Fracciones,Potencias,N cientifica

Unidad :

Conjuntos Numéricos

Objetivo : i) Aplicar propiedades de las potencias
en IR.

18) Recuerde que a n = a·a·a·....·a (n veces a) ;

una

potencia es el producto de un número “a” por si mismo
base
“n” veces; donde “a” se llama __________ ; “n” es el
exponente
______________ y el producto a obtener es la
potencia
______________.
Calcular:
a) 26 = 2·2·2·2·2·2 = 64

2−3 ) = (-3)·(-3) = 9
b) (

c) 53 = 5·5·5 = 125

d) (−4)3 = (-4)·(-4)·(-4) = -64

e) 44 = 4·4·4·4 = 256

f) (−2)4 = (-2)·(-2)·(-2)·(-2) = 16

g) 35 = 3·3·3·3·3 = 243

h) (−6)3 = (-6)·(-6)·(-6) = -216

Si el exponente es par, la potencia es siempre de signo
positivo
______________.
Si el exponente es impar, la potencia conserva el signo
base
de la ________ .
19) Aplicar elteorema a m ⋅ a n = a m+n en:
a) 33 ⋅ 32 = 33 + 2 = 35 = 243
3 ⋅ (−2)2 ⋅ (−2)1= (−2)3 +2 +1 = (−2)6 = 64
b) (−2)
2 ⋅ (−5)1 (−5)2+1 = (-5)3 = -125
=
c) (−5)
d) x 4 ⋅ x 5 = x 4 +5 = x 9
e) 5 x2 ⋅ 4 x 3 = 20 x 2+ 3 = 20 x 5
f) −3 x 2 y⋅ 2 xy 2 = −6 x 2+1 y1+ 2 = −6 x 3 y 3

20) Aplicar el teorema a m : a n = a m− n en:
a) 58 : 55 = 58 −5 = 53 = 125
b) (−7)5 : (−7)2 = (−7)5 −2 = (−7)3 = -3431
6
c) (−2)7 : (−2) = (−2)7 −1 = (−2) = 64

12−7 = x5
d) x12 : x7 = x

e) 18 x15 : 3 x 9 = 6 x15 − 9 = 6 x 6
f) −45 x5 y 3 : 5 xy 2 = −9 x5 −1 y3 −2 = −9 x 4 y

( )

m n = a m⋅n en:
21) Aplicar el teorema a

( )

2 3 = 32·3 = 6
a) 3
3 = 729

( )

(

)

3 2 = (−2)3⋅2 = (−2)6 = 64
b) (−2)

4 5 = 4⋅5
c) x
x
= x 20

(

)

3 )4 5 = a 3·4·5 = 60
d) ((a)
a

( )8 1/4 =
e) (6)

1
8⋅
6 4 = 62 = 36

(

)

6 1/2 =
f) (−5)

1
6⋅
2 = (−5)3 = -125
(−5)

22) Aplicar el teorema (a⋅ b)n = a n ⋅ bn en:
12
5 a ) = 52 ⋅ a 2 = 25a2
a) (
13

b) ( −7 x ) = (−7)3 ⋅ x 3 = −343 x3

(

)

2b 3 = 33 ⋅ (a 2 )3 ⋅ b3 = 27a 6b3
c) 3 a 1

(

)

3y 2z1 4 = (−2)4 ⋅ (x 3 )4 ⋅ (y 2 )4 ⋅ z 4 =16x12 y8 z 4
d) -2x
2b3 c5 3 = (−5)3 (a 2 )3 (b3)3 (c5 )3 = −125a 6b 9c15
e) -5a

(

(

)

)

3b5c 2d 4 2 = (8)2(a 3 )2(b5 )2 (c2 )2 (d 4 )2 =
f) 8 a
= 64a 6b10c 4d 8

23) Aplicar el teorema a n ⋅ bn = (a ⋅ b)n en:
a) a 4 ⋅ b4 = (ab)4
b) 23 ⋅ 33 = (2·3)3 = 63 = 216
2
2
c) ( −5 ) ⋅ ( −3 ) = (−5·−3)2 = 152 = 225

( )
2b 2 2ab 2 2 = (8 a 3 b3 )2 = 64a 6b6
e) 4a
( )( )
2y 2 2 xy 2z 2 -2 yz2 2 =
f) -2x
(8 x 3 y4 z3 )2 =( )( )( )
2 3 ⋅ 2 x 3 = (6 x 3 )3 = 216 x 9
d) 3 x
( )

= 64x 6 y8 z6

n an
a
24) Aplicar teorema   =
en:
b
bn
7 2 72 49

a)   = 2 =
25
 2
5

5 (−2)5 −32
 2
b)  −  =
=
243
 3
35
3
 3a 2b 
(3a 2b)3
27a 6b3
c) 
=
 =
 4c3 
(4c3 )3
64c9


4
 4 x 3y 2 
(-4 x 3 y2 )4 256x12y 8
d)  =
 =
4 )4
 5 z4 
(5 z
625 z16

25) Aplicar el teorema a n : bn = (a : b)n en:
a) a 7 : b7 = (a : b)7
b)154 : 54 = (15 : 5)4 = 34 = 81
c) (−18)3 : (−3)3 = (−18 : −3)3 = 63 = 216
d) (36a 5 )3 : (9a 2 )3 = (4a 3 )3 = 64a9
2 2
e) (-45 x 6 y2 )2 : (5 x 4 y)2 = (-9x y) = 81x 4 y 2

f) (108a 8b6 c3 )2 : (12a 5b4 c2 )2 =(9 a 3b 2c)2 = 81a 6b4c2

1
26) Aplicar el teorema a - n =
an

a - n  b n
o   =   en:
b
 a 

1
1
a) 5−3 =
=
53 125
3 -2
b) (−8 a ) =

1
1
=
(-8 a 3 )2 64a6

3
53
125
8  −3  5 
c)  − 
=−
= −  = −
 5
64
 8


83
2c3 -3
 5b
(4a 4 )3
64a12
d) 
=
 =
 4 a4 
(5 b2c3 )3 125b6c9



27) Aplicar el teorema a 0 = 1 en:
a) 70 + (−5)0 = 1 + 1 = 2
b) 20 − (30 − [40 + 30 ]) = 1 − (1 − [1 + 1]) =
= 1 – (1 – 2)) = 1 – (-1) = 1 + 1 = 2
c) 7x 0 + 8 y0 − 5 z0 = 7·1 + 8·1 – 5·1
= 7 + 8 - 5 = 10
d) 35a 3b2 : 5 a 3b2 = 7 a 0b0
= 7·1·1 = 7

28) ¿Cuál es el número cuyo cuadrado es igual a 22 ⋅ 52 ?

A) 10
B) 25
C) 50
D) 100
E) 425

22 ⋅ 52 = 4 · 25 = 100
El número cuyo cuadrado
es igual a 100 es 10

29) Si P = 5 con Q = P 3 - 25 ; R = Q2 - 1 ; entonces el
valor de R es:
A) 99
B) 999

3
Si P = 5 ⇒ Q = P - 25...