Soluções_Expoente 10 (Manual e Caderno de Actividades)

Prévia do material em texto

Expoente10 2 
 
 
 
 Expoente10 3 
 
Tema I – Introdução à lógica bivalente e à teoria de conjuntos 
Unidade 1 – Proposições 
Páginas 13 a 29 
1. 
a) “32” 
b) “32 = 6” 
c) “2 é 
d) r A s 
e) “ 2 > ” 
 
2. 
b) “32 = 6” 
c) “2 é 
e) “ 2 > ” 
 
3. 
a) p 
 ~p “ 
b) p “ Memorial do Convento 
~p “ Memorial do Convento 
c) p “ ” 
 ~p “ 
 
4. 
a) p ” – 
 ~p “ ” – p 
b) p 3 = 9” – 
 ~p 3 9” – 
c) p 2 ” – 
 ~p 2 ” – 
d) p “ – 
 ~p “ – 
e) p “ – 
 ~p “ – 
 
5. p “ ” 
 q “ ” 
a) “ ” 
b) (~ ) “ ” 
 
6. 
a) “ ” p (~q) 
b) “ ” (~p) (~q) 
 Expoente10 4 
 
7. p “ ” 
 q “ ” 
a) “ ” 
b) (~ ) “ ” 
c) (~ ) (~ ) “ ” 
 
8. p “ – 
 q “ – 
 
 ~p 
 ~q 
 ( F) 
 ~ (F F) F 
 ~ ~ (F ) 
 
9. 
- p q 
 
 
a) p 
b) ~q 
c) 
 (( ) 
d) ~( ) 
 ( ) 
e) ~ 
 ( F) 
f) F uma 
( F) 
g) F 
 p ( F) 
 
10. 
a) 
 
 
b) “2 é a 
 
c) 
 
 Expoente10 5 
 
d) " > 3,14" 1 < 
> 3,14 1 < 2” é uma 
11. 
a) ~ 
 
 
 
 
 
 
b) ~ ( ) 
 
 
 
 
 
 
c) ~ 
 
 
 
 
 
 
 
12. (~ ) 
 
 
 
 
 
 
(~ ) -
p q 
 
13. ( ( )) (( ) ( )) p q r 
 
 
P q r q r p (q r) p q p r (p q) (p r) 
 V V 
 F V F V 
 F V F V 
 F F F F F F F 
F F F F F 
F F F F F F 
F F F F F F 
F F F F F F F F 
 
 - ( ) ( ) ( ) 
( ( )) (( ) ( )) 
 
14. 
a) (~ ) 
 ( ~ ) 
 F F 
 
 
 
 ~ 
 F V 
 F F F 
F V 
F F V 
 ( )
 F 
 F F 
F F 
F F F V 
 ~ 
 F F 
 F V 
F F F 
F F F 
 ~ (~ ) 
 F V 
 F F F V 
F V 
F F V 
 Expoente10 6 
 
b) (~ )
 ( ~ ) 
 q 
 
c) (~ ) 
 ( ~ ) ( ) 
 F ( ) 
 
 
15. ~(p q) (~p ~q) p q. 
 
 
 
 
 
 
 
 
 
- ~ ( )p q ( ~ )p q 
~ ( )p q ( ~ )p q 
 
16. 
a) ~( ) (~ ) 
b) ~( ) ( ~ ) 
c) ~ ( ) (~ ~( ~ )) (~ (~ )) 
 
17. p “ ” q “ 
a) ~( ) (~ ~ ) 
 “ ” 
b) ~( ) (~ ~ ) 
 “ ” 
c) ~(~ ) ( ~ ) 
 “ ” 
d) ~(~ ~ ) ( ) 
 “ ” 
 
18. “ ” 
 “ ” 
 “ ” 
a) i. “ 
 ii. “ ” ~ 
 iii. “ 
 ( ) ~ 
b) i. (~ ) ~ 
“ 
” 
 
p q p q ~(p q) ~p ~q ~p ~q 
 F F F F 
 F F F F 
F F F F 
F F F V V 
 Expoente10 7 
 
 ii. ( ~ ) 
 “ ” 
 
19. ) é 
 
a) 
b) 
c) 
u ) 
d) ~ ~ ) 
e) ~ 
( e ~ 
f) 
( 
g) ~ ~ é 
(~ ~ 
 
20. ( ) ( ) ( ) 
 
P q r p q q r (p q) (q r) p r ((p q) (q r)) (p r) 
 V 
 F F F F V 
 F F F V 
 F F F F F V 
F V 
F F F F V 
F F V 
F F F V 
 
 
21. - ) (~ ) (~ ) ( ) 
 ~ 
 
22. ( “ ” 
 ~( ) ( ~ “ ” 
 
23. 
a) ~( ) 
( ~( )) 
( ) 
b) ~( ) 
( ~ ) 
c) ( ) 
( ~ ) 
d) ( ) 
( ~( )) 
( (~ ~ )) 
 
 
 
 Expoente10 8 
 
24. 
a) ( ) 
(~ ( )) 
((~ ) ) 
 
 
b) ( ) 
( ~ ( )) 
 ((~ ) (~ )) 
( (~ )) 
(~ ) 
c) ( ) ( ) 
( ~( ) ( )) 
((~ ~ ) ( )) 
((~ ) (~ )) 
 ( ) 
 
 
25. “ ” “ ” “ ” 
 
a) i. “ 
 ii. “ 
b) i. ( ) “ ” 
 ii. ( ~ ) “ ” 
 
26. 
a) 
C 
b) - ) 
 
c) -
 
d) é -
 
e) ~( ~ ) -
( ~ 
~ 
f) ~ -
(~ e 
g) ~ é -
 e ~ 
 
 
 
 
 
 Expoente10 9 
 
27. ( ) ( ) 
 (~p q) q p) 
 (( ~ ) ) (( ~ ) ~ ) ) 
 (( ) (~ )) (( ~ ) (~ ~ )) 
 (( ) F) (F (~ ~ )) 
( ) (~ ~ ) 
( ) (~ ~ ) 
 
28. ~(p q) ((p ~q) (q ~p)) 
 
p q p q ~(p q) ~p ~q p ~q q ~p (p ~q) (q ~p) 
 F F F F F F 
 F F V F F V 
F F V F F V
F F F F F F 
 
- ~(p q) (p ~q) q ~p) 
 ~(p q) ((p ~q) (q ~p 
 
29. ( ) 
 ~( ) 
 (( ~ ) ( ~ )) 
 [(( ~ ) ) (( ~ ) ~ )] 
 [(( ) ( ~ )) ((~ ) (~ ~ ))] 
 [(( ) V) (V (~ ~ ))] 
 [( ) (~ ~ )] 
 [ ( )] [ (~ ~ )] 
 [( ) ] [( ~ ) ~ ] 
 ( ) (V ~ ) 
 ( ) V 
 
 
30. 
a) 
b) ( ) 
c) ~ 
d) ( ~ ) 
e) (~ ) 
 
 Expoente10 10 
 
Unidade 2 – Condições e conjuntos 
Páginas 30 a 61
31. 
a) = 4 
 
 
b) = 
 
 
 
32. “ 2 + 1 > 9 “3 + = " “ é ” “ ] , 2]” 
 
 
33. 
a) p( x “ 2x ” 
 
 ( ) - 2 
 ( ) - 2 
b) p( x 2 6 0x x ” 
 
 = ( ) - 2)2 2) – 
 ( ) - 2 – – 
c) p( x y x y > x ” 
 
 = = 2 ( , ) - (1 
 = = 3 ( , ) - 3) < 1 
d) p( x “ 2x 1 é um ” 
 
 
e) p( x 2 3x x ” 
 
 
 
34. 
a) “ 3 10x ” 
b) 
c) ” 
d) “ + ” 
e) “ < 5 ” 
f) “ {1, 3, 5, 15}” 
 
 Expoente10 11 
 
35. 
a) 10 > 1 
10 > 1 
10 > 1 >
1
10
 
b) 13 
13 
 13 13 13
c) , = 13 
= 13 
 = 13 = 13 = 13 
 
36. 
a) + 2 = 0 ( + 2) = 0 = 0 + 2 = 0 = 0 = 2 
+ 2 = 0 
+ 2 = 0 
b) + 2 < 0 < 2 
+ 2 < 0 é 
+ 2 < 0 
c) ( 3)( + 1) = 0 3 = 0 + 1 = 0 = 3 = 1 
[, ( 3)( + 1) = 0 
[ 
( 3)( + 1) = 0 
 
37. 2 < 10 < 5 
a) “2 < 10” 
b) ] , 5[ “2 < 10” 
c) {5, 7, 10} “2 < 10” é um 
 
38. 
a) ( 1)( + 1) = 1 
 ( 1)( + 1) = 1 
b) = 0 = 0 
 = 0 é 
c) + 1 = 0 = 1 
 + 1 = 0 
d) < 0 > 0 
 < 0 
 
 Expoente10 12 
 
39. 
a) 
 - (6 é 
 ( ) 
 - 
 ( F ) F) 
b) 
 ( , ) = ( - 
( ( ( ) 
 ( , ) - 
(2 (F F) F) 
 
40. 
a) 
 - (2 é 
 ( F) 
 -
 (F F) F) 
b) 
 ( , ) -
 ( F) 
 , ) a- 
(2 (F F) F) 
c) 
 - 
(~(2 > 2)) (~F 
 - 
 F) 
 
41. 
a) “ = ” 
 “ ” 
 “ ” 
 “ ” é 
 “ ” 
 “ ” 
b) i. “ = ” 
= ( ) 
ii. “ ” 
 
iii. “ ” 
( ) 
 Expoente10 13 
 
iv. “ ” 
) 
v. “ ” é 
 ( 
 
42. 
a) “ = 0 < 0” -
 
b) “ + 1 = 0 = 0” -
 ( + 1 = 0 ( = 0 
c) “ + 1 = 0 ( 1)( + 1) = 1” -
 ( + 1 = 0
 (( 1)( + 1) = 1 
d) “ + 1 = 0 < 0” -
 ( + 1 = 0 ( < 0 
e) “ + 1 = 0 ( 1)( + 1) = 1” -
 ( + 1 = 0
 (( 1)( + 1) = 1 
 
43. 
a) = 2 - 2 
 - 
= 4 = 2 
b) 1 2 6 {1, 2, 3} -
- 
- 2 6 {1, 2, 3} é 
 
 
44. 
a) = 4 = 2 = 2
= 4 - 
= 2 - 
b) = 4 = 2 = 2 = 4 e = 2 = 2 
 
c) 2 6 {1, 2, 3} = 0
2 6 -
{1, 2, 3} - 
d) 2 6 ] , 3] 2 6 e ] , 3] 
 
 
45. 
a) < 2 < 4 
< 2 < 4. 
 Expoente10 14 
 
b) < 4 < 2 
 < 4 < 2. = 3, 
< 4 - < 2 -
 
c) = 1 = 1 
= 1 = 1. 
d) = 1 = 1 
 = 1 = 1. 
 = 1 = 1 - = 1 
- 
 
46.
a) = 3 = 9 
b) = 9 = 3 
c) = 9 = 3 = 3 
d) > 3 > 9 
e) = 3 = 27 
 
47. 
a) “ 2 1 1x x ” 1 1)2 = 1 
 
b) “ 4 16 2x x ” é 2 e ( 2) = 16 2 = 2 
 
c) “ 2 9 3x x ” 4 e ( 4) > 9 4 > 3 
 
d) “ 1 2 1 2x x ” é 1 1 1 2 1 1 2 é uma 
 
 
48. 
a) 
“ x x ” 
b) 
 “ x x ” 
c) 
 
“ ) 
 
49.
a) > 3 > 9 “ > 3 ” “ 2 9x 
b) ( 1) = 0 = 1 “ ( 1)” “ = 1” 
c) = = = “ = ” “ = = ” 
 
 Expoente10 15 
 
50. 
a) , + 1 2 
“ - + 1 2.” – p 
b) , + 1 2 
“ - + 1 2.” – 
c) x 0x 
 
” – 
 
51. 
a) , 2 + 1 
 , 2 + 1 
b) , | | 
 , | | 
 
52. “
” 
4 = 2 × 2 
k 4 = 10 × 
 
53. > 8 - > 0. 
× > 8 × + > 8 + + > 9 
> 8 + > 9 > 8, + > 9 . 
 
54. 
a) 
 
b) , > 0 < 0 0 
0 > 0 0 < 0 
c) 
 
 
55. 
a) : 9 = 0 
“ 2 9 0x ” – 
b) : 9 = 0 
“ 2 9 0x ” – 
 
56. “ , 0 1 
“ < ". 
 > 0 × < 1 × < 
 < < , 0 
1. 
 Expoente10 16 
 
57. 
a) : 2 = + 1 
 : 2 = + 1 
b) : = 12 
 : = 12 
 
58. 
a) “ ” 
: < 10 
b) “ ” 
 , > 0 
c) “ ” 
: 2 =
2
 
59. {1,2,3},3x x 
 “3 6 ” – 
 
60. {1,2,3}: 3x x é 
 “3 6 ” – 
 
61. 
a) ~( {1, 2, 3}, 3 é ímpar) {1, 2, 3}, 3 não é ímpar) – 
b) ~ : + 3 = : + 3 – 
c) ~( , + 1 > 0) ( , + 1 0) – 
 
62. 
a) p “ ” 
 ~p “ ” 
b) p “ ” 
 ~p “ ” 
c) p “ ” 
 ~p “ ” 
d) p “ ” 
 ~p “ ” 
 
63. = {6, 7, 8, 10} 
 ( ) 
 ( ): “ 
a) , ( ) – 
 , ( ) – 
 : ( ) – 
 : ( ) – 
b) ~ ( , ( ))x U a x : ~ ( ) – “ U um 
” 
 Expoente10 17 
 
 ~( , ( )) : ~ ( ) – “ U 
” 
 ~ ( : ( ))x U a x : ~ ( ) – “ U 
 ~ ( : ( ))x U b x , ~ ( ) – “ U 
” 
c) ( ) U 
 ( ) U 
 ~ ( ) U 
 ~ ( ) U 
 
64. 
a) , 2, 3 , é natural
“ 2 3 ” 
b) ~( 2 3 
 2 3 
 , 2, 3 , não é natural 
 
65. = 0, 2, { }, {1} 
a) 0A 
b) 1 A 
c) 2 A 
d) { } A 
e) {1} A 
 
66. 
a) 
b) {1, 2, 3, 6, 9, 18} 
 
67. 
a) {1, 3, 5, 7, 9, … } 
b) {5, 10, 15, 20, 25, … } 
c) {25, 36, 49, 64, 81, 100} 
 
68. 
a) { : 2 + 1 = 0 
 2 + 1 = 0 
 2 + 1 = 0 = 
± × ×( )
 = 
±
 = 1 = 
1 
 2 + 1 
 
 Expoente10 18 
 
b) { : + + 1 = 0 
 + + 1 = 0 
 + + 1 = 0 = 
± × ×
 = 
±
 3 
 + + 1 = 0 
 
c) { : 2 4 < 6 2 4 < 6 
 2 4 < 6 2 < 10 < 5 
 2 
 3 
 
 
d) { : |1 2 | < 4 |1 2 | < 4 
 
|1 2 | < 4 1 2 < 4 1 2 > 4 
 2 < 3 2 > 5 
 > < 
 1 > 1 < 2 > 2 < , 
 
69. 
a) = 2 = 0 
= = = 
 
 = + = + 
 
 = 2 = 1 
- 
2 = 0 =
1 ± 1 4 × ( 2)
2
 
 = 
±
 
 = 2 = 1 
- 
 -
 
b) 10 = 0 10 = 0 
10 = 0 = 10 = 10 = 10 
 O - 10, 10} 
 10 = 0 = 10 - 10} 
 10
 
 Expoente10 19 
 
70. 
a) { : + 3 = 4 } 
+ 3 = 4 4 + 3 = 0 = 
± ×
 = 
±
 = 3 = 1 
- + 3 = 4 é {1, 3}. 
{ : + 3 = 4 } = {1, 3} 
{ : é ímpar < 5} 
{ : é ímpar 
< 5} = {1, 3} 
 { : + 3 = 4 } = {1, 3} = { : é ímpar < 5} 
b) {1, 2, 3, 4} = {2, 1, 3, 4} = {4, 2, 3, 1, 2, 3} 
 
71. = 5, 5, 1, 0, 1, 5, 5, 25 
a) = { 5, 1} 
b) 5, 1, 0, 1, 5 
c) = {0, 1, 5, 25} 
d) = {25, 5, 1, 0, 625} 
 
72. 
a) {3 : } { : é múltiplo de 3} 
b) { : | | = 5} = { : = 5 = 5} 
c) { : é divisor de 10} 
d) { : | | < 0} 
e) { : | | 0} 
 
73. A B B C 
 
a) 1 A B 
b) 2 
c) 3 
d) 
e) , 
 
f) 
g) 6 , , 
 
 
 
 Expoente10 20 
 
74. A B C D a(n b(n c(n d(n 
= = {2, 3, 5, 7, 11, 13, … } {1, 2, 3, 4, 5, 6, 7, 8, 9} = {2, 3, 5, 7} 
= = {3, 6, 9, 12, 15, 18, … } {1, 2, 3, 6, 9, 18} = {3, 6, 9, 18} 
= = {1, 2, 3, 6, 9, 18, … } {1, 2, 3, 4, 5, 6, 7, 8, 9} = {1, 2, 3, 6, 9, 18, 4, 5, 7, 8} 
 = = {4, 5, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, … } {1, 2, 3, 4, 5, 6, 7, 8, 9} 
 = {4, 5, 7, 8} 
 
75. = ] , 5] = ] , ] = , +
a) = ] , 5] 
b) =] , + [ = 
c) =] , ] 
d) = , 5 
e) ( ) =] , 5] , = , 
f) = ] , 5] = ]5, + [ 
g) = , + = , 
h) \ =] , 5]\ ] , ] = ] , 5] 
i) \( ) =] , 5]\ = 
 
76. 
a) = { : 1 > 0 3 6 < 12} 
1 > 0 1 > 
- 1 > 0 
3 6 < 12 3 < 18 < 6 
- 3 6 < 12 
= {1, 2, 3, 4, 5} = {1, 2, 3, 4, 5} 
b) = { : 2 = 0} 
2 = 0 = 
± ×( )
 = 
±
 = 2 = 1 
- 2 = 0 = {2, 1} 
c) \ = {1, 2, 3, 4, 5}\(] , 1[ ]3, + [) = {1, 2, 3} 
d) = {1, 2, 3, 4, 5} {2, 1} = {2} 
e) = {2, 1} (] , 1[ ]3, + [) = { 1} 
f) = {1, 2, 3, 4, 5} {2, 1} = { 1, 1, 2, 3, 4, 5} 
 
Aprende Fazendo 
Páginas 62 a 72 
1. 
a) p “ ” :q “O ” 
 :p q “ ” 
 Opção (B) 
 Expoente10 21 
 
b) ~ “ ” ~ “ ” 
~ ~ : “ ” 
Opção (D) 
 
2. - a b
a b
 
 a b a b 
 
 F F 
F F
F F 
 
a b a b 
Opção (C) 
3. 
 
 Opção (B) 
 
4. A 
 B 
 C = { : } 
 A = B A C. 
 Opção (D) 
 
5. 
 
(A) ( ) (~ ~ ) 
 
 
p q ~p ~q p q ~p ~q p q (~p ~q) 
 F F F F 
 F F F F 
F F F F 
F F F F 
(B) ~( ) ( ) 
 
 
p q p q ~(p q) p q ~(p q) (p q) 
 F 
 F F 
F F 
F F F F F 
 
 
(C) ( ) 
 
 
p q p q p (p q)
 
 F 
F 
F F F 
 
 
(D) ( ~ ) ( ~ ) 
 
 
 
 
 
 
p ~p p ~p p ~p (p ~p) (p ~p) 
 F F F 
 F F F 
F F F 
F F F 
 Expoente10 22 
 
- p (p q
p q 
 
Opção (C) 
 
6. , ( ) ( ) , ~ ( ) ~ ( ) 
 
Opção (D) 
 
7. 
 
Opção (C) 
 
8. 
 
Opção (A) 
 
9. - é a b a é 
b - 
(A) 
(B) 
(C) ~ ~ 
 a b é 
a b 
Opção (B) 
 
10. -
 ( ) -
p ( ) 
 - q) é 
r 
Opção (D) 
 
11. - ~ , ( ) ( ) é 
: ~ ( ) ( )
: ( ) ~ ( ) 
 Opção (A) 
 
12. (i) 
( ) 
 ( ) 
 
 Expoente10 23 
 
( ) ( ) ( ) ( ), 
 ( ) 
( ) 
( ) ( F) 
(ii) 
( ) 
 ( ) uma 
 
( ) ( ) ( ) ( ), , 
 ( ) 
 ( ) 
(F ) (F F) 
Opção (B) 
 
13. 
 A B , 
 B C , 
 Opção (B) 
 
14. 
a) “5 × ( 3)” 
b) “ + = ” é 
c) 
d) A B C” é . 
e) 
f) “ 3 > + 1” 
g) “ 5 ” 
h) “ ” 
i) “{1, 2, 3, 6} 
j) 
 
15. 
 
16. p “ ” ~p “ ” 
 “ ” ~ “ ” 
 “ ” ~ “ ” 
a) : “ ” 
b) “ ” 
c) ~ ~ “ ” 
d) ~ : “ ” 
e) ~ ( ): “ ” 
f) ~ ~ : “ ” 
 Expoente10 24 
 
17. 
a) p a 
q 
b) p a q é 
 
c) ~ ~p p 
q 
 
18. 
a) ~ 
 
p q ~p ~p q 
 F F 
 F F F 
F 
F F F 
 
b) ~( ) 
 
 
p q p q ~(p q) 
 F 
 F F 
F F 
F F F 
 
c) (~ )p p q 
 
 
p q ~p ~p q p (~p q) 
 F F 
 F F F 
F 
F F F F 
 
19. 
a) p ” 
 ~p 
b) p “ ” 
 ~p “ ” 
c) p “ ” 
 ~p “ ” 
d) p “ ” 
 ~p “ ” 
 
 
20. 
a) ( ): 5 + 1 0 
 ( ) 
 ( ): | | < 0 
 ( ) 
 ( ): ( 2) = 0 
 ( ) 
 ( ): ( + 2) = 0 
 ( ) 
 ( ): 2 0 
 ( ) é 
 Expoente10 25 
 
b) ~ ( ): 5 + 1 < 0 
 ~ ( ): | | 0 
 ~ ( ): ( 2) 0 
 ~ ( ): ( + 2) 0 
 ~ ( ): 2 < 0 
c) (i) ( ) ( ) - 
p( q( 
(ii) ( ) ( ) -
q( r( 
 (iii) ( ) ( ) - 
 
(iv) ( ) ( ) -
t( 
 
21. 
a) (i) : ( ) 
 “ 
 (ii) , ( ) 
 “ ” 
 (iii) , ( ) ( ) 
 “ ” 
 (iv) : ( ) ~ ( ) 
 “ ” 
b) (i) “ ” 
 : ~ ( ) ~ ( ) 
 (ii) “ ” 
 : ( ) ( )x q x r x 
 (iii) “ ” 
 , ( ) ( ) 
 (iv) “ ” 
 , ( ) ~ ( ) 
22. 
a) p “ ” 
 ~p “ ” 
 b) p “ ” 
 ~p “ ” 
 
23. 
a) { : } = {1, 5, 25} 
b) { : } = { 1} 
c) { : > 10} = {5, 25} 
d) { : 2 0} = { 1, 0} 
e) { : é um número irracional} = 
 Expoente10 26 
 
24. 
 a) {3, 3} = { : = 9} = { : = 3 = 3} 
b) {1, 3, 5, 7, 9, 11, 13, 15, … } = { : é ímpar} 
 c) {1, 4, 9, 16, 25, 36, … } = { : } 
 
25. 
a) (i) “ 3,14 ” q r 
 (ii) “ = 3,14” p q 
 (iii) “ = 3,14 ” q ~r 
b) (iv) ~ “9 ” 
 (v) ~( ) “ = 3,14” 
 (vi) ~ (~ ): “ 3,14 ” 
c) p q r - p q r 
 q r, 
 ~p q, F 
 A q ~r, F) 
 p ~r, F 
 ~( ), F) 
 ~ (~ ) , 
(F F)) (F 
 
26. 
a) ~ ( ~ ) 
 
 
p q ~q p ~q ~ (p ~q) 
 F F V 
 F F 
F F F 
F F F 
 
b) ( ) 
 
 
p q p q p (p q) 
 V 
 F V 
F F 
F F F V 
 
c) ( ) (~ ) 
 
 
p q p q ~p ~p q ( ) (~ ) 
 F V 
 F F F F V 
F V 
F F V 
 
27. 
a) ( ( )) 
 
p q p q p (p q) 
 
 F F 
F F F 
F F F F 
 
 - p p (p q 
 Expoente10 27 
 
b) ( ) 
( V) ( ) p p 
( V ) 
V 
 
 
28. 
a) q (p ~q) 
 p (q ~q) 
 p 
 
b) ( ~ ) 
 ( ~ ) 
 F 
 F 
c) (~ ) 
 ( ~ ) ( ) 
 F (q p) 
 q p 
d) ~ ( ) ~ 
 [(~ ) (~ )] ~ 
(F (~ )) ~ 
(~ ) ~ 
~ ( ~ ) 
 ~p F 
 F 
e) (~ ) ~ 
 [( ~ ) ( )] ~ 
 (V ( )) ~ 
 ( ) ~ 
 ( ~ ) 
 V 
29. 
a) ~( ) ~ 
b) ~(~ ~ ) 
c) ~( (~ )) ~ ~(~ ) 
 ~ ( ~ ) 
d) ~(~ ) ~ ~ 
e) ~ ( ) ~( ) 
 ~ ~ 
f) ~(( ) ) ( ) ~ 
 
 Expoente10 28 
 
g) ~(( ) ) ~( ) ~ 
 ~ ~ 
h) ~( ~ ) ~( ~ ~ ) 
 ~( ~ ) ~(~ ) 
 ( ) (~ ~ ) 
 
30. 
a) q p q 
p 
b) ( F F) 
c) ~( ) F) ~F 
d) ~ ~ ( ) 
e) ~ ( F) 
f) ~ q p F) 
g) é uma F) 
h) ~ F) 
 
31. 
a) ( ) ~(~p ~q) 
b) ( ) (~ ) ~( ~ ) 
c) ( ) (( ) ( )) 
 ((~ ) (~ )) 
 (~( ~ ) ~( ~ )) 
 
32. 
a) ~( ) 
p q r p q p q 
- p q 
b) ~( ) 
r ~( ) ~( ) 
 
p q é 
 
 
33. 
a) p “U ” q ” 
 “ - (p q) 
 Negação da implicação: “ ” 
(p ~q) 
Implicação contrarrecíproca: “ 
” q ~p) 
 Expoente10 29 
 
b) :" 3"p x 
 2:" 9"q x 
 2:"3 9"p q x x 
 Negação da implicação: 2" 3 9"x x (p ~q) 
 Implicação contrarrecíproca: 2" 9 3"x x (~q ~p) 
c) p: “n ” 
 q: “n ” 
 p q “ n ” 
 Negação da implicação: “n (p ~q) 
 Implicação contrarrecíproca: “ n é um m ” 
 (~q ~p) 
34. 
a) p “ ” 
~p “ ” 
b) p “ 2 0 - ” 
 ~p “ 2 0.” 
c) p “ 3 25x ” 
 ~p “ 3 25x ” 
 
35. 
a) > 
 ~( > ) ( , ) 
b) , 2 0 
 ~( , 2 0) ( 2 < 0) 
c) = 25 
 ~( : = 25) ( , 25) 
 
36. 
a) 
 
b) 
 
c) 
12 2 1 
 
37. 
a) { : ( ) ( )}P n p n r n n n 
b) { : ( ) ( )}Q n q n r n n n n 
 = 
c) { : ( ) ~ ( )}R n q n p n n n 
 = 
 
 
 Expoente10 30 
 
38. 
a) { 4 < 0 36 = 0} 
 
 
 
 
b) { 4 < 0 36 = 0} 
c) { 4 < 0 36 = 0} = ]4 + [ { 6, 6} = {6} 
d) { 4 < 0 36 = 0} = ] , 4 
 
39. = {0, 1, 2, 3} B C = 
a) = {0, 1, 2, 3} ]2, + [ = [2, + [ {0, 1} 
b) = ]2, + [ ]1, 6] = ]1, + [ 
c) = {0, 1, 2, 3} ]1, 6] = {2, 3} 
d) = ]2, + [ ]1, 6] = ]2, 6] 
e) ( ) =
 )
{2, 3} ]2, + [ = {3} 
f) = \{0, 1, 2, 3} 
g) = \]2, + [ = ] , 2] 
h) = \ , 1] ]6, + [ 
i) \ = {0, 1, 2, 3}\]2, + [ = {0, 1, 2} 
j) \ = ]2, + [ \{0, 1, 2, 3} = ]2, + [\{3} = ]2, 3[ ]3, + [ 
k) \( ) = ]1, 6]\({0, 1, 2, 3} ]2, + [) = ]1, 6]\{3} = ]1, 3[ ]3, 6] 
 
40. 
 
 
 
 
 
 
 
 
 
41. ( ~ ) (~ ) 
- ~
~ p p ~ 
q 
q q 
~ q r é uma 
r r é uma 
 
 
 
 
 
 ~ ~ ( ~ ) ( ) 
 F F V 
 F F V 
 F F F F F 
 F F F V 
F F V 
F F F F F 
F F F F V 
F F F F F F 
Cálculos auxiliares 
4 < 0 
 = 36 = 6 = 6 
 Expoente10 31 
 
42. 
a) ~ (~ ) 
 (~ (~ )) 
 ~(~ (~ )) 
 (~(~ ) ~(~ )) 
 ( ( )) 
 ( ) ( ) 
 ( ) ( ) 
 ( ) 
b) ( ~ ) ( ) (~ ~ ) ( ) 
 
43. 
a) : “ ” 
 : “ ” 
 
 
 
 q p 
b) : “ ” 
 : “ ” 
 : “ ” 
 
” 
 
” 
 
~ ( ~ ) 
 
44. 
a) 
 
 
 
 
 
 
 
b) p q p q 
p q ( ) ( ) 
c) p q 
 
 
 
 
 
 
 
 F 
F 
F 
F F F
 ~( ) 
 F 
 F F 
F F 
F F F 
 Expoente10 32 
 
(i) (p q) 
 
 
 
 
 
p q 
( ) - ~(p q p q
~( ) ( ) 
(ii) (p q) 
 
 
 
 
p q 
( ) - ~(p q p q
~( ) ( ) 
(iii) (p q) 
 
 
 
 
p q
( ) - ~(p q p q
~( ) ( ) 
 
(iv) (p q) 
 
 
 
 
p q 
( ) - ~(p q p q
~( ) ( ) 
 
45. 
a) 
 p( x ) “S ” q( x ) “ ” 
 
q( x ) p( x )
b) 
 ( )p x “ ” ( ) :q x “ ” 
 ” 
 p( x ) q( x ) 
 
 
 F F 
F F 
F F F 
 
 
 F 
F 
F F F 
 
 
 F F 
F 
F F 
 
 
 F F 
F F 
F F 
 Expoente10 33 
 
c) 
 ( , ) :" 0"p x y x y ( , ) :" 0 0q x y x y ” 
 
 p q 
 
46. 
 
a) ( ) é uma , ( ) 
a 
 
 : ~ ( ) 
 ~ ( ) ~ ( ) 
- ( ) ~ ( ) 
 
b) ( ) é uma : ( ) 
 
 
 , ~ ( ) ~ ( ) é 
 
 - ( ) ~ ( ) 
 
 
47. 
a) 
 
b) 2 > 3 
2)2 > ( 3)2 
 
48. n 
n 
 n 
- 
: = 12 
 
: = 3 × 4
 , 
 
 
: = 3 × ( = 4 ) 
 n 
 
49. 
 , é par é par 
(1) é par é par 
 
 
 Expoente10 34 
 
n = 2 
= (2 ) 
 = 4 
 = 2 × 2
 , 
 = 2 × ( = 2 ) 
(2) é par é par 
 
 é ímpar é ímpar n = 2 + 1 
 
= (2 + 1) 
 = 4 + 4 + 1 
 = 2 (2 + 2 )
 , 
+ 1
 = 2 + 1 ( = 2 + 2 ) é 
é par é par n 
 
 
 
Desafios 
Página 73 
1. 
a) 
 ,x R x B x A 
b) e , 
 
c) 
 
 “N 
d) 
~ ( ) ( ).x A x B x A x B x A x B 
 
2. 
a) 
 ~ , ~ , .s v s r v r 
b) 
s v r ~v ~r s vv s ~r s r 
 F F F F 
 F F F 
 F F F 
 F F F 
F F F 
F F F F 
F F F 
F F F F F 
 
x A x B x B x A
 Expoente10 35 
 
c) 
 
s v r ~v ~r s vv s ~r s r 
 F F F F 
 F F F 
 F F F 
 F F F 
F V V F F V V V 
F F F F 
F F V V F V V V 
F F F F F 
 
d) 
 
 
 Expoente10 36 
 
Tema II – Álgebra 
Unidade 1 – Radicais 
Páginas 76 a 100 
1. 
a) a b a < b b < a. 
 
( b) × ( b) < ( a) × ( a) 
 
b2 < a2 
 
a2 > b2 
b) a b a < b < 0, b < a. 
, b2 < a2. 
 
b2 × b2 < a2 × a2 
 
b4 < a4 
 
a4 > b4 
c) a b a < b n an > bn. 
a2 > b2. an > bn a2 > b2 an, bn, a2 b2 
an × a2 > bn × b2 an + 2 > bn + 2. 
 
2. 
 a b a < b n 0 < b < a. 
 , 0 < n <
 
( ) ( )n n n nb a b a , n
 > 
 < 
 a b a < b n 0 < b < a. 
, 0 < n <
 
( ) ( )n n n nb a b a n 
 > 
 
3. 
a) = , então = 36 = 6 
b) = , então = 8 = 2 
c) = 9 = 9 
 > 0 = 9 = 3 . 
 Expoente10 37 
 
4. 
a) = 32 = 32 = 2 
C.S. = { 2}
b) = 16 = ± 16 = ±2
C.S. = { 2, 2} 
c) = 0 = 0 = 0 
C.S. = {0} 
 
5. 
a) = 100 = ± 100 = ±10 
C.S. = { 10, 10} 
b) = 1000 = 1000 = 10 
C.S. = {10} 
c) = 10 000 = ± 10 000 = ±10 
C.S. = { 10, 10} 
d) = 10 = 10 
C.S. = { 10} 
e) = 10 
E 
C.S. = { } 
f) = 0 = 0 = 0 
C.S. = {0} 
g) = 1 = ± 1 = ±1 
C.S. = { 1, 1} 
h) 3 = 0 = 3 = ± 3 
 C.S. = { 3, 3} 
i) + 3 = 0 = 3 
 E 
 C.S. = { } 
j) 9 24 = 1 9 = 25 = = ± = ± 
 C.S. = , 
 
6. = 300 6 = 300 = 50 = ± 50
> 0 = 50 
 
7. 
a) 2 3 + 4 3 3 = 5 3 
b) 2 6 2 = 5 2 
c) 2 + = + = 2 
 Expoente10 38 
 
d) 3 2 + 4 3 2 + = 3 2 2 4 3 + = 2 2 + + = 2 2 + 
8. 
a) 2 × 3 × 5 = 30 
b) 5 2 × 2 2 = 10 4
c) 6 2 × +2 4 = 4 + 2 4 = 3 4 + 2 4 = 5 4 
d) 2 + 5 2 5 = 2 5 = 4 5 = 1 
9. 
 
5 = + 3 5 = + 3 = 2 = ± 2 
> 0 = 2. 
a) = (2 3 + 2 2 
b) = 3 × 2 = 6 cm 
 
10. n m 
 = . 
 
= × 
 = × 
= × 
= . 
 
11. 
a) 5 + ( 5) = 5 + | 5| = 5 + 5 = 10 
 
b) 5 + ( 5) = 5 + ( 5) = 0 
 
c) 3 5 = 3 × 5 = 9 × 5 = 45 
d) 5 2 = 5 4 5 + 4 = 5 4 5 + 4 = 9 4 5 
 
 
12. 
a) 3 × (2 5) + = 3 × (4 4 5 + 5) + = 3 × (9 4 5) + 15 
= 9 3 4 15 + 15 = 9 3 3 15 
 Expoente10 39 
 
b) 5 12 3 = 5 12 ( 3) = 5 4 
c) × = × = × = × = 1 = 1 
13. = 1 = 4. 
 + = 1 + 4 = 5 
+ = 1 + 4 = 1 + 2 = 3 9) 
5 3 + = + 
 
14. 
a) + = 5 + 2 4 5 = 5 + 2 4 5 = 2 3 5 
b) 3 = 3 5 2 4 5 = 3 5 2 + 4 5 = 2 + 7 5 
c) ( ) = 5 2 4 5 = 5 2 + 4 5 = 5 5 2 = 5 5 20 5 + 4 
 = 25 × 5 20 5 + 4 = 129 20 5 
d) × = 5 2 4 5 = 2 5 4 × 5 = 2 5 4 × 5 = 20 + 2 5 
e) = 5 = 5 
f) + = 5 + 5 = 5 + 5 = 2 5 
g) + = 2 4 5 + ( 5) = 2 4 5 + 5 = 7 4 5 
 
15. 
a) 9 = 3 9 = 81 = 3 9 = 729 = 3 
9 = 9 = 9 . 
b) 2 = 64 = 8 
2 = 2 × 2 × 2 = 2 × 2 × 2 = 8 
2 = 2 × 2 × 2 = 2 × 2 × 2 = 8 
2 = 2 = 2 = 8. 
 
16. 
a) 22 24 4 44 43 2 3 2 9 2 18 
b) 3 2 26 6 63 6 65 4 5 4 25 4 100 
c) 2 32 2 2 42 3 44 6 82 4 8 16 2 2 2 2 2 2 2 2 = 4 2 
 
17. > 0. 
8 6x a = 
 ( ) = ( ) 
 
 
=
 = 
= . 
 
 
 Expoente10 40 
 
Outro processo 
28 6 3 2 34 2 4a a a =
 
 
 
18. , 
×
×
= ( ) = > 0 
 
Outro processo 
 , , 
 
= = 
 ( ) = ( ) 
 = ( ) > 0 
 = 
= 
= . 
 
19. 
a) 486 + 4 = 3 × 2 + 2 
 = 3 2 + 2 
 
 
 = + 
 
 = 
 
 
b) 5 = 
×
 5 
 
 =
×
 5 
 
 =
×
 5
= × 5 5
 
 = 2 × 5 5 
 
 = 2 5 5 
 
 = 5 
 
 
 486 2 
 
 81 
 27 
 9 
 
 1 
4096000 2
 2048000 2 
 1024000 2 
 512000 2 
 256000 2 
 128000 2 
 64000 2 
 2 
 16000 2 
 8000 2 
 4000 2 
 2000 2 
 1000 2 
 500 2 
 250 2 
 125 5 
 25 5 
 5 5 
 1 
 Expoente10 41 
 
 
 
c) 275 × 5 + = 5 × 11 × 5 + 
 = 5 × 11 × 5 + 
 = 5 × 11 × 5 + 
 = 5 × 11 + 
 =
×
+ 
 
= 
 
 
d) 5 × 5 ×35 + = 5 × 5 × 35 + = 5 × 5 × 35 + = 5 × 5 × 7 × 5 + 
 
= 5 × 7 + 
= 5 7 + 
 
=
20 7
4
+
7
4
 
 
=
21 7
4
 
e) 
×
 3 = 
×
×
 3 
 
=
2 × 3
2 × 3
3 
 
=
2 × 3
2 × 3
3 
 
= 2 × 3 3 
 
= (2 × 3) 3 
= 2 × 3 3 
 
= 2 3 3 
 
= 2 3 3 
= 3 
 
f) 2 5 × 5 × 5 2 1 = 2 × 5 × 5 × 5 2 2 2 + 1 
 = 2 × 5 × 5 × 5 4 + 2 2 1 
= 2 × 5 4 + 2 2 1 
 = 2 × 5 4 + 2 2 1 
= 2 × 5 4 + 2 2 1 
= 5 2 4 + 2 2 1 
= 7 2 4 1 
 
 
 
 
 275 5 
 55 5 
 11 11 
 1 
 2 
 18 2 
 9 
 
 1 
 Expoente10 42 
 
g) 2 9 + 3 3 × 162 4 = 2 9 + 3 3 × 162 (2 ) 
= 5 9 × 2 × 3 2 
= 5 3 × 2 × 3 2 
= 5 3 × 2 2 × 2 
= 5 × 3 × 2 2 2 
= 15 2 2 2 
= 13 2 
 
 
20. 
a) 250 = 2 × 5 × 5 = 5 10 
b) 250 = 2 × 5 = 5 2 
 
 
 
c) 1008 = 2 × 2 × 3 × 7 = 12 7
 
 
 
 
 
d) 1008 = 2 × 2 × 3 × 7 = 2 126 
e) 1008 = 2 × 3 × 7 = 2 63 
 
21. 
a) = = 
b) = = = 2 3 
c) =
×
= 
d) =
×
= = 
e) =
×
= =
×
= 
f) = = = =
 
g) = = = = = 
 = = 
h) = = =
 162 2 
 81 
 27 
 9 
 
 1 
 250 2 
 125 5 
 25 5 
 5 5 
 1 
1008 2 
 504 2 
 252 2 
 126 2 
 
 21 
 7 7 
 1 
 12 2 
 6 2 
 
 1 
 Expoente10 43 
 
Cálculos auxiliares 
> 0, = 24, = 2 6 
 4 2 = 2 2 +
32 = 8 + 
24 = 
= ± 24 
i) = =
× ×
= = = 
 
22. 5 + 2 3 = 0 5 + 2 = 3 5 + 2 = 3 
 = 
 = 
 = 
 = 3 5 6 
 C.S. = 3 5 6 
 
23. 
a) 2 180 + = 2 × 6 5 + 
 = 5 12 5 + 2 5 
 = 9 5 
 
 180 = 6 5 
b) 1875 + 12 × = 5 3 + 12 × 
 = 5 3 + 
 = 5 3 + 3 = 6 3 
1875 = 3 × 5 
24. 
a) 
 = 4 + 4 = 16 + 16 = 32 = ± 32 
 > 0, = 32 = 4 2 
b) = 3 × 4 2 = 12 2 
c) = 
×
 
=
4 2 × 2 6
2
 
=
8 12
2
 
= 4 12 
= 4 × 2 3 
= 8 3 cm 
d) = 2 × = 2 × × 4 2 × 4 = 2 × × 16 × 2 × 4 = cm 
 180 2 
 90 2 
 45 
 15 
 5 5 
 1 
 1875 
 625 5 
 125 5 
 25 5 
 5 5 
 1 
 2 
 16 2 
 8 2 
 4 2 
 2 2 
 1 
 
 Expoente10 44 
 
 
25. 
a) 9 + 4 5 = 2 + 5 9 + 4 5 2 + 5, 
2 + 5 = 9 + 4 5. 
 2 + 5 = 2 + 2 × 2 5 + 5 = 4 + 4 5 + 5 = 9 + 4 5 
 9 + 4 5 = 2 + 5 
 
b) 29 + 12 5 = 3 + 2 5 29 + 12 5 
3 + 2 5 3 + 2 5 = 29 + 12 5. 
3 + 2 5 = 3 + 2 × 3 × 2 5 + 2 5 = 9 + 12 5 + 20 = 29 + 12 5
 29 + 12 5 = 3 + 2 5 
 
26. 
a) + = 11 6 2 + 2 + × = 11 6 2 
 ( + ) + 2 = 11 6 2 
 
 + = 11
2 = 6 2
 
 
 
+ = 11
2 = 6
= 2
 
 
 
+ 2 = 11
= 3
___________
 
 
 
______________
= , pois 0
_______________
 
 
 
+ 2 = 11
___________________
___________________
 
 
 
+ 2 = 11
___________________
___________________
 
 
9 + 2( ) = 11
___________________
___________________
 
 
 
2( ) 11 + 9 = 0
___________________
___________________
 
 
 
=
± ( ) × ×
×
___________________
___________________
 
 
 
=
±
___________________
___________________
 
 
 Expoente10 45 
 
 
=
___________________
___________________
 
= 1
_________
__________
 
 
= 1
=
_______
 
= 1
=
______
 
 
 
= 1
= 3
= 2
 
= 1
= 3
= 2
 
 
 + = 3 + 2 + = 3 2 
 
3 + 2 11 6 2
11 6 2 = 3 2. 
b) + = 9 + 4 2 + 2 + × = 9 + 4 2 
 
( + ) + 2 = 9 + 4 2 
 
+ = 9
2 = 4 2
 
+ = 9
2 = 4
= 2
 
 
+ 2 = 9
= 2
___________
 
 
2
+ 2 = 9
=
2
 , pois 0
___________________
 
 
4
+ 2 = 9
___________________
___________________
 
 
4
+
2( )
=
9
___________________
___________________
 
 
4 + 2( ) = 9
___________________
___________________
 
 
2( ) 9 + 4 = 0
____________________
____________________
 
 
=
9 ± ( 9) 4 × 2 × 4
4
_____________________________
_____________________________
 
 
=
9 ± 49
4
___________________
___________________
 
 
= . 
 Expoente10 46 
 
=
±
___________________
___________________
 
 
= 4
___________
___________
 
=
1
2
__________
__________
 
 
= 2
=
2
2
= 2
 
= 2
=
2
2
= 2
 
 
= 2
= 1
= 2
 
= 2
= 1
= 2
 
+ = 1 + 2 2 + = 1 2 2 
 
1 2 2 9 + 4 2. 
9 + 4 2 = 1 + 2 2. 
 
 
Unidade 2 – Potências de expoente racional 
Páginas 101 a 105 
27. 
a) 10 ÷ (2 × 5 ) × ÷ 5 = 10 ÷ 10 × 10 ÷ 5 = 10 × 10 ÷ 5 = 10 ÷ 5 = 2 = 16 
b) 
 × 
: [( ) : ( ) ]
 = 
 
[ ] [ : ]
=
÷
÷
=
÷
= = 6 = 36 
 
 
28. 
a) 16 = 16 = 4 
b) 64 = 64 = 4 
c) 81 = 81 = 3 
d) 27 = 27 = (3 ) = (3 ) = 9 
e) 16 = 16 = (2 ) = (2 ) = 8 
f) 16 = 16 = (2 ) = (2 ) = 2 = 128 
 
29. 
a) 25 = = = 
b) 8 = = = 
c) 81 = = = 
d) 64 = = =
( )
= = = = 
 
= . 
 Expoente10 47 
 
30. 
a) × = × = × = = 
b) × = × = × = × = = 
c) = = = = 
d) 
,
,
= = = = 3 = 3 = 3 , 
e) = = = = = 
 
31. 
a) × =
( ) ×
 =
×
( )
= = = 2 = 2 = 2 
b) × ÷ + 5 = ÷ + = + = 5 + = 5 + = 5 + 
 = 5 + = + = = = 
c) 
, ×
+ = 
×
 + 
×
 = + = 2 + = 2 + = × + 
 =
×
+ = × = = 
= = = = × 
 
 = 2 × 2 + 10 2 = 2 + 10 2 = 2 + 10 2 
 
Aprende Fazendo 
Páginas 106 a 113 
1. (A) 7 10 = 7 10, 
(B) 14 10 = 7 10, 
 
(C) 7 10 2 10 7 10 × 2 10 = 14 100 = 140, 
14 10 
(D) 10 10 + 4 10 = 14 10 14 20 = 10 10 + 4 10) 
 
Opção (B) 
 
 Expoente10 48 
 
2. (A) 8 40 = 4 40 = 4 × 2 10 = 8 10, 8 5 . 
 (B) 2 12 2 8 2 12 + 2 8 = 2 × 2 3 + 2 × 2 2 = 4 3 + 4 2, 
8 5. 
(C) 2 10 2 × 2 10 = 4 10, 5. 
(D) 2 10 2 2 2 10 × 2 2 = 4 20 = 4 × 2 5 = 8 5. 
 Opção (D) 
 
3. = , = 2 + 6. 
 Á = 2 + 6 = 2 + 2 × 2 × 6 + 6 = 2 + 2 12 + 6 = 8 + 2 × 2 3 = 8 + 4 3 
 Opção (C) 
 
4. × = = = = 
Opção (C) 
 
5. 
a b c 
= 4, = 3 e = 3 - 
 4 3 = 4 3 = 2 3 (F ) 
 4 + 3 = 4 + 3 (F ) 
 ( 3) = 3 (F ) 
 ( ) = . 
Opção (D) 
 
6. 3 = 3 = 3 × × = 3 
Opção (C) 
 
7. 3 + 23 + 6 + 4 = 3 + 23 + 6 + 2 = 3 + 23 + 8 = 3 + 23 + 2 = 3 + 25 
 = 3 + 5 = 8 = 2 2 
Opção (B) 
 
8. [ ] = 
×
=
×
 
 = 
 =
×
 = 
Opção (A) 
 Expoente10 49 
 
Cálculo auxiliar 
= + 
= 1 + 5 1 
= 1 + 5 2 × 5 + 1 
= 1 + 5 2 5 + 1 
= 7 2 5 
 
9. [ ] = = 7 2 5 
 
 
 
 
Opção (A) 
 
10. 2 × 5 = 2 ×× × 5 ×× = 8 × 25 = 200 
Opção (C) 
 
11. 
 
 
 
 
 
 
 
Opção (B) 
 
12. = [ ] 
 
 =
 × 
× 
 
 
 
 = × 
 
 
 
 
 =
( ) 
 
 
 
=
7 49
2
 
Opção (D) 
 
13. = e = . 
S - = 2 × , 
 = 2 × = 2 = 2 = 2 
 Opção (C) 
 
14. = × × 
=
1
3
× ×
2
2
 
=
2
6
 
 
 
 
 
 
 
 72 2 
 2 
 18 2 
 9 
 
 1 
72 = 6 2 
= (6 ) + (6 ) = 36 + 36 
 = 72 
 = 72 , pois > 0 
 = 72 
 = 6 2 , > 0 
+ = 
+ 7 = 
= 7 
= 7, pois > 0
Cálculo auxiliar 
 
 
 Expoente10 50 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Opção (B) 
 
15. = ×
 
 ×
=
× ×
×
=
×
×
= 2 × × 
= 2 = 2 × × - 
1
2
1
=
1
3
1
3
2
= 0
1
6
=
1
2
=
1
3
= 6
= 6
 
, = 6. 
Opção (D) 
16. 
a) 
×
= = = 2 3 
b) 5 2 × 6 = 12 5 12 = 4 12 = 4 × 2 3 = 8 3 
c) 48 + 10 3 = 48 + 10 3 
 = 2 3 + 10 3 
 = 12 3 
d) 24 2 = 24 2 24 × 2 + 2 = 24 2 48 + 2 
 = 26 2 × 4 3 
 = 26 8 3 
e) 48 + 2 8 3 12 = 4 3 + 2 2 2 3 × 2 3 
 = 4 3 6 3 2 
 = 2 3 2 
f) 1 2 3 2 + 3 = 1 2 3 = 1 (4 3) = 1 1 = 0 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 48 2 
 24 2 
 12 2 
 6 2 
 
 1 
48 = 4 3 
48 = 2 3 
Como > 0, =
3
2
. 
Como > 0, =
2
2
. 
Cálculos auxiliares 
 
 
( ) + = ( ) = ( ) =h 
+ = ( ) + = = = 
 
 
 
 Expoente10 51 
 
 
 
 
 
g) 2 + 32 + 8 3 98 + 4 50 
= 2 + 4 2 + 2 2 3 × 7 2 + 4 × 5 2 
= 7 2 21 2 + 20 2 
= 6 2
 
 
h) 7 20 45 + 3 5
= 7 × 2 5 3 5 + 3 5 
= 14 5 
 
 
i) 12 6 3 + 27 48 
= 2 3 6 3 + 3 3 × 4 3 
= 3 2 3 
= 3 
 
 
j) 17 5 + 245 + 5 180 
= 17 5 + 7 5 + 5 × 6 5 
= 10 5 + 30 5 
= 20 5 
 
 
 
k) 375 24 + 192 81 
= 5 3 2 3 + 4 3 3 3 
= 4 3 
 
 
 
 
 
 
l) 4 2048 5 512 6 256 
= 4 × 8 4 5 × 8 6 × 4 4 
= 32 4 24 4 40 
= 8 4 40 
 
 
 
 
 
 
 
 
 2 
 16 2 
 8 2 
 4 2 
 2 2 
 1 
32 = 4 2 
 
 98 2 
 49 7 
 7 7 
 1 
 
 
98 = 7 2 
 
 50 2 
 25 5 
 5 5 
 1 
 
 
50 = 3 2 
 
 45 
 15 
 5 5 
 1 
45 = 3 5 
 20 2 
 10 2 
 5 5 
 1 
20 = 2 5 
 27 
 9 
 
 1 
 245 5 
 49 7 
 7 7 
 1 
245 = 7 5 
 180 2 
 90 2 
 45 
 15 
 5 5 
 1 
 180 = 6 5 
 
 125 5 
 25 5 
 5 5 
 1 
 
375 = 5 3 
 24 2 
 12 2 
 6 2 
 
 1 
 
 24 = 2 3 
 192 2 
 96 2 
 48 2 
 24 2 
 12 2 
 6 2 
 
 1 
192 = 4 3 
 
 81 
 27 
 9 
 
 1 
 
81 = 3 3 
 
 2048 2 
 1024 2 
 512 2 
 256 2 
 128 2 
 64 2 
 2 
 16 2 
 8 2 
 4 2 
 2 2 
 1 
2048 = 8 4; 256 = 4 4; 512 = 8 
 
27 = 3 3 
 Expoente10 52 
 
 
17. 
a) × = 1 + 3 × 2 = 2 + 6 
b) = 1 + 3 2 = 1 + 2 3 + 3 2 = 1 + 2 3 + 3 2 = 2 + 2 3 
c) = =
×
×( )
= 
d) = =
( )
×
= = = = 
e) + 3 = 2 + 3 2 = 2 + 3 2 = 4 2 
f) + = 1 + 3 2 + 2 = 1 + 2 3 + 3 2 + 2 
 = 1 + 2 3 + 3 4 + 2 = 2 + 2 3 
 
18. 
a) =
×
= 
b) =
×
= = 
c) =
×
×
=
×
= 
d) = = = = = 2 13 + 11 
 = 2 13 + 2 11 
e) = = =
×
×
= 
f) = = =
×
×
= = 5 + 2 
g) =
×
= = 
h) =
×
= = 
i) =
×
×
= = = 16 
j) =
×
= = 
 
19. 
a) 
 
 
 
 
 = 625 = 625 
= 625 = 625 
= 5 625 = 5 = 5 
 625 5 
 125 5 
 25 5 
 5 5 
 1 
 Expoente10 53 
 
b) 8 × 8 = 8 = 8 = = =
( )
= = = 
c) 216 = 216 = 6 = 1296
 
 
 
 
d) 343 = 343 = 7 = = 
 
 
 
 
 
 
20. Á = [ ] + [ ] [ ] 
=
×
2
+ 6 3
×
2
 
=
6 3 × 4 3
2
+ 36 × 3
6 3 × 6 3
2
 
=
24 × 3
2
+ 108
36 × 3
2
 
= 36 + 108 54 
= 90 
 
 
 
21. . 
a) d 
= + = 2 
Como > 0, = 2 = 2 ( > 0) 
 
b) D 
= + = 2 + = 2 + = 3 
 > 0, = 3 = 3 ( > 0) 
 
22. 4 + 1 = 0 
 = 2 + 3, 
 2 + 3 4 2 + 3 + 1 = 0 2 + 4 3 + 3 8 4 3 + 1 = 0 
 4 + 4 3 + 3 8 4 3 + 1 = 0 
 0 = 0 
 2 + 3 
 
 
 216 2 
 108 2 
 54 2 
 27 
 9 
 
 1 
 
216 = 2 × 3 = 6 7 
 49 7 
 7 7 
 1 
343 = 7 
= 10 3 6 3 = 4 3 
Cálculo auxiliar 
 
 
 
 Expoente10 54 
 
 = 2 3, 
 2 3 4 2 3 + 1 = 0 2 4 3 + 3 8 + 4 3 + 1 = 0 
 4 4 3 + 3 8 + 4 3 + 1 = 0 
 0 = 0 
 2 3 
 
23. 7 5 = 3 7 3 = 5 
 7 3 = 5 
 = 
 = 
 = 
 = 
 = 
 
24. 
a) 2 + 4 = 2 + (2 ) = 2 + 2 = 2 + 2 = 2 + 2 = 2 2 
b) 
×
=
×
=
×
=
×
 = 8 × = 8 × = 8 × = 8 × 
 = 8 × 
×
= = = = 2 
c) 5 × 5 = 5 × 5 = 5
3
2 × 5
10
3 = 5
3
2
+
10
3 = 5
9+20
6 = 5
29
6 = 5 = (5 ) × 5 
= 5 5
d) + = + = + = + = + = ( + ) 
 
 
25. 
a) = = = = =
( )
 
= 
b) =
( )
=
( )
( )
=
( )
 
=
( )
=
( )
( )
 =
( )
( )( )
= 
 
( ) 
 Expoente10 55 
 
c) = = = = 
d) = = ×
×
=
( )( )
= 
 
 
 
 
26. 
 + = + = = = 
 
 
 
 
27. . 
 + = (2 ) 2 = 4 = 2 
C > 0, = 2 , = 2 , com > 0. 
 
 
28. 384 cm 
 = 64 cm = 64, = 8 
 
 = 
 × × 8 = 8 = 192 
 Como > 0, = 192 = 8 3. 
 8 3 
 
 
29. -
2 -
- 
. = × 2, = . 
= 
×
= = = , 
30. 
a) = = 
b) = 6
C = , = 6 × = 6 × = 6 × 
 
 
 
 
 192 2 
 96 2 
 48 2 
 24 2 
 12 2 
 6 2 
 
 1 
> 
, 
> 0, = , = , > 0. 
 
 Expoente10 56 
 
31. cm . 
a) . T = e = 4 . 
 = × = × 4 = × 4 × = 4 = 4 . 
b) = 32 
4 = 32 = 8 = 8 
Logo, = 8 = 64 = 64 = 4 
 
32. 
a) . 
 
 
- = , = = = . 
b) = = × = × = 
 
33. > 0, 1 + 4 + 4 + = 1 + × (4 + 4 + ) = 1 + × ( + 2) 
2 21 ( 2)x x 
= 1 + × ( + 2) 
 
 
> 0 
= + 2 + 1 
= ( + 1) 
= + 1 
 
 
 > 0 
 
34. 
a) = = = =
×
×
= 
 
=
27 + 9 + 3 + 1
3 1
=
27 + 9 + 3 + 1
3 1
=
27 + 9 + 3 + 1
2
 
b) =
×
× ×
 = = 
= 4 2 + 2 12 + 2 9 
 
 
 
 
Considerando a sugestão 
 Expoente10 57 
 
35. 
 × = . 
 
 
= 2
×
3
4
2
=
3
16
 
A = 12 = 12 × = . 
 
 
. 
 
 
- 
 
= = . 
, 
= 6 ×
×
3
4
2
=
3 3
4
 
 
= 6 ×
×
3
2
2
=
3 3
2
 
 
 = =
3 3
2
3 3
4
=
6 3 3 3
4
=
3 3
4
 
 
36. 
 
 
 
 
 
Cálculo auxiliar 
cos 60 = = = 
= 2 × + 2 = 2 2 
 Expoente10 58 
 
 
 
= 2 2 + 2 + 2 + 2 = 5 2 . 
 
=
2 2 + 2
2
×
6
2
=
3 2 × 6
4
 
=
3 12
4
 
=
3 × 2 3
4
 
 
= cm , . 
 
37. . 
 = 3 - 
= = . 
- = . L , = × . A 
=
4
3
×
3 3 ×
8
=
3
2
=
2
3
=
2
3
 
 
Unidade 3 – Divisão inteira de polinómios 
Páginas 114 a 141 
 
32. + 3 . 
 
33. 
a) 5 6 + 7 = 6 + 5 + 7 
b) 3 + 12 + 3 = + 12 
c) 3 = + 3 
 
d) (3 ) = 9 6 + = 6 + 9 
 
34. 
a) 2016 
b) 2 + 5 
 
35. 
a) ( ) + ( ) = 2 + 2 + ( 2 + 10) 
= 2 +
3
+ 2 + 2 10 
 
= 2 + 2 8; 
b) ( ) + ( ) = ( 2 + 10) + + 
 
 = 2 + + 10 
 
 = 10 
Cálculo auxiliar 
 = = cm 
 
 Expoente10 59 
 
c) ( ) ( ) = ( 2 + 10) + 
= + 2
5
2
+ + 10 
 
= 2 + 2 10 
d) ( ) ( ) + ( ) = 2 + 2 10
 )
= + 2 + 2 + 10 
 = + 2 + 12; 
 
36. 
a) 6 2( ) ( ) (3 ) ( 2 1)A x B x x x x 8 7 63 6 3x x x 
b) 2 3( ) ( ) ( 2 1) ( 2 )B x C x x x x x
 
5 3 4 2 32 2 4 2x x x x x x 
 5 4 3 22 4 2x x x x x 
c) 6 3( ) ( ( )) (3 ) ( 2 )A x C x x x x 9 73 6x x 
d) 2 2( ) ( ) ( 2 1) ( 2 1)B x B x x x x x
 
 
4 3 2 3 2 22 2 4 2 2 1x x x x x x x x 
4 3 24 6 4 1x x x x 
e) 6 2 3( ) ( ( ) ( )) (3 ) (( 2 1)( 2 ))A x B x C x x x x x x 6 3 23 ( 1)x x x
 
 
9 8 63 3 3x x x 
 
37. 
a) 3 2 5( ) ( ) ( 3 2) (4 1)A x B x x x x x
 
 
3 5 2 5 5(4 1) 3 (4 1) 2(4 1)x x x x x x x x
8 4 3 7 3 2 54 12 3 3 8 2 2x x x x x x x x 
8 7 5 4 3 24 12 8 2 3 2 2x x x x x x x 
b) 2( ) ( ) ( 3 2) (4 1)n mA x B x x x x x 
2(4 1) 3 (4 1) 2(4 1)n m m mx x x x x x x x 
1 2 3 24 12 3 3 8 2 2n m n n m mx x x x x x x x 
 ( ) × ( ) n + m, ( ), n, 
 ( ), m. 
 
38. ( + 3)( + ) + = + + 3 + 3 + = + ( + 3 ) + 3 + 
+ ( + 3 ) + 3 + + 2
= 1 + 3 = 1 3 + = 2. 
 
= 1
+ 3 = 1
3 + = 2
= 1
+ 3 = 1
3 + = 2
= 1
= 2
6 + = 2
= 1
= 2
= 4
 
 
 
 
 
 
 
 Expoente10 60 
 
39. 
a) 
 
 
 
b) 
 
 
c) 
 
 
 
 
 
 
 
 
40. 
a) (2 + 5 + ) ( + 4) 
 
 
 
2 + 5 + = ( + 4)( + 1) 2 
 
b) (3 3 + ) ( ) 
 
 
3 3 + = ( ) × ( + 2) + ( 2 + 3) 
 
 
 
 
 
 
 
 
 
 Expoente10 61 
 
c) ( 1) ( 1) 
 
 1 = ( 1)( + + + + 1) 
 
d) + 2 20 + 10 + 3 
 
 + 2 20 + 10 = + 3 × + 
 
41. 
 
 
+ + 12 2 + 1, 
 
29 + = 0 = 29 
 
 
 
 Expoente10 62 
 
42. 
a) 1074)( 23 xxxxA 2)( xxB 
 
 
( ) = 2 11 = 12 
 
b) 1074)( 23 xxxxA )3(3)( xxxB 
 
 
( ) = 7 + 14 = 32 
 
c) 1074)( 23 xxxxA 1)( xxB 
 
 
( ) = 3 10= 0 
 
d) 1074)( 23 xxxxA 
2
1
2
1)( xxxB 
 
 
( ) =
9
2
19
4
 =
99
8
 
 
43. ( ) = + + + ; 0; , , , 1)( xxB 
 
( ) = + ( + ) + ( + + ) = + + + 
( ) 
A( x B( x RxQxBxA )()()( 
dcbacbaxbaaxxdcxbxax 223 1 . 
RxQxB )()( 
2
3 2 2
3 2 2 2
3 2
( ) ( ) 1
( ) ( ) ( ) ( )
B x Q x R x ax a b x a b c a b c d
ax a b x a b c x ax a b x a b c a b c d
ax ax bx ax bx cx ax ax bx a b c a b c d
ax bx cx d
 
( )A x 
 
 Expoente10 63 
 
44. 
a) ( ) = 3 + + 1 ( ) = + = 
 
 
( ) = 3 2 + = 
 
b) ( ) = 3 + + 1 ( ) = 3 + 2 = 3 + 
( ) + , 
3 2 + = 
 
 ( ) 3 + 
( )
 + . 
 
45. 
a) ( + 6) = ( + + 6) 6 
 
 
( ) = + = 6 
 
 
b) (4 6) (2 1) 
 
 
( ) =
4
2
+
2
2
+
1
2
= 2 + +
1
2
 =
11
2
 
 
c) (8 5 + 1) (4 + 1) 
 
 
 
( ) =
8
4
7
4
= 2
7
4
 =
11
4
 
 
 
46. ( + + 12) ( + 2) 
 
 
 
 Expoente10 64 
 
+ + 12 + 2 = 0. 
–2 – 20 = 0 –2 = 20 = –10 
47. ( ) = 3 7 + 6 ( ) = 4 = ( 2)( + 2) 
 
 
1( )Q x
2x + 5 x 1( )Q x + 2 
 
1( ) ( 2)(3 1) 5Q x x x 
 
2( ) ( 2) (3 5 3) 12A x x x x ( 2) [( 2)(3 1) 5] 12x x x 
 
 ( 2) ( 2)(3 1) 5( 2) 12x x x x 
 
 
2
2
( ) ( )
( 4)(3 1) 5 10 12
( 4) (3 1) (5 2)
Q x R x
x x x
x x x
 
 
13)( xxQ 25)( xxR . 
 
48. ( ) = 3 + 2 + 1 
a) ( ) 1 (1) = 3 × 1 1 + 2 × 1 + 1 = 5. 
b) ( ) + 2 ( 2) = 3 × ( 2) ( 2) + 2 × ( 2) + 1 = 31. 
c) ( ) (0) = 3 × 0 0 + 2 × 0 + 1 = 1. 
d) ( ) 3 + 1 
1
3
= 3 ×
1
3
1
3
+ 2 ×
1
3
+ 1 
 = 3 × +1 
 (× 3) (× 9) 
 = + = 
 
49. ( ) = 3 + 3 
a) ( 1) = ( 1) 3 × ( 1) ( 1) + 3 × ( 1) = 1 + 3 1 3 = 0 
(0) = 0 3 × 0 0 + 3 × 0 = 0 
(1) = 1 3 × 1 1 + 3 × 1 = 1 3 1 + 3 = 0 
 = 3 × +3 × 
 = 3 × + 
 (× 2) (× 4) (× 8) 
 = + = 
 Expoente10 65 
 
 
2 = 2 3 × 2 2 + 3 × 2 = 2 3 × 2 2 2 + 3 2 = 2 3 2
(3) = 3 3 × 3 3 + 3 × 3 = 81 81 9 + 9
 ( 1) = 0, (0) = 0, (1) = 0, (3) = 0, 0 2 0 -
( ) 
b) - ( ) , ( ) 
( ) . 
 ( ) 2 
( ) ( ) = + 1, ( ) = , ( ) =
1 ( ) = 3 ( ) = ( ) = 2. 
 
50. 
a) ( ) = + + 3 
( ) 1 - (1) = 2 
 1 × 1 + 1 + 3 = 2 1 + 4 = 2 = 3 = 3 
b) ( ) + 2 ( 2) = 0. 
 
 ( 2) × ( 2) + ( 2) + 3 = 0 16 4 2 + 3 = 0 4 = 17 = 
 
51. ( ) = 2 + + . 
( ) + 1 ( 1) = 0 ( ) 2 
 (2) = 6. 
 
 
 
3 2
3 2
( 1) 0 1 2 01 2 1 1 0
(2) 6 8 8 2 62 2 2 2 6
A a ba b
A a ba b
 
 
 3 _____ ____ 3 1 4
2 3 6 3 3 1 1 1
b a b b
a a a a a a
 
 = 1 = 4. 
 
52. ( ) = + 3 9 + 5 
a) (1) = 1 + 3 × 1 9 × 1 + 5 = 1 + 3 9 + 5 = 0 ( ). 
b) ( ) 
 
( ) (1) = 1 + 4 × 1 5 = 0 
 
 
 
 Expoente10 66 
 
 ( ) ( ) x 
 
( ) = ( 1) × ( + 4 5) = ( 1)( 1)( + 5) = ( 1) ( + 5) 
 + 5 - ( ). 
53. ( ) = 2 2 10 + 2 + 16 + 8 
a) 
 
 
- ( ) ( ) = ( 0) × (2 2 10 + 2 + 16 + 8), 
 2 2 10 + 2 + 16 + 8
 ( ). 
b) 
 
 
- ( ) Q1( x Q2( x ). 
- ( ). 
c) 
 
 
- ( ) Q1( x Q2( x ), Q ( x ) . 
- ( ). 
 Expoente10 67 
 
54. 
a) -
( ) 1, 2, 4. 
(1) = 3 × 1 + 1 4 = 3 + 1 4 = 0 
( 1) = 3 × ( 1) + ( 1) 4 = 3 + 1 4 = 6 
(2) = 3 × 2 + 2 4 = 24 + 4 4 = 24 
( 2) = 3 × ( 2) + ( 2) 4 = 24 + 4 4 = 24 
(4) = 3 × 4 + 4 4 = 48 + 16 4 = 60 
( 4) = 3 × ( 4) + ( 4) 4 = 48 + 16 4 = 36, 
 
( ) 
 
b) ( ) 1, 2, 4, 6, 12. 
(1) = 6 
( 1) = 4 
(2) = 8 
( 2) = 0 
(3) = 0 
( 3) = 42 
(4) = 84 
( 4) = 196 
(6) = 816 
( 6) = 1224 
 P(12) = 17 892 
 ( 12) = 
( ) 
 
55. ( ) ( ) 
 ( ) ( ) ( ) = ( 0) ( + 1) ( 2) ×
( ) ( ) 
 - 
 
 
- ( ) = ( + 1) ( 2) × 2 m = 1, n p ( ) = 2. 
 
 Expoente10 68 
 
56. ( ) 
- ( ) = ( 5) a 
( ) 3 -
 (3) = 8 
× (3 5) = 8 × 4 = 8 = 2 
 ( ) = 2( 5) ( ) = 2 20 + 50. 
 
57. 
a) 16 = \ 4 = ( 4) × ( + 4) 
b) 8 + 16 = 2 × 4 + 4 = ( 4) = ( 4)( 4) 
c) 9 16 = 3 (4 ) = (3 4 )(3 + 4 ) 
d) 16 = ( 16) 
e) + 3 10 = ( + 5)( 2) 
 
 
 
 
 
f) 2 + 6 20 = 2 ( + 3 10)
 
 ( )( )
= 2( + 5)( 2) 
g) + 1 = ( + 3) 
 
 
 
 
 
 
 
 
 
58. 
a) + 6 7 = ( + 6 7) = ( 1)( + 7) 
 
 
 
 
 
 
 
 
 
b) 4 11 3 = (4 11 3) = × 4 × + ( 3) = 4 + ( 3) 
 
 
 
 
 
Cálculo auxiliar 
+ 3 10 = 0 = 
± × ×( )
×
 = 3± 49
2
 
 = = = 2 = 5 
8
3
+ 1 = 0 =
8
3
±
8
3
4 × ( 1) × 1
2 × ( 1)
=
8
3
±
64
9
+ 4
2
=
8
3
±
100
9
2
 
Cálculo auxiliar 
= = = 3 = 
+ 6 7 = 0 =
6 ± 6 4 × 1 × ( 7)
2 × 1
=
6 ± 64
2
 
Cálculo auxiliar 
 
= = = 1 = 7
4 11 3 = 0 =
11 ± ( 11) 4 × 4 × ( 3)
2 × 4
=
11 ± 169
8
 
Cálculo auxiliar 
= = = 3 = 
 Expoente10 69 
 
 
c) + 5 4 20 = ( + 5)( 4) = ( + 5)( 2)( + 2) 
 
 
 
 
 
 
 
d) 8 + 1 = + (8 4 + 2) 
 
 
 
 
 
 
 
 
 
 8 4 + 2 
59. ( ) = 2 13 + 25 14 2 7 = 2 
 
 
 
 
 
 
 
 
 
 
 
 
( ) , 2 ( ) ( ) = 2 ( 2)( 1). 
60. 
a) 3 6 = 3 ( 2) = 3 × 2 = 3 2 ( + 2) 
b) + 5 + 8 4 = ( 1)( 1)( 1)( + 4 + 4) 
= ( 1)( 1)( 1)( + 2)( + 2) 
 
 
 
 
 
 
 
 
 
 
 
 
+ 5 4 20 = ( + 5)( 4) 
Cálculo auxiliar 
 
8 4 + 2 = 0
4 ± ( 4) 4 × 8 × 2
2 × 8
=
4 ± 48
16
 
Cálculos auxiliares 
 
( ) = 2
7
2
( 3 + 2) 
Cálculos auxiliares 
- ( ). 
 
3 + 2 = 0 = 
± ( ) × ×
×
 = 
±
 = = = 2 = 1 
Cálculos auxiliares 
 
 
( ) = + 4 + 4 
+ 4 + 4 = ( + 2) 
 Expoente10 70 
 
c) 4 + 5 1 = ( 1)( 4 + 1) = ( 1)( + 1) × ( 4) + 
 = 4( 1)( + 1) + 
 
 
 
 
 
 
 
 
 
 
 
61. 
a) 6 + 9 = 0 = 
± ( ) × ×
×
 = 
±
 = 3 
 
b) 2 + 3 4 = 0 = 
± × ×( )
×
 = 
±
 = = 
 
c) 2 + 3 + 4 = 0 = 
± × ×
×
 = 
±
= 
 
d) 2 + 2 12 = 0 2 ( + 6) 2 = 0 + 6 = 0 
 = 0 = 
± × ×( )
×
 
 = 0 = 
±
 
 = 0 = = 
= 0 = 2 = 3
 
e) 1 = 0 = 1 = 1 = 1 
 
62. 4 + 3 = 0 
 = 4 + 3 = 0 = 
± ( ) × ×
×
 
 = 
±
 
 = = 
= 3 = 1
 
1 = 0 = 1 = 1 = 1 
Cálculos auxiliares 
 
( ) = 4 + 1 
4 + 1 = 0 = = = 
4 + 1 = 4 + 
 Expoente10 71 
 
= , = 3 = 1 = 3 = 3 = 1 = 1. 
 
C.S.= 3, 1, 1, 3 
 
 
63. ( ) = 6 + 31 + 10 = 0 
a) ( ) 2 (2) = 6 × 2 + 2 31 × 2 + 10 = 0. 
b) ( ) = 0 6 + 31 + 10 = 0
 
 
 
 
 
 
 
( 2)(6 + 13 5) = 0 
2 = 0 6 + 13 5 = 0 
= 2 =
13 ± 13 4 × 6 × ( 5)
2 × 6
 
= 2 =
13 ± 289
12
 
= 2 =
13 + 17
12
=
13 17
12
 
= 2 =
4
12
=
30
12
 
= 2 = = 
C.S. 2, , 
 
64. ( ) = 0 + 5 + 15 = 0 
 
 
 
 
 
 
 
 
 ( + 3)( + 2 5) = 0 
 + 3 = 0 + 2 5 = 0 
 = 3 = 
± × ×( )
×
 
 = 3 = 
± 
 = 3 = = 
 = 3 = = 
 = 3 = 1 + 6 = 1 6 
 
C.S. 3, 1 + 6, 1 6 
 
 
Cálculo auxiliar 
Cálculo auxiliar 
Cálculo auxiliar 
 
 Expoente10 72 
 
65. 
a) 4 0 ( 4) 0 
 
 
 
 
 
 
 
 
 
 
 4 0 = 0 4. C.S. = {0} [4, + [ 
b) < < 0 (1 ) < 0 
 
 
 
 
 
 
 
 
 
 
 
 < < 0 > 1. C.S. = ]1, + [ 
c) 5 5 5 + 5 0 
 ( 1)(4 5) 0 
 ( 1)(( + 1)( + 5) 0 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
A 5 + 5 0 1 1 5. C.S. = [ 1, 1] [5, + [ 
= 0 = 0
1 = 0 = 1 
Cálculos auxiliares 
 
 
= 0 = 0
4 = 0 = 4 
Cálculos auxiliares 
 
 
( ) = ( 1)( 4 5) 
Cálculos auxiliares 
 
- ( ) = 3 5 2 + 5 (1) = 1 – 5 12 – 1 + 5 = 0, 
( ) 1 
 
 
4 5 = 0 = 
± × ×( )
×
 = ± = ± = 5 = 1 
 
( ) = ( 1)( 4 5) = ( 1)( + 1)( 5). 
 
 
 Expoente10 73 
 
d) 3 + 20 27 + 10 < 0 ( 1)( 3 + 17 10) < 0 
3( 1)
2
3
( 5) < 0 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 3 + 20 27 + 10 < 0 < < 1 > 5. C.S. = , 1 ]5, + [ 
66. 
a) ( 1) = ( 1)4 + 2 ( 1) – 16 ( 1)2 – 2 ( 1) + 15 = 1 – 2 – 16 + 2 + 15 = 0 
 ( ). 
4 + 2 – 16 2 – 2 = 81 + 54 – 144 – 6 + 15 = 0 
( ). 
b) ( ) = 0 + 2 16 2 + 15 = 0 
 ( + 1)( 3)( + 4 5) = 0 
 + 1 = 0 3 = 0 + 4 5 = 0 
 = 1 = 3 = 
± × ×( )
×
 
 = 1 = 3 = 
± 
 = 1 = 3 = = 
 = 1 = 3 = 1 = 5 
 
C.S. = { 5, 
 
 
 
3 + 17 10 = 0 =
17 ± 17 4 × ( 3) × ( 10)
2 × ( 3)
=
17 ± 169
6
=
17 ± 13
6
 
Cálculos auxiliares 
- ( ) = 3 3 + 20 2 27 + 10 
(1) = 1 + 20 12 – ( ) 1 
 
( ) = ( 1)( 3 + 17 10) 
= = = = 5 
 
( ) = ( 1)( 3 + 17 10) = ( 1) × ( 3) ( 5) = 3( 1) ( 5). 
 Expoente10 74 
 
c) ( ) 0 + 2 16 2 + 15 0 
 
 
 
 
 
 
 
 
 
 
 
 
 ( + 1)( 3)( + 4 5) 0 
 ( + 1)( 3)( + 5)( 1) 0 
 
 
 
 
 ( ) 0 5 1 1 3 
C.S. = [ 5, 1] 
 
 
Aprende Fazendo 
Páginas 142 a 150 
1. ( ) = 5 ( ) = = . 
Opção (D) 
 
2. ( ) = (12 ) × × ( + 4) = 12 + 48 4 = + 8 + 48
 Opção (A) 
3. 1 + 4 = 5. 
Opção (C) 
 
4. 
 
 
 
- ( ) 
 
 
Cálculo auxiliar 
 
( ) ( ) + 1 3. 
 Expoente10 75 
 
Cálculo auxiliar 
2 + 3 = 0 = 
± ( ) ×( )× 
×( )
 = 1 = 3 
 
 
 
 
Opção (D) 
 
5. ( ) = 2 + + 3 + 1. 
 ( ) 2 (2) = 7. 
2 × 2 + × 2 + 3 × 2 + 1 = 7 16 + 4 + 6 + 1 = 7 4 = 16 = 4. 
Opção (B) 
 
6. ( ) = 3 + + 2( 1). ( ) + 1 ( 1) = 0. 
( 1) 3 × ( 1) + ( 1) + 2( 1) = 0 1 + 3 + 1 + 2 2 = 0 2 = 1 =
1
2
 
 Opção (D) 
 
7. (2 4 + 
 
( ) = + + = + + = . 
Opção (A) 
 
8. - ( 1) 0 ( 3) 0 ( 5) 0. 
( 1) × ( 3) × ( 5) 0, . -
( 1) × ( 3) × ( 5) < 0 . 
Opção (D) 
 
9. 2 + 3 = ( 1)( + 3) 
 
 
 
 
 
 
 
 
C.S. = ] , 3] [1, 4] 
 
 
2 – 
 Expoente10 76 
 
- A( 
 
- A( ) = 4. 
Opção (B) 
 
10. 
 
P( - 
 
4 + = 0
3 2 + = 0
 
 
 
 
= 4
3 2 + 4 = 0
_____________
= 1
= 5
= 1
 
 
Opção (C) 
 
11. P( x ) = nx + 1, n . 
 P( x x P( 1). 
 P( 1) = ( 1)n 
 n P( 1) = 1 + 1 = 0. 
 n P( 1) = 1 + 1 = 2. 
Opção (B) 
 
12. - P( x
P( x 
Opção (D) 
 
13. 
 ( ) 4 (4) = 0. 
 ( ) 1 ( ) – 
( ) = ( 1) × ( ) 21. 
- ( ) 4 (4). 
( ) = ( 1) ( ) – 
(4) = (4 1) (4) – 21 (4) – 21
 (4) = 
 (4) = 7 
Opção (C) 
 Expoente10 77 
 
14. 
a) 5 + + 2 
 
5; ; 2 
 
b) 4 + + 3 11 
 
–11; 0 
 
 
15. 
a) + + 2 + + = + 2 + + 
 = 2 + + + 
 = + + + 
 = + + 
 = + + 
b) + 2 + + = + +2 + 
 = + 2 + + 
 = 3 + + 
 = 3 + 
c) (2 1)(2 + 1) = (2 ) 1 = 4 1 
d) ( )( 2 + 4) = 2 + 4 + 2 4 = 2 6 + 4 
e) ( 2)( + 1)( + 4) = ( + 2 2)( + 4) = ( 2)( + 4) 
 = + 4 4 2 8 = + 3 6 8 
f) ( 2)( + + 1) = + + 2 2 2 
= + 2 2 2 
= 3 2 
 
 Expoente10 78 
 
16. 
a) ( ) = 2 + 2 + 3 ( ) = + + 2 
 
 2 + 2 + 3 = ( + + 2) × 2 + ( 1)
Q R = 1 
 
b) ( ) = 1 ( ) = 1 
 
 1 = ( 1) + ( 1) 
Q( x ) = x R( x ) = x – 1 
 
c) ( ) = 3 6 ( ) = 3 
 
3 6 = ( 3) 6 
 Q( x ) = R = 6 
 
d) ( ) = 3 6 ( ) = + 2 
 
3 6 = ( + 2)( 3) + ( 2 ) 
Q( x ) = x – R( x ) = 2 x 
 
e) ( ) = + + 2 + + 3 ( ) = + + 1 
 
+ + 2 + + 3 = ( + + 1)( + 1) + 2 
Q( x ) = + 1 R = 2 
 
 
 
 Expoente10 79 
 
f) ( ) = 2 + 4 + + 8 4 ( ) = + 2 1 
 
 2 + 4 + + 8 4 = ( + 2 1)(2 + 4) + 0 
Q( x ) = 2 + 4 R = 0 
 
g) ( ) = + + ( ) = 
 
 + + = + 3 + 5 + 0 
Q( x ) = +3 + 5 R = 0 
 
17. 
a) ( ) = 5 + 3 1 ( ) = + 1 
 
5 + 3 1 = ( + 1)(5 2) + 1 
Q( x ) = 5 2 R = 1 
 
b) ( ) = 2 4 ( ) = 2 
 
2 4 = 2 (2 + 2 2) 
Q( x ) = 2 + 2 2 R = 0 
+ 1 = ( 1) 
 Expoente10 80 
 
 
c) ( ) = 2 4 10 + 24 ( ) = 2 
 
 2 4 10 + 24 = ( 2)(2 10) + 4 
Q( x ) = 2 10 R = 4 
 
d) ( ) = 2 + 4 + 5 ( ) = + 
 
2 + 4 + 5 = +
1
2
2 + +
7
2
+
13
4
 
Q( x ) = 2 + + R = 
 
e) ( ) = 4 + 5 4 + 5 ( ) = 
 
4 + 5 4 + 5 = (4 + 5 4) + 5 
Q( x ) = 4 + 5 4 R = 5 
 
f) ( ) = + 6 + 2 + 36 5 ( ) = + 2 
 
 + 6 + 2 + 36 5 = ( + 2)( + 4 8 + 18 ) + ( 5) 
 Q( x ) = + 4 8 + 18 R = 5 
 
18. 
a) A(1) = 2 14 – 12 – 4 – 1 – 4 + 
b) A( 1) = 2 ( 1)4 – ( 1)2 – 4 ( – 
c) A(0) = 2 04 – 02 – 4 – 0 – 
d) A( 3) = 2 ( 3)4 – ( 3)2 4( 3 9 – 4 3 4 3 
 
 
+
1
2
=
1
2
 
= 0 
+ 2 = ( 2) 
 Expoente10 81 
 
19. 
a) + 12 = ( + 4)( 3) 
 
 
 
 
 
 
 
b) 3 + 5 22 = 3 + ( 2) 
 
 
 
 
 
 
c) 2 15 = 2( 3) + 
 
 
 
 
 
 
d) 9 = ( 9) = ( 3)( + 3) 
e) + 10 + 25 = ( + 10 + 25) = ( + 5) = ( + 5)( + 5) 
f) 5 3 = ( + 1)( 2 3) = ( + 1)( 3)( + 1) 
 
 
 
 
 
 
 
g) 3 + 4 + 5 6 = (3 + 6 + 9) = 3 ( + 2 + 3) 
 
 
 
 
 
 
 
 
 
 + 2 + 3 
 
 
Cálculo auxiliar 
+ 12 = 0 = 
± ×( )
 = 
±
 
= 
±
 = 4 = 3 
Cálculo auxiliar
3 + 5 22 = 0 = 
± × ×( )
 = 
± 
= 
±
 = = 2
Cálculo auxiliar 
2 15 = 0 = 
± × ×( )
 = 
± 
 = 
±
 = 3 = 
Cálculos auxiliares 
 
2 3 = 0 = 
± ×( )
 = 
±
 
 = 
±
 = 3 = 1 
Cálculo auxiliar 
 
+ 2 + 3 = 0 = 
± × ×
 = 
±
 
. 
 Expoente10 82 
 
h) 2 + 7 5 4 = ( 1)(2 + 9 + 4) 
= ( 1)2( + 4) +
1
2
 
= 2( 1)( + 4) +
1
2
 
 
 
 
 
 
 
 
 
 
i) 7 + + 6 = ( + 1)( 1)( 6) 
= ( + 1)( 1)( + 2)( 3) 
 
 
 
 
 
 
 
 
 
 
j) 2 3 12 + 7 + 6 = ( 1)( 3)(2 + 5 + 2) 
 = ( 1)( 3)2( + 2) + 
 = 2( 1)( 3)( + 2) + 
 
 
 
 
 
 
 
 
 
k) 2 + 2 1 = ( 1)( + 1)( 2 + 1) 
 = ( 1)( + 1)( 1) 
 = ( + 1)( 1)( 1)( 1) 
 
 
 
Cálculos auxiliares 
 
 
2 + 9 + 4 = 0 = 
± × ×
= 
±
 
 = 
±
 = 4 = 
Cálculos auxiliares 
 
 
6 = 0 = 
± ×( )
 = 
±
 
 = 
±
 = 3 = 2 
Cálculos auxiliares 
 
2 + 5 + 2 = 0 = 
± × ×
 = 
±
 
 = 
±
 = 2 = 
 Expoente10 83 
 
 
 
 
 
 
l) 3 12 + 9 + 12 12 = ( 2)( 2)(3 3 ) 
 = ( 2)( 2)3 ( 1) 
 = ( 2)( 2)3 ( 1)( + 1) 
 = 3 ( 2)( 2)( 1)( + 1) 
 
 
 
 
 
 
 
 
 
20. 
a) P( x P( x ) = ( x + 2)( x + 1), 0. 
P( x x – P 
(1 + 2)(1 + 1) = 18 2 = 18 6 = 18 = = 
P( x x + 2)( x + 1) x 2 + 2 x + x + 2) x 2 x + 2) = x 2 + 9 x + 6. 
b) P( x 
P( x ) = ( x 5)( x 5)( x 0) ( 0) = x ( x 5)( x 5) 
P( x x – 
P(4) = 8 4 (4 5) (4 5) = 8 4 ( 1) ( 1) = 8 4 = 8 = 2 
P( x ) = 2 x ( x 5)( x 5) = 2 x ( x 2 10 x + 25) = 2 x 20 x 2 + 50 x . 
 
21. 
a) P( – × ( 2 9 × ( = 27 – × 9 + 27 + 27 = 0 
P( P( x ). 
b) 
 
3 9 + 27 = ( + 3)( 6 + 9) = ( + 3)( 3) = ( + 3)( 3)( 3) 
 P( x ). 
c) P( x ) = 0 ( x x x 
 x x – x – 
 x = x x 
C.S. = { 
 
Cálculo auxiliar 
 
 
1 
Cálculo auxiliar 
 
 Expoente10 84 
 
d) ( ) 0 ( + 3)( 3)( 3) 0 3 
 
C.S. = [ [ 
 
22. 
a) ( ) = 2 + 6 = 2( + 3) 
 
 
Q1( x ) = 2 x 2 – 2 x – 2 R = 6 
2 x + 4 x 2 – 8 x = ( x x 2 – 2 x – 2) + 6 2 x + 4 x 2 – 8 x = 2( x ( ) + 6 
A( x B( x 
Q( x ) = x 2 – x – R = 6 
 
b) ( ) = 3 + 1 = 3 + 
 
 
Q1( x ) = 6 x – 2 x 2 + x + R = 
6 x 4 – x 2 + x + 1 = + 6 – 2 + x + + 
 6 x 4 – x 2 + x + 
 – 
 + 
A( x B( xQ( x ) = 2 x – x 2 + x + R = 
c) ( ) = 2 2 2 = 2( 2) 
 
 
Q1( x ) = 2 x + x 2 – 2 x + 1 R = 0 
2 x 4 – x – 8 x 2 x – 2 = ( x – 2)( 2 x + x 2 – 2 x + 1)
 2 x 4 – x – 8 x 2 x – 2 = 2( x – 2) ( ) 
 
+ 3 = 0 = 3 
3 = 0 = 3 
Cálculos auxiliares 
+ 3 = ( 3) 
+
1
3
=
1
3
 
8 + 2 = 2 2 + 2 = 2 
Cálculo auxiliar 
 Expoente10 85 
 
A( x B( x 
Q( x ) = x + x 2 x + R = 0 
 
23. 
a) n1, n2 n x 1, x 2 x , n1 + n2 
+ n n 7 n x 
b) x A( x ) = ( x – x 1)2( x – x 2) ( x – x )Q( x ), 
Q( x ) , 
A( x . 
 
24. 
P( x ) = ( x 1)( x + 1)( x + 2)( x + 2) ( 0). 
P( x x + 
P = 27 ( 1)( + 1)( + 2)( + 2) = 27
 × × × × = 27 
 = 27 
 = 
×
 
 = 16 
 P( x ) = 16 ( x 1)( x + 1)( x + 2)( x + 2) 
 = 16( x 2 1)( x 2 + 4 x + 4) 
 = 16( x 4 + 4 x + 4 x 2 – x 2 4 x 4) 
 = 16( x 4 + 4 x x 2 4 x 4) 
 = 16 x 4 – 64 x – 48 x 2 + 64 x + 64 
25. 
a) ( ) = 4 8 = 4( 2) 
 ( ) ( ) (2) 
 (2) = 2 22 – 5 2 + 4 = 8 + 12 – 10 + 4 = 2 
b) ( ) = 3 2 = 3 
 ( ) ( ) 
 = +3 × 5 × +4 
 = + +4 
 = + + 
 = 
 Expoente10 86 
 
c) ( ) = 4 + 2 = 4 + = 4 + 
 ( ) ( ) 
 = +3 × 5 × +4 
 = +3 × + +4 
 = + + + = + + + = 
26. P( x x P( x P( 1) = 0. 
O P( x x – P 
 
( 1) = 0
(3) = 40
× ( 1) + 2 × ( 1) + × ( 1) + 2 = 0
× (3) + 2 × (3) + × 3 + 2 = 40
+ 2 + 2 = 0
27 + 18 + 3 + 2 = 40
= 4
27 + 3 = 20
+ = 4
27 + 3 = 20
= 4
27 + 3(4 ) = 20
__________
27 + 12 3 = 20
____________
24 = 8
___________
=
8
24
= 4
1
3
=
1
3
=
11
3
=
1
3
 
= = 
 
27. 2 5 = 2 
P( x x – . P = 0. 
6 21 + 3 + 30 =
5
2
(6 6 12) 
=
5
2
6( 2)( ( 1)) 
= 6
5
2
( 2) ( 1) 
 
 
+
1
2
=
1
2
 
 Expoente10 87 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
28. 
a) P(1) = 6 1 – 7 12 – 14 1 + 15 = 6 – 7 – 14 + 15 = 0 
P P( x ). 
b) ( ) = 6 7 14 + 15 = ( 1)(6 15) = ( 1)6 + 
 = 6( 1) + 
 
 
 
 
 
 
 
 
 
 
c) P( x ) = 0 6( 1) + = 0 x 1 = 0 = 0 + = 0 
 x = 1 = = 
C.S. = , 1, 
d) P( x ) < 0 6( 1) + < 0 ( x 1) + < 0 
 x < 1 < x < 
 
 
 
 
Cálculos auxiliares 
 
 
6 x 2 – x 15 = 0 x = ± × ×( ) x = ± x = ± 
 
 x = x = x = x = 
Cálculos auxiliares 
 
6 6 12 = 0 = 
± × ×( )
 
 = 
±
 
 = 
±
 
 = 
±
 
 = 2 = 1 
 Expoente10 88 
 
 
 
 
 
 
 
 
 
 
 
C.S. = , 1,
 
29. 
a) P( x ) = 2 x – 6 x + 4. 
 
 
 
 
 
 
 
P( x ) = 2 x – 6 x + 4 = ( x – 1)(2 x 2 + 2 x – 4) = ( x – 1)2( x + 2)( x – 1) 
 = 2( x – 1)( x – 1)( x + 2) 
2 x – 6 x = –4 2 x – 6 x + 4 = 0 2( x – 1)( x – 1)( x + 2) = 0 
 x – 1 = 0 x – 1 = 0 x + 2 = 0 
 x = 1 x = 1 x = 2 
C.S. = {1, –2} 
b) P( x ) = 2 x – 7 x 2 + 9. 
 
 
 
 
 
 
 
P( x ) = 2 x – 7 x 2 + 9 = ( x – x 2 – x – = ( x – x – )( x + 1) 
 = 2( x – x – )( x + 1) 
2 x – 7 x 2 + 9 = 0 2( x – x – )( x + 1) = 0 x – x – = 0 x + 1 = 0 
 x x = x = –1 
C.S. = 1, , 3 
 
Cálculos auxiliares 
 
x 1 = 0 x = 1 = 0 x = + x = . 
Cálculos auxiliares 
 
2 x 2 + 2 x – 4 = 0 x = ± × ×( ) x = ± x = ± x = 2 x = 1 
Cálculos auxiliares 
2 x 2 – x – x = ± × ×( ) x = ± x = ± x = x = – 1 
 Expoente10 89 
 
c) P( x ) = x 4 – 2 x + 2 x 1. 
P( x ) = x 4 – 2 x + 2 x – 1 = ( x + 1)( x 1)( x 2 2 x + 1) 
 = ( x + 1)( x 1)( x 1)2 
 = ( x + 1)( x 1)( x 1)( x 1) 
x 4 – 2 x + 2 x – 1 = 0 ( x + 1)( x – 1)( x – 1)( x – 1) = 0 
 x + 1 = 0 x – 1 = 0 x – 1 = 0 x – 1 = 0 
 x = –1 x = 1 x = 1 x = 1 
C.S. = {–1, 1} 
d) P( x ) = 4 x 4 – 12 x + x 2 + 12 x + 4. 
P( x ) = 4 x 4 – 12 x + x 2 + 12 x + 4 
 = ( x – 2)( x – 2)(4 x 2 + 4 x + 1) 
 = ( x – 2)( x – 2)(2 x + 1)(2 x + 1) 
 
 
4 x 4 – 12 x + x 2 + 12 x + 4 = 0 ( x – 2)( x – 2)(2 x + 1)(2 x + 1) = 0 
 x – 2 = 0 x – 2 = 0 2 x + 1 = 0 2 x + 1 = 0 
 x = 2 x = 2 x = x = 
C.S. = , 2 
 
e) P( x ) = x 4 + 5 x + 4 x 2. 
P( x ) = x 4 + 5 x + 4 x 2 = x 2( x 2 + 5 x + 4) 
 = x 2( x + 4)( x + 1) 
 = x × x ( x + 4)( x + 1) 
 
 
 
 
x 4 + 5 x + 4 x 2 = 0 x x ( x + 4)( x + 1) = 0 
 x = 0 x = 0 x + 4 = 0 x + 1 = 0 
 x = 0 x = 0 x = –4 x = –1 
C.S. = {0, –4, –1} 
30. 
a) ( ) = . 
( ) = = ( 1) = × ( 1) 
 < < 0 × ( 1) < 0 
 < 0 0 < < 1 
 
C.S.= ] , 0[ ]0, 1[ 
Cálculo auxiliar 
Cálculo auxiliar 
 
Cálculo auxiliar 
x 2 + 5 x + 4 = 0 x = ± × × 
 x = ± 
 
 x = ± 
 
 x = 4 x = 1 
= 0
1 = 0 = 1 
Cálculos auxiliares 
 
 Expoente10 90 
 
b) ( ) = 3 + 2 5. 
( ) = 3 + 2 5 = ( 1)(3 + 3 + 5) 
 
 
 
 
 
 
 
 3 + 2 5 < 0 1 < 0 < 1 
C.S. = ] , 1[ 
 
c) ( ) = + 4 16 12 = ( + 3)( + 2)( 2) 
 = ( + 3)( + 2)( 2)( + 1) 
 
 
 
 
 
 
 
 
 + 4 16 12 0 ( + 3)( + 2)( 2)( + 1) 0 
 3 2 1 2 
 
C.S. = ] , 3] [ 2, 1] [2, + [ 
 
d) ( ) = 5 + 6 = ( 5 + 6) = × × ( 3)( 2) 
 
 
 
 
 5 + 6 > 0 × × ( 3)( 2) > 0 
 < 0 0 < < 2 > 3 
 
C.S. = ] , 0[ ]0, 2[ ]3, + [ 
3 + 3 + 5 = 0 =
3 ± 9 4 × 3 × 5
6
=
3 ± 51
6
 
Cálculos auxiliares 
 
. 
2 = 0 =
1 ± 1 4 × ( 2)
2
=
1 ± 9
2
=
1 ± 3
2
= 2 = 1 
Cálculos auxiliares 
 
5 + 6 = 0 =
5 ± 25 4 × 6
2
=
5 ± 1
2
= 3 = 2 
Cálculo auxiliar 
 Expoente10 91 
 
e) ( ) = + 3 4 = ( + 3 4) 
 = ( + 1)( + 4 4) 
 = ( + 1)( 4 + 4) 
 = ( + 1)( 2) 
 = ( + 1)( 2)( 2) 
 
 + 3 4 < 0 ( + 1)( 2)( 2) < 0 
 ( + 1)( 2)( 2) > 0 
 < 1 0 < < 2 > 2 
 
 
C.S. = ] , 1[ ]0, 2[ ]3, + [ 
 
31. 
a) (2 3)(1 2 ) ( ) < 0 < < 1 < < 2 
 
 
 
C.S. = , 1 , 2 
 
b) ( + 2 ) ( ) 0 ( + 2) ( ) 0 0 1 
 
 
C.S. = ] , 0] [1, + [ 
 
c) ( + 6 9) ( ) 0 ( 6 + 9) ( ) 0 
 ( 3) ( ) 0 
 ( 3)( 3) ( ) 0 
 1 2 = 0 = 3 
 
 [1, 2] 
Cálculo auxiliar 
2 3 = 0 2 = 3 =
3
2
 
1 2 = 0 1 = 2 =
1
2
 
Cálculos auxiliares 
 
= 0 
+ 2 = 0 = 2 = 2 
Cálculos auxiliares 
 Expoente10 92 
 
 
32. ( ) ( ) 
 ( ) ( ) = ( ) × ( ) + . 
 
 = × × + = 0 × + = 
 ( ) . 
 
33.
a) ( ) ( ) 
, ( ) = ( ) × ( ), ( ) 
 
 = × × = 0 × = 0 
 ( ) = 0. 
b) ( ) = 0. 
( ) ( ) . 
, ( ) = ( ) × ( ) + . 
 = × × + = 0 × + = 
( ) ( ) = ( ) × ( ) + . 
 = 0 ( ) = ( ) × ( ) , ( ) t. 
 
34. 
a) . 
 = 0, . 
 . 
b) ( ) + 
 + . . 
 ( ) = 0 ( ) = ( 1) × = ( 1) = ( 0) 
 ( 1) = 1 
n ( 1) = 1, 
( ) + .
 ( ) = ( 1) × 
 =
 é 
1 × = = 0 
 + . 
 + 
 Expoente1093 
 
c) ( ) + + 
+ + – + 
 ( ) + = 0 ( 1) × = ( 1) = ( 0) 
 ( 1) = 1 
 ( 1) = 1 
( ) + + . 
 + ( ) + = ( 1) × + 
 =
 é 
1 × + = + = 0 
+ + . 
+ + 
 
35. 
a) ( ) + ( ) = + 1 + ( ) ( ) + + 1 
 = + 1 + ( 1) × ( 1) × + + 1 
 = + 1 + ( 1) × 1 × + + 1 
 = + 1 + + 1 
 = 2 2 . 
Nota: 
2 + 1, 2 , 
 
 
b) 
 , × (1 1) × (1 + (1) = 0 × 0 × (1 + 1) = 0. 
 , 2 × 1) × + (1) = 0 × 0 × (1 + 1) = 0. 
 - 
1 = ( 1)( + + + + 1) 
+ + + + 1 1 1 + 1 + + 1 + 1 = × 1 = 
- 1. 
+ 1 1 1 + 1 = 2. 
( ) = ( 1)( 1)( + + + + 1)( + 1) 
 = ( 1) ( + + + + 1)( + 1) 
. 
 
36. 
a) 19 + 48 = 0 ( ) 19 + 48 = 0 
 y 
19 + 48 = 0 = 
± ×
 = 
±
 = 
±
 = 16 = 3 
b) y = 16 = 3 = 4 = 4 = 3 = 3 
C.S. = 4, 4 3, 3 
 
 
 Expoente10 94 
 
37. 
a) 13 + 36 = 0 ( ) 13 + 36 = 0 
 y 
13 + 36 = 0 = 
± ×
 = 
±
 = 
±
 = 9 = 4 
 
 y = 9 = 4 = 3 = 3 = 2 = 2 
 2} 
b) 26 + 25 = 0 ( ) 26 + 25 = 0 
 y 
26 + 25 = 0 = 
± ×
 = 
±
 = 
±
 = 25 = 1 
 
y = 25 = 1 = 5 = 5 = 1 = 1 
C.S. = {5, 5, 1, 1} 
c) 2 26 + 80 = 0 2( ) 26 + 80 = 0 
 y 
2 26 + 80 = 0 = 
± × ×
 = 
±
 = 
±
 = 8 = 5 
 
 y 
 = 8 = 5 = 8 = 8 = 5 = 5 
 = 2 2 = 2 2 = 5 = 5 
C.S. = 2 2, 2 2, 5, 5 
d) 3 4 = 0 ( ) 3 4 = 0 
 y 
 3 4 = 0 = 
± ×( )
 = 
±
 = 
±
 = 4 = 1 
 
y = 4 = 1
 
 = 2 = 2 
C.S. = {2, 2} 
 
 
 Expoente10 95 
 
Desafios 
Página 151
1. 
a) ( ) = ( 1)( 2). 
b) 
( ) 
 
2. 
a) ( ) = 
( )
 . 
b) ( ) = 
( )( )
. 
c) 
 
 
 
, 
d) ( ) = 2 ( ) + ( ) + 6 ( ) = ( 1)( 2) ( 2) + 3 ( 1) = 3 4 + 2 
 
3. 
-1, 
- 
 
( ) =
( 1)( 2)
6
+
( + 1)( 1)( 2)
2
( + 1) ( 2)
2
+ 13
( + 1) ( 1)
6
 
P(0) 2C(0) 1B(0) 6A(0) 2 1 1 0 6 0 2
P(1) 2C(1) 1B(1) 6A(1) 2 0 1 1 6 0 1
P(2) 2C(2) 1B(2) 6A(2) 2 0 1 0 6 1 6
96 
 Expoente10 
Tema III – Geometria analítica 
Unidade 1 – Geometria analítica no plano 
Páginas 154 a 181 
1. 
a) (1, 2) ( 3, 1) 
( , ) = ( 3 1) + 1 ( 2) = ( 4) + 3 = 16 + 9 = 25 = 5 
b) , 3 (0, 0) 
( , ) = 0 + 0 ( 3) = + 3 = + 9 = = 
c) (2016, 5) (2016, 4) 
( , ) = (2016 2016) + (4 5) = 0 + ( 1) = 0 + 1 = 1 = 1 
d) 0, 2 1, 2 3 
 ( , ) = (1 0) + 2 3 2 = 1 + ( 3) = 1 + 9 = 10 
e) 5, 3 3, 0 
( , ) = 5 3 + (3 0) = 5 + 3 + 9 = 5 + 2 5 × 3 + 3 + 9 
= 5 + 2 15 + 3 + 9 = 17 + 2 15 
f) ( , ) ( , ) 
( , ) = ( ) + ( ) = 2( ) = | | 2 
 
2. (5, 3) (3, 0) ( 1, 2) (1, 1) 
= (3 5) + (0 3) = ( 2) + ( 3) = 4 + 9 = 13 
= ( 2 0) + ( 1 3) = ( 2) + ( 4) = 4 + 16 = 20 = 2 5 
= 1 ( 2) + 1 ( 1) = 3 + 2 = 9 + 4 = 13
= (5 1) + (3 1) = 4 + 2 = 16 + 4 = 20 = 2 5 
= e = , , 
 
 
3. = = = = | | 
= = = = | | 
= 
 
4. 
a) (2, 7) (6, 11) 
A [ ] , = ,
 
97 
 Expoente10 
 
b) 1, 5 2, 3 5 
A [ ] 
( )
, = , = , 2 5 
c) ( + 2, 3 1) (5 + 4, 3) 
A [ ] 
 , = , = (3 + 3, 2 2) 
 
5. ( 3, 4) (1, 5) 
[ ] , = , = 1, 
 
6. (2, 4) 
a) = 
 - P 
= 
b) = 4 
 - P 
= 4 
c) = 0 
 - P 
= 0 
d) =
 - P 
= 
 
7. ( 3, 1) 
a) 0 
 - 1 P 
 0 
b) > 3 
 - 3 > P 
> 3 
c) 3 
 - 3 P 
3 
d) > 
 - 3 > P 
> 
 
8. 
a) 
 = 4 
98 
 Expoente10 
b) 
 = 0; > 0 
c) 
 = 5; 5 
 
 
9. 
a) 
 = 2 
 
b) 
 = 
 
c) 
< 3 
 
 
 
10. 
a) > 2 
 
 
b) 2 
 
 
c) 2 < 4 < 2 + 4 > 2 4 
 
 
 
 = 2 4 
 4 
 1 
99 
 Expoente10 
 
d) 3 0 3 3 
 
e) 2 > 1 
 
 
f) < 2 < 3 
 
g) 2 < 4 > 4 2 > 0 < 2 + 4 > 4 > 2 
 > 2 4 > 4 < 2 
 
h) < 0 ( < 0 > 0) ( > 0 < 0) 
 
i) = 0 ( )( + ) = 0 = 0 + = 0 = = 
 
j) 4 > 0 ( 2 )( + 2 ) > 0 
( 2 > 0 + 2 > 0) ( 2 < 0 + 2 < 0) 
( 2 > 2 > ) ( 2 < 2 < ) 
(2 < 2 > ) (2 > 2 < ) 
 < > > < 
 
 
11. 
a) ~( < 3) 3 
 
 
 
 = 2 4 
 4 
 1 
100 
 Expoente10 
b) ~( 1 > 0) > 1 0 
 
 
 
 
c) ( > 2 ) 2 < 
 
 
d) ( 4 < < 1) ( > 4 < 1) 4 1 
 
 
12. 
a) 2 > 1 
b) 1 < < 1 
c) 1 1 2 3 
d) 4 < < 4 
 
13. (1, 3) ( 4, 2) (2, ), 
= (2 1) + ( 3) = 2 ( 4) + ( 2) 
1 + ( + 3) = 6 + ( 2) 
1 + + 6 + 9 = 36 + 4 + 4 
10 = 30 
= 3 
 
14. 
a) ( 3, 2) (1, 0) 
( 3) + ( 2) = ( 1) + ( 0) + 6 + 9 + 4 + 4 = 2 + 1 + 
 4 = 8 12 
 = 2 + 3 
[ ] = 2 + 3 . 
 
b) (1, 7) (4, 7) 
A B AB
AB 
AB , = , 7 
[ ] = . 
 
101 
 Expoente10 
 
15. (2, 0) ( 5, 1) 
a) ( 2) + ( 0) = ( + 5) + ( 1) 4 + 4 + = + 10 + 25 + 2 + 1 
 2 = 14 + 22 
 = 7 + 11 
[ ] = 7 + 11 . 
 
b) PQ P 
Q 
 
(1 2) + (11 0) = (1 + 5) + (11 1) ( 1) + 11 = 6 + 10 
1 + 121 = 36 + 100 
122 = 136 
PQ 
 
c) = 7 + 11 PQ 
 = 7 -
PQ 
 = 7 - 
PQ 
 = 7 -
PQ 
 
16. A B C( 
a) 
i. = ( 1 0) + (4 5) = ( 1) + ( 1) = 1 + 1 = 2 
A ( 2) + ( 3) = 2 
ii. B A 
= (2 0) + (3 5) = 2 + ( 2) = 4 + 4 = 8 = 2 2 
B A + ( 5) = 8 
iii. C 
C 
C 
( + 1) + ( 4) = 16 
b) P 
 
P 
i. (1 2) + (2 3) = 2 ( 1) + ( 1) = 2 1 + 1 = 2 2 = 2
P A 
ii. 1 + (2 5) = 8 1 + ( 3) = 8 1 + 9 = 8 10 = 8
P B A 
102 
 Expoente10 
iii. (1 + 1) + (2 4) = 16 2 + ( 2) = 16 4 + 4 = 16 8 = 16
P C 
 
 
17. 
a) ( 1) + ( 2) = 9 
(1, 2) 
9 = 3 
b) + + 3 = 2 
, 3 
2 
 
c) + = 1 
(0, 0) 
1 = 1 
 
d) + 2 4 4 = 0 2 + 1 + 4 + 4 = 4 + 1 + 4 
( 1) + ( 2) = 9 
(1, 2) 
9 = 3 
e) 2 + 2 + 8 12 + 16 = 0 + + 4 6 + 8 = 0 
 + 4 + 4 + 6 + 9 = 8 + 4 + 9 
 ( + 2) + ( 3) = 5 
( 2, 3) 
5 
f) + + = 0 + + + = + 
 + + = 
0, 
 = 
 
18. 
a) ( 1) + ( + 2) 9 C 
b) - 
 
 (0 1) + (3 + 2) 9 ( 1) + 5 9 1 + 25 9 26 9
A 
 (1 1) + (1 + 2) 9 0 + 3 9 9 9
B 
103 
 Expoente10 
 1 + + 2 9 + 9 + 9 9
D 
 
19. 
a) ( + 1) + ( 2) 4 2 
 
 
 
 
b) + > 1 < 0 < 0 
 
c) ( 3) + < 9 1 < < 5 
 
d) 4 + 16 
 
 
e) ( + 1) + ( + 1) 25 ( 1) + ( 1) 1 
 
 
20. 
a) 1 < + < 4 
b) ( 2) + ( 4) 4 5 
c) ( + 36 0 0) ( + 36 3) 
 
104 
 Expoente10 
 
21. 
a) ( 3, 5) (1, 1) 
AB 
( + 3) + ( 5) = ( 1) + ( 1) 
+ 6 + 9 + 10 + 25 = 2 + 1 + 2 + 1 
8 = 8 32 
= + 4 
 
b) (2, 3) 
C ( 2) + ( + 3) 16 
 
c) - P( , ) 
( , ) = 2 ( , ) ( + 5) + ( 4) = 2 ( 1) + ( 4) 
( + 5) + ( 4) = 4(( 1) + ( 4) ) 
+ 10 + 25 + 8 + 16 = 4( 2 + 1 + 8 + 16) 
+ 10 + 25 + 8 + 16 = 4 8 + 4 + 4 32 + 64 
 3 + 18 3 + 24 27 = 0 
6 + 8 + 9 = 0 
6 + 9 + 8 + 16 = 16 
( 3) + ( 4) = 16 
D(
E C 
 
22. 
a) + = 1 
= 25 = 9 = 5 = 3 2 = 10 2 = 6
= 25 9 = 16 = 4 2 = 8
 
 
b) + = 1 
= 16 = 3 = 4 = 3 2 = 8 2 = 2 3 
= 16 3 = 13 2 = 2 13 
2 3
2 13 13 13 
23. 
a) 
 
2 7 c = 7 
A( 7 B( 7 
 
 
105 
 Expoente10 
b) c = 7 
-
a 
= 4 7 = 16 7 = 9 = 3 
 + = 1 
24. A(
B 
 
2 = 7 = = 
= 4 = = 
 + = 1 
25.
a) + = 1 
b) + = 1 = 9 = 3 2 = 6
= 1 = 1 2 = 2 2 2, 0 2 2, 0 
= 9 1 = 8 = 2 2 
c) 6 + 9 = 18 + = 1 2 3 = 3, 
 = 3 2 = 2 3 2 2 = 2 = 2 2 = 2 2
(1, 0) ( 1, 0) = 3 2 = 1 
d) 25 + 4 = 100 + = 1 10 = 25
= 5 2 = 10 4 = 4 = 2 2 = 4 0, 21
0, 21 = 25 4 = 21 
e) 4 + 4 = 24 + = 6 6 
 
 
Aprende Fazendo 
Páginas 182 a 189 
1. (4, 4) ( 1, 5) ( 2, 0)
( , ) = 4 ( 1) + (4 5) = 5 + ( 1) = 25 + 1 = 26 
( , ) = 4 ( 2) + (4 0) = 6 + 4 = 36 + 16 = 52 
( , ) = 1 ( 2) + (5 0)= 1 + 5 = 1 + 25 = 26 
( , ) = ( , ) ( , ) ABC
 
106 
 Expoente10 
 
( , ) = 52 = 52 
( , ) + ( , ) = 26 + 26 = 26 + 26 = 52 
( , ) = ( , ) + ( , )
ABC -
 
Opção (A) 
 
2. = 
-
3 
3 3 = 
Opção (B) 
 
3. (2, 5) ( 2, 3) 
(0, 8) 
( , ) = (0 2) + (8 5) = ( 2) + 3 = 4 + 9 = 13 
( , ) = (0 + 2) + (8 3) = 2 + 5 = 4 + 25 = 29 
 P AB ( , ) ( , ) 
 , 
( , ) = 2 + 5 = + = + = = 
 
( , ) = + 2 + 3 = + = + = = 
Q AB ( , ) ( , ) 
(5, 2) 
( , ) = (5 2) + (2 5) = 3 + ( 3) = 9 + 9 = 18 
( , ) = (5 + 2) + (2 3) = 7 + ( 1) = 49 + 1 = 50 
 R AB ( , ) ( , ) 
(0, 4) 
( , ) = (0 2) + (4 5) = ( 2) + ( 1) = 4 + 1 = 5 
( , ) = (0 + 2) + (4 3) = 2 + 1 = 4 + 1 = 5 
 S AB ( , ) = ( , ) 
Opção (D) 
 
4. + 2 + ( ) = 2, 
 = = = 
Opção (A) 
 
5. A 
Opção (D) 
107 
 Expoente10 
 
6. (3, 2 ) (1, 5) 
( , ) = 13 (3 1) + ( 2 + 5) = 13 
 2 + (3 ) = 13 
 (3 ) = 9 
 3 = 3 3 = 3 
 = 0 = 6 
Opção (B) 
 
7. ( + 3) + ( 2) = 9 = 1 ( 1 + 3) + ( 2) = 9 
 2 + ( 2) = 9 
 ( 2) = 5 
 2 = 5 2 = 5 
= 2 + 5 = 2 5
 = 1 
( + 3) + ( 2) = 9 1, 2 + 5 
1, 2 5 
Opção (B) 
 
8. = A 
 
2 + 9 = 5 2 + 9 5 = 0 
= 
± × ×( )
 
= 
±
 
 = 
±
 
 = 5 = 
Opção (C) 
 
9. 
+ 9
= = 
+ 9 
Opção (C) 
10. -
2 = 16 = 8 = 64 c 
= 8 + 6 = 100 = 10 = 100
 + = 1 
Opção (D) 
108 
 Expoente10 
 
11. + = 16 
 + = 1 2 = 2 25 = 10 2 = 2 16 = 8
 
Opção (D) 
 
12. + + 6 + 9 = 0 + ( + 3) = 0 
Opção (B) 
 
13. P + = 14 
A B 
Opção (C) 
 
14. 
a) (2, 4) (6, 15) 
 ( , ) = (2 6) + (4 15) = ( 4) + ( 11) = 16 + 121 = 137 
b) ( 2, 3) (6, 5) 
 ( , ) = ( 2 6) + 3 ( 5) = ( 8) + 8 = 64 + 64 = 128 
c) (1, 4) ( 2, 0) 
 ( , ) = 1 ( 2) + ( 4 0) = 3 + ( 4) = 9 + 16 = 25 = 5 
d) ( 2, 2) ( 6, 5) 
 ( , ) = 2 ( 6) + ( 2 5) = 4 + ( 7) = 16 + 49 = 65 
 
15. 
a) ( 2, 1) ( 3, 1) 
= 2 ( 3) + ( 1 1) = 1 + ( 2) = 1 + 4 = 5 
b) (2, 2) (0, 5) 
= (2 0) + 2 ( 5) = 2 + 7 = 4 + 49 = 53 
16. A B C D 
= (1 6) + 2 ( 1) = ( 5) + ( 1) = 25 + 1 = 26 
= (6 9) + ( 1 3) = ( 3) + ( 4) = 9 + 16 = 25 = 5 
= (9 4) + (3 2) = 5 + 1 = 25 + 1 = 26 
= (4 1) + 2 ( 2) = 3 + 4 = 9 + 16 = 25 = 5 
= = ABCD 
109 
 Expoente10 
17. 
a) (2, 4) (2, 15) 
A B AB
= 2 AB M 
M = , = 2, 
AB =
b) ( 2, 3) (6, 3) 
A B AB
= 3 AB M 
M = , = (2, 3) 
AB = 2 
c) (1, 2) ( 2, 0) 
( 1) + ( + 2) = ( + 2) + 2 + 1 + + 4 + 4 = + 4 + 4 + 
4 = 6 1 
= AB] 
d) ( 2, 2) ( 1, 3) 
( + 2) + ( + 2) = ( + 1) + ( 3) 
+ 4 + 4 + + 4 + 4 = + 2 + 1 + 6 + 9 
10 = 2 + 2 
= + AB] 
 
18. A 3 B D C 
( , ) = (1 2) + 3 0 = ( 1) + 3 = 1 + 3 = 4 = 2 
( , ) = (2 2) + (2 0) = 0 + 2 = 0 + 4 = 4 = 2 
( , ) = (0 2) + (0 0) = ( 2) + 0 = 4 + 0 = 4 = 2 
A B D - C
C 
 
19. 
a) O B C D 
OB 
BC 
CO 
DO 
b) (i) = 2 2 4 
(ii) = 3 
(iii) ( 4) + ( 2) = 9 
(iv) 0 4 0 2 
110 
 Expoente10 
 
20. 
a) ( 1) + ( + 1) = 4 (= 4 
b) + ( + 3) = 8 2 2 (= 8 
c) ( + 4) + ( 3) =
d) ( 5) + > 20 2 5
(= 20 
e) ( + 3) + ( + 5) < 9 
3 (= 9 
f) 2 ( 2) + 4 2, 
2 
 
 
21. 
a) P( Q 
 ( , ) = ( 2 3) + ( 3 3) 
 = ( 5) + ( 6) 
 = 25 + 36 
 = 61 
b) PQ , = , 0 
c) PQ PQ
P Q
PQ + = 
e) ( + 2) + ( + 3) = ( 3) + ( 3) 
+ 4 + 4 + + 6 + 9 = 6 + 9 + 6 + 9 
12 = 10 + 5 
= + PQ] 
f) ( 2 1 3 3) ( 3 0 ) 
 
22. 
a) = 4 = 16 
 = 3, = 9 
 + = 1 
b) = 5, = 25 
= 1, = 1 
+ = 1 
 
111 
 Expoente10 
c) = 2, = 4 
= 7, = 49 
+ = 1 
 
23. 
a) + = 1 
= 16 = 4 2 = 8 = 9 = 3 2 = 6 
 = 16 9 = 7
7, 0 7, 0 
b) + = 1 
= 20 = 20 = 2 5 2 = 4 5 = 11 = 11 2 = 2 11
= 20 11 = 9 = 3 4 5
2 11 ( 3, 0) (3, 0) 
 
 
24. = = A B C 
 = 5 ( 3) + 1 1 + 2 3 = 5 ( 1) + (1 ) 
 ( 2) + 2 3 = ( 4) + (1 ) 
 4 + 12 = 16 + (1 ) 
 (1 ) 
 1 = 0 
 = 1 
C 
 
25. 
a) 0 5 
b) ( 1 0 1 0) (0 1 0 1) 
c) ( 2) + ( 1) 4 1 
d) ( > 0) ( < 0) ( < 0) ( > 0) 
e) + ( 1) 1 + ( 2) 4 
f) ( + 6) + = 1 + = 1 ( 6) + = 1 
 
26. 
a) < 7 2 
 
112 
 Expoente10 
b) ( 2 1) ( = > 2) 
 
 
c) ( 3) + ( + 2) 4 ( 3) + 4 
 
 
 
 
d) + > 1 + < 4 0 
 
 
e) + = 1 < 0 
 
f) ( 3 < < 3 ) + = 4 
 
 
 
27. 
a) + + 2 10 + 14 = 0 + 2 + 1 + 10 + 25 = 14 + 1 + 25 
 ( + 1) + ( 5) = 12 
12 = 2 3 
113 
 Expoente10 
b) = 1 
(1 + 1) + ( 5) = 12 ( 5) = 8 5 = 8 5 = 8 
 = 5 + 2 2 = 5 2 2 
1, 5 + 2 2 1, 5 2 2 
c) A 
 (2 + 1) + (3 5) = 3 + ( 2) = 9 + 4 = 13 
 13 A - 
B - 
( 1 + 1) + (4 5) = 0 + ( 1) = 0 + 1 = 1 
 B - 
d) 
= 0 
( + 1) + (0 5) = 12 ( + 1) + 25 = 12 
 ( + 1) = 13 
 
 
= 0 
(0 + 1) + ( 5) = 12 1 + ( 5) = 12 
 ( 5) = 11 
 5 = 11 5 = 11
 = 5 + 11 = 5 11 
0, 5 + 11 0, 5 11 
 
e) ( 2) + ( 3) = ( + 1) + ( 4) 
4 + 4 + 6 + 9 = + 2 + 1 + 8 + 16 
2 = 6 + 4 
= 3 + 2 AB] 
f) - 
 
AB 
 
28. 
a) 2 + + 8 = 8 2 + 1 + + 8 + 16 = 8 + 1 + 16 
 ( 1) + ( + 4) = 9 
 
b) 2 2 + 2 + 2 = 7 + + = + + + + = + + 
 + + = 4 
, 
 
114 
 Expoente10 
 
c) + + + + + 9 
0, 
d) + 12 + + 28 > 0 + 12 + 36 + > 28 + 26 ( + 6) + > 8 
2 2 
e) 25 + 36 = 900 + = 1 
 = 2 × 36 
= 2 × 25 11, 0 11, 0 
 
 
 
f) 4 + 8 = 32 + = 1 
4 2 (= 2 × 8 
= 2 × 4 ( 2, 0) (2, 0) 
 
 
 
 
29. 
a) 2 = 6 = 3 
 2 = 12 = 6 
 = 6 3 = 36 9 = 27 
 + = 1 
b) 2 = 10 = 5 
 + = 1 
 
8
+
3
25
= 1
64
+
9
25
= 1 1600 + 9 = 25 16 = 1600 = 100 
+ = 1 
c) = 8 
= = 10 = 100 
= 100 64 = 36 = 6 = 36 
 + = 1 
 
Cálculo auxiliar 
 = 36 25 = 11 
Cálculo auxiliar 
 = 8 4 = 4 = 2 
115 
 Expoente10 
d) + = 1 
- 
 
( )
+ = 1
+ = 1
+ = 1
= 1
+ = 1
= 4
 
 =
= 4
 
 
 =
= 4
 
 
+ = 1 
 
29. 
a) A B AB 
 ( 2) + ( + 3) = ( + 1) + ( 1) 
4 + 4 + + 6 + 9 = + 2 + 1 + 2 + 1 
8 = 6 11 
= AB] 
 
b) C(
C ( + 3) + < 25 
 
c) A(
B A B 
2 = 9 = = = 9 = 
L + = 1 
d) P A B
 
( , ) = ( , ) ( 1) + ( 3) = ( 1) + ( 6) 
 2 ( 1) + ( 3) = ( 1) + ( 6) 
 4(( 1) + ( 3) ) = ( 1) + ( 6) 
 4( 2 + 1 + 6 + 9) = 2 + 1 + 12 + 36 
 4 8 + 4 24 + 40 = 2 + 12 + 37 
 3 + 3 6 12 = 3 
 + 2 4 = 1 
 2 + 1 + 4 + 4 = 1 + 1 + 4 
 ( 1) + ( 2) = 4 
 
116 
 Expoente10 
31. B = 8 = B , 
( , ) = (0 ) + 2 = + 
 = 8 + 
 = 8 + 256 256 +64
2
8
 
 = 8 + 4 4 + 
 = + 4 + 4 
 = ( + 2) 
 = + 2 
 
32. P P 
2 P AB 
( , ) = ( , ) ( 1) + (2 1) = ( 3) + (2 3) 
( 1) + (2 1) = ( 3) + (2 3) 
2 + 1 + 4 4 + 1 = 6 + 9 + 4 12 + 9 
12 = 16 
= 
P , . 
 
33. 
a) + + 10 4 + = 0 + 10 + 25 + 4 + 4 = + 25 + 4 
 ( + 5) + ( 2) = 29 
29 > 0 < 29. 
b) 
29 = 0 = 29 
c) 
29 < 0 > 29 
 
34. 
a) A 
OA 
 + = ( + 3) + ( + 3) + = + 6 + 9 + + 6 + 9