TEMA2_HOJA2_RADICALES_SOLUCIONES
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(2) IES Juan García Valdemora Departamento de Matemáticas. o). 3 5. 92. 3. p). 5. 3 92 3. q). =. =. 3. 3 5. 1 3. 3. =. 34. 3. 5. 1 3. =. 34. =3. 4 5. 3. 1 4 − 3 5. 2. =. =3. 4 5. =3. −. 1 5. 7 15. 2 ⋅ 3 32. 23 ⋅ 25. 3. 2. =. 4. s). 31. 64 = 6 2 6 = 2 2. r). =. (32 ) 2. 5. 3. TEMA 3. HOJA2.RADICALES. 3º ESO. 6. 21. =. 28. =. 8. 26. 21 4. =2. −. 1 3. 23. 2. 81 = 6 3 4 = 3 6 = 3 3. 3. 4. t). 2 8. 25 = 5 = 5 = 5 8. 2. 1 4. 1 1 3. 5. 4 − − 3 3 34 = = 3 = 34 2 = 3 4 3 27 3 32. u). 1. v) 4 3. 1. =. 12. 1024. 1. =. 210. 2. =2. 10 12. −. 5 6. 1. 5 ⋅ 5 15 = 45. 3. w). 36. x). 6 ⋅ 3 27. 55 ⋅ 3 ⋅ 5. =. 15. =. 32 ⋅ 5. 3 ⋅ 56. 3⋅5. 2 2 ⋅ 32. 3. 1 3. 81−5. =. 1. =. (3 4 ) −5. 3. 3 − 20. 6. 1. = 3. −. 1 2. =. 1. =. 2. 315 ⋅ 5 5 3 ⋅5 1. 2 2 ⋅ 32 6. 23 ⋅ 36. 1 2. =. =3. −. 14 15. 2 2 ⋅ 32 1 2. ⋅5. −. 1 10. 3 2. = 2 ⋅3. 2 ⋅3. 1 7 6. 2 ⋅3 ⋅ 2⋅3 = 2 ⋅3 = 2 ⋅3. 3. 2. 1 3. 3 ⋅5 1. 23 ⋅ 33 ⋅ 33. 12 ⋅ 18 = 2 ⋅ 3 ⋅ 2 ⋅ 3 = 3. 2. =. 1 2. 6. 315 ⋅ 515. 2 2 ⋅ 32. =. 2 ⋅ 3 ⋅ 3 33. 3. y) z). 3 5. 20 3. 3. 2. 6. 7. 5. 5 6. 20. =33. 2. Ordena de mayor a menor: a). 4. 2,. 3. 36. 2 9 , 36 312 , 36 4 6 , 36 6 3 y. 3,. Por tanto,. 6. 3. 4,. 12. 3 >. 6 y. 6. 9. 4 >. 5. 9. 36. 54 → 36 512 , 36 531441, 36 4096 , 36 216 y. 5 >. 4. 2 >. 12. 36. 625. 6. 2.
(3) IES Juan García Valdemora Departamento de Matemáticas. b). 3. 0,01 y. 3. 10 −2 y 10 −1 → 6 10 −4 y. 0,1. 4 y. 5. 5. 4. 4 y. 5. 5 → 20 4 5 y 4. 20. 54 → 20 1024 y. 6. 3 y. 8. 7. 12. 5,. 6. 3 y. 8. 7 → 24 5 2 , 24 3 4 y. 8. 7 >. 6. 3 >. 12. 24. 5. (−2) 2 ,. 45. (− 4)7. y. 9. 8. 3. −2,. 5. (−2) 2 ,. 45. (− 4)7. y. 9. 8 → − 3 2 ,. 5. ( − 2) 2 >. 45. (− 4)7. >. 3. 2,. 9. −3,. 10. 23 y. 15. 16. 6. 2,. 9. −3,. 10. 23 y. 15. 16 → 6 2 , − 9 3 ,. 10. 23 >. 15. 343. 16 >. 6. 2 >. 9. 5. 2 2 , − 45 214 y. 9. 23 → − 45 215 , 45 218 , − 45 214 y. 45. 215. −2. 6. Por tanto,. 24. 5. −2,. 8 >. 625. 73 → 24 25 , 24 81 y. 3. 9. 20. 5. 5,. Por tanto,. f). 5. 10 −3. 12. Por tanto,. e). 4 >. 6. 0,01. 4. Por tanto,. d). 3. 0,1 >. Por tanto,. c). TEMA 3. HOJA2.RADICALES. 3º ESO. 10. 23 y. 15. 24 → 90 215 , − 90 310 ,. 10. 2 27 y. 15. 2 24. −3. 3. Extrae todos los factores posibles de los siguientes radicales: a). 9a 4 bc 5 = 32 a 4 bc 5 = 3a 2 c 2 bc. b). 18a 12 b 9 c 7 = 2 ⋅ 32 ⋅ a12 b 9 c 7 = 3a 6 b 4 c 3 2bc. c). 3. 864a 4 b 9 c 24 = 3 2 5 ⋅ 33 ⋅ a 4 b 9 c 24 = 2 ⋅ 3 ⋅ a ⋅ b 3 ⋅ c 8 3 2 2 a = 6ab 3 c 8 3 4a. d). 7. 3072a 12 b14 c 26 = 7 210 ⋅ 3 ⋅ a 12 b14 c 26 = 2 ⋅ a ⋅ b 2 ⋅ c 3 7 2 3 ⋅ 3 ⋅ a 5 c 5 = 2ab 2 c 3 7 24a 5 c 5. e). 8x 3 y = 125 z 9. 23 x 3 y 2x = 4 3 9 5 z 5z. 2 xy 5z. 3.
(4) IES Juan García Valdemora Departamento de Matemáticas. f). 3. h). 108a 9bc16 3 2 2 ⋅ 33 ⋅ a 9bc16 3a 3c 5 3 2 2 bc 3a 3c 5 3 4bc = = = 875d 4e 53 ⋅ 7 ⋅ d 4e 5d 7 de 5d 7 de 16a 4 24 a 4 22 a 2 = 3 6 = 27bc 6 3 bc 3c 3. g). 3. TEMA 3. HOJA2.RADICALES. 3º ESO. 1 4a 2 = 3b 3c 3. 1 3b. 8a 9 c10 2 3 a 9 c 10 2a 3 c 3 3 3 = = c 125d 3 e 9 53 d 3e 9 5de 3. 4. Introduce en el radical los factores que multiplican a las raíces siguientes: a) 3a 2 b ab = 3 2 a 4 b 2 ab = 9a 5 b 3 b) pq 3 ⋅ 3 pq =. p 2 q 6 ⋅ 3 pq = 3 p 3 q 7. c) 2a 3b ⋅ 3 3a 2 b = 3 2 3 a 9 b 3 ⋅ 3a 2 b = 3 24a 11b 4 d). 2 2 2 xy xy = 5 5. 2 2 ⋅ x 2 ⋅ y 4 ⋅ 2 xy 8x 3 y 5 = 125 52 ⋅ 5. 2 x 2 a 2ax 2 5 ⋅ x 10 ⋅ a 5 ⋅ 2ax 64 x11 a 6 5 5 5 e) = = y3 5y2 y 15 ⋅ 5 y 2 5 y 17 f) 2ab ⋅ 4 3a 2 b = 4 2 4 ⋅ a 4 ⋅ b 4 ⋅ 3a 2 b = 4 48a 6 b 5 g). 10 x 3 c 5c 10 2 ⋅ x 6 ⋅ c 2 ⋅ 5c 500 x 6 c 3 = = 9e 3 92 ⋅ e2 ⋅ 3 243e 2. 5. Escribe tres radicales equivalentes a los siguientes: a). 3. 2 = 6 2 2 = 9 2 3 = 12 2 4. 5 = 4 5 2 = 6 53 = 8 54. b) c). 5. − 5 = − 10 (−5) 2 = 15 (−5) 3 = − 20 (−5) 4. d). 4. 36 = 8 312 = 12 318 = 16 324. e). 5. (−2 ) 3 = − 10 (−2) 6 = 15 (−2) 9 = − 20 (−2)12. 8 = 4 8 2 = 6 83 = 8 8 4. f). 6. Simplifica los siguientes radicales: a). 36. 245 = 4 25 = 4 32. b). 12. 625a 8 b 20 = 12 5 4 a 8 b 20 = 3 5a 2 b 5. m.c.d .(36,45) = 9 m.c.d .(12,4,8,20) = 4 4.
(5) IES Juan García Valdemora Departamento de Matemáticas. TEMA 3. HOJA2.RADICALES. 3º ESO. c). 6. 512 = 6 2 9 = 2 3 = 8. d). 60. 384 b12 = 5 37 b = 5 2187b m.c.d .(60,84,12) = 12. e). 12. 1728 = 12 2 6 ⋅ 33 = 4 2 2 ⋅ 3 = 4 12 m.c.d .(12,6,3) = 3. f). 4. 324a 6 b14 = 4 2 2 ⋅ 3 4 ⋅ a 6 b14 = 2 ⋅ 3 2 ⋅ a 3b 7 = 18a 3 b 7. m.c.d .(6,9) = 3. m.c.d .(4,2,6,14) = 2. 7. Calcula y simplifica: a). 3. 2 ⋅ 3 16 ⋅ 3 128 = 3 2 ⋅ 3 2 4 ⋅ 3 2 7 = 3 2 ⋅ 2 4 ⋅ 2 7 = 3 212 = 2 4 = 16. b). 3. 18 ⋅ 5 4 ⋅ 6 36 = 3 2 ⋅ 3 2 ⋅ 5 2 2 ⋅ 6 2 2 ⋅ 3 2 = 30 210 ⋅ 3 20 ⋅ 30 212 ⋅ 30 210 ⋅ 310 = 30 2 32 ⋅ 330 = 15 216 ⋅ 315 =. = 2 ⋅ 315 2 = 615 2. c). 1000 ⋅ 4 100 ⋅ 1000 = 5 10 3 ⋅ 4 10 2 ⋅ 10 3 = 20 1012 ⋅ 20 1010 ⋅ 20 10 30 = 20 10 52 = 5 1013 = 10 2 ⋅ 5 10 3 =. 5. = 100 ⋅ 5 1000. d). e). f). 5 4. 3. 24. 3. 3. g). 3. 4. 5. =2. 37 12. = 12 2 37 = 2 3 ⋅ 12 2 = 8 12 2. = 3 24 : 3 = 3 8 = 2. 9. =. 25. 3. =. 32. =. 3. 3. 52. 4. 2. 6. 34. 6. 3. =. 3. =6 3. 12. 58. 12. 6. = 12 5 2 = 6 5. 4. 25. 6. 108 6 22 ⋅ 33 18 26 ⋅ 39 18 35 18 243 = = = = 24 16 288 9 25 ⋅ 32 18 210 ⋅ 34. 9. 1 3 5 + + 3 2 4. 6 3 3 6 33 6 −1 6 1 1 1 ⋅ 6 35 243 =3 =3 = 3 = =6 = = 3 3 9 3 racionalizar 6 3 ⋅ 6 35 32 34. 3. 3. i). 3 2. 2 ⋅ 8 ⋅ 32 = 2 ⋅ 2 ⋅ 2 = 2 ⋅ 2 ⋅ 2 = 2 3. 3. 4. 3. h). 1 3. 3. 5. 5. 5.
(6) IES Juan García Valdemora Departamento de Matemáticas. 3. j). 1200 ⋅ 5 125 3 24 ⋅ 3 ⋅ 52 ⋅ 5 53 15 220 ⋅ 35 ⋅ 510 ⋅ 15 59 15 220 ⋅ 35 ⋅ 519 15 15 3 18 = 15 = = 15 = 2 ⋅3 ⋅5 = 15 15 5 simplificar 1440 25 ⋅ 32 ⋅ 5 2 ⋅ 32 ⋅ 5 25 ⋅ 32 ⋅ 5. = 5 2 5 ⋅ 3 ⋅ 56. k). 4 x ⋅ 2 3 x2 6. x. 5. = 2 ⋅ 5 ⋅ 5 3 ⋅ 5 = 10 5 15. extraer factores. =. 12 x 3 ⋅ 4 9 y 2. l). TEMA 3. HOJA2.RADICALES. 3º ESO. 6. 4 xy. 8 ⋅ 6 x3 ⋅ 6 x 4 6. x. 5. = 8 6 x2. 2 2 ⋅ 3 ⋅ x 3 ⋅ 4 32 y 2. =. 6. 2 2 xy. =. simplificar. = 12. 83 x. 212 ⋅ 3 6 ⋅ x18 ⋅ 3 6 ⋅ y 6 212 ⋅ 312 ⋅ x18 ⋅ y 6 12 8 12 16 4 12 = = 2 ⋅3 ⋅ x ⋅ y = 24 ⋅ x 2 ⋅ y 2 24 ⋅ x 2 ⋅ y 2. = 3 2 2 ⋅ 33 ⋅ x 4 ⋅ y = 3 x 3 4 xy. xy ⋅ 8 x 3 y. 3. m). 3. 2 xy. =. 6. 2− 3 xa 5 ⋅ 3 128a 2. n) 4 3. = 12. 1 3 a x ⋅ 16 x −3 8. xy ⋅ 2 3 x 3 y 3. 2 xy. 4. = 12. =6. x ⋅ y ⋅ 29 ⋅ x 9 ⋅ y 3 6 7 8 2 = 2 ⋅ x ⋅ y = 2x 6 2x 2 y 2 2 2 2 2 ⋅x ⋅y. 2 −3 xa 5 ⋅ 3 27 a 2 2− 3 a 3 x ⋅ 24 x − 3. = 12. 2 −9 x 3a15 2 28 a 8 12 219 x 3a 23 12 219 x 3a 23 x17 = = = 2 − 3 a 3 x 2 24 x −18 2 21 a 3 x −17 2 21 a 3. 219 x 20 a 23 12 x 20 a 20 6 x10 a10 x4a 4 6 = = = xa = 2 21 a 3 22 2 2. o). 4. 125 3 ⋅ 3 − 58 = − 4 5 9 ⋅ 3 58 = −12 5 27 ⋅ 12 5 32 = −12 5 59 = −5 4 12 511 = −625 12 511. p). 3. 125 5 ⋅ 7 −. q). 3. a 2 ⋅ 4 a 3 ⋅ b −3 ⋅ 3 − a = − 12 a 8 ⋅ 12 a 9 ⋅ b −9 ⋅ 12 a 4 = − 12 a 21 ⋅ b −9 = − 12. 1 = − 3 515 ⋅ 7 5 −6 = − 21 5105 ⋅ 5 −18 = − 21 587 = −5 4 6 5. 21. 5 3 = −625 21 125. a 21 a7 4 = − −a b9 b3. 4. a3 b3. r) 3 2 ⋅ 3 3 = 3 6 23 ⋅ 32 = 3 6 72. 6.
(7) IES Juan García Valdemora Departamento de Matemáticas. 4. s). 3. 12. 4. 3. 7938 xy 4. 33 ⋅ 6 2 ⋅ 3 3. a5. 6048 x 7 y 3. =. 3. =. 2. 4. 2. 6. a. = 12. a5. 25337 x 7 y 3 3. 2 ⋅ 347 2 xy 4. 3. 5. 39 ⋅ 2 2 ⋅ 3 2 12 311 ⋅ 2 2 12 5 2 12 = = 3 ⋅ 2 = 243 ⋅ 4 = 12 972 36 36. a3 ⋅ a ⋅ 4 a3 12. ( a ) ⋅ (a ⋅ a ) = 3. v). =. a ⋅ 3 a ⋅ 4 a3. t). u). 27 ⋅ 6 6. TEMA 3. HOJA2.RADICALES. 3º ESO. 3. =. 6. =. a. a 4 ⋅ 4 a3 12. a5. =. 12. a 8 ⋅ a 9 12 12 = a =a a5. 2153973 x 21 y 9 6 213 ⋅ 3 ⋅ x19 y 2⋅3⋅ x ⋅ y 6 xy = = 2 2 x 3 12 = 4 x 3 12 2 8 4 2 8 237 x y 7 7 7. a8 ⋅ a 6 ⋅ a 3 6. 6. 5. =. a 6 ⋅ 6 a16 ⋅ a 9 6. a. 5. = a 6 6 a 20 = a 6 3 a10 = a 6 ⋅ a 3 3 a = a 9 3 a. 8. Calcula y simplifica: a) 3 2 − 5 8 + 7 32 = 3 2 − 5 2 3 + 7 2 5 = 3 2 − 5 ⋅ 2 2 + 7 ⋅ 2 2 2 = 3 2 − 10 2 + 28 2 = 21 2. b) − 2 48 −. 1 1 1 5 1 2 1 1 243 − 75 = −2 2 4 ⋅ 3 − 3 − 5 ⋅ 3 = −2 ⋅ 2 2 3 − ⋅ 3 2 3 − ⋅ 5 3 = 3 5 3 5 3 5. = −8 3 − 3 3 − 1 3 = −12 3. c) 5 48 −. 1 3 1 2 3 2 1 3 12 + 75 = 5 2 4 ⋅ 3 − 2 ⋅3 + 5 ⋅ 3 = 5 ⋅ 22 3 − ⋅ 2 3 + ⋅ 5 3 = 8 5 8 5 8 5. = 20 3 −. d). 1 91 3 +3 3 = 3 4 4. 2 + 8 − 5 18 + 32 = 2 + 2 3 − 5 2 ⋅ 3 2 + 2 5 = 2 + 2 2 − 5 ⋅ 3 2 + 2 2 2 =. = 2 + 2 2 − 15 2 + 4 2 = −8 2. e). 28 + 3 63 − 5 7 + 175 = 2 2 ⋅ 7 + 3 3 2 ⋅ 7 − 5 7 + 5 2 ⋅ 7 = 2 7 + 3 ⋅ 3 7 − 5 7 + 5 7 = = 2 7 + 9 7 − 5 7 + 5 7 = 11 7. f). 24 + 7 6 − 2 486 = 2 3 ⋅ 3 + 7 2 ⋅ 3 − 2 2 ⋅ 35 = 2 2 ⋅ 3 + 7 2 ⋅ 3 − 2 ⋅ 3 2 2 ⋅ 3 =. 7.
(8) IES Juan García Valdemora Departamento de Matemáticas. TEMA 3. HOJA2.RADICALES. 3º ESO. = 2 6 + 7 6 − 18 6 = −9 6. g). 3. 108 − 2 3 32 −. = 33 4 − 43 4 −. 2 3 23 1 25 + − = 3 2 33 2 3 ⋅ 5 2. 2 3 2 2 1 22 + ⋅ − ⋅ 3 2 3 3 2 5. 2 = 3. 2 2 2 2 8 2 + − = 3 3 5 3 5 3. 3 3 8 1 3 27 1 3 2 3 1 33 1 3 2 1 1 3 1 1 − +43 = 3 4 − 3 3 + 43 6 = ⋅ 3 − ⋅ 3 + 4⋅ 2 = 5 81 2 24 64 5 3 2 2 ⋅3 5 3 3 2 2 3 2 2. =. j). 53 8 4 =− 3 4 3 3. 2 3 8 1 32 + − = 3 2 27 2 75. h). i). 13 1 1 500 = 3 2 2 ⋅ 33 − 23 25 − 3 2 2 ⋅ 53 = 3 3 2 2 − 2 ⋅ 2 3 2 2 − ⋅ 5 3 2 2 = 3 3 3. 3. 23 1 33 1 7 1 − +1 = − 3 +1 5 3 4 3 20 3. 16 − 2 3 250 −. 13 1 1 54 = 3 2 4 − 2 3 2 ⋅ 53 − 3 2 ⋅ 33 = 2 3 2 − 2 ⋅ 5 3 2 − ⋅ 3 ⋅ 3 2 = 3 3 3. = 2 3 2 − 10 3 2 − 1 3 2 = −9 3 2. 48 −. k). =4 3−. 8 3 2 19 10 2+ 3− 2= 3− 2 3 4 3 4 3. 128 27 23 9 3 4 4 2 512 + 648 − = 2 + 2 ⋅3 − = 2 2 + 2⋅3 2 − 2 81 34 3. l) =. m). 128 3 3 8 27 3 3 23 23 2 3 3 2 + − = 24 ⋅ 3 − + − 2 = 22 3 − + − 2= 3 4 9 3 4 3 4 3 3. 2 = 16 2 + 18 2 −. 8 2= 9. 298 2 9. 75 −. 18 3 12 2 32 ⋅ 2 3 2 2 ⋅ 3 2 3 2 3⋅ 2 3 1 + − = 3 ⋅ 52 − + − 2 =5 3− + − 2= 3 4 25 3 4 3 4 5 5. =5 3− 2+. 3 1 13 6 2 3− 2= 3− 2 5 2 5. 8.
(9) IES Juan García Valdemora Departamento de Matemáticas. n). TEMA 3. HOJA2.RADICALES. 3º ESO. 1 1 5 1 32 − 3 27 − 5 12 + 50 = 2 − 3 33 − 5 2 2 ⋅ 3 + 2 ⋅ 5 2 = ⋅ 2 2 2 − 3 ⋅ 3 3 − 5 ⋅ 2 3 + 5 2 = 3 3 3 =. 4 19 2 − 9 3 − 10 3 + 5 2 = 2 − 19 3 3 3. 40 − 5 90 − 5 10 = 2 3 ⋅ 5 − 5 2 ⋅ 3 2 ⋅ 5 − 5 2 ⋅ 5 = 2 2 ⋅ 5 − 5 ⋅ 3 2 ⋅ 5 − 5 2 ⋅ 5 =. o). = 2 10 − 15 10 − 5 10 = −18 10. 108 x3 − 300 x 5 +. p). 36 x 2 2 ⋅ 32 x = 2 2 ⋅ 33 ⋅ x 3 − 3 ⋅ 22 ⋅ 52 ⋅ x 5 + 363 3 ⋅ 112. =. simplificamos el radicando del tercer radical. 22 ⋅ 3 ⋅ x 2 = 2 ⋅ 3 ⋅ x ⋅ 3x − 2 ⋅ 5 ⋅ x 2 3x + 3x = 2 11 11. = 2 2 ⋅ 33 ⋅ x 3 − 3 ⋅ 2 2 ⋅ 5 2 ⋅ x 5 + 2 = 6 x − 10 x 2 + 3 x 11 . q). 6. 8 + 4 4 − 7 72 = 6 2 3 + 4 2 2 − 7 2 3 ⋅ 3 2. =. simplificar. 2 + 2 − 7 23 ⋅ 32. =. extraer factores. 2 + 2 − 7⋅2⋅3 2 =. = 2 + 2 − 42 2 = −40 2. r). 6. 6561a 2 + 3 3993a − 3 3a 4 = 6 38 a 2 + 3 3 ⋅ 113 a − 3 3a 4. =. simplifica r. 3. 3 4 a + 3 3 ⋅ 113 a − 3 3a 4. =. extraer factores. = 3 3 3a + 11 3 3a − a 3 3a = (14 − a ) 3 3a. s). t). u). 3. 0,04 ⋅ 0,2 = 3. ( 2) ⋅ 3. 4. 3. 2 = 6. 4 2 3 1 1 3 1 1 6 1 6 1 6 1 1 1 ⋅ = ⋅ = ⋅ = ⋅ 3 = = = 6 2 4 7 6 7 100 10 25 5 5 5 5 5 5 5 5 5. 23 ⋅ 3 2 6 29 ⋅ 22 6 211 28 22 4 6 6 = = = = 2 =26 3 3 3 3 3 3 2 ⋅3 2 ⋅3 3 3 27 2⋅3. 4 ⋅ 6 8 ⋅ 8 81 = 4 22 ⋅ 6 23 ⋅ 8 34. =. simplificar. 2 ⋅ 2 ⋅ 3 = ( 2 )2 ⋅ 3 = 2 3. 9.
(10) IES Juan García Valdemora Departamento de Matemáticas. v). 2⋅. 3. 3. 2. 4. TEMA 3. HOJA2.RADICALES. 3º ESO. 2 ⋅ 6 22 6 23 ⋅ 22 6 4 3 2 3 = = 2 = 2 = 4 6 2 2. =. w) − 3 − 16 − 3 − 54 − 3 − 250 = 3 16 + 3 54 + 3 250 = 3 2 4 + 3 2 ⋅ 33 + 3 2 ⋅ 53 = 2 3 2 + 3 3 2 + 5 3 2 = 10 3 2. 9. Calcula: a). (2. 3+ 5. ). 2. = (2 3 ) 2 + 2 ⋅ 2 3 ⋅ 5 + ( 5 ) 2 = 12 + 4 15 + 5 = 17 + 4 15. (. b) 2 6 ⋅ 2 5 − 2. ). 2. = 2 6 ⋅ [(2 5 ) 2 − 2 ⋅ 2 5 ⋅ 2 + ( 2 ) 2 ] = 2 6 ⋅ (20 − 4 10 + 2) =. = 2 6 ⋅ (22 − 4 10 ) = 44 6 − 8 60 = 44 6 − 8 2 2 ⋅ 3 ⋅ 5 = 44 6 − 8 ⋅ 2 2 2 ⋅ 3 ⋅ 5 = 44 6 − 16 15. c). (. ). d). (2. 2. 2 + 1 ⋅ 3 = [( 2 ) 2 + 2 ⋅ 2 ⋅ 1 + 12 ] ⋅ 3 = (2 + 2 2 + 1) ⋅ 3 = (3 + 2 2 ) ⋅ 3 = 3 3 + 2 6. (. ). 2 − 3 ⋅ ( 3 + 3) = 2 6 + 6 2 − 3 − 3 3. [. ). ]. [. ]. 2 e) 2 − 1 − 1 ⋅ 2 = ( 2 ) 2 − 2 ⋅ 2 ⋅ 1 + 12 − 1 ⋅ 2 = 2 − 2 2 + 1 − 1 ⋅ 2 = (2 − 2 2 ) ⋅ 2 = 2 2 − 4 . f). (1 + 2 )⋅ (1 − 2 ) = 1. g). (2. 3− 6. h). (2. 5 − 3 10. 2. ). 2. − ( 2 ) 2 = 1 − 2 = −1. = (2 3 ) 2 − 2 ⋅ 2 3 ⋅ 6 + ( 6 ) 2 = 12 − 4 18 + 6 = 18 − 4 3 2 ⋅ 2 = 18 − 4 ⋅ 3 2 = 18 − 12 2. ). 2. = (2 5 ) 2 − 2 ⋅ 2 5 ⋅ 3 10 + (3 10 ) 2 = 4 ⋅ 5 − 12 50 + 9 ⋅ 10 = 20 − 12 50 + 90 =. = 110 − 12 5 2 ⋅ 2 = 110 − 12 ⋅ 5 2 = 110 − 60 2. i). (1 + 3 3 )⋅ (2. j). (. 7 − 18. ). 2. ). 3 + 4 = 2 3 + 4 + 6 ⋅ 3 + 12 3 = 2 3 + 4 + 18 + 12 3 = 14 3 + 22. + 3 56 = [( 7 ) 2 − 2 ⋅ 7 ⋅ 18 + ( 18 ) 2 ] + 3 56 = [7 − 2 7 ⋅ 2 ⋅ 3 2 + 18] + 3 2 3 ⋅ 7 =. = [25 − 2 ⋅ 3 7 ⋅ 2 ] + 3 ⋅ 2 2 ⋅ 7 = (25 − 6 14 ) + 6 14 = 25. 10.
(11) IES Juan García Valdemora Departamento de Matemáticas. k). (3. )(. TEMA 3. HOJA2.RADICALES. 3º ESO. ). 3 − 2 8 ⋅ 4 + 3 6 = 12 3 + 9 18 − 8 8 − 6 48 = 12 3 + 9 3 2 ⋅ 2 − 8 2 3 − 6 2 4 ⋅ 3 =. = 12 3 + 9 ⋅ 3 2 − 8 ⋅ 2 2 − 6 ⋅ 2 2 3 = 12 3 + 27 2 − 16 2 − 24 3 = 11 2 − 12 3. l). (1 + 3 3 )⋅ (2 (. ). 3 + 4 = 2 3 + 4 + 6 ⋅ 3 + 12 3 = 2 3 + 4 + 18 + 12 3 = 14 3 + 22. )(. ). m) 3 + 4 2 ⋅ 3 − 4 2 = (3) 2 − (4 2 ) 2 = 9 − 16 ⋅ 2 = 9 − 32 = −23. n). (2 − 3 2 ). o). (3 − 2 5 )⋅ (4 + 3 5 ) = 12 + 9. p). (3. 2. = 2 2 − 2 ⋅ 2 ⋅ 3 2 + (3 2 ) 2 = 4 − 12 2 + 9 ⋅ 2 = 4 − 12 2 + 18 = 22 − 12 2. )(. 5 − 8 5 − 6 ⋅ 5 = 12 + 9 5 − 8 5 − 30 = 5 − 18. ). 5 + 4 2 ⋅ 3 5 − 4 2 = (3 5 ) 2 − (4 2 ) 2 = 9 ⋅ 5 − 16 ⋅ 2 = 45 − 32 = 13. 10. Calcula: a). 4 ⋅ 3 8 ⋅ 4 256 = 2 2 ⋅ 3 2 3 ⋅ 4 2 8 = 2 2 ⋅ 3 2 3 ⋅ 2 2 = 2 2 ⋅ 3 2 5 =. b). 9 ⋅ 3 ⋅ 27 = 3 2 ⋅ 3 ⋅ 3 3 = 3 2 ⋅. c). 3⋅ 3⋅ 3 = 3⋅. d). 2 ⋅ 3 8 ⋅ 32 = 2 ⋅ 3 2 3 ⋅ 2 5 = 2 ⋅ 3. 2 6 ⋅ 2 5 = 2 ⋅ 6 211 =. e). 3 ⋅ 3 9 ⋅ 27 = 3 ⋅ 3 3 2 ⋅ 33 = 3 ⋅ 3. 3 4 ⋅ 33 = 3 ⋅ 6 3 7 =. f). 4 ⋅ 3 8 ⋅ 4 256 ⋅ 4 2 ⋅ 3 16 = 2 2 ⋅ 3 2 3 ⋅ 4 2 8 ⋅ 4 2 ⋅ 3 2 4 = 2 2 ⋅ 2 ⋅ 2 2 ⋅ 4. 3 2 ⋅ 33 = 3 2 ⋅ 4 35 =. 3 2 ⋅ 3 = 3 ⋅ 4 33 =. 4. 3. 4. 2 6 ⋅ 2 5 = 6 211 = 2 6 2 5 = 2 6 32. 38 ⋅ 35 = 8 313 = 3 8 35 = 3 8 243. 3 4 ⋅ 33 = 8 3 7 = 8 2187. 6. 6. 2 6 ⋅ 211 = 12 217 = 2 12 2 5 = 2 12 32. 3 6 ⋅ 3 7 = 12 313 = 3 12 3. 3. 23 ⋅ 2 4 =. = 2 5 ⋅ 12 2 7 = 12 2 30 ⋅ 12 2 7 = 12 2 37 = 2 3 ⋅ 12 2 = 8 12 2. 11.
(12) IES Juan García Valdemora Departamento de Matemáticas. TEMA 3. HOJA2.RADICALES. 3º ESO. 6. 6. g) 4 ⋅ 3 16 = 2 2 ⋅ 3 2 4 = . 5. h). i). 4 ⋅ 3 2 ⋅ 128 =. 22 ⋅ 3. 2 ⋅ 2 = 6. 4. 2 2 ⋅ 27 =. 5. ( 2 ) =2 6. 6. 10. 2 2 ⋅ 6 29 =. 10. 5 6. = 1024. 212 ⋅ 2 9 = 60 2 21. =. 20. Simplifica r (:3 ). 2 7 = 20 128. 4x + 8 4 2 − 2 ⋅ 9 x + 18 = 3 ⋅ ⋅ ( x + 2) − 2 ⋅ 9 ⋅ ( x + 2) = 3 ⋅ x + 2 − 2⋅3 x + 2 = 25 25 5. 3⋅ =. 5. 6. 3. 6 24 6 x + 2 − 6 x + 2 = − 6 x + 2 = − x+2 5 5 5 . 11. Racionaliza los denominadores de las siguientes expresiones: 3. a). b). c). d). e). f). g). 6. =. 3⋅ 6 6⋅ 6. =. 3 6 6 = 6 2. 3 − 3 (3 − 3 ) ⋅ 6 3 6 − 18 3 6 − 2 ⋅ 32 3 6 − 3 2 6− 2 = = = = = Simplifica r (: 3) 6 6 6 2 6 6⋅ 6. 6 9. 32. =. 1+ 3 3 −1. 6 9. 5+ 3 3 −5. 6 ⋅ 9 24 9. 25 ⋅ 9 2 4. ( 3 − 1) ⋅ ( 3 + 1). =. =. (5 − 10 ) ⋅ 5 5⋅ 5. 6 ⋅ 9 16. =. (1 + 3 ) ⋅ ( 3 + 1). =. 5 − 10 5. 25. =. 9. =. =. (5 + 3 ) ⋅ ( 3 + 5) ( 3 − 5) ⋅ ( 3 + 5). 29. =. 6 ⋅ 9 16 = 3 9 16 2. 3 +1+ 3 + 3 ( 3 ) 2 − (1) 2. =. 4+2 3 4+2 3 = = 2+ 3 Simplifica r (: 2) 3 −1 2. 5 5 − 50 5 5 − 2 ⋅ 5 2 5 5 − 5 2 = = 5 5 5. =. 5 3 + 25 + 3 + 5 3 ( 3 ) − (5) 2. 2. =. =. Simplifica r (: 5). 5− 2. − 14 − 5 3 28 + 10 3 28 + 10 3 = = 3 − 25 − 22 Simplificar (: − 2) 11. 2 2 ⋅ (2 3 + 2 ) 4 3+2 2 4 3+2 2 4 3+2 2 = = = = = 2 2 4⋅3 − 2 10 2 3 − 2 (2 3 − 2 ) ⋅ (2 3 + 2 ) (2 3 ) − ( 2 ). 12.
(13) IES Juan García Valdemora Departamento de Matemáticas. Simplifica r (:2 ). 2 3+ 2 5. 1+ 3 2. 1+ 3 2. =. h). i). 3. 1− 2 2. l). m). n). ñ). (1 + 3 2 ) ⋅ 3 2 22 ⋅ 3 2. 3. (1 − 2 2 ) ⋅ (3 8 + 1) (3 8 − 1) ⋅ (3 8 + 1). =. =. 3. 2 +3 4 3. 23. =. 2 +3 4 2. 3. 3 8 + 1 − 6 16 − 2 2 (3 8 ) 2 − (1) 2. 3 23 + 1 − 6 ⋅ 4 − 2 2 = = 9 ⋅8 −1. 3 ⋅ 2 2 + 1 − 24 − 2 2 6 2 − 23 − 2 2 4 2 − 23 = = 72 − 1 71 71. 18 + 5 2 ⋅ 32 18 + 5 ⋅ 3 2 18 + 15 2 = = = −6−5 2 Simplificar (:−3) −3 −3 24 − 27 a + b ( a + b) ⋅ ( a + b) ( a + b )2 ( a ) 2 + ( b ) 2 + 2 a b a + b + 2 ab = = = = a−b a − b ( a − b) ⋅ ( a + b) ( a − b) ⋅ ( a − b) ( a )2 − ( b )2. 3 +1 2. =. 1− 5 2 5 +1. 7 3⋅3 2. 3. =. 2 3. 6+ 2 2. (1 − 5 ) ⋅ (2 5 − 1) (2 5 + 1) ⋅ (2 5 − 1). 7 3 3⋅ 3 ⋅ 3 2. 22 ⋅ 3. =. =. 2⋅ 2. 3. =. 2− 2. ( 3 + 1) ⋅ 2. =. 2 3. o). 22. =. 3 6 − 2 3 (3 6 − 2 3 ) ⋅ (2 6 + 3 3 ) 6 ⋅ 6 + 9 18 − 4 18 − 6 ⋅ 3 36 + 5 18 − 18 = = = = 4⋅6 − 9⋅3 2 6 − 3 3 (2 6 − 3 3 ) ⋅ (2 6 + 3 3 ) (2 6 ) 2 − (3 3 ) 2. =. k). 3. =. 3 8 −1 =. j). 4. =. TEMA 3. HOJA2.RADICALES. 3º ESO. =. =. 7 3 3⋅ 3 2. 3 4. 22 ⋅ 3. (2 − 2 ) ⋅ 3 2 3⋅ 3. =. =. =. =. 2 5 −1− 2 ⋅ 5 + 5 (2 5 ) − (1) 2. 7 3 ⋅ 3 22 3 ⋅ 3 2 ⋅ 3 22. 3 ⋅ 4 2 2 ⋅ 33 4. 2 2 ⋅ 3 ⋅ 4 2 2 ⋅ 33. =. 2. =. 2 5 − 1 − 10 + 5 3 5 − 11 = 4 ⋅ 5 −1 19. 7 6 33 ⋅ 6 2 4. =. 3 3 23. 4. 7 6 33 ⋅ 2 4 7 6 432 = = 3⋅ 2 6. 3 2 ⋅ 4 2 2 ⋅ 33 4. 2 4 ⋅ 34. =. 4. 2 2 ⋅ 3 5 3 4 2 2 ⋅ 3 4 12 = = 2⋅3 2⋅3 2. 2 3− 6 2 3− 6 = 2⋅3 6 13.
(14) IES Juan García Valdemora Departamento de Matemáticas. 3. p). q). 6 −2 3. 9. =. 6 −2. 3. 3. 3 3−2 2. =. (3 6 − 2) ⋅ 3 3 3. =. 32 ⋅ 3 3. 3. 18 − 2 3 3 3. 18 − 2 3 3 = 3 3. 33. (3 3 − 2 2 ) ⋅ (2 3 − 3 2 ). =. 2 3 +3 2 =. 32. TEMA 3. HOJA2.RADICALES. 3º ESO. (2 3 + 3 2 ) ⋅ (2 3 − 3 2 ). =. 6⋅3− 9 6 − 4 6 + 6⋅2 (2 3 ) − (3 2 ) 2. 2. =. 18 − 13 6 + 12 = 4⋅3− 9⋅2. 30 − 13 6 30 − 13 6 30 − 13 6 =− = 12 − 18 −6 6. 12. Calcula y simplifica: a). 3. −. 3 4. ( 3). ⋅9. −3. 3 2. ⋅ 3 81. =. 3. 3 4. −. ⋅ (3 ) 2. 3 2. −3. 12 3 4 3 ⋅ 3 . =. 3 3. −. −. 3 4 3 2. ⋅3. 3 4. ⋅ 33. =. 3 3. 3 − +3 4. =. 3 4 − + 2 3. 3 3. 9 4. −. 1 6. 9 1 + 6. = 34. b). ( ). c). 5 9 5 5 5 9 5 32 + 7 27 − 3+ 18 = 2 + 7 33 − 3+ 2 ⋅ 32 = 2 2 3 2 2 3. 3. 20 5 ⋅ = 10 3. =. d). ( ). 9 1 − − 6. = 34. 53 ⋅ 3 22 ⋅ 5 6 59 ⋅ 6 24 ⋅ 52 5 ⋅ = = 6 2⋅5 23 ⋅53 3. 3⋅. 3. 3. 3. (. =2 5⋅. 9. =. 2 ⋅5 3. 3. = 6 2 ⋅ 58 = 5 ⋅ 6 2 ⋅ 52 = 5 ⋅ 6 50. 3 ⋅ 6 32 6 33 ⋅ 6 32 6 35 6 4 3 2 = = 6 = 3 = 3 =3 9 6 6 Simplifica r (: 2 ) 3 3 3. ) (. 20 − 8 45 −. (2 (. 2. )(. 3+ 5 ⋅. ). 3− 5 =. ) ( 3 + 5 )⋅ ( 3 − 5 ) = 2 5 ⋅ (2. ⋅ 5 − 8 32 ⋅ 5 −. ) ( 3 + 5 )⋅ ( 3 − 5 ) = −44( 5 ) a + b )⋅ ( a − b − a + b ) = ( a − b ) − (. = 2 5 ⋅ − 22 5 −. f). 6. 24 ⋅ 511. 5 2 9 5 9 33 ⋅2 2 + 7⋅3 3 − 3 + ⋅ 3 2 = 10 2 + 21 3 − 3 + 5 2 = 15 2 + 3 2 2 3 2 2. e) 2 5 ⋅. (. 6. 29. = 3 12. 2. a−b +. 3+ 2 5− 6. +. 2. ) (. 5 − 24 5 −. )(. 3+ 5 ⋅. ). 3− 5 =. − [( 3 ) 2 − ( 5 ) 2 ] = −44 ⋅ 5 − (3 − 5) = −220 + 2 = −218 a+b. ). 2. = (a − b) − (a + b) = a − b − a − b = −2b. 3− 2. = (− 15 − 3 2 − 10 − 2 3 ) + (− 15 + 3 2 + 10 − 2 3 ) = −2 15 − 4 3 5 + 6 ( ∗). 14.
(15) IES Juan García Valdemora Departamento de Matemáticas. (∗). (∗). 3− 2 5+ 6. 5+ 3 3 3 −5 5. (∗) =. +. ( 3 − 2 )( 5 − 6 ) ( 5 + 6 )( 5 − 6 ). 5− 3 3 3+5 5. =. 3 3 −5 5. 5− 3 3 3 +5 5. 15 + 18 + 10 + 12 ( 5)2 − ( 6)2. =. 15 + 3 2 ⋅ 2 + 10 + 2 2 ⋅ 3 = 5−6. =. 15 − 18 − 10 + 12 ( 5)2 − ( 6)2. 15 − 3 2 ⋅ 2 − 10 + 2 2 ⋅ 3 = = 5−6. =. − 4 15 − 17 − 4 15 + 17 − 4 15 − 17 − 4 15 + 17 − 8 15 + = = 49 49 49 49. ( 5 + 3 )(3 3 + 5 5 ) (3 3 − 5 5 )(3 3 + 5 5 ). =. 3 15 + 5( 5 ) 2 + 3( 3 ) 2 + 5 15 (3 3 ) − (5 5 ) 2. 2. =. 3 15 + 25 + 9 + 5 15 = 9 ⋅ 3 − 25 ⋅ 5. =. ( 5 − 3 )(3 3 − 5 5 ) (3 3 + 5 5 )(3 3 − 5 5 ). =. 3 15 − 5( 5 ) 2 − 3( 3 ) 2 + 5 15. =. 3 15 − 25 − 9 + 5 15 = 9 ⋅ 3 − 25 ⋅ 5. (3 3 ) 2 − (5 5 ) 2. 8 15 − 34 8 15 − 34 − 4 15 + 17 = = (: − 2 ) 27 − 125 49 − 98. 3 2 +2 3. (∗). +. 5 6 − 12 − 5 6 − 12 5 6 − 12 − 5 6 − 12 − 24 + = = = −4 6 6 6 6 2 3 − 3 2 ( ∗). 3− 2 3 2+2 3. 3+ 2. =. =. ( 3 − 2 )(3 2 − 2 3 ) (3 2 + 2 3 )(3 2 − 2 3 ). =. 3 6 − 2( 3 ) 2 − 3( 2 ) 2 + 2 6 (3 2 ) 2 − (2 3 ) 2. =. 3 6 − 2 ⋅ 3 − 3 ⋅ 2 + 2 6 3 6 − 6 − 6 + 2 6 5 6 − 12 = = 9 ⋅ 2 − 4 ⋅3 18 − 12 6. (∗) =. =. 5+ 3. 3− 2. =. ( 5 − 6 )( 5 + 6 ). =. 8 15 + 34 8 15 + 34 − 4 15 − 17 = = 27 − 125 − 98 (:−2) 49. (∗) =. ( 3 + 2 )( 5 + 6 ). 15 − 3 2 − 10 + 2 3 = − 15 + 3 2 + 10 − 2 3 −1. =. h). 5− 6. =. 15 + 3 2 + 10 + 2 3 = − 15 − 3 2 − 10 − 2 3 −1. =. g). 3+ 2. TEMA 3. HOJA2.RADICALES. 3º ESO. 3+ 2 2 3 −3 2. =. ( 3 + 2 )(2 3 + 3 2 ) (2 3 − 3 2 )(2 3 + 3 2 ). =. 2( 3 ) 2 + 3 6 + 2 6 + 3( 2 ) 2 (2 3 ) 2 − (3 2 ) 2. ==. 2 ⋅ 3 + 3 6 + 2 6 + 3 ⋅ 2 6 + 5 6 + 6 5 6 + 12 − 5 6 − 12 = = = 4⋅3− 9⋅2 12 − 18 −6 6. 15.
(16) IES Juan García Valdemora Departamento de Matemáticas. i). TEMA 3. HOJA2.RADICALES. 3º ESO. 5 3+ 6 2 +1 3 2 5 2 + 2 7 2 + 9 − 3 2 + 12 − + = − + = 4 17 7 2 6 5 − 2 2 1+ 8 119(5 2 + 2) − 28(7 2 + 9) + 68(−3 2 + 12) 595 2 + 238 − 196 2 − 252 − 204 2 + 816 = = 476 476 195 2 + 802 = 476. =. (∗) (∗). =. 5 3+ 6 2 6 2 +1 5−2 2. =. =. (5 3 + 6 ) ⋅ 6 2 6⋅ 6. =. ( 2 + 1)(5 + 2 2 ) (5 − 2 2 )(5 + 2 2). 5 18 + ( 6 ) 2 2( 6 ) 2 =. =. 5 2 ⋅ 3 2 + 6 5 ⋅ 3 2 + 6 15 2 + 6 5 2 + 2 = = = (:3) 2⋅6 12 12 4. 5 2 + 2( 2 ) 2 + 5 + 2 2 (5) − (2 2 ) 2. 2. =. 5 2 + 2⋅2 +5+ 2 2 7 2 + 4 + 5 = = 25 − 4 ⋅ 2 25 − 8. 7 2 +9 17. (∗). 3 2 3 2 (1 − 8 ) 3 2 − 3 16 3 2 − 3 ⋅ 4 3 2 − 12 − 3 2 + 12 = = 2 = = = 1− 8 −7 7 1 + 8 (1 + 8 )(1 − 8 ) (1) − ( 8 ) 2. 16.
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