Ejercicio 248

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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