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Leyes de los exponentes Expresar usando exponentes positivos 1. 3−1 5. a−3 2. 6−3 6. m−1 3. 2−4 7. x −4 −1 4. 4 8. n−6 Simplificar. Expresa sin usar exponentes. 1. 4−2 4. 5−3 −1 2. 8 5. 1−4 3. 7−1 6. 5 0 Simplificar 1. (−2)4 (−2)2 (−3)6 3. 4 (−3) 2 2. (−5) (−5) (−10)7 4. 6 (−10) Simplificar. Expresar utilizando exponentes. 1. 2 4⋅23 75 2. 35⋅32 23. 3. 8 5⋅89 72 3 20 4. n ⋅n 47 24. 5. x 4⋅x 3 43 6. y 7⋅y 9 7. n3⋅n 812 7 7 25. 8. z ⋅z 86 3 9. x ⋅x 10. a 6⋅a 8 915 7 0 26. 11. m ⋅m 92 12. p⋅p⋅p 13. x 4⋅x 2⋅x 64 2 4 3 27. 14. y ⋅y ⋅y 64 3 4 15. a ⋅a ⋅a⋅a 16. b⋅b 5⋅b 2⋅b2 27 3 6 5 28. 17. ( a b )( a b) 27 2 5 2 18. ( x y )(x y ) 19. y9 2 3 2 3 29. ( p q r )( pqr ) y5 20. ( x 7 y 4 z 4 )( x 2 y 5 z 8) x 12 2 3 2 30. 21. (5s t )(5s t ) x 11 2 2 2 22. ( 2xy )(2x y )

9. 10. 11. 12.

3a−1 (3x )−1 ( 2y)−1 4x−3

13. 14. 15. 16.

5c−4 8m−1 (3a )−1 cd −2

7. 8. 9.

2−4 1−3 n0

10. 6−2 11. 100 12. x 0

5.

43 45

6.

34 36

7.

(−2)2 5 (−2)

8.

(−5)3 4 (−5)

31.

a6 a4

39.

x2 y5 y3

32.

n8 n4

40.

m6 n4 m3

33.

x4 x2

41. 42.

34.

y9 y6 g5 g5

43.

35.

36.

h4 h

44.

38.

m m8 x7 x5

a 3 b4 ab x8 y x7 y

45.

43 x3 42 x

46.

64 a5 b 62 a2 b

8

37.

a 6 b9 a5 b5 p5 q7 pq 4

Simplificar. Expresar utilizando exponentes (a) negativos y (b) positivos. y x3 a2 2. 1. 3. 4 y x7 a6 Evaluar cada expresión 1. x 5⋅x 3 con x = 2 2. 10 m⋅10n con m = 2 y n = 4 Simplificar 4 2⋅4 5 1. 43 5

3.

a 2⋅b3 a 2⋅b5

4

5

6

4.

3. 4.

a3⋅a 2⋅a con a = – 2 2a⋅2b⋅2c con a = 3, b = 2 y c = 2

5.

4−1⋅4 5

6.

(−3) (−3)−4

2 ⋅3 m ⋅m 2. 4. 2 2 2 ⋅3 m2⋅m2 m m m 9. ¿Es ( a+ b) =a + b verdadero para todos los números? Expresar: 1. 16 como potencia de 2. 2. 43 como potencia de 2 3. 8 2 como potencia de 2 Más sobre exponentes. A Simplificar. 1. ( 25 )2 13. 4 3 2. (3 ) 14. 2 3 3. (5 ) 15. 8 9 4. (6 ) 16. 5 9 5. ( y ) 17. 3 5 6. ( x ) 18. 8 4 7. ( x ) 19. 5 12 8. ( n ) 20. 6 5 9. ( a ) 21. 7 7 10. ( y ) 22. 10 10 11. ( p ) 23. 12 7 12. ( w ) 24.

4. 5. 6.

(3y)4 ( 2t)5 (7y)3 (8x )4 (5m)2 ( 4y)5 (7x)4 (12a )3 ( 2m 2)2 ( 4n 3)2 (5y 4 )3 (3x 5 )4

25. (−6t 2 )3 26. (−10b6 )2 27. (8k 4 )3 28. (7x 5 )3 29. ( 2x8 y 3)2 30. (3mn 4)3 31. (−2x 2 y 4 )3 4

7.

x 6⋅x−2 x2

8.

a −2⋅b−3 a 4⋅b−1

43⋅8⋅16 como potencia de 2 28⋅163⋅64 como potencia de 2 9⋅27⋅3⋅81 como potencia de 3

34. ( 2m5 n 4 p 3 )3 3

35.

3 a2

( )

7

36.

7 x7

( )

37.

x2 4

( )

39.

m4 n2

( )

40.

a8 b4

( )

B Simplificar 41.

2 3

( ) xy z

42.

ab4 c

3

( )

43.

(

2

y5 3

38.

2 2

32. (3m n ) 33.

m5 m10

−2x 2 y 6 5

2

)

3

( )

44.

45.

46.

4

(

3x 3 y 3 2

(

−4m 2 n 5 3 a8 b4

)

51.

x2 y z

3

m n4 p

3

57. (7a )( 4a)−(3a )2

( )

58. (−2y 2 )3+ 4y( 2y5 )

3

)

52.

( )

59. (−3z4 )2−(−z 2)4

3

( )

53. 5

(

−3a 2 b 4 4c 3

(

2m 5 n5 p4

60. (6cd 2 )2 + 3cd(cd 3)

2

)

61. −3z 3 (2z 4 )−(−5z 3)2

6

47. [(−x )]

54.

18 2

48. [(− y) ] 3

49.

( −x3y )

50.

2c −y

4 55. ( 2n)

4

( )

3

)

3 n 2

62. b2 ( a3 b)2 + a 2 (a 2 b 2)2 63. (3c 4)2 (2c)

3

( )

64. (−2x 2 y 3 )4 ( xy)3

56. ( 4x 3)2+ (2x 2 )3

65. (−3a 2 b4 )3 (4a 3 b)2

Simplificar 66. 67.

x 2a x 4 x 3a x 2b

68. 69.

x a+ 4 x 3 x 5 x 2a−4

70. ( a n+ 1 b m+ 2 )3 71. ( x a y a−3 )3

72. ( c3 d )a (cd 7 )a 73. x 2 ( x a+ 2 y 3)

Multiplicar 1. 2. 3. 4. 5. 6. 7. 8. Dividir 25.

26.

27.

(6x 2 )(7) (5y 3)(−2) (−x 3)(−x) (− y 4 )( y 2 ) (−x 5)( x 3 ) (−m6 )(−m2 ) (3a 4 )(2a 2 ) (5x 3 )(4x 5 ) x6 x2 a7 a 4x 5 2x 2

9. 10. 11. 12. 13. 14. 15. 16.

28.

−6a3 6a 4

29.

12m 4m 4

(7t 5 )(4t 3 ) (−3b3 )(5b5) (3g 4 )(−6g3 ) ( h5)(−7h 3 ) (−6x 3)( x 8 ) (−8m7 )(−4m3 ) (−5n 4 )(−5n 4) (−x 7)(5x 12 )

30.

31.

32.

17. 18. 19. 20. 21. 22. 23. 24.

( x 3 y 4)( x 4 y 2 ) ( 2m3 n 2 )(−3m 6 n 5) ( 4a 4 b 8 )(2a 4 b 2 ) (−2x 3 y)(−6x 9 y 8) ( y 5 )( 2y)(3y 2 ) (3x 4 )( x 4)(5x 2) (−4m 2 )(5m 4 )(−2m3 ) (9b 2)(2b5 )(−3b7 )

−4x 6 −2x 6

33.

5a3 a7

−h 5 2h 4

34.

k3 3k 8

15y8 3

35.

36.

37.

38.

2x 10 8x5

39.

3m 5 6m 7

50.

43.

6

40.

16x 2 −4x 2

6m 2m 2 5

41.

−25a 7 −25

10x y 2x 3 y

44.

4

45.

Simplificar. 49.

42.

45x 3 15x 2

−12m7 n8 4m 2 n5 24a 6 b9 −6a 6 b3 48x 6 y 7 12xy 5

65.

a 4 b5 3a 2 b6

66.

2x 6 y 4 8x 4 y 7

3 2

67.

−2 m3 4m 4 n6

3 2

68.

−4ab3 −8a 2 b 4

59. ( ab2 )3 (a 3 b4 )2 3

2

5 3

60. ( m n) (mn )

52. ( m2)4 ( m3 )2

2 2

2

3 2

4

4 2

61.

53. ( 2x )(3x )

54. (3y )(5y )

62.

55. (3x 4 )2 (2x 5 )2 56. ( 4y)3 (−2y 2 )2

63.

48.

6x 13 y 4 24x 5 y 7

(5m)4 2 2 (−25m )

p( p )

51. ( a 3 )2 ( a4 )3

47.

5a 11 b7 −7a 5 b9

64.

58. (−3mn 4 )(4mn 2 )

4 3

2x 12 y 5 3x 4 y 2

−12p 8 r 3 4p6 r 4

57. ( 2x 2 y)(3x 4 y 5 )

x 3 ( x 4) 2

46.

(−2x ) 3 x (3a ) 2 18a

( 4y ) 2 2 (4y )

En los siguientes ejercicios efectúa los productos indicados.

( )( ) 3

−a b 5

2

a bc 8

1.

−(4 )

6.

2.

(−4)4

7. 3x4 x5

3.

(−2)2 (−2)5

8.

( y−18)12( y −8)6

4.

7z

9.

8 w7 ( 2w2 ) w

10.

c 3 c8 c 4

11.

(−2z+ 5)10(−2z+ 5)11

4

(

)

1 11 z 6z9 14 3

5.

( −32 ) ( −32 )

2

12.

13.

a6 9 (−12a ) 4 (

16. (cd 10 )(−6c 2 d )(−6c 2 d ) 17. (( a−2)7 (3b+ 7))((a−2) 4 (3b+ 7)13)

−ab ab −ab )( )( ) 3 3 3

(

5 8 x y 14. 8x(−3y ) 18

(

9 15. (−st )

)(

18. 1 2 17 x y 10

)(

−3 21 −1 4 5 st s t 4 12

)

(

7a5 b c 2 d 4 8

)(

−6ab 6 d 15 28

)(

−4cd 5

)

19. (6a 5 b 2 c d 4 e3 )(−3ab c 2 d 3 e6 )(−b3 c 2 df )

)

8 9 20. ( √ 7 x y z )(

4 7 4 3 x y z w)(−7x 2 y 3 z 6) √7

En los ejercicios de 1 al 38 simplifica las expresiones. 1.

(4 2)3

2.

(

9.

1 3 ) 24

3.

((−0.3)3 )2

4.

((−1)5)9

5.

((−10)4)2

6.

((0.1)3 )1

7. 8.

16. 16 (−s 12)4

10. −12(w 7)5

17. (( y 3 −3y+ 1)4 )8

11. ( z 8 )7

18. (( 4a 5 + 7a 3 )7)9

12. −1.5(b 5)2

19. (( x 2 + x+ 1)10)3

13.

24 (( y+ 3)5 )2 25

20. ( w−1)( 2(w−1)5 )4 21. 6z8 (3z 7 )4

14.

7 8 10 (a ) 3

15.

9 ((x−8)3 )6 4

−6( x 2 )4 (−c)10

(−x 11)6

2

22. ( y 5 )2 y (5 ) 23. ( z 4 )3 ( z )4

3

6

24. ( w)2 (w 2 )6 25. (( y 4−3y2+ 2)5)6 (( y 4−3y 2+ 2)6 )2

31. ((5a 2−b3 )5)2 ((5a 2−b 3)3)7

26. (( w5 + 8)4 )7 ((w 5+ 8)9 )3 32. 27. (( x 2 + 1)7 )9 (( x 2 + 1)10 )9 28. (( z+ 9)5)5 (( z+ 9)7)7 8 3 3 8 3 3 29. (( y−4) ) (( −4) ) 5 5y

30. (( x+

1 9 11 1 44 ) ) (( x+ ) ) 2 2

(( y 5+ y−3)6)3 (( y 5+ y−3)3 )9

33. (( z−7)14 )11(( z −7)3)11 34. (( a4 + 3b 2−b)5)9 ((a 4+ 3b2 −b)4 )12 35. ((7x 2 + 5y3)5 )6 ((7x 2 + 5y 3 )10)7 36. ((3w 4 + w 3−2w )8)4 ((3w 4+ w3 −2w)5 )9

37. (( r 4 s 9 t −8s 4 t 7 )19)4 ((r 4 s9 t −8s 4 t 7)2)17

38.

En los ejercicios 1 al 37, simplifica las expresiones dadas. 1. (11y2 )2 19. 2. (−7a 2 )3 20. 9 2 3. (−8k ) 21. 5 2 4. −1.5(6b ) 22. 2 5 5. −12(a b) 6.

23.

3 (2z 8)4 8

(( a8−5b5 c 4 )10 )11 (( a8−5b5 c 4 )10 )14

(9xy3 z 2 )2

7( x 6 y 11 z 14)10 (5 a 4 b10 c12 d )4

(7a ( bc)9 d 4)2 (r 21 s30 t 19)7 14

24. 5a 5 ( a 2 b4 )3 (a 5 b 3)6

7.

3 − (ab)6 5

8.

−7( r 4 s 5)3

26. (1.1 x 14 y 19 z 16)2

9.

(−5t 4 )3

27. (3z 2 c 7 )3 (−a 9 b8)6

10.

(0.5a b 4)4

28.

12 a 7 c 8 (2 a10 b15 c)3

11.

8 a(b2c )7 9

29.

(−2a 2 )3 (−2a 2)2

25.

2

8 2 3 3 12. (6a b c )( a ) 5

9 −1 3 2 2 4 s( r s t ) 13. 11 3 14. (−cd )7 ( 2ce) 15. (−c 3 d 6)5 (c 2 d 6)

−xy 3 (−x 6 y 7 )5 ( x 2 y 5)7

30. (0.1 a 5 b4 c 7)4 (abc)6 31.

(

c5 d 3 e 4 2

32. −

(c 5 d 6)8 (−c 4 d )9 32

(

17. −

18.

−a 5 b 7 c 9 2

)(

(

)

(

)

4

x 5 y 12 z 6 33. 24 x z 2 8

2

a6 d 4 e 6

27 x 5 y 12 z 6 8 3 6

16.

4

4

)

34. 2 a 4 (a 6 b c 2 )3 ( b5 c 7 d 2 )2

5

)

(10 c 5 d 15 e 21 )6 64

35. 6(r 4 s6 t )11 8(r s2 t 3 )7 36. ( r s 3 t )7 ( rs3 t )4 (rs 3 t )8

37. (( x 3+ 7x−4)6 ( z 4 −z 2+ 13)8 )12 Simplifica las siguientes expresiones 1.

−

60 12

9.

a8 (−a 2)4

17.

7r 3 s 7 42 r 3 s7

18.

(5c 4 d ) d 2 2 15(cd ) −12ab 2 3 a b

25.

2.

24 14

10.

26 y 18 13 y 8

3.

0.25 1.5

11.

5x 8 20 x 11

19.

12.

24 z 6 4z 10

20.

81 a 5 b 4 9 a 10 b2

13.

(3c)6 3 (3c)

21.

(2w 3 z 4 )3 5 2 (8 w z )

4.

32 − 64 4

5.

6.

7.

8.

32.

33.

34.

35.

36.

(6) 3 (15) b4 b7

12

7

26. (−8a 3 b4 c 2 d 3 )2 4 5 9 10 −16 a b c d 27.

17

4x 10 y 7 z 8 24 x 3 y 5 z 3

56 w15 x 4 y 6 z 17 (2w 5 xyz 4 )4

28. 24x 8 y 2 (3x 4 y 3 )3 2 6 4 ( 4xy) (−3y )

8

29. (15 a 5 b7)2 (a 9 c 4)5 3 2 11 2 (12 a b c )

22.

11 x y z 44 x 7 z 13

15.

a 4 b9 a 6 b7

23.

−35 a 4 b 2 c 5 a5 b 2

16.

45 x 6 y 8 −25x 9 y 4

24.

75c 5 e15 f 11 (−5 cd 6 e 7)3

37.

10a (3a 2 b8 )(a5 c7 )4 (18a 3 b c 4) 4 7 5 3 3 (6a b )(5 b c ) (abc)

(c 2 d 4 )9 (ce 3)7 (3c d e)5 11 6 2 12 5 6 (9d e ) (c e )

38.

(9c 4 d 7 e 5)4 (5c6 d 9 e)(12c d e)3 2 3 2 7 5 7 3 3 3 5 2 2 (3c d e ) (20 c d e ) (15 c d e )

45(2x 7 y 4 z 5)4 (10x 3 y 7 z 9 )3 6 3 2 3 9 3 5 3 (8xy ) (5x y z ) (6y z )

39.

(16 x 8 y 4 z 2)3 (75 x 5 y 7 z 9)4 ( 2y)3 10 9 8 3 3 4 4 4 (30 x y z ) (40x y z )

(r 2 s)6 (14t 2 )(3rt 3)4 ( s5 t 2 )8 4 7 3 3 2 8 6 2 (6r s t )(7r t ) (9s t )

40.

(8w 8 x 10 )2 (3w 2 y 6 z 9 )7 7 5 5 9 8 6 15 2 9 (18w z )(2x y z ) (5w y z )

−

d 13 d 20

c9 c6

14.

(9d ) 5 (−9d )

12b3 (−a 5 b3 c 6 )4 a c 8 2 2 5 7 3 132(a b ) (−b c)

(2 x 7 y 9 z 6)5 (7x 2 y 5 )4 ( 4x 8 y 2 z 3 )3 5 3 3 6 3 4 (14x y z) (8y z )

30. (2r 5 s 7 t 3 )5 (6rs8t )4 6 2 10 3 (9rst ) (4r st) 31. (11x 7 z 9 )2 (6rs8t )4 15 14 3 12 66x y (55y z )

41.

(5a7 b 9 c 10 d 17)5 ( 42a 16 b6 c 26 )6 (8a 3 c 9 d 2 )12 12 14 16 4 4 7 17 7 7 (35 a b c ) (48 b c d )

42.

( 4r 6 s12 t 15)21 (21rs7 t 5)13 (3rs5 t )5 8 8 18 19 2 16 6 10 12 3 9 (6 r s t ) (28 r s t ) (14 r s t )