Frac Pot Sol

3º ESO Fracciones y potencias – Soluciones y resueltos

Pedro Castro Ortega Profesor del IES “Fernando de Mena”

Soluciones a los ejercicios de Fracciones Nota: en algunos ejercicios se da solamente la solución final. En otros se hace el desarrollo completo hasta llegar a la solución. 1. Resolver las siguientes operaciones con fracciones, simplificando en todo momento los pasos intermedios y el resultado: a)

1 1 6 13 + ⋅ = 4 3 5 20

1 1 6 7 b)  +  ⋅ =  4 3  5 10 2 1 13 c) 1 − ⋅ = 3 5 15

 2 1 1 d)  1 −  ⋅ =  3  5 15 2 4 1 e) − + ⋅ = 0 3 3 2

1 1 6  6 3 2 6 5 6 30  f)  −1 + −  ⋅ =  − + −  ⋅ = − ⋅ = − = −1 2 3 5  6 6 6 5 6 5 30  2 1 4 1 6 2 4 6 2 4 2 6 4 6 8 g) − + ⋅ − ⋅ = − + − = − + − = − + − = − 5 3 5 3 5 5 15 15 5 15 5 15 15 15 15

1 4 2 4 2  2 1 4 1 6  6 5  4 6 h)  − +  ⋅ − ⋅ =  − +  ⋅ − = − ⋅ − = − − = 15 5 5 75 5  5 3  5 3 5  15 15  5 15 =−

i)

1 1 4 1 5 8 1 4 1 40 1 4 1 10 + ⋅ − + ⋅ = + − + = + − + = 2 3 3 12 4 3 2 9 12 12 2 9 12 3 =

j)

4 30 34 − =− 75 75 75

18 16 3 120 151 + − + = 36 36 36 36 36

 1 1  4 1 5 8  3 2  4 1 40 5 4 1 10 20 1 10 = ⋅ − + = − + =  + ⋅ − + ⋅ =  + ⋅ − +  2 3  3 12 4 3  6 6  3 12 12 6 3 12 3 18 12 3 =

10 1 10 40 3 120 157 − + = − + = 9 12 3 36 36 36 36

 1 1 2 1 k)  1 − +  ⋅ =  2 3 5 3 1 1 2 19 l) 1 − + ⋅ = 2 3 5 30 1 4 2 1 5 4 2 5 1 m) − ⋅ − + ⋅ = − − + = − 2 7 14 2 7 14 14 14 14

1

3º ESO Fracciones y potencias – Soluciones y resueltos

Pedro Castro Ortega Profesor del IES “Fernando de Mena”

1 4 2  1 5 1 8 2 5 1 6 5 1 3 5 n) − ⋅  −  + ⋅ = − ⋅  −  + = − ⋅ + = − ⋅ + = 2  7 14  2 7 2  14 14  14 2 14 14 2 7 14 =−

ñ)

o)

3 5 2 1 + = = 14 14 14 7

17 15 4  1 2 1  14 16 17 4  3 10 1  14 − + : + −  + : = −3+ : + −  + : 2 = 9 5 3  5 3 15  3 8 9 3  15 15 15  3 =

17 4 12 14 17 4 4 7 17 20 7 −3+ : + = −3+ : + = −3+ + = 9 3 15 6 9 3 5 3 9 12 3

=

17 5 7 17 27 15 21 26 −3+ + = − + + = 9 3 3 9 9 9 9 9

1 4 5  1 3 10  1 24  1 30  1 8 1 5  + : ⋅ − ⋅ + 4 = + ⋅ − + 4 = + ⋅ − + 4 = 3 3 6 2 2 9  3 15  2 18  3 5 2 3  1 8  3 10 24  1 8 17 1 136 1 68 10 136 146 73 = + ⋅ − +  = + ⋅ = + = + = + = = 3 5  6 6 6  3 5 6 3 30 3 15 30 30 30 15

p)

q)

4 7 3 1  1 7 6 4 21 1  4 1  7 20 − ⋅ + ⋅  2 +  − + 4: = − + ⋅  +  − + = 5 3 7 5  2 3 5 5 21 5  2 2  3 6 =

4 1 5 7 10 4 5 7 10 4 1 7 10 − 1 + ⋅ − + = −1 + − + = −1 + − + = 5 5 2 3 3 5 10 3 3 5 2 3 3

=

24 30 15 70 100 39 13 − + − + = = 30 30 30 30 30 30 10

2 5  3 4  5  3  12 2 5  3 2  5 3 12 + ⋅ +  − +  : 4 + = + ⋅ +  − + + = 3 4  5 10  4  5  5 3 4  5 5  4 20 5 =

2 5 5 5 3 12 2 5 5 3 12 2 5 5 3 12 + ⋅ − + + = + ⋅1 − + + = + − + + = 3 4 5 4 20 5 3 4 4 20 5 3 4 4 20 5

=

40 75 75 9 144 193 + − + + = 60 60 60 60 60 60

1  7 2 5 1  7 1 5 1  12 14 3 10  r) 2 + :  2 + − +  = 2 + :  2 + − +  = 2 + :  + − +  = 5  3 4 3 5  3 2 3 5 6 6 6 6 1 33 1 11 2 110 2 112 = 2+ : = 2+ : = 2+ = + = 5 6 5 2 55 55 55 55

 2 4 2  3 7 4  2 4 1  3 49  40 112 35  3 49 s)  − +  ⋅ − : =  − +  ⋅ − = − + = ⋅ −  7 5 8  2 5 7  7 5 4  2 20  140 140 140  2 20 =−

37 3 49 111 49 111 686 797 ⋅ − =− − =− − =− 140 2 20 280 20 280 280 280

2

3º ESO Fracciones y potencias – Soluciones y resueltos

t)

u)

3 1 4  4 2 15  3 4  4 30  3 2  4 5  − ⋅ :  − ⋅ + 1  = − :  − + 1 = − :  − + 1 = 2 2 3 3 3 8  2 6  3 24  2 3  3 4  3 2  16 15 12  3 2 13 3 24 117 48 69 23 = − : − +  = − : = − = − = = 2 3  12 12 12  2 3 12 2 39 78 78 78 26 2   3 1  2   9 2  2  7  2  12 7  + 1 −  −   = + 1 −  −   = +  1 −  = +  −  = 3   4 6   3   12 12   3  12  3  12 12  =

v)

Pedro Castro Ortega Profesor del IES “Fernando de Mena”

2 5 8 5 13 + = + = 3 12 12 12 12

2  3 1  2 1   6 1  3  1 1  − − − − + −  − + −  = 3  2 5  5 3   5 2   4  2 3  =

2  3 1  6 5   12 5   3  3 2  − − − −  + −  − + −  = 3  2 5  15 15   10 10   4  6 6 

=

2  3 1 1 7  3 1 2  45 6 2 21  3 1 − − − + − + = − − − + − + = 3  2 5 15 10  4 6 3  30 30 30 30  4 6

=

2 58 3 1 2 29 3 1 40 116 45 10 111 37 − − + = − − + = − − + =− =− 3 30 4 6 3 15 4 6 60 60 60 60 60 20

5  5   7  2 1  5 6  7  8 w) 2 +  − 3  −  −  +   = 2 +  −  −  −  +   =  2  10  5 4    2 2  10  20 20  

1  7 13  1  14 13  1 1 40 10 1 29 = 2− −  −  = 2− −  −  = 2− − = − − = 2 10 20  2  20 20  2 20 20 20 20 20 4  1 2  1  4 4  5 4  1  4 6  1  1 x) 2 −  −  +  −  −  + 2  − = 2 −  −  +  −  −  +  − =  5  3  2 5  3  3  3  10 10  3   3 3  5

 4 9 1  10 1  40 27 10  10 1 = 2−  − −  − − = 2−  − −  − − =  3 10 3  3 5  30 30 30  3 5 = 2−

3 10 1 1 10 1 60 3 100 6 49 − − = 2− − − = − − − =− 30 3 5 10 3 5 30 30 30 30 30

 4 −1    5 2   7  12 −1    15 8   7 y)  −  +  2 −  − +   − =  −  +  2 −  − +   − =  3 9    4 3   2  9 9    12 12   2 7 7  12 −1    15 8   7 13  =  −  + 2 −  − +  − = + 2 +  − =  9 9    12 12   2 9  12  2

=

13  24 7  7 13 31 7 52 93 126 19 + + − = + − = + − = 9  12 12  2 9 12 2 36 36 36 36

 4 1 7   4 5    1 1   4 1   4 5    1 1  z)  +  :  −   ⋅  +  =  +  :  −   ⋅  +  =  6 2   3 12    6 15   6 14   3 12    6 15 

3

3º ESO Fracciones y potencias – Soluciones y resueltos

Pedro Castro Ortega Profesor del IES “Fernando de Mena”

 28 3   16 5    15 6   31 11  21 372 7 =  +  :  −   ⋅  +  =  :  ⋅ = ⋅ =  42 42   12 12    90 90   42 12  90 462 30 =

372 7 62 7 434 31 ⋅ = ⋅ = = 462 30 77 30 2310 165

 3 

1   

5 7

1 

 3 8

1    8

5   28 1  

α)  − +  4 −   −  2 −  +  −   =  − +  −   −  −  +  −  = 2    4   2 8    8  2 2    4 4   8 8    8 

8  3 7   3 27   3 28   6 27  25 33 =  − +  −  +  = − +  −  +  = − = − = −1 8  8 2 4 8   8 8  8 8  8 8 1 + 2 5   5 12   1 3  7 5  1 4   1  β)  −  ⋅  − 1 ⋅ 3 − =  −  ⋅  −  ⋅ 3 − =  3   15 15   3 3  3   3 5   3 

γ)

=−

7  2 7 7  6 7 7  7 7  30 7  ⋅  − ⋅ 3 −  = − ⋅  − −  = − ⋅  −2 −  = − ⋅  − −  = 15  3 15  15  3 15  15  15  15  15 15 

=−

7  37  259 ⋅ −  = 15  15  225

4 12  1 2  3   1  2  4  3  1 4  3   1  5 2  :   +  −  − 3  : 1 −  = :   +  −  − 3  :  −   = 5 16  6 3  8   6  5  5  4  6 6  8   6  5 5 

=

4  3 5 3 5 4  5 3  15  1 3  4  15 3  : ⋅ −  −3 :  = :  −  −3 = : −  − = 5 4 6 8  6 5  5  24 8  18 5  8 8  18

=

4 2 15 32 15 16 15 288 75 213 71 : − = − = − = − = = 5 8 18 10 18 5 18 90 90 90 30

4

3º ESO Fracciones y potencias – Soluciones y resueltos

Pedro Castro Ortega Profesor del IES “Fernando de Mena”

Soluciones a los ejercicios de Potencias Nota: en algunos ejercicios se da solamente la solución final. En otros se hace el desarrollo completo hasta llegar a la solución. 2. Calcular las siguientes potencias de exponente natural (sin usar calculadora): a) (–2)4 = 16 e) −2−3 = −

1 8

b) (–2)3 = −8

c) −22 = −4

1 4

g) (–2)−3 = −

f) (–2)−2 =| −2

−4

1 8

h) –32 = −9

 −1  k)   = 16  2 

l) (−4)2 = 16

−4

1 j)   = 4 2

i) (–1)−7 = −1

d) (–3)2 = 9

 −1  m)   = 81  3 

4 n)   = 1 5

ñ) 1−37 = −1

o) –52 = −25

p) (–1)523 = −1

q) 10 = 1

r) 2350 = 1

s) (–1)0 = 1

0

t) (0,75)0 = 1

3. Expresar como una única potencia de base entera o racional: 2 a)   5

c)

2

−1

2 2 :  =   5 5

3

35 ⋅ 3−7 3−2 = 2 = 3−4 2 3 3

e) ( 22 ⋅ 2−3 ) = ( 2−1 ) = 24 −4

−4

3

4

2

5 2 2 :  =   = 2 5 5

2

4

j)

−2

4

 3  3  3  3 ⋅ −  =   ⋅  =    2  2  2  2

2 4 ⋅ 4 −2 2 4 ⋅ 2 −4 = = 2 −6 2 6 8 2

125

2

2 k)   3

2

−1

2 d)   5

2 3

 1 3   1 6 i)    =    2    2 

3

1 1 :   =   = 22  2  2

(5 ) h)

4

−2

3

f)

 3  3  3 3 g)  −  ⋅  −  =  −  =    2  2  2 2

5

1 b)   2

=

55 = 52 53

2 2 = 6 = −2−5 3 (−4) −2 −2

 5 −3   5  −2  5  6  5  −2  5 8 l)    :   =   :   =    3  3 3  3    3 

4. Aplica las propiedades de las potencias y simplifica todo lo que puedas:

−32 −9 a) = = −1 2 (−3) 9 b)

2 −5 ⋅ 42 ⋅ 32 2−5 ⋅ 24 ⋅ 32 34 81 −4 4 = = 2 ⋅ 3 = = 23 ⋅ 9−1 23 ⋅ 3−2 2 4 16

5

3º ESO Fracciones y potencias – Soluciones y resueltos

1 c)   3

3

Pedro Castro Ortega Profesor del IES “Fernando de Mena”

2

1 1 42 16 1 :  = 3 : 2 = 3 = 3 27 4 3 4

d)

3 ⋅ (−3)2 ⋅ 42 3 ⋅ 32 ⋅ 24 33 ⋅ 24 2 2 = = = 2 ⋅ 3−4 = 4 = 3 2 3 4 3 3 4 6 ⋅9 (2 ⋅ 3) ⋅ 3 2 ⋅3 ⋅3 3 81

e)

23 ⋅ (−3) 2 ⋅ 42 23 ⋅ 32 ⋅ 24 32 ⋅ 27 23 8 3 −5 = = = 2 ⋅ 3 = = 3 2 3 4 3 3 4 5 6 ⋅9 (2 ⋅ 3) ⋅ 3 2 ⋅3 ⋅3 3 243

f)

2 −4 ⋅ 42 ⋅ 3 ⋅ 9−1 2−4 ⋅ 24 ⋅ 3 ⋅ 3−2 20 ⋅ 3−1 22 4 2 −5 = −5 3 2 2 = −2 4 = 2 ⋅ 3 = 5 = −5 2 2 ⋅8 ⋅9 ⋅ 3 2 ⋅ 2 ⋅3 ⋅3 2 ⋅3 3 243

2 g)   3

5

−4

0

1

2 2 2 2 ⋅  ⋅  =   = 3 3 3 3 −2

 5 −3   5  −2  5 6  5  −2  5 8 58 390625 h)    :   =   :   =   = 8 = 6561 3  3 3 3  3    3  2

2

i)

 1 3   1 3   1 6  1 6 1 1  − 1  =  −   =  −  =   = 6 = 64  2    2    2   2  2

j)

(−1)3 ⋅ 23 ⋅ 84 −1⋅ 23 ⋅ 212 −215 = −2 6 = 4 = −211 = −2048 −2 6 2 ⋅2 2 ⋅2 2

k)

(2

l)

  1 2  −1   1 4  −1    3  −1    1  −1   −   =  −   =  −   =  −   =  6    6 3    6 6    2  

4

⋅ 2 −5 ) : 2 3 = 2 −4 =

1 1 = 4 2 16

−5

−5

−5

−5

5

1 1  1 = −  = − 5 = − 2 32  2

34 ⋅ 3−5 3−1 1 1 = 3 = 3−4 = 4 = m) 2 3⋅3 3 3 81 2

2  52  54 54 ⋅ 24 ⋅ 52 ⋅ 7 2  25  2 2 4 2 2 ⋅ 2 ⋅ 5 ⋅ 7 = ⋅ 2 ⋅ 5 ⋅ 7 = = n)   ⋅ 202 ⋅ 7 2 =  ( )  22 ⋅ 7 2 22 ⋅ 7 2  14   2⋅7  2

= 56 ⋅ 22 ⋅ 7 0 = 62500

(3 ⋅ 2 ) ( −2 ⋅ 3 ) −2

o)

−4

4 −1 2 2

32 ⋅ 2−4 24 16 −2 4 3 2 = ⋅ = = 2 −8 ⋅ 34 32 9

=

3−2 ⋅ ( −7 2 ⋅ 32 )

3

p)

(−3)2 ⋅ 73 

2

=

3−6 ⋅ ( −76 ⋅ 36 ) 34 ⋅ 7 6

=

−30 ⋅ 7 6 1 1 = −3−4 = − 4 = − 4 6 3 ⋅7 3 81

6

3º ESO Fracciones y potencias – Soluciones y resueltos −2

−1

Pedro Castro Ortega Profesor del IES “Fernando de Mena” −2

−1

−2

−1

 3 3 1 7 6 3 3 7 3  4 q)  −  ⋅  −  =  −  ⋅  −  =   ⋅  −  =  2 4 3 9 4 4 9 9 4  9 4 =  3

2

1

 9  16  9  ⋅  −  = ⋅  −  = −4 9  4  4 −4

r)

s)

4   2  2  2  −3  2   ⋅      4    5   5   16 5 2  = =  = − 2 0  2  −1  2    2   5  625   :      5  5   5  

 2  2 2   ⋅    3  3  

3 9

2 5   32 3 2  = =  = 4 5 3 2  2   2    2   3  243 3   :       3   3  

5. Simplifica: 2

2

 1 3  3 3 9 3 6 3 a)  1 +  − =   − = − = =  2 4  2 4 4 4 4 2 2

2

2

1 4 1 5 1 1 2 1 1 b) 4  −  + 1 = 4  −  + 1 = 4   + 1 = 4 + 1 = + 1 = + 1 = 16 16 4 4 2 4 4 4 4 4  1  −1   8  1 1 64 1 1 64 1 256 45 301  + = + = c)  4 −  ⋅ +   =   ⋅ + = ⋅ + = 3  5  2   3  5 4 9 5 4 45 4 180 180 180  2

2

2

2

3

2

9 9 32 23 1 1  1 3  3 d)  :  −  +  =   − 23 = − 8 = − =− 4 4 4 4  2 3  2 2  2 1 1 e)  −  3 6

2

2

1 1  2 1 : −  =  −   6 3  6 6 2

f)

2

2

1 2 1 : −  =   6 6 6

2

2

2

 1 1 :−  =    6  6

2

1  1 1 2 1 4 9 20 29 + 1 −  +  + + 5  3 5 3 5 9 45 45 = 45 = 1044 = 116 = = = 2 2 25 72 25 97 4365 485 1   5 2+ + 2 +  − 1 2+−  36 36 36 36 6   6

7

2

1 :  =1 6