ALGEBRA PREUNIVERSITARIA

1. 2020 . .
POTENCIAS, RADICALES Y ECUACIONES
EXPONENCIALES
Ing. WIDMAR AGUILAR, Msc
Noviembre 2020

2. 2020 . .
1)
= 1 + 2 + 6 1 + 6
= 1 + 4 + 6 − 6 + 5
$ = %3 + 2'( = 9 + 16 /
$ = 25
+
, = 5
+ $= 5+5=10
2)
=
,-..,.+ ,/.,.-
. 00
1 0-.0,.0.-

3. 2020 . .
=
,-..,.+ ,/.,.-
. ,2
[ 1 0,
]
=
,-.,/.,.+.,.-
. ,2
2,
=
,0
. ,2
2,
=
1. ,2
2,
=
05
2,
= 36 7
3)
=(
8
6
/
9
.:,5;
.
+
5
=
8
6
/
9
. , .+
=
8
6<
/
9
=
.
+
,
=
8
6<
9
/
=
+
,
= /
6
0,
+
,
,,
+
,
= /
6
0
, = 2'
0
, = 28
=64
4)
=>
8?. 8@. ?. ,@
A?. A@
@
=
. ?. 8@. ?. ,@
A?. A@
+
@

4. 2020 . .
= [
?. ?. ?. ,@. 8@
A?. A@
+
@ =
?. , ?B@ . 8@
?.C?. @.C@
+
@
= (
?.C?.C@. 8@
?.C?. @.C@
+
@ =
8 D
+
@
= 8
5)
E F =(√F
0 6H
+
0.H
,
0
= √F
0 6HI H
.
-
0
si : x= 3
f x L M
= 36 .
0
0
= 36 .+
f x L M
=
+
0 =
.+
+
0
= 36
+
0
6)

5. 2020 . .
= >FD.F
@
?
?
= >FDN
@
?
?
= >F
?@B@
?
?
= F
?@B@
?,
ODND
O,
= 4 ;
Q FON D?,
= √F ODND?,
= F
, ?@B@
?,
= F '
= FA
------------el exponente es 8.
7)
R =
6S
= >
O
O
+
T
U
= [
O
O
+
T
]
+
U = 6V
+
U
=
T.+
T
.
+
U =
+
+.U
.+
+
+.U
.
+
U
=
+.+BU
+.U
+
+.U
.
+
U
= S.
+
U
IO

6. 2020 . .
8)
=
>
√3
0 √
0 UB0√0
0 U
= √3
0 √
0 UB0 +
√0
0 U
= > √3
0 √
0 U
. √
0 0< √0
0
=
U
= > √3
0 √
0 U
.
< √0
0
=
U
=W √3
0 . √
0 U
X
+
< √0
0
=
U
= √3
0
= 3
9)
√Y = F
QF√F = F
0
/
>FQF√F = F
2
1
-----

7. 2020 . .
------
ZF>FQF … … . √F = F
,U.+
,U
: ZF >F … … … … . QF √F
UUUU
= F
,U.+
,U
n=100
= F
,+]].+
,+]]
10)
4H
. 8H
. 8 = 2 H
. 26
. 16 H
. 166
2 H
. 2 H
. 2 = 2 H
. 26
. 2' H
. 2' 6
2 H
. 2 = 26
. 26 H
. 26A
0-
+,-
= 26
. 26A
. 26
26CH
= 26
-9x = -12 : 3x = 4
X =
'

8. 2020 . .
11)
2' ,_
= 2 ',_
2'. ,_
= 2 .',_
; 4.3 `
= 3. 4 `
,_
',_
= <'
=
'
H
=
'
; 2x =1
x = →
b
c
; d = 1 e f = 2
p+q = 1+2 =3
12)
√3
√
-
= 3
√3
-
,
= 3 ; √3
-
, = 3
3
-
, = 3

9. 2020 . .
H
= 2
X = 4
13)
FH
= 6 . 8
FH
= 6 .8,
= 6 8
FH
= 36 8
; F = 36
E = √F +
H
'
= √36 +
8
'
E =6+9 ; E = 15
14)
FH
=
+
9 = =
0
0
.
+
9 =
FH
= [ ]
+
0
.
+
9 =
7
+
0
.
+
9 =
7
+
,2
Por analogía:
X =
7
= 0
= 36

10. 2020 . .
15)
<
H-6H+0
H026H-=
+
-.+0
=
H
H
-
-.+06H
+0
-.+0
H
02
-.+06H
+0
-.+0
=
H
H H
-
-.+06H
+0
-.+0
H
02
-.+06H
+0
-.+0
= 1
H
-
-.+0
B+
6H
+0
-.+0
B+
H
02
-.+06H
+0
-.+0
= 1
F
,-.+0
-.+0 − F
-
-.+0 = F
02
-.+0 − F
+0
-.+0
F
,-.+0
-.+0 = F
02
-.+0
Luego:
H6
H6
=
7
H6
; 2x = 50
X = 25
16)

11. 2020 . .
= -(1)-4(1)+( + <
g
A
=
= -5 +
7
A
+
g
A
=
6'hN 7Ng
A
=
A
A
= 1
17)
= [ ]6[
+
9
][
+
0
]
.[
+
9
][0]
= [ ]6[
+
9
][
+
0
].0
= [ ]
6<
+
9
= 0
= < =
6
= 3 = 27
18)

12. 2020 . .
=
[g/-.0]
[[gj],k
],0.k
'g6H
. 5
5
=
g/-.0./5.-
.g0,
gj ,k.,0.k
=
g/-.0B5.-
.g0,
gj ,kB0.k =
g/,
.g0,
gj.1
=5 8N 6'A
= 5h
= 1
19)
= [(8
,UB+ 5U
+]U]0.+
= [ 8
UB+ 5U
5U.,U]
+
0
= [ 8
UB+ .U
]
+
0
= (8
U. . .U +
0 = 8
+
0 = 64
+
0
= 4
20)
Haciendo: √F
5
= l ; F = lg

13. 2020 . .
P(x) = [ml6 n5
0
5
o
n
]n
= [ l6n0 n,
]
P(x) = l6n5
=
np5 =
H
+
5 -
P(x) =
H
-
5
; q 5 = ?
P(5) =
g
5
5
=
g
= 56
21)
= >FD6 . F
s
?
?
= FD6 N
s
?
+
? = F
@?.?Bs
?,
ODNt6O
O,
= 5 ; ab+c =a+5
ab+c = a(5a+1)
En: Q FgON O?@Bs
= F
5?,B?
?@Bs = F
5?,B?
5?,B?
= F
El exponente es ----- 1

14. 2020 . .
22)
= >
O
O
U.+
U
U
=
O
+
U
O.O
.
+
U
+
U
=
O
+
U
O
+
U.O
.
+
U,
=
O
.
+
U,
=
+
U,
= √
U,
23)
=
>
√5
5 √g
5 UB5√5
5 U
=
>
√5
5 √g
5 U
. √g
5 5√5
5 U
= >
√5
5 √g
5 U
.g
√5
5 U
= >
√5
5 g. √g
5 U√5
5 U
= √5
5 g. √g
5 U +
√5
5 U
= 5
+
5 g
= 5

15. 2020 . .
24)
=[ + 1 ON
]
+
???
= [ + 1 ON
]
+
??B+
= [ O O?
]O. ?B+
= O.O?.O.?.O.+
= O+B?.?.+
= O]
= =
25)
3'
. 36H
. 3 8NH
. 3 h6H
= 3' 'NH
3'
. 36H
. 3 . 3 H
. 3 h
. 36 H
= 3 8
.3'H
3 H6H6 H
. 3 N'N h
= 3 8
. 3'H
3'86'H
= 3 8N'H
46-2x = 16+4x
6x = 30
X =5
26)

16. 2020 . .
3' ,-
= 3 ',-
3'. ,-
= 3 .',-
4. 3 H
= 3. 4 H
,-
',-
= <'
=
'
-
= <'
= ; 2F = 1
X =
27)
4H
. 4 = 2'
. 3 − 2 H
. 2
2 H
. 2 = 2'
. 3 − 2 H
. 2
4.2 H
+ 8. 2 H
= 2'
. 3
12. 2 H
= 48
2 H
=22
2x =2
X =1
28)

17. 2020 . .
5
+
j
-
,
= 5
+
,
5
0
-
,
j = 5
+
, ;
-
,
8
=
3
H
= 3
H
= 1
X =2
29)
2 H
= 2
2x = 32
X =16
30)
g+56g-
g-.g.+6g/
= 58
5 g
− 5H
= 58
. 5H
. 56
− 58
. 5'
5 g
+ 5 h
= 58NH6
+ 5H
5 h
5g
+ 1 = 5gNH
+5H

18. 2020 . .
5 h
5g
+ 1 = 5g
. 5H
+5H
5 h
5g
+ 1 = 5H
5g
+ 1 ; 5 h
= 5H
X =10
31)
FH
= >g
5/
FH
= Q 56 g5/.5
FH
= Q 56 g55
FH
= 56g
+
55
; FH
= 56g g.5
X = 56g
32)

19. 2020 . .
M =
U. .'.,U.'NA.U.A,
8. .0U
=
U. . , .,U.'N A.U.A,
8. .0U
=
UB+./UB,N .0U. j
/. .0U
=
0.0UN j.0U
/.0U
= 26 '6 S
.( 2 6 S
+ 286 S
= 26'N S
. 2 6 S
+ 286 S
= 26'N
. 2 S6 S
+ 286'
. 2 S6 S
= 26
+ 2 = + 4
M = 4,5
33)
=
-B
0
,.'
.
-
/-
.+.
+
- -,
=
-.
0
,. , .
+
/
.+. -
=
0
,.
.
+
,
.+
= 2. 2
0
,.26
+
,
= 2 N
0
,
6
+
, = 2
= 4

20. 2020 . .
34)
M = √F
- +B-+B-
. √FHH+B-+B-
= F
+
- +B-+B-
. (FHH
+
+B-+B-
= F
+
- +B-+B-
. F
--
+B-+B-
= F
+
- +B-+B- N
--
+B-+B-
= F
+B--.-
- +B-+B-
M = F
:+B-+B-;
-:+B-+B-;
`= Y
+
_
M = √Y
_
35)
M= [√8
√A
√A
]√
√, √,.j
= [√8
√A
√A
]√
, .j√,

21. 2020 . .
= [√8
√
√
]√
, .j√,
= √8
√
,√,
.√
,.j√,
= √8
,√,.√
,√,
.√
,.j√,
2 √ = [ √2 '
]√ = √2
'√
M= √8
,√,.√
,√,
.√
,
.√
.j√,
M = √8
,√,.√
,./√,
= √8
√
/√,
.√
,./√,
M= √8
√
/√,B,./√,
.
= √8
√
,
.
M= √8 = 8
36)
USAR: Q √ = √ . Q√
= > Q2 √C
0+B √0
0√00
.
Z > Q2 √C
0+B √0
0√00√0
0
= Q2 √ ,0√00 +B √00
. > Q2 √ ,0+B √0
0√0
0 ,
= > 2 √ ,0
+
√0
0
+B √0
0
.
Z
2 √ ,0
+
< √0
0
=
,
+B √0
0

22. 2020 . .
= Q2 √
0+B √0
0
. √2
+B √0
0
= Q2 √ N
0+B √0
0
= 2
+B √0
0
+B √0
0
= 2
37)
27
0 -B+
= 3
, -.,
27
0-B0
= 3
,-./
3
0-B0
= 3
,-./
3
0-. /
= 3
,-./
3
0-B/
= 3
,-./
; 3 HN'
= 3 H6'
3F + 4 = 2F − 4
X = -8
38) Resolver:
>
u,k
k.u
k.+
= 3
Resolviendo:
>
u,k.,B,
kB+.+.u
k.+
= 3
>
u,.u, k.+
k.+. u
k.+
= 3 ; >
u.u, k.+
k.+.8
k.+
= 3

23. 2020 . .
:u.u, k.+ ;
+
k.+
8. k.+
+
k.+
= 3 ;
u
+
k.+.u,
8
+
k.+
.+,
= 3
e
N
+
k.+ = 36. 6
+
k.+
e
,k.+
k.+ = 6 . 6
+
k.+
e
,k.+
k.+ = 6
N
+
k.+
e
,k.+
k.+ = 6
,k.+
k.+
Finalmente: y =6
39)
-.+
, .
,-B+
0 .
,.0-
/ = 1
Considerando: v
= 1
-.+
, .
,-B+
0 .
,.0-
/ = v
-.+
,
N
,-B+
0
N
,.0-
/ = v
H6
+
HN
+
6 H
'
= 0
mcm = 12
6(x-1) +4(2x+1)+3(2-3x) = 0
6x-6+8x+4+6+9x =0
5x = -4

24. 2020 . .
F = −
'
g
40)
4H,
. 4
+
, = 3H,
.36
+
, + 3H,
. 3
+
,
4H,
. 4
+
, =
-,
+
,
+ 3H,
. 3
+
,
4H,
. 4
+
, =
-,
√
+ 3H,
. √3
2. 4H,
=
-,
N . -,
√
2. 4H,
=
'. -,
√
4H,
=
. -,
√
;
'-,
-, =
√
' H,
=
'
+
,
+
,
;
' H,
=
' +
,
F = ; F =
√
; F = −
√
H+
H,
=
+
√,
6
+
√,
= −1

25. 2020 . .
41)
Si: x -√2 = t → F = l + √2
x -√2 = √2
:-.√,;
,
t = √2
p,
l = 2
+
p,
; ln,
= 2
ln,
= [√2] Q
,
t = √2
x = √2+√2
F = 2√2
42)
FHH HU.-
= FH--U

26. 2020 . .
FH-.HU.-
= FH--U
; FH-.HU.-
= FH--U
FH-BU.-
= FH--U
FHU
= FH--U
FS
= FH-U
w = FHU
; wS
= FHU S
wS
= FS.HU
wS
= FS HU
n = FS
x = w
+
U → F = √w
U
43)
4`,6 `
=
g
_
0
g
4`,6 `
= 5
_
0
6
4`,6 `
= 5
_.0
0
4 ≠ 5 → yYqz y {y| }y$y |y~ •y~z
Y − 3Y = 0 ó
`6
= 0
Resolviendo se tiene:
X -3=0
X=3

27. 2020 . .
44)
4`,6 `
=
7
_
,
7
4`,6 `
= 7
_
,
6
4`,6 `
= 7
_.,
,
4 ≠ 7 → yYqz y {y| }y$y |y~ •y~z
Y − 2Y = 0 ó
`6
= 0
Resolviendo se tiene:
X -2=0
X=2
45)
[ ] H
. [ ]6
. [
C
'
]CH
. [
C
'
]6'
=[ ]6 CH
. [
A
7
] 7

28. 2020 . .
[ ] H
. [ ] . [
C
'
]CH
. [
'
C
]'
=[ ]6 CH
. [
A
7
] 7
0
0
.
'/
C/
.
,-
,-
.
C9-
'9-
=
+9-
+9-
.
A,2
7,2
0
0
.
'/
C/
.
,-
,-
.
+1-
+1-
=
+9-
+9-
.
A,2
7,2
0
0
.
'/
C/
. [2 H6 AH]. [3 AH6 H
] =
+9-
+9-
.
A,2
7,2
0
0
.
1
C/
. [26 8H]. [3 8H
] =
+9-
+9-
.
A,2
,2.C,2
5
.0
. [26 8HN CH]. [3 8H6 CH
] =
A,2
,2.C,0
5
.0
. [2 H]. [36 H
] =
,2.0
,2. ,0.,
0-
0-
=
2j
2j
H
= 78
3x =76
F =
78
46)
= √FV . >QF•. √F‚50
= √FV. >F•. QHƒ5j

29. 2020 . .
= √FV . QF•
j
Q√F‚5j
= F
T
,.F
„
j . F
ƒ
0]
De la condición dada, se tiene:
→
V
+
•
8
+
‚
h
= 1
3R + … =
‚
g
15R + 5…+ † = 30 -----(a)
5(3R + …) = †
De (a) se tiene:
5(3R + … + † = 30
† + † = 30
2 † = 30
† = 15
47)
Si O
= 2 , hallar el valor de E= O?B+
+
g-B/6g-B,Ng-B06g-B+
g-B06g-B+
a) 6 b) 15 c)10 d)12
O?B+
= O?.O
= O O?
= 2 = 4
E = 4+
g-.g/6g-.g,Ng-.g06g-.g+
g-.g06g-.g+
E= 4+
g- g/6g,Ng06g
g- g06g
E = 4+
g g06g g,6
g g,6
E = 4+
g6gN g6
'

30. 2020 . .
E = 4+
gh68
'
= 4+
''
'
E= 4+6
E = 10
48)
>FS QF‡
√Fˆs@?
= F
[ U@ B‰]sBŠ
?@s
√F
0
= F
,
0
>F QF
00
= F
A
C
>F QF √F
000
=F
,j
,2
-------
-------- =F
0U.+
0U
Luego se tiene:
‹F ZF >F QF … … … … … . . √F
00000
= F
0U.+
0U

31. 2020 . .
F
0U.+
0U
= FŒ
Reemplazando:
S 0U.+
0U
=
S 1]
0U
U6
U
=
Ah
U
3S
− 1 = 80
3S
= 81
3S
= 3'
n =4
reemplazando: x=
S
=
'
= 2
n +x = 4+2
n+x = 6
49)
y`N
= Y√A`
. y
y`
. y = Y√A`
. y ; y`
= Y√A`

32. 2020 . .
y`
= Y√'. .`
; y`
= Y √ .√`
X = 2√
y
√,
= 2√ √ . √,
y
√,
= 2√ . √ . √,
y
√,
= 2'. √,
E = 2'
E =16
50)
A = mY NH+B-....
o
+
,
= F
+
,
NH
+
,B-...
A = F
+
,
N•
A = F
+
,. F•
Miembro del lado izquierdo:
A= mY NH+B-....
o
+
,

33. 2020 . .
= Y NH+B-....
= Y N•,
= F . F•,
A = F. F•, +
,
A = F
+
,. F
Ž,
,
Igualando las dos ecuaciones de A:
F
+
,. F•
= F
+
,.F
Ž,
,
F•
= F
Ž,
,
X = F
Ž,
,
+
Ž
F = F
Ž
,
Igualando los exponentes: 1 =
•
A=2
De: A = F
+
,.F•
2 = F
+
,.F
2 = F
+
,
N
= F
5
,
F = 2
,
5
F = √2
5
F = √4
5

34. 2020 . .
51)
Haciendo: t = √F
0
; F = l
l = √64
j.ppp,
l = 64
+
j.ppp,
; t = 28
j.+
ppp,
t = 28.8.+.n.pp,
t = 2n.pp,
lnpp,
= 2n.pp,
npp,
lnpp,
= 2
lnpp,
= √2 = √2 √
,
lnpp,
= :√2;
Q √, ,
lnpp,
= :√2;
Q √, √,
,
lnpp,
= :√2;
Q √, √,
,
Por analogía, se tiene:
l = √2

35. 2020 . .
F = l
X = √2 = √8
52)
Encuentre el valor de X
FH/
= 4
Si : F'
= → F =
+
/
FO
= 4
+
/ O
= 4
[
?
/]'
= 4'
O
= 4'
X=Por analogía:
a = 4 ; F =
+
/
x = 4
+
/
F = √4
/
53)
Si: √F = l → F = l

36. 2020 . .
l = √5
,pp5
l = 5
+
,pp5
l = 5
+
pp5
lnp5
= [ 5
+
pp5
]np5
lnp5
= 5
lnp5
= √5
5 g
= √5
5 √g
5 5
= √5
5 √g
5 √5
5 5
Por analogía, se tiene:
l = 55
F = l
F = √5
5
54)
26 .,.+
= 26
.
+
,
= 2
6
+
,
.
+
, = 2
6
+
√, = +
√,
=
√
√,
= [ >2 √ ,√,√,
√,
]
+
√,√,

37. 2020 . .
=
>
2
• √
√,
. √
√,
•.
+
√,√,
√,
√,
=
>
2
• √
√,
•
√,
√,
= [2
• √
√,
•
]
+
√,
√,
√2
√,
= 2
+
√,
.√,
√, = 2
√,
, = 2
+
, √ I √
√,
= 2√
√,
+
√,
√,
= 21
=2
55)
F6H,-,
= Q√2
√,
F6 H, -,
= Q√2
√,

38. 2020 . .
[F6 H, -,
]6
= < Q√2
√,
=
6
[F H, -,
] = √2
6 +
√,=(
√
+
√,
[F H, -,
] = (
√
.
√
√
+
√,
[F H, -,
] = (
√
+
√,
[F H, -,
] = (
√ √,
,
,.
+
,
= . =
'
=
√
[F H, -,
] = (
√ √,
,
,
√,
,
,
X =
√
56)
FH-
= 36
Q0,0
0
FH-
= 36
,
0
.+

39. 2020 . .
FH-
=
.
+
0
FH-
=
.+
+
0
FH-
=
+
0
+
0
Luego se tiene:
X =
57)
= 2 w − w = w 2 − 1
w = 2 w − = 2w − 1
y =
'‰6'U
√
U U ,‰.+
6 √
‰ ‰ ,U.+
y =
'‰6'U
U ,‰.+
U 6
‰ ,U.+
‰
y =
'‰6'U
,‰.+6 ,U.+
y =
'‰6'U
,‰. .+6 ,U. .+
y =
'‰6'U
.+. ,‰6 ,U
y =
'‰6'U
, ‰6 , U

40. 2020 . .
y =
'‰6'U
'‰6'U
y = 2