1. Educación

MATEMÁTICAS 4º ESO
TIMONMATE
EJERCICIOS RESUELTOS DE POTENCIAS Y RADICALES
POTENCIAS Y RADICALES
Notas teóricas
-
Operaciones con potencias:
I.
am
a : a = n = a m −n
a
II.
(am ) = am⋅n
III.
ap ⋅ bp = (a ⋅ b)
IV.
(a p ⋅ b q )
m
n
n
VII.
a−1 =
1
a
VIII.
a− b =
1
ab
−1
a
1 b
  = =

a a
 b
b
p
m
IX.
= a p⋅m ⋅ bq⋅m
X.
-
V.
a0 = 1
VI.
a1 = a
n
−n
 
1
 b 
=
n =
 a 
 a 
 
 b 
p
q
 
 a 
 b 
Operaciones con radicales:
a =a
XI.
XII.
n
m
1
2
a =a
XIV.
n
m
n
p
q
a ⋅ a = a ⋅a =
=
m
n
m
nq
amq +np =
nq
amq ⋅ a np
1
XIII.
-
p
 p 1  n
n m p
mn

m
a = (a )  = a


Racionalizar:
Racionalizar es quitar del denominador las raíces. Se pueden presentar
dos casos:
a) En el denominador hay sólo una raíz. en este caso, la raíz se elimina
multiplicando el numerador y el denominador el mismo número de
veces que el radical de la raíz.
1/6
Potencias y radicales resueltos
TIMONMATE
b) En el denominador hay una raíz y otro término que la suma o la resta.
En este caso, las raíz o raíces se eliminan multiplicando el numerador
y el denominador por el conjugado del denominador.
-
La jerarquía que hay que seguir a la hora de operar con radicales :
Simplificar
Operar dentro del paréntesis
Cálculo de potencias y raíces
Multiplicaciones y divisiones de izquierda a derecha
Sumas y restas en el orden que aparecen
Ejercicios resueltos
Opera con las siguientes potencias y raíces
−2
3
1.
16−2 ⋅ 4 3 = ( 2 4 ) ⋅ ( 2 2 ) = 2−8 ⋅ 2 6 = 2−8+6 = 2−2 =
2.
(7 2 )
3.
−3
⋅ 7 3 = 7 2⋅(−3) ⋅ 7 3 = 7 −6 ⋅ 7 3 = 7 −6+3 = 7 −3 =
1
73
1
(3−2 : 33 )⋅ 3−2 = 3−2−3 ⋅ 3−2 = 3−5 ⋅ 3−2 = 3−5+(−2 ) = 3−5−2 = 3−7 = 37
2
4.
1
4
3
2
2
2
4 2 ⋅ 12 3 ⋅ 152 (2 ) ⋅ (2 ⋅ 3) ⋅ (3 ⋅ 5)
2 4 ⋅ 2 6 ⋅ 3 3 ⋅ 32 ⋅ 52 2 10 ⋅ 3 5 ⋅ 52
=
=
=
=
3
2
9 3 ⋅ 8 2 ⋅ 33
36 ⋅ 2 6 ⋅ 33
2 6 ⋅ 39
(3 2 ) ⋅ (2 3 ) ⋅ 33
= 2 4 ⋅ 3−4 ⋅ 52
4
5.
2
3
−3
3
2
2
8 4 ⋅ 153 ⋅ 182 ⋅ 12−3 (2 ) ⋅ (3 ⋅ 5) ⋅ (2 ⋅ 3 ) ⋅ ( 2 ⋅ 3)
=
3
2
20 3 ⋅ 27 2 ⋅ 3−3
(2 2 ⋅ 5) ⋅(33 ) ⋅ 3−3
2/6
=
TIMONMATE
Potencias y radicales resueltos
=
2 12 ⋅ 3 3 ⋅ 5 3 ⋅ 2 2 ⋅ 3 4 ⋅ 2−6 ⋅ 3−3
2 8 ⋅ 3 4 ⋅ 53
=
= 2 2 ⋅ 3 = 12
6
3
6
−3
6
3
3
2 ⋅5 ⋅3 ⋅3
2 ⋅3 ⋅5
−1
 23 
−1
3
27 ⋅ 81 ⋅ 3 ⋅   ⋅ 2 3
(3 3 ) ⋅ 3 4 ⋅ 3 4 ⋅ 2 3 ⋅ 2 3 36
 3 
=
= 6 =1
−2
2
33
3
 1  4 27 0 −2
2
2
2 2
3 ⋅2 ⋅3 ⋅ ⋅ 4 ⋅1
36 ⋅   ⋅ ⋅ ⋅ (2 )
3 2
 3  3 16
−1
6.
4
3
7.
−6
5
(−27 ) ⋅ 32−5 ⋅(−8) ⋅ (252 )
4
4
(−72) ⋅(−50 3 )
=
1
3
3 1
+
5
3⋅5 1⋅2
+
10
3
15
= 2 10
5
9.
−6
5
=
39 ⋅ 2−25 ⋅ 2 15 ⋅ 5−24
3
= 34 48
8
12
24
12
3 ⋅2 ⋅5 ⋅2
2 ⋅5
22 ⋅25 = 22
8.
−5
3
(33 ) ⋅ (2 5 ) ⋅ ( 2 3 ) ⋅ ( 54 )
=
4
3 4
(32 ⋅ 2 3 ) ⋅ (52 ⋅ 2) 
= 2 10
3
+
2
10
= 19
15+ 2
10
17
= 2 10 = 10 2 17
5⋅4 3⋅3
−
12
5 3
−
4
19 5 : 4 19 3 = 19 3 : 19 4 = 19 3
11
12
=2
= 19 12
20
= 19 12
−
9
12
= 19
20−9
12
=
= 12 19 11
1
55 ⋅ 5 2
55 ⋅ 5
=
= 55−(−3) = 55+3 = 58
−3
−
3
5⋅5
5 ⋅5
10.
1
5
11.
3
2 ⋅2 ⋅2
2 3—2
1
12.
22 ⋅ 2
−
−
1
2
25
125
1
3
22 ⋅ 2
⋅ 22
1
2
1
5
=
3
2 ⋅ 2 ⋅2
23— 2
=2
−
1
3
2
1
2
=2
1
5
1
=
−
1
3
−
1
2
1
=
2
=
1
2
=
1
2
1
3
2
3
4 3
(33 )
27
3
39
3
3
12
13. 3
=
= 12
=
= 12 4 = 12
4
4
8
2
3
2
2 ⋅3
2
16
18
2⋅ 3
(2 ⋅ 3 )
4
14.
4
−80 : 3 18 =
−4 24 ⋅ 5
3
2 ⋅ 32
=−
24 5
3
2 ⋅ 32
2 4 53
=
4
3/6
4
(2 ⋅ 32 )
= 2⋅ 4
53
=
2 4 ⋅ 38
Potencias y radicales resueltos
TIMONMATE
4
2
75
4 3
⋅
5
=
2
2 ⋅3
9
=
3
3

1  
 = −
15. 15 −

243  
15
3
 1 
1 
15 

=
−
 5  = −
3 5 
3
5
1
1
1
=− 3 =−
15
3
3
27
2
16.
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
17.
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
18.
3 4
19.
3 2
=
8
64 4 =
2
2
4
21.
3
3
( )
34
= 15
12
2
4
4 5
3 24
=
3 ⋅ 32
=3
32
15
34 ⋅ 3 32
5
15
325
9 5 3  ⋅3




=
312
2
=
 4 1
(3 ) 4 


3
5
( 3 4 ) ⋅ (3 2 )
15
=
32
15 22
312 ⋅ 310
3
3 22
15
=
=
=
32
32
330


3


2
3 9
1
4
14

⋅ (325 )5 


3
4
( 2)
4


 2

1
1 4 2
4






=
1 15
1 9
5



2 ) ⋅2
(
(2 ) ⋅ 2 2
23.
=
=
9
=
1
38
( 3)⋅
22.
4
3 4 ⋅ 3 36
2
( 3 ) ⋅( 3 )
5
4
=
6
4
32 ⋅ 2
1 4 32 ⋅ 2 1 4 9
=
=
22
2 22
2 2
6
3
12
= 2⋅3⋅4 2 24 = 24 2 24 = 2
32 ⋅ 2
1
=
2
2⋅2
2
( 3 ) ⋅( 3 )
20.
( 3)
4
4
(2 6 )
3 4
6
9
2


 ⋅3


⋅2
1
2
=
3
1
4⋅ ⋅4
4
3
2
2
+1
3
2
1
2
4/6
=
2
2
⋅3
11
⋅ ⋅15
59
11
25⋅ ⋅
54
=
⋅3
5
3
1
2
5
=2
5 1
−
3 2
34 ⋅ 3 4
5
4
= 35
3 ⋅3
=2
10−3
6
7
6
= 2 = 6 27 = 2 6 2
TIMONMATE
Potencias y radicales resueltos
4
4
( 5)⋅
24.
2
4
4 5
520
15
 3 5 5  ⋅ 25




=
1
1 4

⋅ (520 )5 


15
 1 1
 5 3 
5   ⋅ 52
  


3
a 3 −2 b 3
2a
b
a
25.
 2 1 
(5 ) 4 


3
=
2 ab2
a
2a−2  
 b
52 ⋅ 5
=
=1
5 ⋅ 52
2
b3
a
4ab 2
3
=
 −2  a 3  b 3
 2a    ⋅

 b   a


4ab 2
=
2
12
=
26.
27.
 −2  a 3  b3
4a
 2a    ⋅
12
 b   a

3
1 12 4a
1 12 4


= b =
=
6 3
2
2b a 5 b 3
2b a 2b a b
4ab
8 − 50 −
1
1 2
98 = 2 2 ⋅ 2 − 2 ⋅ 52 −
7 ⋅ 2 = 2 2 − 5 2 − 7 2 = −10 2
2
2
1
3
1
3 2
1
3⋅ 5
3 − 12 −
75 =
3 − 22 ⋅ 3 −
5 ⋅3 =
3 −2 3 −
3=
2
4
2
4
2
4
1
15
21
=
3 −2 3 −
3 =−
3
2
4
4
21
28.
= 3 xy −
29.
21
6
(xy)
(xy)
xy
xy
1
9xy +
+ 3 3 = 3 xy −
−6
= 3 xy −
6
xy
2
4xy
2 xy
(x 3 y 3 )
1
3
xy − xy =
xy
2
2
1 4 81y 2
3
− x 225 y = x ⋅ 2 8 y + x ⋅ 4 y 2 − x ⋅ 32 ⋅ 52 y =
−4
3
3 x
= 16x ⋅ y + x ⋅ y − 15x ⋅ y = 2x ⋅ y
256x 2 y +
Racionaliza
30.
1
2⋅ 3 5
=
1
3
2⋅ 3 5
3
5
5
3
3
5 3 25 3 25
=
=
2⋅5
10
5
5/6
2
(xy)
xy
3
− 6 (xy) =
Potencias y radicales resueltos
TIMONMATE
4
4
1  5 x 4 

=
 =
x 4 5 x 4  5 x 4 
1
31.
5
3
32.
6
5
4
(x)
(x)
5
4
5
4
5
=
x4
5
( )
( )
3
x 6 x5
x
x  6 x 5 

=
 =
6
5
6 5 

6 5
x
x  6 x 5 
x
3
 4 1 
(x )5 


16
5 16
5 15
x5
x
x ⋅ x x3 ⋅ 5 x 5 x
= 4 = 4 =
=
=
x
x
x4
x4
x
1
=
1
x 3 (x 5 )6
x5
×5
1
25
2+25
27
6 27
6 3
x3 ⋅ x 6
x 6
x6
x
x
=
=
=
=
=
5
5
5
5
x
x
x
x
x
33.
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
34.
2+ 3
=
2− 3
2
2
( 2 + 3) = ( 2 + 3) = − 2 + 3 2
2+ 3 2+ 3
⋅
=
(
)
2
2
2−3
2 − 3 2 + 3 ( 2 ) −( 3 )
2
2
2 3 + 2 (2 3 + 2 )⋅ (2 3 + 2 ) (2 3 ) + 2 ⋅ 2 3 + ( 2 )
4⋅ 3 + 4 3 + 2
35.
=
=
=
=
2
2
4
⋅
3
−
2
2 3 − 2 (2 3 − 2 )⋅ (2 3 + 2 )
2
3
−
2
( ) ( )
=
7+2 6
5
***
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