Soluciones a los ejercicios de Fracciones

(1)

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

b) 1 1 6 7 4 3 5 10

 ₊ _{⋅ =}

 

 

c) 1 2 1 13 3 5 15 − ⋅ =

d) 1 2 1 1 3 5 15

 ₋ _{⋅ =}

 

 

e) 2 4 1 0

3 3 2

− + ⋅ =

f) 1 1 1 6 6 3 2 6 5 6 30 1

2 3 5 6 6 6 5 6 5 30

_{− + −} _{⋅ = − + −} _{⋅ = − ⋅ = −} _{= −}

   

   

g) 2 1 4 1 6 2 4 6 2 4 2 6 4 6 8

5 3 5 3 5 5 15 15 5 15 5 15 15 15 15

− + ⋅ − ⋅ = − + − = − + − = − + − = −

h) 2 1 4 1 6 6 5 4 6 1 4 2 4 2

5 3 5 3 5 15 15 5 15 15 5 5 75 5

_{− +} _{⋅ − ⋅ = −} ₊ _{⋅ −} _{= − ⋅ − = −} _{− =}

   

   

4 30 34

75 75 75

= − − = −

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 =

18 16 3 120 151

36 36 36 36 36

= + − + =

j) 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

= − + = − + =

k) 1 1 1 2 1 2 3 5 3

 _{− +} _{⋅ =}

 

 

l) 1 1 1 2 19

2 3 5 30

(2)

n) 1 4 2 1 5 1 8 2 5 1 6 5 1 3 5 2 7 14 2 7 2 14 14 14 2 14 14 2 7 14

   

− ⋅_ − _+ ⋅ = − ⋅_ − _+ = − ⋅ + = − ⋅ + =

   

3 5 2 1

14 14 14 7

= − + = =

ñ) 17 15 4: 1 2 1 14 16: 17 3 4: 3 10 1 14: 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

= − + + = − + + =

o) 1 4 5: 1 3 10 4 1 24 1 30 4 1 8 1 5 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) 4 7 3 1 2 1 7 4:6 4 21 1 4 1 7 20

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

= − + − + = =

q) 2 5 3 4 5 3: 4 12 2 5 3 2 5 3 12

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

= + − + + =

r) 2 1: 2 7 2 5 2 1: 2 7 1 5 2 1: 12 14 3 10

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

= + = + = + = + =

s) 2 4 2 3 7 4: 2 4 1 3 49 40 112 35 3 49 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

(3)

t) 3 1 4: 4 2 15 1 3 4: 4 30 1 3 2: 4 5 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

 

= − _ − + _= − = − = − = =

 

u) 2 1 3 1 2 1 9 2 2 1 7 2 12 7

3 4 6 3 12 12 3 12 3 12 12

         

+ −_ _ − __= + −_ _ − __= + −_ _= +_ − _=

       

   

2 5 8 5 13

3 12 12 12 12

= + = + =

v) 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

= − − + = − − + = − − + = − = −

w) 2 5 3 7 2 1 2 5 6 7 8 5

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

   

= − −_ − _= − −_ − _= − − = − − =

   

x) 2 4 1 2 1 4 2 1 2 4 5 4 1 4 6 1

3 2 5 3 3 5 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

   

= −_ − − _− − = −_ − − _− − =

   

3 10 1 1 10 1 60 3 100 6 49

2 2

30 3 5 10 3 5 30 30 30 30 30

= − − − = − − − = − − − = −

y) 4 1 2 5 2 7 12 1 2 15 8 7

3 9 4 3 2 9 9 12 12 2

−   −  

 ₋ _{+ − − +}  _{− =} ₋ _{+ − −} ₊  _{− =}

         

         

12 1 15 8 7 13 7 7

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

(4)

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

= ⋅ = ⋅ = =

α) 3 4 1 2 5 7 1 3 8 1 8 5 28 1

8 2 4 2 8 8 2 2 4 4 8 8

_{− +} ₋  ₋  ₋  ₊ ₋  _{= − +} ₋  ₋  ₋  ₊ ₋ ₌

           

 _ _ _ _{ } _  _ _ _ _{ } _

       

3 7 3 27 3 28 6 27 25 33 8

1

8 2 4 8 8 8 8 8 8 8 8

       

= − +_ _{ }− + _{ }= − + _{ }− + _= − = − = −

       

β) 1 4 1 1 3 1 2 5 5 12 1 3 3 7 5

3 5 3 3 15 15 3 3 3

 +   

 ₋  _⋅ _{− ⋅ −} ₌ ₋  _⋅ ₋ _{⋅ −} ₌

         

         

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

 

= − ⋅ −_ _=

 

γ) 4: 12 1 2 3 3 1: 1 2 4: 3 1 4 3 3 1: 5 2

5 16 6 3 8 6 5 5 4 6 6 8 6 5 5

  ₊ ₋ ₋   ₋ ₌   ₊ ₋ ₋   ₋ ₌

       

 _ _   _ _  _ _   _ _

       

4 3 5 3 1 3 4 15 3 5 4 5 3 15

: 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

(5)

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 b) (–2)3 = −8 c) −22 = −4 d) (–3)2 = 9

e) −2−3 = 1 8

− f) (–2)−2 =|1

4 g) (–2) −3

= 1 8

− h) –32 = −9

i) (–1)−7 = −1 j) 2 1 2 −    

  = 4 k)

4 1 2 − −    

  = 16 l) (−4) 2 = 16 m) 4 1 3 − −    

  = 81 n) 0 4 5    

  = 1 ñ) 1 −37

= −1 o) –52 = −25

p) (–1)523 = −1 q) 10 = 1 r) 2350 = 1 s) (–1)0 = 1 t) (0,75)0 = 1

3. Expresar como una única potencia de base entera o racional:

a)

2 1 3

2 2 2

:

5 5 5

−

    ₌       

      b)

3 5 2

2

1 1 1

: 2

2 2 2

−

    ₌  ₌            

c)

5 7 2

4

2 2

3 3 3

3 3 3 − − − ⋅ ₌ ₌ d)

2 3 1

2 2 2 5

:

5 5 5 2

−

    ₌  ₌            

e)

(

22⋅2−3

) ( )

−4 = 2−1 −4 =24 f)

4 2 4 4

6

2 6

2 4 2 2

2 8 2 − − − ⋅ ₌ ⋅ ₌ g)

3 4 4

3 3 3 3

2 2 2 2

₋  _{⋅ −}  _{= −}  ₌         

        h)

( )

3 2 ₅ 2 3 5 ₅ 5 125 =5 =

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

j) 2 ₃ 2₆ 2 5 ( 4) 2

−

= = −

− −

k)

2 4 2 4 2

2 3 3 3 3

3 2 2 2 2

−

  _{⋅ −}  ₌   _⋅ ₌                      l)

2

3 2 6 2 8

5 5 5 5 5

: :

3 3 3 3 3

− − − − _{ }  _{ } _{   } _{ } = = _{ }  _{ } _{   } _{ }              

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

a) 2

2

3 9

1 ( 3) 9

− ₌ − _{= −}

−

b)

5 2 2 5 4 2 4

4 4

2 4 3 2 2 3 3 81

2 3

− −

−

(6)

c)

3 2 ₂

3 2 3

1 1 1 1 4 16

: :

3 4 3 4 3 27

    = = =    

   

d)

2 2 2 4 3 4

4

3 2 3 4 3 3 4 4

3 ( 3) 4 3 3 2 3 2 2 2

2 3

6 9 (2 3) 3 2 3 3 3 81

−

⋅ − ⋅ ₌ ⋅ ⋅ ₌ ⋅ _{= ⋅} ₌ ₌

⋅ ⋅ ⋅ ⋅ ⋅

e)

3 2 2 3 2 4 2 7 3

3 5

3 2 3 4 3 3 4 5

2 ( 3) 4 2 3 2 3 2 2 8

2 3

6 9 (2 3) 3 2 3 3 3 243

−

⋅ − ⋅ ₌ ⋅ ⋅ ₌ ⋅ _{= ⋅} ₌ ₌

⋅ ⋅ ⋅ ⋅ ⋅

f)

4 2 1 4 4 2 0 1 2

2 5

5 2 5 3 2 2 2 4 5

2 4 3 9 2 2 3 3 2 3 2 4

2 3

2 8 9 3 2 2 3 3 2 3 3 243

− − − − − − − − − ⋅ ⋅ ⋅ ₌ ⋅ ⋅ ⋅ ₌ ⋅ _{= ⋅} ₌ ₌ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ g)

5 4 0 1

2 2 2 2 2

3 3 3 3 3

−      _⋅ _⋅ ₌  ₌                 h) 2

3 2 6 2 8 ₈

8

5 5 5 5 5 5 390625

: :

3 3 3 3 3 3 6561

− − − − _{ }  _{ } _{   } _{ } = = = =  _{ }   _{ }    _{   }  _{ }     i) 2 2

3 3 6 6

6

1 1 1 1 1 1

1

2 2 2 2 2 64

_ _  _ _  _ _ _{ } − = − = − = = = _ _  _ _  _ _ _{ }                 j)

3 3 4 3 12 15

11

2 6 2 6 4

( 1) 2 8 1 2 2 2

2 2048

2− 2 2− 2 2

− ⋅ ⋅ ₌− ⋅ ⋅ ₌ − _{= −} _{= −}

⋅ ⋅

k)

(

24 2 5

)

: 23 2 4 1₄ 1

2 16

− −

⋅ = = =

l)

5 5 5 5

1 1 1 1

1 2 1 4 3 1

6 3 6 6 6 2

− − − − − − − − _ _  _ _  _ _  _ _  − = − = − = − = _ _  _ _  _ _  _ _                          5 5

1 1 1

2 2 32

 

= −_ _ = − = −  

m)

4 5 1

4

2 3 4

3 3 3 1 1

3

3 3 3 3 81

− −

−

⋅ ₌ ₌ ₌ ₌

⋅

n)

( )

2

2 2 4 4 4 2 2

2

2 2 2 2 4 2 2

2 2 2 2

25 5 5 5 2 5 7

20 7 2 5 7 2 5 7

14 2 7 2 7 2 7

  ⋅ ⋅ ⋅

 _{ ⋅ ⋅ =} _⋅ _⋅ _{⋅ =} _{⋅ ⋅ ⋅ =} ₌

 

  _⋅ _⋅ _⋅

   

6 2 0

5 2 7 62500

= ⋅ ⋅ =

o)

(

)

(

)

1

2 4 ₂ ₄ ₄

2 4

2 8 4 2

4 2

3 2 _{3 2} ₂ ₁₆

3 2

2 3 3 9

2 3 − − ₋ − − − ⋅ _⋅ = = ⋅ = = ⋅ − ⋅

p)

(

)

(

)

3

2 2 2 6 6 6 ₀ ₆

4

2 4 6 4 6 4

2 3

3 7 3 3 7 3 _{3 7} ₁ ₁

3

3 7 3 7 3 81

( 3) 7

(7)

q)

2 1 2 1 2 1

3 3 1 7 6 3 3 7 3 4

2 4 3 9 4 4 9 9 4 9

− − − − − −

 ₋  _⋅ ₋  ₌ ₋  _⋅ ₋  ₌  _{⋅ −}  ₌

           

           

2 1

4 9 16 9

4

3 4 9 4

     

=_{  }⋅ − _ = ⋅ −_ _= −

     

r)

4

2 3 ₄

4

2 0

1

2 2 ₂

5 5 ₅ ₂ ₁₆

5 625 2

2 2

:

5

5 5

− −

_{   }  _{ }

⋅

   _{   }  _{ }

   

  ₌  ₌ ₌

   

_{   }  

     _{   } _{ }

 

 

s)

3

2 ₉

5

2 4

5 3

2 2 ₂

3 3 ₃ ₂ ₃₂

3 243 2

2 2

:

3

3 3

 _{ }  _{ }

⋅

  _{ }  _{ }

   

  ₌  ₌ ₌

   

_{   }   

     _{   }  _{ }

 

 

5. Simplifica:

a)

2 2

1 3 3 3 9 3 6 3

1

2 4 2 4 4 4 4 2

 ₊  _{− =}  _{− = − = =}

   

   

b)

2 2 2

1 1 2 1 1 1 4 1 5

4 1 4 1 4 1 4 1 1 1

2 4 4 4 4 16 16 4 4

 ₋  _{+ =}  ₋  _{+ =}   _{+ =} _{+ =} _{+ = + =}

     

     

c)

2 2 2

4 1 1 8 1 1 64 1 1 64 1 256 45 301

4

3 5 2 3 5 4 9 5 4 45 4 180 180 180

−

 ₋  _{⋅ +}  ₌  _{⋅ + =} _{⋅ + =} _{+ =} ₊ ₌

     

     

d)

2 3 2

3

1 1 1 3 3 9 9 32 23

: 2 8

2 3 2 2 2 4 4 4 4

  ₋ ₊  ₌  _{− = − = −} _{= −}

     

     

e)

2 2 2 2 2 2 2 2

1 1 1 1 2 1 1 2 1 1 1 1

: : : : 1

3 6 6 3 6 6 6 6 6 6 6 6

 ₋   ₋  ₌ ₋   ₋  ₌  ₋  ₌    ₌

               

               

f)

2 2

1 1 1 2 ₁ ₄ ₉ ₂₀ ₂₉

1

1044 116

5 3 5 3 ₅ ₉ ₄₅ ₄₅ ₄₅

25 72 25 97 4365 485

1 5 ₂

2 1 2

36 36 36 36

6 6

   

+ −_ _ +_{ } ₊ ₊

  ₌   ₌ ₌ ₌ ₌ ₌

    ₊ ₊

+_ − _ + −_ _