Ejercicio 212

CAPITULO XXVIII

Problemas sobre ecuaciones simultaneas
Ejercicio 212
Hallar el
  1. 3° término de (xy ) 5
    Sea: n=5 r=3 t 3 = 5 × 1 × 2 x 5(31 ) y 31 t 3 =10 x 3 y 2
  2. 4° término de (a4b ) 7
    Sea: n=7 r=4 t 4 = 7 × 6 × 5 1 × 2 × 3 a 7(41 ) (4b ) 41 t 4 =35 a 4 (4b ) 3 t 4 =2240 a 4 b 3
  3. 5° término de (1+x ) 11
    Sea: n=11 r=5 t 5 = 11 × 10 × × 8 1 × 2 × 3 × 4 1 11(51 ) ( x ) 51 t 5 =330 x 4
  4. 4° término de (3x2y ) 6
    Sea: n=6 r=4 t 4 = 6 × 5 × 4 1 × 2 × 3 (3x ) 6(41 ) (2y ) 41 t 4 =20(27 x 3 ) (8 y 3 ) t 4 =4320 x 3 y 3
  5. 5° término de ( a 2 2b ) 9
    Sea: n=9 r=5 t 5 = 9 × 8 × 7 × 1 × 2 × 3 × 4 ( a 2 ) 9(51 ) (2b ) 51 t 5 =126( a 10 ) (16 b 4 ) t 5 =2016 a 10 b 4
  6. 6° término de (2a b 2 ) 8
    Sea: n=8 r=6 t 6 = 8 × 7 × 6 × 5 × 4 1 × 2 × 3 × 4 × 5 (2a ) 8(61 ) ( b 2 ) 61 t 6 =56(3 a 3 ) ( b 5 32 ) t 6 =14 a 3 b 5
  7. 7° término de ( x 2 2y ) 10
    Sea: n=10 r=7 t 7 = 10 × × 8 × 7 × 6 × 5 1 × 2 × 3 × 4 × 5 × 6 ( x 2 ) 10(71 ) (2y ) 71 t 7 =210( x 8 ) (64 y 6 ) t 7 =13440 x 8 y 6
  8. 8° término de (x y 2 ) 11
    Sea: n=11 r=8 t 8 = 11 × 10 × × 8 × 7 × 6 × 5 1 × 2 × 3 × 4 × 5 × 6 × 7 ( x ) 11(81 ) ( y 2 ) 81 t 8 =330 x 4 ( y 14 ) t 8 =330 x 4 y 14
  9. 10° término de ( a 2 +b ) 15
    Sea: n=15 r=10 t 10 = 15 × × 13 × 12 × 11 × 10 × 9 × 8 × 7 1 × 2 × 3 × 4 × 5 × × 7 × 8 × 9 ( a 2 ) 15(101 ) ( b ) 101 t 10 =5005 a 12 b 9
  10. 9° término de (1 x 2 ) 12
    Sea: n=12 r=9 t 9 = 12 × 11 × × 9 × 8 × 7 × 6 × 5 1 × 2 × 3 × 4 × 5 × 6 × 7 × 8 ( 1 ) 12(91 ) ( x 2 ) 91 t 9 =495 x 16
  11. El penúltimo término de (2a b 2 ) 6
    Sea: n=6 r=6 t 6 = 6 × 5 × 4 × 3 × 2 1 × 2 × 3 × 4 × 5 (2a ) 6(61 ) ( b 2 ) 61 t 6 =6(2a ) ( b 10 ) t 6 =12a b 10
  12. El término del medio de (3 x 2 y 2 ) 8
    Sea: n=8 r=5 t 5 = × 7 × 6 × 5 1 × 2 × 3 × 4 (3 x 2 ) 8(51 ) ( y 2 ) 51 t 5 =70(81 x 8 ) ( y 8 ) t 5 =5670 x 8 y 8

Ejercicio 211

CAPITULO XXVIII

Problemas sobre ecuaciones simultaneas
Ejercicio 211
Desarrollar, hallando los coeficientes por el triángulo de Pascal:
Para resolver este grupo de ejercicios, hay que tener en cuenta el tringulo de Pascal que se detalla a continuación:

1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 1 8 28 56 70 56 28 8 1 1 9 36 84 126 126 84 36 9 1 1 10 45 120 210 252 210 120 45 10 1
  1. (a+2b ) 6 = a 6 +6 a 5 (2b ) +15 a 4 (2b ) 2 +20 a 3 (2b ) 3 +15 a 2 (2b ) 4 +6a (2b ) 5 + (2b ) 6 = a 6 +12 a 5 b+60 a 4 b 2 +160 a 3 b 3 +240 a 2 b 4 +192a b 5 +64 b 6
  2. (2 m 2 3 n 3 ) 5 = (2 m 2 ) 5 5 (2 m 2 ) 4 (3 n 3 ) +10 (2 m 2 ) 3 (3 n 3 ) 2 10 (2 m 2 ) 2 (3 n 3 ) 3 +5(2 m 2 ) (3 n 3 ) 4 (3 n 3 ) 5 =32 m 10 240 m 8 n 3 +720 m 6 n 6 1080 m 4 n 9 +810 m 2 n 12 243 n 5
  3. ( x 2 + y 3 ) 6 = ( x 2 ) 6 +6 ( x 2 ) 5 ( y 3 ) +15 ( x 2 ) 4 ( y 3 ) 2 +20 ( x 2 ) 3 ( y 3 ) 3 +15 ( x 2 ) 2 ( y 3 ) 4 +6( x 2 ) ( y 3 ) 5 + ( y 3 ) 6 = x 12 +6 x 10 y 3 +15 x 8 y 6 +20 x 6 y 9 +15 x 4 y 12 +6 x 2 y 15 + y 18
  4. (3 y 7 ) 7 = ( 3 ) 7 7 ( 3 ) 6 ( y 7 ) +21 ( 3 ) 5 ( y 7 ) 2 35 ( 3 ) 4 ( y 7 ) 3 +35 ( 3 ) 3 ( y 7 ) 4 21 ( 3 ) 2 ( y 7 ) 5 +7( 3 ) ( y 7 ) 6 ( y 7 ) 7 =21875103 y 7 +5103 y 14 2835 y 21 +945 y 28 189 y 35 +21 y 42 y 49
  5. (2 x 3 3 y 4 ) 6 = (2 x 3 ) 6 6 (2 x 3 ) 5 (3 y 4 ) +15 (2 x 3 ) 4 (3 y 4 ) 2 20 (2 x 3 ) 3 (3 y 4 ) 3 +15 (2 x 3 ) 2 (3 y 4 ) 4 6(2 x 3 ) (3 y 4 ) 5 + (3 y 4 ) 6 =64 x 18 576 x 15 y 4 +2160 x 12 y 8 4320 x 9 y 12 +4860 x 6 y 16 2916 x 3 y 20 +729 y 24
  6. ( x 2 2 + y 3 ) 5 = ( x 2 2 ) 5 +5 ( x 2 2 ) 4 ( y 3 ) +10 ( x 2 2 ) 3 ( y 3 ) 2 +10 ( x 2 2 ) 2 ( y 3 ) 3 +5( x 2 2 ) ( y 3 ) 4 + ( y 3 ) 5 = x 10 32 + 5 x 8 y 3 16 + 5 x 6 y 6 4 + 5 x 4 y 9 2 + 5 x 2 y 12 2 + y 15
  7. ( a 3 3 b ) 6 = ( a 3 ) 6 6 ( a 3 ) 5 ( 3 b ) +15 ( a 3 ) 4 ( 3 b ) 2 20 ( a 3 ) 3 ( 3 b ) 3 +15 ( a 3 ) 2 ( 3 b ) 4 6( a 3 ) ( 3 b ) 5 + ( 3 b ) 6 = a 6 729 2 a 5 27b + 5 a 4 3 b 2 20 a 3 b 3 + 135 a 2 b 4 486a b 5 + 729 b 6
  8. (1 x 4 ) 8 = ( 1 ) 8 8 ( 1 ) 7 ( x 4 ) +28 ( 1 ) 6 ( x 4 ) 2 56 ( 1 ) 5 ( x 4 ) 3 +70 ( 1 ) 4 ( x 4 ) 4 56 ( 1 ) 3 ( x 4 ) 5 +28 ( 1 ) 2 ( x 4 ) 6 8( 1 ) ( x 4 ) 7 + ( x 4 ) 8 =18 x 4 +28 x 8 56 x 12 +70 x 16 56 x 20 +28 x 24 8 x 28 + x 32
  9. ( 2 3x 3 2y ) 7 = ( 2 3x ) 7 7 ( 2 3x ) 6 ( 3 2y ) +21 ( 2 3x ) 5 ( 3 2y ) 2 35 ( 2 3x ) 4 ( 3 2y ) 3 +35 ( 2 3x ) 3 ( 3 2y ) 4 21 ( 2 3x ) 2 ( 3 2y ) 5 +7( 2 3x ) ( 3 2y ) 6 ( 3 2y ) 7 = 128 2187 x 7 224 243 x 6 y + 56 9 x 5 y 2 70 3 x 4 y 3 + 105 2 x 3 y 4 567 8 x 2 y 5 + 1701 32x y 6 2187 128 y 7
  10. ( 2 m m 2 2 ) 7 = ( 2 m ) 7 7 ( 2 m ) 6 ( m 2 2 ) +21 ( 2 m ) 5 ( m 2 2 ) 2 35 ( 2 m ) 4 ( m 2 2 ) 3 +35 ( 2 m ) 3 ( m 2 2 ) 4 21 ( 2 m ) 2 ( m 2 2 ) 5 +7( 2 m ) ( m 2 2 ) 6 ( m 2 2 ) 7 = 128 m 7 224 m 4 + 168 m 70 m 2 + 35 m 2 2 21 m 8 8 + 7 m 11 32 m 14 128
  11. ( x 3 +mn ) 8 = ( x 3 ) 8 8 ( x 3 ) 7 (mn ) +28 ( x 3 ) 6 (mn ) 2 56 ( x 3 ) 5 (mn ) 3 +70 ( x 3 ) 4 (mn ) 4 56 ( x 3 ) 3 (mn ) 5 +28 ( x 3 ) 2 (mn ) 6 8( x 3 ) (mn ) 7 + (mn ) 8 = x 18 8 x 21 mn+28 x 18 m 2 n 2 56 x 15 m 3 n 3 +70 x 12 m 4 n 4 56 x 9 m 5 n 5 +28 x 6 m 6 n 6 8 x 3 m 7 n 7 + m 8 n 8
  12. (3 b 2 3 ) 9 = ( 3 ) 9 9 ( 3 ) 8 ( b 2 3 ) +36 ( 3 ) 7 ( b 2 3 ) 2 84 ( 3 ) 6 ( b 2 3 ) 3 +126 ( 3 ) 5 ( b 2 3 ) 4 126 ( 3 ) 4 ( b 2 3 ) 5 +84 ( 3 ) 3 ( b 2 3 ) 6 36 ( 3 ) 2 ( b 2 3 ) 7 +9( 3 ) ( b 2 3 ) 8 ( b 2 3 ) 9 =1968319683 b 2 +8748 b 4 2268 b 6 +378 b 8 42 b 10 + 28 b 12 9 4 b 14 27 + b 16 243 b 18 19683
  13. (1 1 x ) 10 = ( 1 ) 10 10 ( 1 ) 9 ( 1 x ) +45 ( 1 ) 8 ( 1 x ) 2 120 ( 1 ) 7 ( 1 x ) 3 +210 ( 1 ) 6 ( 1 x ) 4 252 ( 1 ) 5 ( 1 x ) 5 +210 ( 1 ) 4 ( 1 x ) 6 120 ( 1 ) 3 ( 1 x ) 7 +45 ( 1 ) 2 ( 1 x ) 8 10( 1 ) ( 1 x ) 9 + ( 1 x ) 10 =1 10 x + 45 x 2 120 x 3 + 210 x 4 252 x 5 + 210 x 6 120 x 7 + 45 x 8 10 x 9 + 1 x 10
  14. (2 m 2 5 n 5 ) 6 = (2 m 2 ) 6 6 (2 m 2 ) 5 (5 n 5 ) +15 (2 m 2 ) 4 (5 n 5 ) 2 20 (2 m 2 ) 3 (5 n 5 ) 3 +15 (2 m 2 ) 2 (5 n 5 ) 4 6(2 m 2 ) (5 n 5 ) 5 + (5 n 5 ) 6 =64 m 12 960 m 10 n 5 +6000 m 8 n 10 20000 m 6 n 15 +37500 m 4 n 20 37500 m 2 n 25 +15625 n 30
  15. (4 x 5 4 ) 7 = ( 4 ) 7 7 ( 4 ) 6 ( x 5 4 ) +21 ( 4 ) 5 ( x 5 4 ) 2 35 ( 4 ) 4 ( x 5 4 ) 3 +35 ( 4 ) 3 ( x 5 4 ) 4 21 ( 4 ) 2 ( x 5 4 ) 5 +7( 4 ) ( x 5 4 ) 6 ( x 5 4 ) 7 =163847168 x 5 +1344 x 10 140 x 15 + 35 x 20 4 21 x 25 64 + 7 x 30 1024 x 35 16384

Ejercicio 210

CAPITULO XXVIII

Problemas sobre ecuaciones simultaneas
Ejercicio 210
Desarrollar:
  1. (x2 ) 4 = x 4 4 x 3 ( 2 ) + 4.3 2 x 2 ( 2 ) 2 6.2 3 x ( 2 ) 3 + 2 4 = x 4 8 x 3 +24 x 2 32x+16
  2. (a+3 ) 4 = a 4 +4 a 3 ( 3 ) + 4.3 2 a 2 ( 3 ) 2 + 6.2 3 a ( 3 ) 3 + 3 4 = a 4 +12 a 3 +54 a 2 +108a+81
  3. (2x ) 5 = 2 5 5 ( 2 ) 4 x+ 5.4 2 ( 2 ) 3 x 2 10.3 3 ( 2 ) 2 x 3 + 10.2 4 ( 2 ) x 4 x 5 =3280x+80 x 2 40 x 3 +10 x 4 x 5
  4. (2x+5y ) 4 = (2x ) 4 +4 (2x ) 3 (5y ) + 4.3 2 (2x ) 2 (5y ) 2 + 6.2 3 (2x ) (5y ) 3 + (5y ) 4 =16 x 4 +160 x 3 y+600 x 2 y 2 +1000x y 3 +625 y 4
  5. (a3 ) 6 = a 6 6 a 5 ( 3 ) + 6.5 2 a 4 ( 3 ) 2 15.4 3 a 3 ( 3 ) 3 + 20.3 4 a 2 ( 3 ) 4 15.2 5 a ( 3 ) 5 + 3 6 = a 6 18 a 5 +135 a 4 540 a 3 +1215 a 2 1458a+729
  6. (2ab ) 6 = (2a ) 6 6 (2a ) 5 ( b ) + 6.5 2 (2a ) 4 ( b ) 2 15.4 3 (2a ) 3 ( b ) 3 + 20.3 4 (2a ) 2 ( b ) 4 15.2 5 2a ( b ) 5 + b 6 =64 a 6 192 a 5 b+240 a 4 b 2 160 a 3 b 3 +60 a 2 b 4 12a b 5 + b 6
  7. ( x 2 +2 y 3 ) 5 = ( x 2 ) 5 +5 ( x 2 ) 4 (2 y 3 ) + 5.4 2 ( x 2 ) 3 (2 y 3 ) 2 + 10.3 3 ( x 2 ) 2 (2 y 3 ) 3 + 10.2 4 ( x 2 ) (2 y 3 ) 4 + (2 y 3 ) 5 = x 10 +10 x 8 y 3 +40 x 6 y 6 +80 x 4 y 9 +80 x 2 y 12 +32 y 15
  8. ( x 3 +1 ) 6 = ( x 3 ) 6 6 ( x 3 ) 5 ( 1 ) + 6.5 2 ( x 3 ) 4 ( 1 ) 2 15.4 3 ( x 3 ) 3 ( 1 ) 3 + 20.3 4 ( x 3 ) 2 ( 1 ) 4 15.2 5 x 3 ( 1 ) 5 + 1 6 = x 18 6 x 15 +15 x 12 20 x 9 +15 x 6 6 x 3 +1
  9. (2a3b ) 5 = (2a ) 5 5 (2a ) 4 (3b ) + 5.4 2 (2a ) 3 (3b ) 2 10.3 3 (2a ) 2 (3b ) 3 + 10.2 4 (2a ) (3b ) 4 (3b ) 5 =32 a 5 240 a 4 b+720 a 3 b 2 1080 a 2 b 3 +810a b 4 243 b 5
  10. ( x 4 5 y 3 ) 6 = ( x 4 ) 6 6 ( x 4 ) 5 (5 y 3 ) + 6.5 2 ( x 4 ) 4 (5 y 3 ) 2 15.4 3 ( x 4 ) 3 (5 y 3 ) 3 + 20.3 4 ( x 4 ) 2 (5 y 3 ) 4 15.2 5 x 4 (5 y 3 ) 5 + (5 y 3 ) 6 = x 24 30 x 20 y 3 +375 x 16 y 6 2500 x 12 y 9 +9375 x 8 y 12 18750 x 4 y 15 +15625 y 18
  11. (2x y 2 ) 6 = (2x ) 6 6 (2x ) 5 ( y 2 ) + 6.5 2 (2x ) 4 ( y 2 ) 2 15.4 3 (2x ) 3 ( y 2 ) 3 + 20.3 4 (2x ) 2 ( y 2 ) 4 15.2 5 2x ( y 2 ) 5 + ( y 2 ) 6 =64 x 6 96 x 5 y+60 x 4 y 2 20 x 3 y 3 + 15 4 x 2 y 4 3 8 x y 5 + y 64 6
  12. (3 x 2 3 ) 5 = ( 3 ) 5 5 ( 3 ) 4 ( x 2 3 ) + 5.4 2 ( 3 ) 3 ( x 2 3 ) 2 10.3 3 ( 3 ) 2 ( x 2 3 ) 3 + 10.2 4 ( 3 ) ( x 2 3 ) 4 ( x 2 3 ) 5 =243135 x 2 +30 x 4 10 3 x 6 + 5 27 x 8 x 10 243
  13. (2 m 3 3 n 4 ) 6 = (2 m 3 ) 6 6 (2 m 3 ) 5 (3 n 4 ) + 6.5 2 (2 m 3 ) 4 (3 n 4 ) 2 15.4 3 (2 m 3 ) 3 (3 n 4 ) 3 + 20.3 4 (2 m 3 ) 2 (3 n 4 ) 4 15.2 5 (2 m 3 ) (3 n 4 ) 5 + (3 n 4 ) 6 =64 m 18 576 m 15 n 4 +2160 m 12 n 8 4320 m 9 n 12 +4860 m 6 n 16 2916 m 3 n 20 +729 n 24
  14. ( x 2 3 ) 7 = ( x 2 ) 7 7 ( x 2 ) 6 ( 3 ) + 7.6 2 ( x 2 ) 5 ( 3 ) 2 21.5 3 ( x 2 ) 4 ( 3 ) 3 + 35.4 4 ( x 2 ) 3 ( 3 ) 4 35.3 5 ( x 2 ) 2 ( 3 ) 5 + 21.2 6 ( x 2 ) ( 3 ) 6 ( 3 ) 7 = x 14 21 x 12 +189 x 10 945 x 8 +2835 x 6 5103 x 4 +5103 x 2 2187
  15. (3a b 2 3 ) 5 = (3a ) 5 5 (3a ) 4 ( b 2 3 ) + 5.4 2 (3a ) 3 ( b 2 3 ) 2 10.3 3 (3a ) 2 ( b 2 3 ) 3 + 10.2 4 (3a ) ( b 2 3 ) 4 ( b 2 3 ) 5 =243 a 5 135 a 4 b 2 +30 a 3 b 4 10 3 a 2 b 6 + 5 27 a b 8 b 10 243
  16. ( x 2 +2 y 2 ) 7 = ( x 2 ) 7 +7 ( x 2 ) 6 (2 y 2 ) + 7.6 2 ( x 2 ) 5 (2 y 2 ) 2 + 21.5 3 ( x 2 ) 4 (2 y 2 ) 3 + 35.4 4 ( x 2 ) 3 (2 y 2 ) 4 + 35.3 5 ( x 2 ) 2 (2 y 2 ) 5 + 21.2 6 ( x 2 ) (2 y 2 ) 6 + (2 y 2 ) 7 = x 14 +14 x 12 y 2 +84 x 10 y 4 +280 x 8 y 6 +560 x 6 y 8 +672 x 4 y 10 +448 x 2 y 12 +128 y 14
  17. ( x 3 1 ) 8 = ( x 3 ) 8 8 ( x 3 ) 7 ( 1 ) + 8.7 2 ( x 3 ) 6 ( 1 ) 2 28.6 3 ( x 3 ) 5 ( 1 ) 3 + 56.5 4 ( x 3 ) 4 ( 1 ) 4 70.4 5 ( x 3 ) 3 ( 1 ) 5 + 56.3 6 ( x 3 ) 2 ( 1 ) 6 28.2 7 ( x 3 ) ( 1 ) 7 + ( 1 ) 8 = x 24 8 x 21 +28 x 18 56 x 15 +70 x 12 56 x 9 +28 x 6 8 x 3 +1
  18. ( x 2 y 2 ) 9 = ( x 2 ) 9 9 ( x 2 ) 8 ( y 2 ) + 9.8 2 ( x 2 ) 7 ( y 2 ) 2 36.7 3 ( x 2 ) 6 ( y 2 ) 3 + 84.6 4 ( x 2 ) 5 ( y 2 ) 4 126.5 5 ( x 2 ) 4 ( y 2 ) 5 + 126.4 6 ( x 2 ) 3 ( y 2 ) 6 84.3 7 ( x 2 ) 2 ( y 2 ) 7 + 36.2 8 ( x 2 ) ( y 2 ) 8 ( y 2 ) 9 = x 18 9 2 x 16 y+9 x 14 y 2 21 2 x 12 y 3 + 63 8 x 10 y 4 63 16 x 8 y 5 + 21 16 x 6 y 6 9 32 x 4 y 7 + 9 256 x 2 y 8 y 9 512
  19. (2 m 3 n 4 ) 7 = (2 m 3 ) 7 7 (2 m 3 ) 6 ( n 4 ) + 7.6 2 (2 m 3 ) 5 ( n 4 ) 2 21.5 3 (2 m 3 ) 4 ( n 4 ) 3 + 35.4 4 (2 m 3 ) 3 ( n 4 ) 4 35.3 5 (2 m 3 ) 2 ( n 4 ) 5 + 21.2 6 (2 m 3 ) ( n 4 ) 6 ( n 4 ) 7 =128 m 21 448 m 18 n 4 +672 m 15 n 8 560 m 12 n 12 +280 m 9 n 16 84 m 6 n 20 +14 m 3 n 24 n 28
  20. ( x 2 2 + 2 y 2 3 ) 5 = ( x 2 2 ) 5 +5 ( x 2 2 ) 4 ( 2 y 2 3 ) + 5.4 2 ( x 2 2 ) 3 ( 2 y 2 3 ) 2 + 10.3 3 ( x 2 2 ) 2 ( 2 y 2 3 ) 3 + 10.2 4 ( x 2 2 ) ( 2 y 2 3 ) 4 + ( 2 y 2 3 ) 5 = x 10 32 + 5 x 8 y 2 24 + 5 x 6 y 4 9 + 20 x 4 y 6 27 + 40 x 2 y 8 81 + 32 y 10 243
  21. ( 1 5 5a 2 ) 6 = ( 1 5 ) 6 6 ( 1 5 ) 5 ( 5a 2 ) + 6.5 2 ( 1 5 ) 4 ( 5a 2 ) 2 15.4 3 ( 1 5 ) 3 ( 5a 2 ) 3 + 20.3 4 ( 1 5 ) 2 ( 5a 2 ) 4 15.2 5 ( 1 5 ) ( 5a 2 ) 5 + ( 5a 2 ) 6 = 1 15625 3a 625 + 3 a 2 20 5 a 3 2 + 375 a 4 16 1875 a 5 16 + 15625 a 6 64

Ejercicio 208

CAPITULO XXVIII

Problemas sobre ecuaciones simultaneas
Ejercicio 208
Elevar al cuadrado
  1. x 2 2x+1 = ([ x 2 2x ] +1 ) 2 = ( x 2 2x ) 2 +2( x 2 2x ) +1 = x 4 4 x 3 +4 x 2 +2 x 2 4x+1 = x 4 4 x 3 +6 x 2 4x+1
  2. 2 x 2 +x+1 = ([2 x 2 +x ] +1 ) 2 = (2 x 2 +x ) 2 +2(2 x 2 +x ) +1 =4 x 4 +4 x 3 + x 2 +4 x 2 +2x+1 =4 x 4 +4 x 3 +5 x 2 +2x+1
  3. x 2 5x+2 = ([ x 2 5x ] +2 ) 2 = ( x 2 5x ) 2 +4( x 2 5x ) + 2 2 = x 4 10 x 3 +25 x 2 +4 x 2 20x+4 = x 4 10 x 3 +29 x 2 20x+4
  4. x 3 5 x 2 +6 = ([ x 3 5 x 2 ] +6 ) 2 = ( x 3 5 x 2 ) 2 +12( x 3 5 x 2 ) + 6 2 = x 6 10 x 5 +25 x 4 +12 x 3 60 x 2 +36
  5. 4 a 4 3 a 2 +5 = ([4 a 4 3 a 2 ] +5 ) 2 = (4 a 4 3 a 2 ) 2 +10(4 a 4 3 a 2 ) + 5 2 =16 a 8 24 a 6 +9 a 4 +40 a 4 30 a 2 +25 =16 a 8 24 a 6 +49 a 4 30 a 2 +25
  6. x+2yz = ([x+2y ] z ) 2 = (x+2y ) 2 2z(x+2y ) + z 2 = x 2 +4xy+4 y 2 2xz4yz+ z 2
  7. 3 x 3 x 6 = ([3 x 3 ] x 6 ) 2 = (3 x 3 ) 2 2 x 6 (3 x 3 ) + x 12 =96 x 3 + x 6 6 x 6 +2 x 9 + x 12 =96 x 3 5 x 6 +2 x 9 + x 12
  8. 5 x 4 7 x 2 +3x = ([5 x 4 7 x 2 ] +3x ) 2 = (5 x 4 7 x 2 ) 2 +6x(5 x 4 7 x 2 ) + (3x ) 2 =25 x 8 70 x 6 +49 x 4 +30 x 5 42 x 3 +9 x 2 =25 x 8 70 x 6 +30 x 5 +49 x 4 42 x 3 +9 x 2
  9. 2 a 2 +2ab3 b 2 = ([2 a 2 +2ab ] 3 b 2 ) 2 = (2 a 2 +2ab ) 2 6 b 2 (2 a 2 +2ab ) + (3 b 2 ) 2 =4 a 4 +8 a 3 b+4 a 2 b 2 12 a 2 b 2 12a b 3 +9 b 4 =4 a 4 +8 a 3 b8 a 2 b 2 12a b 3 +9 b 4
  10. m 3 2 m 2 n+2 n 4 = ([ m 3 2 m 2 n ] +2 n 4 ) 2 = ( m 3 2 m 2 n ) 2 +4 n 4 ( m 3 2 m 2 n ) + (2 n 4 ) 2 = m 6 4 m 5 n+4 m 4 n 2 +4 m 3 n 4 8 m 2 n 5 +4 n 8
  11. a 2 b+ c 4 = ([ a 2 b ] + c 4 ) 2 = ( a 2 b ) 2 + c 2 ( a 2 b ) + ( c 4 ) 2 = a 2 4 ab+ b 2 + ac 4 bc 2 + c 2 16 = a 2 4 + b 2 + c 2 16 ab+ ac 4 bc 2
  12. x 5 5y+ 5 3 = ([ x 5 5y ] + 5 3 ) 2 = ( x 5 5y ) 2 + 10 3 ( x 5 5y ) + ( 5 3 ) 2 = x 2 25 2xy+25 y 2 + 2 3 x 50 3 y+ 25 9
  13. 1 2 x 2 x+ 2 3 = ([ 1 2 x 2 x ] + 2 3 ) 2 = ( 1 2 x 2 x ) 2 + 4 3 ( 1 2 x 2 x ) + ( 2 3 ) 2 = x 4 4 x 3 + x 2 + 2 3 x 2 4 3 x+ 4 9 = x 4 4 x 3 + 5 3 x 2 4 3 x+ 4 9
  14. a x 1 3 + x a = ([ a x 1 3 ] + x a ) 2 = ( a x 1 3 ) 2 + 2x a ( a x 1 3 ) + ( x a ) 2 = a 2 x 2 2a 3x + 1 9 +2 2x 3a + x 2 a 2 = a 2 x 2 2a 3x 2x 3a + x 2 a 2 + 19 9
  15. 3 4 a 2 1 2 a+ 4 5 = ([ 3 4 a 2 1 2 a ] + 4 5 ) 2 = ( 3 4 a 2 1 2 a ) 2 + 8 3 ( 3 4 a 2 1 2 a ) + ( 4 5 ) 2 = 9 16 a 4 3 4 a 3 + 1 4 a 2 +2 a 2 4 3 a+ 16 25 = 9 16 a 4 3 4 a 3 + 29 20 a 2 4 3 a+ 16 25
  16. a 2 4 3 5 + b 2 9 = ([ a 2 4 3 5 ] + b 2 9 ) 2 = ( a 2 4 3 5 ) 2 + 2 b 2 9 ( a 2 4 3 5 ) + ( b 2 9 ) 2 = a 4 16 3 10 a 2 + 9 25 + a 2 b 2 18 2 b 2 15 + b 4 81 = a 4 16 3 10 a 2 + a 2 b 2 18 2 b 2 15 + b 4 81 + 9 25
  17. x 3 x 2 +x+1 = ([ x 3 x 2 ] +[x+1 ] ) 2 = ( x 3 x 2 ) 2 +2( x 3 x 2 ) (x+1 ) + (x+1 ) 2 = x 6 2 x 5 + x 4 +2( x 4 + x 3 x 3 x 2 ) + x 2 +2x+1 = x 6 2 x 5 + x 4 +2 x 4 2 x 2 + x 2 +2x+1 = x 6 2 x 5 +3 x 4 x 2 +2x+1
  18. x 3 3 x 2 2x+2 = ([ x 3 3 x 2 ] [2x2 ] ) 2 = ( x 3 3 x 2 ) 2 2( x 3 3 x 2 ) (2x2 ) + (2x2 ) 2 = x 6 6 x 5 +9 x 4 2(2 x 4 2 x 3 6 x 3 +6 x 2 ) +4 x 2 8x+4 = x 6 6 x 5 +9 x 4 4 x 4 +16 x 3 12 x 2 +4 x 2 8x+4 = x 6 6 x 5 +5 x 4 +16 x 3 8 x 2 8x+4
  19. x 4 +3 x 2 4x+5 = ([ x 4 +3 x 2 ] [4x5 ] ) 2 = ( x 4 +3 x 2 ) 2 2( x 4 +3 x 2 ) (4x5 ) + (4x5 ) 2 = x 8 +6 x 6 +9 x 4 2(4 x 5 5 x 4 +12 x 3 15 x 2 ) +16 x 2 40x+25 = x 8 +6 x 6 +9 x 4 8 x 5 +10 x 4 24 x 3 +30 x 2 +16 x 2 40x+25 = x 8 +6 x 6 8 x 5 +19 x 4 24 x 3 +46 x 2 40x+25
  20. x 4 4 x 3 +2x3 = ([ x 4 4 x 3 ] +[2x3 ] ) 2 = ( x 4 4 x 3 ) 2 +2( x 4 4 x 3 ) (2x3 ) + (2x3 ) 2 = x 8 8 x 7 +16 x 6 +2(2 x 5 3 x 4 8 x 4 +12 x 3 ) +4 x 2 12x+9 = x 8 8 x 7 +16 x 6 +4 x 5 22 x 4 +24 x 3 +4 x 2 12x+9
  21. 36a+ a 2 a 3 = ([36a ] +[ a 2 a 3 ] ) 2 = (36a ) 2 +2(36a ) ( a 2 a 3 ) + ( a 2 a 3 ) 2 =936a+36 a 2 +2(3 a 2 3 a 3 6 a 3 +6 a 4 ) + a 4 2 a 5 + a 6 =936a+36 a 2 +6 a 2 18 a 3 +12 a 4 + a 4 2 a 5 + a 6 =936a+42 a 2 18 a 3 +13 a 4 2 a 5 + a 6
  22. 1 2 x 3 x 2 + 2 3 x+2 = ([ 1 2 x 3 x 2 ] +[ 2 3 x+2 ] ) 2 = ( 1 2 x 3 x 2 ) 2 +2( 1 2 x 3 x 2 ) ( 2 3 x+2 ) + ( 2 3 x+2 ) 2 = x 6 4 x 5 + x 4 +2( 1 3 x 4 + x 3 2 3 x 3 2 x 2 ) + 4 9 x 2 + 8 3 x+4 = x 6 4 x 5 + x 4 + 2 3 x 4 + 2 3 x 3 4 x 2 + 4 9 x 2 + 8 3 x+4 = x 6 4 x 5 + 5 3 x 4 + 2 3 x 3 32 9 x 2 + 8 3 x+4
  23. 1 2 a 3 2 3 a 2 + 3 4 a 1 2 = ([ 1 2 a 3 2 3 a 2 ] +[ 3 4 a 1 2 ] ) 2 = ( 1 2 a 3 2 3 a 2 ) 2 +2( 1 2 a 3 2 3 a 2 ) ( 3 4 a 1 2 ) + ( 3 4 a 1 2 ) 2 = a 6 4 2 a 5 3 + 4 a 4 9 +2( 3 a 4 8 a 3 4 a 3 2 + a 2 3 ) + 9 a 2 16 3a 4 + 1 4 = a 6 4 2 a 5 3 + 4 a 4 9 + 3 a 4 4 3 a 3 2 + 2 a 2 3 + 9 a 2 16 3a 4 + 1 4 = a 6 4 2 a 5 3 + 43 a 4 36 3 a 3 2 + 59 a 2 48 3a 4 + 1 4
  24. x 5 x 4 + x 3 x 2 +x2 = ([ x 5 x 4 + x 3 ] [ x 2 x+2 ] ) 2 = ( x 5 x 4 + x 3 ) 2 2( x 5 x 4 + x 3 ) ( x 2 x+2 ) + ( x 2 x+2 ) 2 = ([ x 5 x 4 ] + x 3 ) 2 2( x 7 x 6 +2 x 5 x 6 + x 5 2 x 4 + x 5 x 4 +2 x 3 ) + ([ x 2 x ] +2 ) 2 = ( x 5 x 4 ) 2 +2 x 3 ( x 5 x 4 ) + x 6 2( x 7 2 x 6 +4 x 5 3 x 4 +2 x 3 ) + ( x 2 x ) 2 +4( x 2 x ) + 2 2 = x 10 2 x 9 + x 8 +2 x 8 2 x 7 + x 6 2 x 7 +4 x 6 8 x 5 +6 x 4 4 x 3 + x 4 2 x 3 + x 2 +4 x 2 4x+4 = x 10 2 x 9 +3 x 8 4 x 7 +5 x 6 8 x 5 +7 x 4 6 x 3 +5 x 2 4x+4

Ejercicio 207

CAPITULO XXVIII

Problemas sobre ecuaciones simultaneas
Ejercicio 207
Desarrollar:
  1. (2a+3b ) 3 =8 a 3 +3(4 a 2 ) (3b ) +3(2a ) (9 b 2 ) +27 b 3 =8 a 3 +36 a 2 b+54a b 2 +27 b 3
  2. (4a3 b 2 ) 3 =64 a 3 3(16 a 2 ) (3 b 2 ) +3(4a ) (9 b 4 ) 27 b 6 =64 a 3 144 a 2 b 2 +108a b 4 27 b 6
  3. (5 x 2 +6 y 3 ) 3 =125 x 6 +3(25 x 4 ) (6 y 3 ) +3(5 x 2 ) (36 y 6 ) +216 y 9 =125 x 6 +450 x 4 y 3 +540 x 2 y 6 +216 y 9
  4. (4 x 3 3x y 2 ) 3 =64 x 9 3(16 x 6 ) (3x y 2 ) +3(4 x 3 ) (9 x 2 y 4 ) 27 x 3 y 6 =64 x 9 144 x 7 y 2 +108 x 5 y 4 27 x 3 y 6
  5. (7 a 4 5 a 2 b 3 ) 3 =343 a 12 3(49 a 8 ) (5 a 2 b 3 ) +3(7 a 4 ) (25 a 4 b 6 ) 125 a 6 b 9 =343 a 12 735 a 10 b 3 +525 a 8 b 6 125 a 6 b 9
  6. ( a 8 +9 a 5 x 4 ) 3 = a 24 +3( a 16 ) (9 a 5 x 4 ) +3( a 8 ) (81 a 10 x 8 ) +729 a 15 x 12 = a 24 +27 a 21 x 4 +243 a 18 x 8 +729 a 15 x 12
  7. (8 x 4 7 x 2 y 4 ) 3 =512 x 12 3(64 x 8 ) (7 x 2 y 4 ) +3(8 x 4 ) (49 x 4 y 8 ) 343 x 6 y 12 =512 x 12 1344 x 10 y 4 +1176 x 8 y 8 343 x 6 y 12
  8. (3 a 2 b5 a 3 b 2 ) 3 =27 a 6 b 3 3(9 a 4 b 2 ) (5 a 3 b 2 ) +3(3 a 2 b ) (25 a 6 b 4 ) 125 a 9 b 6 =27 a 6 b 3 135 a 7 b 4 +225 a 8 b 5 125 a 9 b 6
  9. ( 1 2 a+ 2 3 b 2 ) 3 = a 3 8 + 3 ( a 2 ) ( 2 b 2 3 ) + 3 ( a 2 ) ( b 4 ) + 8 b 6 27 = a 3 8 + a 2 b 2 2 + 2a b 4 3 + 8 b 6 27
  10. ( 3 4 a 2 4 5 b 2 ) 3 = 27 a 6 64 3( 9 a 4 ) ( 4 b 2 5 ) +3( 3 a 2 4 ) ( b 4 25 ) 64 b 6 125 = 27 a 6 64 27 a 4 b 2 20 + 36 a 2 b 4 25 64 b 6 125
  11. ( 5 6 a 2 b 3 10 b 4 ) 3 = 125 a 6 b 3 216 3 ( a 4 b 2 ) ( 3 b 4 ) + 3 ( 5 a 2 b ) ( 9 b 8 ) 27 b 12 1000 = 125 a 6 b 3 216 5 a 4 b 6 8 + 9 a 2 b 9 40 27 b 12 1000
  12. ( 7 8 x 5 4 7 y 6 ) 3 = 343 x 15 512 3( x 10 ) ( 4 y 6 7 ) +3( 7 x 5 8 ) ( y 12 ) + 64 y 18 343 = 343 x 15 512 21 x 10 y 6 16 + 6 x 5 y 12 7 + 64 y 18 343
  13. ( x 2y + 3y x 2 ) 3 = x 3 8 y 3 +3( x 2 4 y 2 ) ( 3 y x 2 ) +3( x 2 y ) ( 9 y 2 x ) + 27 y 3 x 6 = x 3 8 y 3 + 9 4y + 27y 2 x 3 + 27 y 3 x 6
  14. ( 2 a 2 5 5 2 b 3 ) 3 = 8 a 6 125 3( a 4 ) ( 5 2 b 3 ) +3( 2 a 2 5 ) ( b 6 ) 125 8 b 9 = 8 a 6 125 6 a 4 5 b 3 + 15 a 2 2 b 6 125 8 b 9
  15. (4 x 4 3x y 3 ) 3 =64 x 12 3(16 x 8 ) ( 3x y 3 ) +3(4 x 4 ) ( 9 x 2 y 6 ) 27 x 3 y 9 =64 x 12 144 x 9 y 3 + 108 x 6 y 6 27 x 3 y 9
  16. ( 3a 2b + 4 b 2 5 ) 3 = 27 a 3 8 b 3 +3( 9 a 2 4 b 2 ) ( 4 b 2 5 ) +3( 3a 2b ) ( b 25 ) + 64 b 6 125 = 27 a 3 8 b 3 + 27 a 2 5 + 72a b 3 25 + 64 b 6 125
  17. ( 7 8 x 4 y 5 ) 3 = 343 512 3( 49 64 ) x 4 y 5 +3( 7 8 ) x 8 y 10 x 12 y 15 = 343 512 147 64 x 4 y 5 + 21 8 x 8 y 10 x 12 y 15
  18. ( 1 6 m 3 6 n 2 m 2 ) 3 = m 9 216 3 ( m ) ( 6 n 2 m 2 ) +3( m 3 6 ) ( n 4 m 4 ) 216 n 6 m 6 = m 9 216 m 4 n 2 2 + 18 n 4 m 216 n 6 m 6

Ejercicio 206

CAPITULO XXVIII

Problemas sobre ecuaciones simultaneas
Ejercicio 206
Desarrollar:
  1. ( a 5 +7 b 4 ) 2 = ( a 5 ) 2 +2( a 5 ) (7 b 4 ) + (7 b 4 ) 2 = a 10 +14 a 5 b 4 +49 b 8
  2. (3 x 4 5x y 3 ) 2 = (3 x 4 ) 2 2(3 x 4 ) (5x y 3 ) + (5x y 3 ) 2 =9 x 8 30 x 5 y 3 +25 x 2 y 6
  3. ( a 2 b 3 a 5 ) 2 = ( a 2 b 3 ) 2 2( a 2 b 3 ) ( a 5 ) + ( a 5 ) 2 = a 4 b 6 2 a 7 b 3 + a 10
  4. (7 x 5 8 x 3 y 4 ) 2 = (7 x 5 ) 2 2(7 x 5 ) (8 x 3 y 4 ) + (8 x 3 y 4 ) 2 =49 x 10 112 x 8 y 4 +64 x 6 y 8
  5. (9a b 2 +5 a 2 b 3 ) 2 = (9a b 2 ) 2 +2(9a b 2 ) (5 a 2 b 3 ) + (5 a 2 b 3 ) 2 =81 a 2 b 4 +90 a 3 b 5 +25 a 4 b 6
  6. (3 x 2 y 3 7 x 3 y 2 ) 2 = (3 x 2 y 3 ) 2 2(3 x 2 y 3 ) (7 x 3 y 2 ) + (7 x 3 y 2 ) 2 =9 x 4 y 6 42 x 5 y 5 +49 x 6 y 4
  7. (xy a 2 b 2 ) 2 = (xy ) 2 2xy a 2 b 2 + ( a 2 b 2 ) 2 = x 2 y 2 2xy a 2 b 2 + a 4 b 4
  8. ( 1 2 x+ 2 3 y ) 2 = ( 1 2 x ) 2 + 2 ( 1 2 x ) ( 2 3 y ) + ( 2 3 y ) 2 = 1 4 x 2 + 2 3 xy+ 4 9 y 2
  9. ( 3 4 a 2 2 5 b 2 ) 2 = ( 3 4 a 2 ) 2 2 ( 3 4 a 2 ) ( 2 5 b 2 ) + ( 2 5 b 2 ) 2 = 9 16 a 4 3 5 a 2 b 2 + 4 25 b 4
  10. ( 5 6 x 3 + 3 5 x y 2 ) 2 = ( 5 6 x 3 ) 2 + 2 ( 5 6 x 3 ) ( 3 5 x y 2 ) + ( 3 5 x y 2 ) 2 = 25 36 x 6 + x 4 y 2 + 9 25 x 2 y 4
  11. ( 1 9 a 5 3 7 a 3 b 7 ) 2 = ( 1 9 a 5 ) 2 2( 1 a 5 ) ( 3 7 a 3 b 7 ) + ( 3 7 a 3 b 7 ) 2 = 1 81 a 10 2 21 a 8 b 7 + 9 49 a 6 b 14
  12. ( 2 5 m 4 5 4 n 3 ) 2 = ( 2 5 m 4 ) 2 2 ( 2 5 m 4 ) ( 5 4 n 3 ) + ( 5 4 n 3 ) 2 = 4 25 m 8 m 4 n 3 + 25 16 n 6
  13. ( x 3 + y 2 4 ) 2 = ( x 3 ) 2 + 2 ( x 3 ) ( y 2 ) + ( y 2 4 ) 2 = x 2 9 + x y 2 6 + y 4 16
  14. ( 2x 3 3y 5 ) 2 = ( 2x 3 ) 2 +2( 2x 3 ) ( 3 y 5 ) + ( 3y 5 ) 2 = 4 x 2 9 4xy 5 + 9 y 2 25
  15. ( a 3 8 + 4 a 2 7b ) 2 = ( a 3 8 ) 2 + 2 ( a 3 8 ) ( 4 a 2 7b ) + ( 4 a 2 7b ) 2 = a 6 64 + a 5 7b + 16 a 4 49 b 2
  16. ( 3 2x 2 x 4 3 ) 2 = ( 3 2x ) 2 2 ( 3 2 x ) ( 2 x 3 ) + ( 2 x 4 3 ) 2 = 9 4 x 2 2 x 3 + 4 x 8 9
  17. ( 5 x 7 6 y 4 3 y 6 10 x 2 ) 2 = ( 5 x 7 6 y 4 ) 2 2 ( 5 x y 4 ) ( 3 y 10 x 2 ) + ( 3 y 6 10 x 2 ) 2 = 25 x 14 36 y 8 x 5 y 2 2 + 9 y 12 100 x 4
  18. ( 3 8 a 6 4 a 2 9 b 5 ) 2 = ( 3 8 a 6 ) 2 2 ( 3 8 a 6 ) ( 4 a 2 b 5 ) + ( 4 a 2 9 b 5 ) 2 = 9 64 a 12 a 8 3 b 5 + 16 a 4 81 b 10

Ejercicio 205

CAPITULO XXVIII

Problemas sobre ecuaciones simultaneas
Ejercicio 205
Desarrollar:
  1. (4 a 2 ) 2 = 4 2 a 2 × 2 =16 a 4
  2. (5a ) 3 = (5 ) 3 a 3 =125 a 3
  3. (3xy ) 3 = 3 3 x 3 y 3 =27 x 3 y 3
  4. (6 a 2 b ) 2 = (6 ) 2 a 2 × 2 b 2 =36 a 4 b 2
  5. (2 x 2 y 3 ) 3 = (2 ) 3 x 2 × 3 y 3 × 3 =8 x 6 y 9
  6. (4 a 2 b 3 c 4 ) 3 = 4 3 a 2 × 3 b 3 × 3 c 4 × 3 =64 a 6 b 9 c 12
  7. (6 x 4 y 5 ) 2 = (6 ) 2 x 4 × 2 y 5 × 2 =36 x 8 y 10
  8. (7a b 3 c 4 ) 3 = (7 ) 3 a 3 b 3 × 3 c 4 × 3 =343 a 3 b 9 c 12
  9. ( a m b n ) x = a mx b nx
  10. (2 x 3 y 5 z 6 ) 4 = (2 ) 4 x 3 × 4 y 5 × 4 z 6 × 4 =16 x 12 y 20 z 24
  11. (3 m 3 n ) 3 = (3 ) 3 m 3 × 3 n 3 =27 m 9 n 3
  12. ( a 2 b 3 c ) m = a 2m b 3m c m
  13. ( m 2 n x 3 ) 4 = ( m 2 ) 4 n 4 x 3 × 4 = m 8 n 4 x 12
  14. (3 a 2 b ) 5 = (3 ) 5 a 2 × 5 b 5 =243 a 10 b 5
  15. (7 x 5 y 6 z 8 ) 2 = 7 2 x 5 × 2 y 6 × 2 z 8 × 2 =49 x 10 y 12 z 16
  16. ( x 2y ) 2 = (x ) 2 2 2 y 2 = x 2 4 y 2
  17. ( 2m n 2 ) 3 = (2m ) 3 n 2 × 3 = 8 m 3 n 6
  18. ( a b 2 5 ) 3 = a 3 b 2 × 3 5 3 = a 3 b 6 125
  19. ( 3 x 2 4y ) 2 = (3 ) 2 x 2 × 2 4 2 y 2 = 9 x 4 16 y 2
  20. ( 2a b 2 3 m 3 ) 4 = (2 ) 4 a 4 b 2 × 4 3 4 m 3 × 4 = 16 a 4 b 8 81 m 12
  21. ( 2 m 3 n 3 x 4 ) 5 = 2 5 m 3 × 5 n 5 3 5 x 4 × 5 = 32 m 15 n 5 243 x 20
  22. ( 3 4 a 3 b 2 ) 2 = ( 3 4 ) 2 a 3 × 2 b 2 × 2 = 9 16 a 6 b 4
  23. ( 1 3 m n 2 ) 4 = ( 1 3 ) 4 m 4 n 2 × 4 = 1 81 m 4 n 8
  24. ( 1 2 a 2 b 4 ) 5 = ( 1 2 ) 5 a 2 × 5 b 4 × 5 = 1 32 a 10 b 20