Bachiller

MATEMÁTICAS 4º ESO EJERCICIOS RESUELTOS DE FRACCIONES, POTENCIAS Y LOGARITMOS

1. Operaciones con fracciones:

 8  24   1 −3 − − − 50 ⋅ 1 −  −3 − −4 − 50 ⋅       2  7 3 6 25   25  −3 −[−6 ]   = = a) = = 3 : −  = −  1 7 1   1  2 7   −4 + − −4 − −  −4 −  − 1   2  2    2 2

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

1 10 − 1 9 9 9 10 = 1− 10 c) 1− = 1− 10 = 1− 10 = 1− 10 = 1 1 1 10 9 − 10 1− 1− 1− 1− 1 10 − 1 9 9 9 1− 10 10 10 1− 9 9 1 81 10 + 81 91 = 1− 10 = 1+ : = 1+ = = −1 10 9 10 10 10 9

2 1 3 + 3− 3 4⋅ 2 3 1 1 − 2+ 5 = d) 4 2 1 5⋅ 2

11 3 12 ⋅ 2 1 11 11 ⋅ 4 3 ⋅ 5 ⋅ 4 5 = 12 2 ⋅ 11 = 11 ⋅ 4 ⋅ 3 ⋅ 5 ⋅ 2 = 1 5 512 ⋅ 2 ⋅ 11 ⋅ 5 2 2

2. Operaciones con potencias:

a) 16−2 ⋅ 4 3 = ( 2 4 ) ⋅ ( 2 2 ) = 2−8 ⋅ 2 6 = 2−8+6 = 2−2 =
2 3 2

−2

3

1 4

2 2 4 2 ⋅ 12 3 ⋅ 152 (2 ) ⋅ (2 ⋅ 3) ⋅ (3 ⋅ 5) 2 4 ⋅ 2 6 ⋅ 33 ⋅ 32 ⋅ 5 2 2 10 ⋅ 35 ⋅ 52 b) = = = = 2 4 ⋅ 3−4 ⋅ 5 2 3 2 3 6 6 3 6 9 2 3 3 2 3 9 ⋅8 ⋅3 3 ⋅2 ⋅3 2 ⋅3 (3 ) ⋅ (2 ) ⋅ 3 3 2 2 8 4 ⋅ 153 ⋅ 182 ⋅ 12−3 (2 ) ⋅ (3 ⋅ 5) ⋅ (2 ⋅ 3 ) ⋅ ( 2 ⋅ 3) c) = 3 2 203 ⋅ 27 2 ⋅ 3−3 (2 2 ⋅ 5) ⋅(33 ) ⋅ 3−3 4 3 2
−3

=

Juan Pascual

1/7

Matemáticas 4ª ESO

Ejercicios Resueltos de Fracciones, Potencias y Logaritmos

=

2 12 ⋅ 3 3 ⋅ 5 3 ⋅ 2 2 ⋅ 3 4 ⋅ 2−6 ⋅ 3−3 2 8 ⋅ 3 4 ⋅ 53 = 6 3 3 = 2 2 ⋅ 3 = 12 2 6 ⋅ 5 3 ⋅ 36 ⋅ 3−3 2 ⋅3 ⋅5
−1 −1 4

d)

 23  27 ⋅ 81 ⋅ 3 ⋅   ⋅ 2 3   3  
−2  1  4 27 36 ⋅   ⋅ ⋅ ⋅ (2 0 )   3  3 16   −2

33 ⋅2 36 23 = = 6 =1 2 2 33 3 32 ⋅ 2 2 ⋅ 32 ⋅ ⋅ 4 ⋅ 1 3 2

(33 )

−1

⋅ 3 4 ⋅ 34 ⋅

3. Haz las siguientes operaciones con radicales:

4

a)

4 3 (33 ) 27 3 39 3 3 = = 12 = 12 4 8 = 12 4 = 12 3 2 3 2 4 2 ⋅3 2 16 18 2⋅ 3 (2 ⋅ 3 )

3

b)

4

−80 : 18 =

3

−4 24 ⋅ 5
3

2 ⋅ 32

=−

24 5
3

2 ⋅ 32

=
4

2 4 53

(2 ⋅ 32 )

4

53 = 2⋅ 4 8 = 2 ⋅3
4

=

42 75 ⋅ 4 53 = 2 2 ⋅3 9
3 3  1  1 15   =−  5 =− 3     35   3

   1    = − c) 15 −      243     

15

5

1 1 1 =− 3 =− 15 3 3 27
2

d) e) f)

3

2 ⋅ 3 16 = 6 2 ⋅ 3 16 = 6 2 ⋅ 16 2 = 6 2 ⋅ (2 4 ) = 6 2 9 = 6 2 6 ⋅ 2 3 = 2 ⋅ 6 2 3 = 2 ⋅ 2 2 ⋅ 3 16 = 6 2 ⋅ 3 16 = 6 2 ⋅ 16 2 = 6 2 ⋅ (2 4 ) = 6 2 9 = 6 2 6 ⋅ 2 3 = 2 ⋅ 6 2 3 = 2 ⋅ 2 3 2 = 8
4
2

332 ⋅ 2 1 = 2 2⋅2 2
3 6 4

32 ⋅ 2 1 32 ⋅ 2 1 9 = 4 2 = 4 22 2 2 2 2

g)

( ) ⋅( 3 ) ( 3)
32
12 4 6

2

=

3 4 ⋅ 3 36
12

3 24

=

3 ⋅ 32 =3 32

h)

( 3 ) ⋅( 3 )
5 3

4

2 5

( )
34
3 4

3

=

3 4 ⋅ 3 32 312

15

=

(3 4 ) ⋅ (32 )
32

3

5

15

=

15 22 312 ⋅ 310 3 322 1 = 2 = 15 30 = 15 8 2 3 3 3 3

i) j)

64 4 =

3 4

(2 6 )

4= 2⋅3⋅4 2 24 = 24 2 24 = 2

8 − 50 −

1 1 2 98 = 2 2 ⋅ 2 − 2 ⋅ 52 − 7 ⋅ 2 = 2 2 − 5 2 − 7 2 = −10 2 2 2

2/7

Juan Pascual

Ejercicios Resueltos de Fracciones, Potencias y Radicales

Matemáticas 4º ESO

k)

1 3 1 3 2 1 3⋅ 5 3 − 12 − 3 − 22 ⋅ 3 − 75 = 5 ⋅3 = 3 −2 3 − 3= 2 4 2 4 2 4 1 15 21 = 3 −2 3 − 3 =− 3 2 4 4
6 (xy) (xy) xy xy 1 9xy + + 3 3 = 3 xy − −6 = 3 xy − 3 3 6 xy 2 4xy2 xy (x y ) 21 21

l)

(xy)
xy

2

− 6 (xy) =

3

= 3 xy −

1 3 xy − xy = xy 2 2

4. Racionaliza:

a)

1 2⋅ 3 5

=

1 2⋅ 3 5

3 3

5 5

3 3

5 3 25 3 25 = = 2⋅5 10 5

 1 1  5 x4    =  b) =  5 4 5 4 5 4  x x  x   

4

(x) (x)
5 4 5 4

4

5

=

 4 1  (x )5       x4
5

4 5 16 5 15 x x x ⋅ x x3 ⋅ 5 x 5 x = 4 = 4 = = = x x x4 x4x 16 5

3

c)

6

3   x 6 x5 x x  6 x5   =  =   6  6 6  x5 x5  6 x5   x5 3

5

( ) ( )

=

x 3 ( x 5 )6 x5

1

1

×5

x ⋅x = x5

1 3

25 6

=

x

2+25 6

x5

6 6 x x 27 x3 = 5 = 5 = x x x

27 6

d)

2 ⋅ ( 3 − 1) 2 ⋅ ( 3 − 1) 2 ⋅ ( 3 − 1) 2 ⋅ ( 3 − 1) 2 = = = = 2 3−1 2 3 + 1 ( 3 + 1)⋅ ( 3 − 1) ( 3 ) − 12 2+ 3 = 2− 3
2+ 3 2+ 3 ⋅ = 2 2...