Ejercicio 252

CAPITULO XXXI

R a d i c a l e s
Resolución de Ecuaciones con radicales que se reducen a primer grado
Ejercicio252
Resolver las ecuaciones:
  1. x + x+5 = 10 x x 2 + x(x+5 ) =10 x+ x(x+5 ) =10 ( x(x+5 ) ) 2 = (10x ) 2 x 2 +5x =10020x+ x 2 5x+20x =100 25x =100 x = 25 x =4
  2. 4x11 +2 x = 55 4x11 (4x11 ) 2 +2 x(4x11 ) =55 4x11+2 x(4x11 ) =55 2 x(4x11 ) =664x 2 x(4x11 ) = 2 (332x ) ( x(4x11 ) ) 2 = (332x ) 2 4 x 2 11x =1089132x+ 4 x 2 132x11x =1089 121x =1089 x = 121 x =9
  3. x x7 = 4 x x 2 x(x7 ) =4 x x 2 7x =4 (x4 ) 2 = ( x 2 7x ) 2 x 2 8x+16 = x 2 7x 8x+7x =16 x =16 x =16
  4. x 2 x +4 = x +1 x +13 ( x 2 ) ( x +13 ) =( x +1 ) ( x +4 ) x 2 +13 x 2 x 26 = x 2 +4 x + x +4 11 x 5 x =26+4 6 x =30 x = 6 (x ) 2 = 5 2 x =25
  5. 6 x+8 = x+8 x 6 = ( x+8 ) 2 x(x+8 ) 6 =x+8 x 2 +8x x 2 +8x =x+86 ( x 2 +8x ) 2 = (x+2 ) 2 x 2 +8x = x 2 +4x+4 8x4x =4 4x =4 x = 4 4 x =1
  6. x3 + 8 x+9 = x+9 (x3 ) (x+9 ) +8 x+9 = x+9 x 2 +6x27 +8 = (x+9 ) 2 x 2 +6x27 +8 =x+9 ( x 2 +6x27 ) 2 = (x+1 ) 2 x 2 +6x27 = x 2 +2x+1 6x2x =27+1 4x =28 x = 4 x =7
  7. x +4 x 2 = x +11 x 1 ( x +4 ) ( x 1 ) =( x +11 ) ( x 2 ) x 2 x +4 x 4 = x 2 2 x +11 x 22 3 x =9 x 22+4 6 x =18 x = 6 (x ) 2 = 3 2 x =9
  8. 2 x+6 4x3 = 9 4x3 2 (x+6 ) (4x3 ) (4x3 ) 2 =9 2 4 x 2 3x+24x18 4x+3 =9 2 4 x 2 +21x18 =93+4x 2 ( 4 x 2 +21x18 ) 2 = 2 (2x+3 ) 2 4 x 2 +21x18 = 4 x 2 +12x+9 21x12x =18+9 9x =27 x = 9 x =3
  9. x 2 x +2 = 2 x 5 2 x 1 ( x 2 ) (2 x 1 ) =(2 x 5 ) ( x +2 ) 2 x 2 x 4 x +2 = 2 x 2 +4 x 5 x 10 2+10 =4 x 12 =4 x ( 4 ) 2 = (x ) 2 x =9
  10. x+14 x7 = 6 x7 (x+14 ) (x7 ) (x7 ) 2 =6 x 2 +7x98 x+7 =6 x 2 +7x98 =67+x ( x 2 +7x98 ) 2 = (x1 ) 2 x 2 +7x98 = x 2 2x+1 7x+2x =98+1 9x =99 x = 9 x =11

Ejercicio 251

CAPITULO XXXI

R a d i c a l e s
Resolución de Ecuaciones con radicales que se reducen a primer grado
Ejercicio 251
Resolver las ecuaciones:
  1. x8 =2 ( x8 ) 2 = 2 2 x8 =4 x =12
  2. 5 3x+1 =0 5 2 = ( 3x+1 ) 2 25 =3x+1 251 =3x 24 =3x 3 =x x =8
  3. 7+ 5x2 3 =9 5x2 3 =97 ( 5x2 3 ) 3 = 2 3 5x2 =8 5x =8+2 x = 5 x =2
  4. 9 x 2 5 3x =1 ( 9 x 2 5 ) 2 = (3x1 ) 2 9 x 2 5 = 9 x 2 6x+1 51 =6x 6 6 =x x =1
  5. x 2 2x+1 =9x (x1 ) 2 =9x x1 =9x x+x =9+1 2x =10 x = 2 x =5
  6. 15 7x1 3 =12 1512 = 7x1 3 3 3 = ( 7x1 3 ) 3 27 =7x1 27+1 =7x 7 =x x =4
  7. x + x+7 =7 ( x+7 ) 2 = (7 x ) 2 x +7 =4914 x + x 749 =14 x 42 =14 x 14 = x 3 2 = x 2 x =9
  8. 3x5 + 3x14 =9 ( 3x5 ) 2 = (9 3x14 ) 2 3x 5 =8118 3x14 + 3x 14 581+14 =18 3x14 72 =18 3x14 18 = 3x14 4 2 = ( 3x14 ) 2 16 =3x14 16+14 =3x 3 =x x =10
  9. x+10 x+19 =1 ( x+10 ) 2 = ( x+19 1 ) 2 x +10 = x +192 x+19 +1 1020 =2 x+19 2 = x+19 5 2 = ( x+19 ) 2 25 =x+19 2519 =x x =6
  10. 4x11 =7 2x29 ( 4x11 ) 2 = (7 2x29 ) 2 4x11 =49(2x29 ) 4x11 =98x1421 4x98x =1421+11 94x =1410 x = 94 x =15
  11. 5x19 5x =1 ( 5x19 ) 2 = ( 5x 1 ) 2 5x 19 = 5x 2 5x +1 191 =2 5x 2 = 5x 1 0 2 = ( 5x ) 2 100 =5x 5 =x x =20
  12. x2 +5 = x+53 ( x2 +5 ) 2 = ( x+53 ) 2 x 2+10 x2 +25 = x +53 10 x2 =53+225 10 x2 =30 ( x2 ) 2 = ( 3 0 1 0 ) 2 x2 =9 x =11
  13. 9x14 =3 x+10 4 ( 9x14 ) 2 = (3 x+10 4 ) 2 9x14 =9(x+10 ) 24 x+10 +16 9x 1416 = 9x +9024 x+10 3090 =24 x+10 ( 24 ) 2 = ( x+10 ) 2 25 =x+10 2510 =x x =15
  14. x16 x+8 =4 ( x16 ) 2 = ( x+8 4 ) 2 x 16 = x +88 x+8 +16 16816 =8 x+8 40 =8 x+8 ( 8 ) 2 = ( x+8 ) 2 25 =x+8 258 =x x =17
  15. 5x1 +3 = 5x+26 ( 5x1 +3 ) 2 = ( 5x+26 ) 2 5x 1+6 5x1 +9 = 5x +26 6 5x1 =268 ( 5x1 ) 2 = ( 6 ) 2 5x1 =9 x = 5 x =2
  16. 13 13+4x =2 x (13 13+4x ) 2 = (2 x ) 2 16926 13+4x +13+ 4x = 4x 26 13+4x =182 ( 13+4x ) 2 = ( 26 ) 2 13+4x =49 4x =4913 x = 4 x =9
  17. x4 + x+4 =2 x1 ( x4 + x+4 ) 2 = (2 x1 ) 2 x 4 +2 x 2 16 +x+ 4 =4(x1 ) 2x+2 x 2 16 =4x4 Dividiendo la ecuación para 2 x+ x 2 16 =2x2 ( x 2 16 ) 2 = (x2 ) 2 x 2 16 = x 2 4x+4 164 =4x 4 =x x =5
  18. 9x+7 x 16x7 =0 ( 9x+7 16x7 ) 2 = (x ) 2 9x+ 7 2 (9x+7 ) (16x7 ) +16x 7 =x x = 2 (9x+7 ) (16x7 ) (12x ) 2 = ( (9x+7 ) (16x7 ) ) 2 144 x 2 = 144 x 2 63x+112x49 49 = 49 x x =1
  19. 9x+10 2 x+3 = x2 ( 9x+10 2 x+3 ) 2 = ( x2 ) 2 9x+104 (9x+10 ) (x+3 ) +4(x+3 ) =x2 9x+104 9 x 2 +27x+10x+30 +4x+12 =x2 4 9 x 2 +37x+30 =x213x22 4 9 x 2 +37x+30 =12x24 Dividiendo la ecuacion para 4 ( 9 x 2 +37x+30 ) 2 = (3x+6 ) 2 9 x 2 +37x+30 = 9 x 2 +36x+36 37x36x =30+36 x =6
  20. 18x8 2x4 2 2x+1 =0 ( 18x8 2x4 ) 2 = (2 2x+1 ) 2 18x82 4(9x4 ) (x2 ) +2x4 =4(2x+1 ) 20x124 9 x 2 18x4x+8 =8x+4 4 9 x 2 22x+8 =20x+12+8x+4 4 9 x 2 22x+8 =12x+16 Dividiendo la ecuación para 4 ( 9 x 2 22x+8 ) 2 = (3x4 ) 2 9 x 2 22x+8 = 9 x 2 24x+16 22x+24x =168 2x =8 x = 2 x =4
  21. 8x+9 18x+34 + 2x+7 =0 ( 8x+9 + 2x+7 ) 2 = ( 18x+34 ) 2 8x+9+2 (8x+9 ) (2x+7 ) +2x+7 =18x+34 10x+16+2 16 x 2 +56x+18x+63 =18x+34 2 16 x 2 +74x+63 =18x10x16+34 2 16 x 2 +74x+63 =8x+18 Dividiendo la ecuación para 2 ( 16 x 2 +74x+63 ) 2 = (4x+9 ) 2 16 x 2 +74x+63 = 16 x 2 +72x+81 74x72x =8163 2x =18 x = 2 x =9
  22. x2 x5 = 4x23 ( x2 x5 ) 2 = ( 4x23 ) 2 x22 (x2 ) (x5 ) +x5 =4x23 2 x 2 7x+10 =4x2x23+7 2 x 2 7x+10 =2x16 Dividiendo la ecuación para 2 ( x 2 7x+10 ) 2 = (x8 ) 2 x 2 7x+10 = x 2 16x+64 16x7x =6410 9x =54 x = 9 x =6
  23. x+6 9x+70 =2 x+9 ( x+6 9x+70 ) 2 = (2 x+9 ) 2 x+62 (x+6 ) (9x+70 ) +9x+70 =4(x+9 ) 2 9 x 2 +70x+54x+420 =4x+3610x76 2 9 x 2 +124x+420 =6x20 Dividiendo la ecuación para 2 ( 9 x 2 +124x+420 ) 2 = (3x+10 ) 2 9 x 2 +124x+420 = 9 x 2 +60x+100 124x60x =420+100 64x =320 x = 64 x =5
  24. xa + x+a = 4x2a ( xa + x+a ) 2 = ( 4x2a ) 2 x a +2 x 2 a 2 +x+ a =4x2a 2 x 2 a 2 =4x2x2a 2 x 2 a 2 =2x2a Dividiendo la ecuación para 2 ( x 2 a 2 ) 2 = (xa ) 2 x 2 a 2 = x 2 2ax+ a 2 2a x = 2 a 2 x =a
  25. x4ab =2b+ x ( x4ab ) 2 = (2b+ x ) 2 x 4ab =4 b 2 4b x + x 4ab4 b 2 =4b x Dividiendo la ecuación para 4b (ab ) 2 = (x ) 2 x = (ab ) 2
  26. x+4a x+2a1 =1 ( x+4a 1 ) 2 = ( x+2a1 ) 2 x +4a2 x+4a +1 = x +2a1 2 x+4a =2a4a11 2 x+4a =2a2 Dividiendo la ecuación para 2 ( x+4a ) 2 = (a+1 ) 2 x+4a = a 2 +2a+1 x = a 2 +2a4a+1 x = a 2 2a+1 x = (a1 ) 2

Ejercicio 250

CAPITULO XXXI

R a d i c a l e s
División de radicales cuando el divisor es compuesto
Ejercicio 250
Dividir:
  1. 2 entre 2 + 3 2 2 + 3 = 2 2 + 3 × 2 3 2 3 = 2 6 23 = 2 6 1 = 6 2
  2. 3 entre 3 2 5 3 3 2 5 = 3 3 2 5 × 3 +2 5 3 +2 5 = 3+2 15 (3 ) 2 (2 5 ) 2 = 3+2 15 34( 5 ) = 3+2 15 320 = 3+2 15 17
  3. 2+ 5 entre1 5 2+ 5 1 5 = 2+ 5 1 5 × 1+ 5 1+ 5 = 2+ 5 +2 5 +5 1 (5 ) 2 = 7+3 5 15 = 7+3 5 4 = 7+3 5 4
  4. 2 + 5 entre 2 5 2 + 5 2 5 = 2 + 5 2 5 × 2 + 5 2 + 5 = ( 2 + 5 ) 2 (2 ) 2 (5 ) 2 = 2+2 10 +5 25 = 7+2 10 3
  5. 2 3 7 entre 3 + 7 2 3 7 3 + 7 = 2 3 7 3 + 7 × 3 7 3 7 = 2 3 2 21 2 21 + 7 2 (3 ) 2 (7 ) 2 = 2( 3 ) 3 21 +7 37 = 63 21 +7 4 = 3 21 13 4
  6. 6 +2 5 entre2 6 5 6 +2 5 2 6 5 = 6 +2 5 2 6 5 × 2 6 + 5 2 6 + 5 = 2( 6 ) +4 30 + 30 +2( 5 ) (2 6 ) 2 (5 ) 2 = 12+5 30 +10 4( 6 ) 5 = 22+5 30 19
  7. 5 2 +3 3 entre3 2 4 3 5 2 +3 3 3 2 4 3 = 5 2 +3 3 3 2 4 3 × 3 2 +4 3 3 2 +4 3 = 15 2 2 +9 6 +20 6 +12 3 2 (3 2 ) 2 (4 3 ) 2 = 15( 2 ) +29 6 +12( 3 ) 9( 2 ) 16( 3 ) = 30+29 6 +36 1848 = 66+29 6 30
  8. 7 2 11 entre2 7 + 11 7 2 11 2 7 + 11 = 7 2 11 2 7 + 11 × 2 7 11 2 7 11 = 2 7 2 77 4 77 +2 1 1 2 (2 7 ) 2 (11 ) 2 = 2( 7 ) 5 77 +2( 11 ) 4( 7 ) 11 = 145 77 +22 2811 = 365 77 17

Ejercicio 249

CAPITULO XXXI

R a d i c a l e s
Racionalización de expresiones conjugadas
Ejercicio 249
Racionalizar el denominador de:
  1. 3 2 + 3 5 = 3 ( 2 + 3 ) 5 × 2 + 3 + 5 ( 2 + 3 ) + 5 = 3 ( 2 + 3 + 5 ) ( 2 + 3 ) 2 (5 ) 2 = 6 + 3 2 + 15 2 +2 6 + 3 5 = 6 + 15 +3 2 6 = 6 + 15 +3 2 6 × 6 6 = 6 2 + 10. 3 2 +3 6 2( 6 ) = 6+3 10 +3 6 2( 6 ) = 3 (2+ 10 + 6 ) 2 = 2+ 10 + 6 4
  2. 2 2 + 3 + 6 = 2 ( 2 + 3 ) + 6 × 2 + 3 6 ( 2 + 3 ) 6 = 2 ( 2 + 3 6 ) ( 2 + 3 ) 2 (6 ) 2 = 2 2 + 6 3. 2 2 2+2 6 +36 = 2+ 6 2 3 2 6 1 × 2 6 +1 2 6 +1 = 4 6 +2( 6 ) 4 18 +2+ 6 2 3 (2 6 ) 2 1 = 5 6 +144 2. 3 3 2 3 4( 6 ) 1 = 5 6 12 2 2 3 +14 23
  3. 2 3 2+ 3 + 5 = 2 3 (2+ 3 ) + 5 × 2+ 3 5 (2+ 3 ) 5 = 4+ 2 3 2 5 2 3 3 2 + 15 (2+ 3 ) 2 (5 ) 2 = 42 5 3+ 15 2 2 +4 3 +35 = 15 2 5 +1 4 3 +2 × 4 3 2 4 3 2 = 4 45 8 15 +4 3 2 15 +4 5 2 (4 3 ) 2 2 2 = 4 5. 3 3 10 15 +4 3 +4 5 2 16( 3 ) 4 = 12 5 10 15 +4 3 +4 5 2 484 = 16 5 10 15 +4 3 2 44 = 2 (8 5 5 15 +2 3 1 ) = 8 5 5 15 +2 3 1 22
  4. 3 + 5 2 + 3 + 5 = 3 + 5 ( 2 + 3 ) + 5 × 2 + 3 5 ( 2 + 3 ) 5 = 6 + 3 2 15 + 10 + 15 5 2 ( 2 + 3 ) 2 (5 ) 2 = 6 +3+ 10 5 2 +2 6 + 3 5 = 6 + 10 2 2 6 × 6 6 = 6 2 + 60 2 6 2( 6 ) = 6+ 2 2 .15 2 6 2( 6 ) = 6+2 15 2 6 2( 6 ) = 2 (3+ 15 6 ) 2 ( 6 ) = 3+ 15 6 6
  5. 6 + 3 + 2 6 + 3 2 = 6 + 3 + 2 ( 6 + 3 ) 2 × 6 + 3 + 2 ( 6 + 3 ) + 2 = ( 6 + 3 + 2 ) 2 ( 6 + 3 ) 2 (2 ) 2 = 6+3+2+2 18 +2 12 +2 6 6+2 18 +32 = 11+2 2. 3 2 +2 3. 2 2 +2 6 7+2 2. 3 2 = 11+6 2 +4 3 +2 6 7+6 2 × 76 2 76 2 = 77+42 2 +28 3 +14 6 66 2 36 2 2 24 6 12 12 7 2 (6 2 ) 2 = 77+28 3 24 2 36( 2 ) 10 6 12 3. 2 2 4936( 2 ) = 77+28 3 24 2 7210 6 24 3 4972 = 5+4 3 24 2 10 6 23 = 10 6 4 3 +24 2 5 23
  6. 2 5 2 + 5 10 = 2 5 ( 2 + 5 ) 10 × 2 + 5 + 10 ( 2 + 5 ) + 10 = 2+ 10 + 20 10 5 50 ( 2 + 5 ) 2 (10 ) 2 = 3+ 5. 2 2 5 2 .2 2+2 10 +510 = 3+2 5 5 2 2 10 3 × 2 10 +3 2 10 +3 = 6 10 +4 50 10 20 9+6 5 15 2 (2 10 ) 2 3 2 = 6 10 +4 5 2 .2 10 5. 2 2 9+6 5 15 2 4( 10 ) 9 = 6 10 +20 2 20 5 9+6 5 15 2 31 = 5 2 6 10 14 5 9 31

Ejercicio 248

CAPITULO XXXI

R a d i c a l e s
Racionalización de expresiones conjugadas
Ejercicio 248
Racionalizar el denominador de:
  1. 3 2 1+ 2 = 3 2 1+ 2 × 1 2 1 2 = 3 2 3 2 + 2 2 1 2 2 = 34 2 +2 12 = 54 2 1 =4 2 5
  2. 5+2 3 4 3 = 5+2 3 4 3 × 4+ 3 4+ 3 = 20+8 3 +5 3 +2 3 2 4 2 3 2 = 20+13 3 +2( 3 ) 163 = 13 3 +26 13 = 13 ( 3 +2 ) 13 = 3 +2
  3. 2 5 2 + 5 = 2 5 2 + 5 × 2 5 2 5 = ( 2 5 ) 2 2 2 5 2 = 2 2 2 2 × 5 + 5 2 25 = 22 10 +5 3 = 72 10 3 = 2 10 7 3
  4. 7 +2 5 7 5 = 7 +2 5 7 5 × 7 + 5 7 + 5 = 7 2 +2 35 + 35 +2 5 2 7 2 5 2 = 7+3 35 +2( 5 ) 75 = 17+3 35 2
  5. 2 3 5 2 2 + 5 = 2 3 5 2 2 + 5 × 2 2 5 2 2 5 = 2 2 2 6 10 10 +3 5 2 (2 2 ) 2 5 2 = 2( 2 ) 7 10 +3( 5 ) 4( 2 ) 5 = 47 10 +15 85 = 197 10 3
  6. 19 5 2 4 3 = 19 5 2 4 3 × 5 2 +4 3 5 2 +4 3 = 19(5 2 +4 3 ) (5 2 ) 2 (4 3 ) 2 = 19(5 2 +4 3 ) 25( 2 ) 16( 3 ) = 19(5 2 +4 3 ) 5048 = 19(5 2 +4 3 ) 2
  7. 3 2 7 2 6 3 = 3 2 7 2 6 3 × 7 2 +6 3 7 2 +6 3 = 21 2 2 +18 6 (7 2 ) 2 (6 3 ) 2 = 21( 2 ) +18 6 49( 2 ) 36( 3 ) = 42+18 6 98108 = 2 (21+9 6 ) = 21+9 6 5
  8. 4 3 3 7 2 3 +3 7 = 4 3 3 7 2 3 +3 7 × 2 3 3 7 2 3 3 7 = 8 3 2 6 21 12 21 +9 7 2 (2 3 ) 2 (3 7 ) 2 = 8( 3 ) 18 21 +9( 7 ) 4( 3 ) 9( 7 ) = 2418 21 +63 1263 = 8718 21 51 = 18 21 87 51 = 3 (6 21 29 ) = 6 21 29 17
  9. 5 2 6 3 4 2 3 3 = 5 2 6 3 4 2 3 3 × 4 2 +3 3 4 2 +3 3 = 20 2 2 24 6 +15 6 18 3 2 (4 2 ) 2 (3 3 ) 2 = 20( 2 ) 9 6 18( 3 ) 16( 2 ) 9( 3 ) = 409 6 54 3227 = 409 6 54 3227 = 14+9 6 5
  10. 7 +3 11 5 7 +4 11 = 7 +3 11 5 7 +4 11 × 5 7 4 11 5 7 4 11 = 5 7 2 +15 77 4 77 12 1 1 2 (5 7 ) 2 (4 11 ) 2 = 5( 7 ) +11 77 12( 11 ) 25( 7 ) 16( 11 ) = 35+11 77 132 175176 =9711 77
  11. 5 + 2 7+2 10 = 5 + 2 7+2 10 × 72 10 72 10 = 7 5 +7 2 2 50 2 20 7 2 (2 10 ) 2 = 7 5 +7 2 2 5 2 .2 2 5. 2 2 494( 10 ) = 7 5 +7 2 10 2 4 5 9 = 3 5 3 2 9 = 3 ( 5 2 ) = 5 2 3
  12. 9 3 3 2 6 6 = 9 3 3 2 6 6 × 6+ 6 6+ 6 = 54 3 18 2 +9 18 3 12 6 2 6 2 = 54 3 18 2 +9 3 2 .2 3 3. 2 2 366 = 54 3 18 2 +27 2 6 3 30 = 48 3 +9 2 30 = 3 (16 3 +3 2 ) = 16 3 +3 2 10
  13. a + x 2 a + x = a + x 2 a + x × 2 a x 2 a x = 2 a 2 +2 ax ax x 2 (2 a ) 2 (x ) 2 = 2a+ ax x 4ax
  14. x x1 x + x1 = x x1 x + x1 × x x1 x x1 = x 2 x(x1 ) x(x1 ) + (x1 ) 2 x 2 (x1 ) 2 = x2 x(x1 ) +x1 x(x1 ) = 2x2 x(x1 ) 1 x x +1 =2x2 x(x1 ) 1
  15. a a+1 a + a+1 = a a+1 a + a+1 × a a+1 a a+1 = ( a a+1 ) 2 (a ) 2 ( a+1 ) 2 = a2 a(a+1 ) +a+1 a(a+1 ) = 2a2 a(a+1 ) +1 a a 1 =2 a(a+1 ) 2a1
  16. x+2 + 2 x+2 2 = x+2 + 2 x+2 2 × x+2 + 2 x+2 + 2 = ( x+2 + 2 ) 2 ( x+2 ) 2 (2 ) 2 = x+2+2 2(x+2 ) +2 x+ 2 2 = x+2 2(x+2 ) +4 x
  17. a+4 a a+4 + a = a+4 a a+4 + a × a+4 a a+4 a = ( a+4 a ) 2 ( a+4 ) 2 (a ) 2 = a+42 a(a+4 ) +a a +4 a = 2a2 a(a+4 ) +4 4 = 2 (a a(a+4 ) +2 ) = a a(a+4 ) +2 2
  18. a+b ab a+b + ab = a+b ab a+b + ab × a+b ab a+b ab = ( a+b ab ) 2 ( a+b ) 2 ( ab ) 2 = a+ b 2 a 2 b 2 +a b a+b(ab ) = 2a2 a 2 b 2 a +b a +b = 2 (a a 2 b 2 ) 2 b = a a 2 b 2 b

Ejercicio 247

CAPITULO XXXI

R a d i c a l e s
Racionalización
Ejercicio 247
Racionalizar el denominador de:
  1. 1 3 = 1 3 × 3 3 = 3 3 2 = 3 3
  2. 5 2 = 5 2 × 2 2 = 5 2 2 2 = 5 2 2
  3. 3 4 5 = 3 4 5 × 5 5 = 3 5 4 5 2 = 3 5 4( 5 ) = 3 5 20
  4. 2a 2ax = 2a 2ax × 2ax 2ax = 2a 2ax (2ax ) 2 = 2a 2ax 2a x = 2ax x
  5. 5 4 a 2 3 = 5 2 2 a 2 3 × 2a 3 2a 3 = 5 2a 3 (2a ) 3 3 = 5 2a 3 2a
  6. 1 9x 3 = 1 3 2 x 3 × 3 x 2 3 3 x 2 3 = 3 x 2 3 (3x ) 3 3 = 3 x 2 3 3x
  7. 3 9a 4 = 3 3 2 a 4 × 3 2 a 3 4 3 2 a 3 4 = 3 3 2 a 3 4 (3a ) 4 4 = 3 9 a 3 4 3 a = 9 a 3 4 a
  8. 6 5 3x 3 = 6 5 3x 3 × (3x ) 2 3 (3x ) 2 3 = 6 (3x ) 2 3 5 (3x ) 3 3 = 9 x 2 3 5( 3 x ) = 2 9 x 2 3 5x
  9. x 27 x 2 4 = x 3 3 x 2 4 × 3 x 2 4 3 x 2 4 = x 3 x 2 4 (3x ) 4 4 = x 3 x 2 4 3 x = 3 x 2 4 3
  10. 1 8 a 4 5 = 1 2 3 a 4 5 × 2 2 a 5 2 2 a 5 = 2 2 a 5 (2a ) 5 5 = 4a 5 2a
  11. 5 n 2 3 mn = 5 n 2 3 mn × mn mn = 5 n 2 mn 3 (mn ) 2 = 5 n 2 mn 3m n = 5n mn 3m
  12. 1 5a 25 x 3 4 = 1 5a 5 2 x 3 4 × 5 2 x 4 5 2 x 4 = 5 2 x 4 5a (5x ) 4 4 = 25x 4 5a(5x ) = 25x 4 25ax

Ejercicio 245

CAPITULO XXXI

R a d i c a l e s
Potenciación de radicales
Ejercicio 245
Desarollar:
  1. (4 2 ) 2 = 4 2 2 2 =16 × 2 =32
  2. (2 3 ) 2 = 2 2 3 2 =4 × 3 =12
  3. (5 7 ) 2 = 5 2 7 2 =25 × 7 =175
  4. (2 43 ) 2 = 2 2 4 2 3 =4 2 4 3 =4 × 2 23 =8 23
  5. (3 2 a 2 b 3 ) 4 = 3 4 (2 a 2 b ) 4 3 =81 2 4 a 8 b 4 3 =81 × 2 a 2 b 2 a 2 b 3 =162 a 2 b 2 a 2 b 3
  6. ( 8 x 3 4 ) 2 = ( 2 3 x 3 ) 2 4 = 2 6 x 6 4 =2x (2x ) 2 =2x 2x
  7. ( 81a b 3 5 ) 3 = ( 9 2 a b 3 ) 3 5 = 9 6 a 3 b 9 5 =9b 9 a 3 b 4 5
  8. (186 ) 3 = 1 8 3 = 18 = 3 2 × 2 =3 2
  9. (4a 2x ) 2 = (4a ) 2 (2x ) 2 = 2 4 a 2 × 2x = 2 5 a 2 x =32 a 2 x
  10. (2 x+1 ) 2 = 2 2 (x+1 ) 2 =4(x+1 ) =4x+4
  11. (3 xa ) 2 = 3 2 (xa ) 2 =9(xa ) =9x9a
  12. (4 9 a 3 b 4 6 ) 3 = 4 3 (9 a 3 b 4 ) 3 = 2 6 3 2 a 3 b 4 = 2 6 × 3a b 2 a =192a b 2 a
    Elevar al cuadrado:
  13. 2 3 = ( 2 3 ) 2 =22 2 × 3 +3 =52 6
  14. 4 2 + 3 = (4 2 + 3 ) 2 = (4 2 ) 2 +8 2 × 3 +3 = 2 4 2+8 6 +3 = 2 5 +8 6 +3 =32+8 6 +3 =35+8 6
  15. 5 7 = ( 5 7 ) 2 =52 5 × 7 +7 =122 35
  16. 5 7 6 = (5 7 6 ) 2 = (5 7 ) 2 2 × 5 7 × 6+36 =25 × 760 7 +36 =17560 7 +36 =21160 7
  17. x + x1 = ( x + x1 ) 2 = (x ) 2 +2 x × x1 + ( x1 ) 2 =x+2 x 2 x +x1 =2x+2 x 2 x 1
  18. x+1 4 x = ( x+1 4 x ) 2 = ( x+1 ) 2 2 x+1 × 4 x + (4 x ) 2 =x+18 x 2 +x +16x =17x8 x 2 +x +1
  19. a+1 a1 = ( a+1 a1 ) 2 = ( a+1 ) 2 2 a+1 × a1 + ( a1 ) 2 =a+ 1 2 a 2 1 +a 1 =2a2 a 2 1
  20. 2 2x1 + 2x+1 = (2 2x1 + 2x+1 ) 2 = (2 2x1 ) 2 +2 × 2 2x1 × 2x+1 + ( 2x+1 ) 2 =4(2x1 ) +4 4 x 2 1 +2x+1 =8x4+4 4 x 2 1 +2x+1 =10x+4 4 x 2 1 3

Ejercicio 244

CAPITULO XXXI

R a d i c a l e s
División de radicales de distinto índice
Ejercicio 244
Dividir:
  1. 23 ÷ 2 23 = 2 2 6 2 = 2 3 6 23 ÷ 2 = 2 2 6 ÷ 2 3 6 = 2 2 2 3 6 = 1 2 6 = 1 2 × 2 5 2 5 6 = 2 5 2 6 6 = 1 2 2 5 6 = 1 2 326
  2. 9x ÷ 3 x 2 3 9x = 3 2 x =3 x =3 x 3 6 3 x 2 3 = (3 x 2 ) 2 6 = 3 2 x 4 6 9x ÷ 3 x 2 3 =3 x 3 6 ÷ 3 2 x 4 6 =3 x 3 3 2 x 4 6 =3 1 3 2 x 6 =3 1 3 2 x × 3 4 x 5 3 4 x 5 6 =3 3 4 x 5 3 6 x 6 6 = 3 3 x 3 4 x 5 6 = 1 x 81 x 5 6
  3. 8 a 3 b 3 ÷ 4 a 2 4 8 a 3 b 3 = 2 3 a 3 b 3 =2a b3 =2a b 4 12 4 a 2 4 = ( 2 2 a 2 ) 3 12 = 2 6 a 6 12 8 a 3 b 3 ÷ 4 a 2 4 =2a b 4 12 ÷ 2 6 a 6 12 =2a b 4 2 6 a 6 12 =2a b 4 2 6 a 6 × 2 6 a 6 2 6 a 6 12 =2a 2 6 a 6 b 4 2 12 a 12 12 = 2a 2a (2a ) 6 b 4 12 = 2a b3
  4. 1 2 2x ÷ 1 4 16 x 4 6 1 2 2x = 1 2 (2x ) 3 6 1 4 16 x 4 6 = 1 4 (2x ) 4 6 1 2 2x ÷ 1 4 16 x 4 6 = 1 2 (2x ) 3 6 ÷ 1 4 (2x ) 4 6 = 1 2 (2x ) 3 6 1 (2x ) 4 6 =2 (2x ) 3 (2x ) 4 6 =2 1 2x × (2x ) 5 (2x ) 5 6 =2 (2x ) 5 (2x ) 6 6 = 2 1 2 x (2x ) 5 6 = 1 x 32 x 5 6
  5. 5 m 2 n 3 ÷ m 3 n 2 5 5 m 2 n 3 = (5 m 2 n ) 5 15 = 5 5 m 10 n 5 15 m 3 n 2 5 = ( m 3 n 2 ) 3 15 = m 9 n 6 15 5 m 2 n 3 ÷ m 3 n 2 5 = 5 5 m 10 n 5 15 ÷ m 9 n 6 15 = 5 5 m 10 n 5 m 9 n 6 15 = 5 5 m n × n 14 n 14 15 = 5 5 m n 14 n 15 15 = 1 n 3125m n 14 15
  6. 18 x 3 y 4 z 5 6 ÷ 3 x 2 y 2 z 3 4 18 x 3 y 4 z 5 6 = (18 x 3 y 4 z 5 ) 2 12 = 1 8 2 x 6 y 8 z 10 12 3 x 2 y 2 z 3 4 = (3 x 2 y 2 z 3 ) 3 12 = 3 3 x 6 y 6 z 9 12 18 x 3 y 4 z 5 6 ÷ 3 x 2 y 2 z 3 4 = 1 8 2 x 6 y 8 z 10 12 ÷ 3 3 x 6 y 6 z 9 12 = 2 2 . 3 4 x 6 y z 10 3 3 x 6 y 6 z 9 12 = 12 y 2 z 12
  7. 3 m 4 3 ÷ 27 m 2 9 3 m 4 3 = (3 m 4 ) 3 9 = 3 3 m 12 9 27 m 2 9 = 3 3 m 2 9 3 m 4 3 ÷ 27 m 2 9 = 3 3 m 12 9 ÷ 3 3 m 2 9 = 3 3 m 3 3 m 2 9 = m 10 9 = m 9 .m 9 =m m9
  8. 4 5 4ab 3 ÷ 1 10 2 a 2 4 5 4ab 3 = 4 5 (4ab ) 2 6 = 4 5 2 4 a 2 b 2 6 1 10 2 a 2 = 1 10 (2 a 2 ) 3 6 = 1 10 2 3 a 6 6 4 5 4ab 3 ÷ 1 10 2 a 2 = 4 5 2 4 a 2 b 2 6 ÷ 1 10 2 3 a 6 6 = 4 5 2 4 a 2 b 2 6 1 2 3 a 6 6 =8 2 4 a 2 b 2 2 3 a 6 =8 2 b 2 a 4 × a 2 a 2 6 =8 2 a 2 b 2 a 6 6 = 8 a 2 a 2 b 2 6

Ejercicio 243

CAPITULO XXXI

R a d i c a l e s
División de radicales del mismo índice
Ejercicio 243
Dividir:
  1. 4 6 ÷ 2 3 4 6 ÷ 2 3 = 2 3 =2 2
  2. 2 3a ÷ 10 a 2 3a ÷ 10 a = 2 3 a a = 1 5 3
  3. 1 2 3xy ÷ 3 4 x 1 2 3xy ÷ 3 4 x = 1 2 3 3 x y x = 2 3 3y
  4. 75 x 2 y 3 ÷ 5 3xy 75 x 2 y 3 ÷ 5 3xy = 1 5 x 2 y 3xy = 1 5 25x y 2 = 1 5 5 y x =y x
  5. 3 16 a 5 3 ÷ 4 2 a 2 3 3 16 a 5 3 ÷ 4 2 a 2 3 = 3 4 a 2 a 2 3 = 3 4 8 a 3 3 = 3 2 a = 3 2 a
  6. 5 6 1 2 ÷ 10 3 2 3 5 6 1 2 ÷ 10 3 2 3 = 5 3 1 2 2 3 = 1 4 3 4 = 1 4 3 2 2 = 1 8 3
  7. 4x a 3 x 2 ÷ 2 a 2 x 3 4x a 3 x 2 ÷ 2 a 2 x 3 = 2 x a 3 x 2 a 2 x 3 =2x a x =2x a x × x x =2x ax x 2 =2 x x ax =2 ax
  8. 2a 3 x 2 3 ÷ a 3 x 2 x 3 3 2a 3 x 2 3 ÷ a 3 x 2 x 3 3 = 2 a 3 a 3 x 2 x 2 x 3 3 =2 x 2 1 x 3 =2 x 2 1 x × x 2 x 2 3 =2 x 2 x 2 x 3 3 =2 x 2 x x 2 3 =2x x 2 3
  9. 1 3 1 2 3 ÷ 1 6 1 3 3 1 3 1 2 3 ÷ 1 6 1 3 3 = 1 3 1 1 2 1 3 3 =2 3 2 3 =2 3 2 × 2 2 2 2 3 =2 12 2 3 3 = 2 2 123 = 123