HOJA2_POTENCIAS_SOLUCIONES
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(2) IES Juan García Valdemora Departamento de Matemáticas. i) j). TEMA1. HOJA 2. 2 EJERCICIOS DE POTENCIAS 2º ESO. 1 1 = 2 5 25 1 1 −2 p) 11 = 2 = 11 121 1 1 q) ( −11) −2 = = 2 (− 11) 121. 1 1 = 5 2 32 (− 2)−5 = 1 5 = 1 = − 1 (− 2) − 32 32. o) 5 −2 =. 2−5 =. 1 1 = 4 3 81 (− 11)−1 = 1 1 = 1 = − 1 (− 11) − 11 11. k) 3−4 = l). 1 1 = 1 13 13 1 1 n) 51 = 1 = 5 5. m) 13−1 =. r). (− 13)−1 =. 1 1 1 = =− 1 (− 13) − 13 13. s). (− 13)−2 =. 1 1 = = 2 (− 13) 169. 3. REDUCE A UNA ÚNICA POTENCIA: a) 6 3 ⋅ 6 ⋅ 6 5 = 63+1+5 = 69 b). (− 7 )9 : (− 7 )5 = (− 7 )9−5 = (− 7 )4. d). [(− 2) ] = (− 2) = (− 12) (− 2) : [(− 2) ⋅ (− 2)] = (− 2) : [(− 2) ] = (− 2) : (− 2). e). 205 : (− 4 ) = [20 : (− 4 )] = (− 5). f). (− 36)4 : (− 9 )4 = [(− 36 ) : (− 9 )]4 = (4)4. c). 4 3. 4⋅3. 9. 12. 2. 4. 5. 5. [. 9. 9. 9. 8. 8. 4+1 2. 9. 5⋅2. 9. = (− 2) : (− 2) = (− 2) 9. 10. 9−10. = (− 2). −1. 5. i). ] (− 5) ⋅ [20 : (− 4) ] = (− 5) ⋅ [20 : (− 4)] = (− 5) ⋅ (− 5) = (− 5) = (− 5) (6 ) ⋅ [(− 7) ⋅ (− 7)] = (6) ⋅ (− 7) = 6 ⋅ (− 7) = [6 ⋅ (− 7)] = (− 42). j). (5 ) ⋅ (5 ). k). (− 5)8 : (− 5)4 ⋅ (− 5) = (− 5)8 − 4 ⋅ (− 5) = (− 5)4 ⋅ (− 5) = (− 5)4 +1 = (− 5)5. l). (8 ) : [(− 2) ⋅ (− 4) ] = (8) : [(− 2)⋅ (− 4)]. g) 129 : (− 3) ⋅ (− 4 ) = 129 : [(− 3) ⋅ (− 4 )] = 129 : (12 ) = 1 h). 3 2. 7 4. 9. 4 3. 5 4. 8. 9. 3⋅2. 5. 9. 5+1. 9. 6. 6. 9+8. 8. 6. 17. 6. = 57⋅4 ⋅ 54⋅3 = 528 ⋅ 512 = 528+12 = 540. 12. 5⋅4. 12. 12. = 820 : (8) = 820−12 = 88 12. m) ( a 5 ) 3 : (a 7 ) 2 = a 5⋅3 : a 7⋅2 = a15 : a14 = a15−14 = a1 = a n) 25 ⋅ 8 ⋅ 27 ⋅16 = 25 ⋅ 23 ⋅ 27 ⋅ 2 4 = 25+3+7+4 = 219 o). (3 ) ⋅ 27 ⋅ 9 2 3. 4. ( ). = 32⋅3 ⋅ 33 ⋅ 32. ( ). p) 315 : 813 = 315 : 34 q) r). (m. 3. 3. 4. = 36 ⋅ 33 ⋅ 32⋅4 = 36 ⋅ 33 ⋅ 38 = 317. = 315 : 34⋅3 = 315 : 312 = 315−12 = 33. ) ( ) = [m ] : m 2. ⋅ m : m2. 4. 3+1 2. 2⋅4. ( ). 2. = m4 : m8 = m4⋅2 : m8 = m8 : m8 = 1. (− 10)20 : [(− 2)9 ⋅ 59 ] = (− 10)20 : [[(− 2) ⋅ 5]9 ] = (− 10)20 : (− 10)9⋅2 = (− 10) : (− 10)18 = (− 10)20−18 = (− 10)2 2. 2. 20.
(3) IES Juan García Valdemora Departamento de Matemáticas. TEMA1. HOJA 2. 2 EJERCICIOS DE POTENCIAS 2º ESO. s). [(− 3) ⋅ (− 2) ] ⋅ (6 ) = [[(− 3)⋅ (− 2)] ] ⋅ 6. t). [a. u). 5 3. 5. ] [. 5 3. 4 2. ] [ ] [. 2. 2. 4⋅2. ]. 2. = 65⋅3 ⋅ 68 = 615 ⋅ 68 = 615+8 = 623. 2. ⋅ a 4 : a15 : a13 = a 3+4 : a15−13 = a 7⋅2 : a 2⋅2 = a14 : a 4 = a14−4 = a10. 3. [(− 30). ][. ]. [. : 519 : (− 2) ⋅ 34 = [(− 30) : 5] : [(− 2) ⋅ 3]. 19. 3. 4. 19. ] = (− 6). 4 3. 19. : (− 6) = (− 6) : (− 6) = (− 6) 4⋅3. 19. 19−12. 12. = (− 6). 7. v) m14 : ( m8 ⋅ m) = m14 : m 8+1 = m14 : m 9 = m14−9 = m 5 w) ( x 2 ) 3 : ( x 3 ) 2 = x 2⋅3 : x 3⋅2 = x 6 : x 6 = 1. (8. ((. ). )(. ). 3. ). 2. ⋅ 22 : 162 23 ⋅ 2 2 : 2 4 23⋅3+2 : 2 4⋅2 211 : 28 211−8 23 = x) = = = 3 = 3 =1 23 23 2 2 (− 10)3 : (− 5)3 [(− 10) : (− 5)]3 3. y) z). (b ) : (b ⋅ b) = b : (b ) = b : b = b : b a ⋅ a : [a : a ] = a : [a ] = a : a = 1 5 3. 4. 3. 3. 4. 12. 3+1 3. 5⋅3. 4+ 4. 4. 4⋅3. 15. 12− 4. 15. 8. 12. = b15−12 = b3. 8. 4. REDUCE A UNA ÚNICA POTENCIA DE EXPONENTE NATURAL: a). 73 ⋅ (7 ⋅ 7 3 ) 5 = 7 3 ⋅ (71+3 ) 5 = 7 3 ⋅ 7 4⋅5 = 7 3 ⋅ 7 20 = 7 3+20 = 7 23. b). (− 6)9 : (− 6)−7 = (− 6)9−(−7 ) = (− 6)9+7 = (− 6)16. c). [(− 5) ]. d). (− 12)3 : [(− 12)4 ⋅ (− 12)]. 4 −3. = (− 5). 4⋅( − 3 ). = (− 5). −12. −1. = (− 12). 3+5. 12. 12. 1 1 = = − −5 5. [. ]. = (− 12) : (− 12) 3. = (− 12) : (− 12). 4+1 −1. = (− 12) : (− 12) = (− 12). 5⋅( −1). 3. 3−( −5 ). −5. 3. = (− 12). 8. e) 155 : (− 3) = (15 : (− 3)) = (− 5 ) 5. f). 5. 5. 814 : (− 9 ) = (81 : (− 9 )) = (− 9 ) 4. 4. 4. [. 9. 9. i). ] (− 5) ⋅ [35 : (− 7 ) ] = (− 5) ⋅ [35 : (− 7 )] = (− 5) ⋅ [− 5] = (− 5) = (− 5) (3 ) ⋅ [(− 5) : (− 5) ] = 3 ⋅ [(− 5) ( ) ] = 3 ⋅ (− 5) = (3 ⋅ (− 5)) = (−15) 6. 8. 8. j). (4 ) ⋅ (4 ). k). (− 3) : (− 3) ⋅ (− 3) = (− 3). l). (− 3)4 : [(− 3)8 ⋅ (− 3)] = (− 3)4 : [(− 3)8 +1 ] = (− 3)4 : (− 3)9 = (− 3)4 − 9 = (− 3)− 5 = . g) 159 : (− 5 ) ⋅ (− 3) = 159 : [(− 5) ⋅ (− 3)] = 159 : [15] = 1 h). 3 2. 9 −3. 4. 9. 8. 6. −1. 5. 7 4. 9. 5− −1. 3⋅2. 6. 6. 6. 14. 6. 6. = 47⋅4 ⋅ 49⋅(−3) = 4 28 ⋅ 4 −27 = 428+(−27 ) = 41 = 4 4 −8. 8. ⋅ (− 3) = (− 3) ⋅ (− 3) = (− 3) −4. − 4 +1. = (− 3). −3. 3. 1 1 = = − −3 3 5. 3. 1 1 = − − 3 3. ( ) : [(− 3) ⋅ (− 4) ] = 12 : [(− 3) ⋅ (− 4)]. 4 m) 12. n). 6 +8. 8. 2. 12. 12. [(−11) ] : [(−11) ] 5 3. 4 −2. 4⋅ 2. 12. = (− 11) : (− 11) 5⋅3. 4⋅( −2 ). 1 = 128 : 1212 = 128 −12 = 12− 4 = 12 . = (− 11) : (− 11) = (− 11) 15. −8. 15−( −8 ). 5. 4. = (− 11). 15+8. = (− 11). 23. =.
(4) IES Juan García Valdemora Departamento de Matemáticas. o). (3 ). p). (2 ⋅ 2) : (2 ) = (2 ). q). 2 −3. 2⋅( −3 ). −4. ⋅ 27 ⋅ 9 = 3 2. 3. 2 4. ( ). 2 −4. ⋅3 ⋅ 3 3. 3+2 2. (− 20)7 : [(− 4)9 ⋅ 59 ]. TEMA1. HOJA 2. 2 EJERCICIOS DE POTENCIAS 2º ESO. −6+3+( −8 ). =3. =3. −11. =3. 1 = 3. : 22⋅4 = 25⋅2 : 28 = 210−8 = 22. = (− 20) : [(− 4) ⋅ 5]. −1. 11. −6+3−8. 9⋅( −1). 7. = (− 20) : [− 20] = (− 20). 7 −( −9 ). −9. 7. = (− 20). 16. r). [(− 2) ⋅ (− 3) ] ⋅ (6 ). s). [40. t). (− 3)4 ⋅ (− 3)7 : [122 : (−4) 2 ] = (− 3)4+7 : [12 : (−4)]2 = (− 3)11 : [− 3]2 = (− 3)11−2 = (− 3)9. 9. 4 −2. 5 3. 5. ][. : 20 : 2 ⋅ 2 9. 4. = [(− 2) ⋅ (− 3)] ⋅ 64⋅(−2 ) = 615 ⋅ 6−8 = 615+(−8 ) = 67 5⋅3. ] = [40 : 20] : [2 ⋅ 2]. 4 3. 4⋅3. 9. ( ). =2 : 2 9. 2 12. 15. = 2 :2 9. 2⋅12. = 2 :2 = 2 9. 24. 9−24. =2. −15. 1 = 2. 5. Simplifica implifica (utilizando las propiedades de las potencias) las siguientes expresiones y después calcula: a). 35 ⋅ 2 2 ⋅ 4 35 ⋅ 2 2 ⋅ 2 2 35 ⋅ 2 2+ 2 35 ⋅ 2 4 2 4 = = = = = 2 4 = 16 33 ⋅ 9 33 ⋅ 32 33+2 35 1. b). 32 ⋅ 27 ⋅ 3 25 ⋅ 33 ⋅ 3 25 ⋅ 33+1 25 ⋅ 34 34 = 3 2 = = = = 34 = 81 3+ 2 5 8⋅4 2 ⋅2 2 2 1. c). 63 ⋅12 4 ⋅ 36 (2 ⋅ 3) ⋅ 2 2 ⋅ 3 ⋅ 2 2 ⋅ 32 23 ⋅ 33 ⋅ 2 2 ⋅ 34 ⋅ 2 2 ⋅ 32 23 ⋅ 33 ⋅ 28 ⋅ 34 ⋅ 2 2 ⋅ 32 23+8+2 ⋅ 33+4+2 = = = = = 3 2 27 3 ⋅ 82 ⋅ 64 33⋅3 ⋅ 23⋅2 ⋅ 2 6 39 ⋅ 26 ⋅ 26 39 ⋅ 2 6+6 33 ⋅ 23 ⋅ 2 6. ( ) ( ) ( ). =. ( ). 4. 3. 4. 213 ⋅ 39 2 = =2 39 ⋅ 212 1. d). 8 ⋅ 64 ⋅ 81 ⋅ 2 4 ⋅ 52 2 3 ⋅ 2 6 ⋅ 34 ⋅ 2 4 ⋅ 5 2 2 3 ⋅ 2 6 ⋅ 34 ⋅ 2 4 ⋅ 5 2 23+6+4 ⋅ 34 ⋅ 52 213 ⋅ 34 ⋅ 52 1 1 = = = = = 2 = 4 2 5 3 4 2 5 3 4 4 4 5+ 4+ 4 2+ 4 3 13 6 3 2 5 3 4 3 ⋅ 2 ⋅ 5 ⋅ 6 ⋅16 3 ⋅ 2 ⋅ 5 ⋅ (2 ⋅ 3) ⋅ 2 3 ⋅ 2 ⋅5 ⋅ 2 ⋅3 ⋅ 2 2 ⋅3 ⋅5 2 ⋅3 ⋅5 3 ⋅ 5 45. e). 8 ⋅ 25 ⋅ 73 ⋅ 34 2 3 ⋅ 5 2 ⋅ 7 3 ⋅ 34 2 3 ⋅ 5 2 ⋅ 7 3 ⋅ 34 23 ⋅ 52 ⋅ 73 ⋅ 34 5 ⋅ 33 = 2 = 2+1 = = =1 20 ⋅ 21 ⋅14 ⋅ 7 2 ⋅ 5 ⋅ 3 ⋅ 7 ⋅ 2 ⋅ 7 ⋅ 7 2 ⋅ 5 ⋅ 3 ⋅ 71+1+1 23 ⋅ 5 ⋅ 3 ⋅ 7 3 1. f). 54 4 ⋅103 ⋅ 52 ⋅152 2 ⋅ 33 ⋅ (2 ⋅ 5) ⋅ 52 ⋅ (3 ⋅ 5) 2 4 ⋅ 33⋅4 ⋅ 23 ⋅ 53 ⋅ 52 ⋅ 32 ⋅ 52 2 4+3312+2 ⋅ 53+2+2 2 7314 ⋅ 57 = = = = = 3 7 1253 ⋅187 53⋅3 ⋅ 2 7 ⋅ 32⋅7 59 ⋅ 27 ⋅ 314 59 ⋅ 27 ⋅ 314 53 ⋅ 2 ⋅ 3 2. (. =. 4. 3. 2. ). 1 1 = 2 5 25. (7 ⋅ 8) 3. g). ) ( ) (. 4. 82 ⋅ 494 ⋅145. (7 ) ⋅ (2 ) = (2 ) ⋅ (7 ) ⋅ (2 ⋅ 7) 3 4. 3 2. 2 4. 3 4. 5. =. 7 3⋅4 ⋅ 23⋅4 712 ⋅ 212 712 ⋅ 212 712 ⋅ 212 2 = = = = 23⋅2 ⋅ 7 2⋅4 ⋅ 25 ⋅ 75 26 ⋅ 78 ⋅ 25 ⋅ 75 26+5 ⋅ 78+5 211 ⋅ 713 7. h). 3−3 ⋅ 4 ⋅ 9 −1 4 ⋅ 35 ⋅ 41 2 2 ⋅ 35 ⋅ 2 2 2 2+ 2 ⋅ 35 2 4 ⋅ 35 2 4 = 3 1 = = = = = 16 3−5 ⋅ 4 −1 3 ⋅9 33 ⋅ 32 33+2 35 1. i). 216 ⋅ 27 −3 ⋅ 64 −1 216 ⋅12 4 ⋅ 361 23 ⋅ 33 ⋅ 2 2 ⋅ 3 ⋅ 2 2 ⋅ 32 23 ⋅ 33 ⋅ 2 2⋅4 ⋅ 34 ⋅ 2 2 ⋅ 32 23+8+2 ⋅ 33+4+2 213 ⋅ 39 = 2 = = = = 12 9 = 2 2 3 12 −4 ⋅ 82 ⋅ 36 −1 8 ⋅ 27 3 ⋅ 641 23⋅2 ⋅ 33⋅3 ⋅ 26 26+6 ⋅ 39 2 ⋅3 23 ⋅ 33 ⋅ 2 6. ( ) ( ) ( ). 4.
(5) IES Juan García Valdemora Departamento de Matemáticas. TEMA1. HOJA 2. 2 EJERCICIOS DE POTENCIAS 2º ESO. 6. Reduce a una única potencia y después calcula: a) b) c) d) e). (a ) ⋅ (a ⋅ a ) = (a ) ( ) ⋅ (a ) = a ⋅ a = a ⋅ a = a = a = 1 a : [a : a ] = a : [a ] = a : [a ] = a : a = a : a = a = a = 1 (x ) ⋅ (x ) : (x ) = x ⋅ x ( ) : x = x ⋅ x : x = x ( ) : x = x : x (b ⋅ b) : (b : b ) = (b ) : (b ) = b : b = b : b = 1 (a ⋅ b) : (a ⋅ b ) = (a ⋅ b ): (a ⋅ b ) = (a ⋅ b ): (a ⋅ b ) = 1 (a ) ⋅ (b ) ⋅ a = a ⋅ b = a ⋅ b = a ⋅ b = a ⋅ b a ⋅b ⋅b ⋅ a a ⋅b ⋅b ⋅ a a ⋅b a ⋅b a ⋅ (b ) ⋅ b 3 −5. 4. 6. 9. 5. 4. 10. 7. 2 3. 4 3. 6⋅3. 3 5. 2. 3⋅4. 7. 4⋅3. 10⋅ 2. 5⋅6. 4. 6⋅5. 2⋅ 2. 7. 4. 20. 14. 12. 2. 6. 0. 6. 6. =1. 30. 4. 4. 0. 4− 4. 4. 18+ −12. 6. 30. 12. −15+15. 15. 2⋅2. −12. 18. 7⋅2 2. −15. 4. 2⋅3. 9−3 5. 4. 3⋅5. 2 2. 3⋅ −4. 5⋅ 4. −4. −15. 4. 4+1 6. 2 2. 7 2. 2+1 5. 7−5 2. 4. 3 −4. 4. 3 4. f). 5 2. 7. 6 3. 3⋅ −5. 5. 2. 20. 4. 12. 14. 12. 14. 12+ 2. 11. 14. 7+ 4. =. a =a 1.
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