 5 3 3 3
33 3 23 2 2 2 2 2 23 2 4 2 2 3 24 3 2 2 2m mn m m n m m nm mn m m n
m mnm m m n mn + + = + + += + + 6.6.6.6.6.( ) ( ) ( ) y xy y x y x
y xy xy xy y xy y xy yy y2 2 3 3 2 2 2 2 22 2 3 2 3 3 2 2 2 2 2 23
21 1 13+ + + + = + + + += +7.7.7.7.7. ( ) ( )( )( )5 2 1 4 65 10 4
4 610 4 46 6222x x x xx x x x xx x x xx x+ + + = + + + + = + =
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aa a+ + + += + + + += + = + += + +5 5 3 2 2 5 45 5 25 3 2 2 4 5 205
5 3 45 5 3 122 5 72 22 22 229.9.9.9.9.( )( ) ( )( ) ( )( )( ) ( )a
b a b a b a b a b a ba ab b a ab b a ab ba ab b a ab b a ab ba ab
b+ + + = + = + + + + += + +4 3 5 2 3 3 64 3 15 2 3 3 64 3 15 2 3 3
614 5 52 2 2 2 2 22 2 2 2 2 22 210.10.10.10.10.( ) ( )( )( ) ( )(
)( )a c a ca c a c a c a ca ac ac c a ac ac ca ac c a ac ac cac+ =
+ + = + + + += + + + + =2 22 2 2 22 2 2 22411.11.11.11.11. ( ) ( )(
)( ) ( )( )( ) ( )( ) ( )3 4 3 33 4 3 33 4 3 33 2 4 2 3 33 6 3 4 8
4 3 32 14 42 22 22 22 2 2 2 2 22 2 2 2 2 22 2 2 2 2 22 2x y x y x
yx y x y x y x y x yx xy xy y x xy xy y x yx xy y x xy y x yx xy y
x xy y x yx xy y+ + = + + + = + + + + + = + + + + = + + + + = + 12.
12. 12. 12. 12.( ) ( ) ( )( )( ) ( )( ) ( )( )( )m n m n m nm n m n
m n m n m n m nm mn n m mn n m mn nm mn n m mn nm mn n+ + + = + + +
+ + = + + + + + += + = +2 2 22 2 2 2 2 22 2 2 22 22 42 2 4 42 4 4 8
162 6 17 4 42 10 1613.13.13.13.13.( ) ( ) ( )( ) ( )( ) ( )( )( )x
a x xa x a x a xax x ax x ax x a x a x a xax x x ax x a x a ax ax
xax x x a a ax ax xx x ax a a+ + + + + = + + + + + + = + + += + + +
= + + 3 1 1 23 3 2 24 3 2 232 522 22 2 2 22 2 22 214.14.14.14.14. (
) ( ) ( ) a b c a b c a b ca b c a b ca b c a b ca ab ac a ab acab
b bc ab b bcac bc c ac bc ca ab ac b bc c a ab ac b bc ca b ca b ca
ab acab b bcac bc ca ab+ + + + ++ ++ ++ ++ + + + + ++ + + + + ++ ++
++ ++ + ++ + ++ +2 2 22 22 22 22 2 2 2 2 222222 2 2 2 2 22 2ac b bc
ca ab ac b bc ca ab ac b bc ca ab ac b bc ca ab ac b bc ca b c ab
ac bc+ + ++ + + + + + + + + + 2 22 2 22 2 22 2 22 2 22 2 222 2 22 2
22 2 22 2 62 2 615.15.15.15.15.( ) ( ) ( ) x x x x x xx x x xx x x
xx x x x x xx x x x x xx x x xx x x x x x x xx xx xx x x x2222222
22 24 3 2 4 3 23 2 3 22 24 3 2 4 3 2224 3 23 2 33 23 23 23 23 3 9 2
2 42 5 6 9 2 3 4 4333+ + + + + + + + + + + + + + ++ + + + 4 3 23 2
4 3 22 4 3 24 3 2 4 3 22 5 6 93 2 3 4 43 3 9 2 5 6 92 5 6 9 2 7 4
14+ + + + + + + + + + + + + +