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Resoluções das Atividades
LIVRO 1 | ÁLGEBRA
SumárioCapítulo 1 – Potenciação................................................................................................................................................................................................................. 1
Capítulo 2 – Radiciação ................................................................................................................................................................................................................... 5
Capítulo 3 – Radicais ....................................................................................................................................................................................................................... 7
Capítulo 4 – Operações com radicais I ........................................................................................................................................................................................... 9
Capítulo 5 – Operações com radicais II ........................................................................................................................................................................................ 12
9o Ano – Ensino Fundamental II | 1
01 a) 34 = 3 · 3 · 3 · 3 = 81b) 83 = 8 · 8 · 8 = 512c) 2.0100 =1d) (–3)2 = (–3) · (–3) = + 9e) (–4)4 = (–4) · (–4) · (–4) · (–4) = + 256
f) 02010 = 0 0 0 0
2 010
0⋅ ⋅ ⋅ ⋅ =…
. fatores
g) −
= −
⋅ −
= +
1
7
1
7
1
7
1
49
2
h) (– 0,666...)2 =
x
x
x
x
= ⋅
=
− =
=
0 666 10
10 6 666
0 666
9 6
,
,
,
Logo, x x= =
÷
÷
6
9
2
3
3
3
( , )− = −
= −
⋅ −
= +0 666 2
3
2
3
2
3
4
92
2
i) (–0,111...)3
x = 0,111... · 10
10 1 111
0 1119 1
x
xx
=
− =
=
,
,,
Logo, x =1
9
02 a) 106 = 1.000.000b) 18 = 1
c) 023 = 0 d) (–1)2 = + 1 e) (–1)13 = – 1 f) (–100)12 = +1 000 0
24
. …
zeros
g) 10243 = 1 000 0243
. …
zeros
h) – (–1)194 = – (+1) = –1 i) – (–1)201 = – (–1) = + 1
( , ...)− = −
=
= −
⋅ −
⋅ −
=
−
0 111 1
9
1
9
1
9
1
9
1
7
3
2
229
03 I. a) ( , ) ( )− = −
= −
= − = −−− −
0 2 2
10
1
55 1253
3 3
2
b) − = − = −−6
1
6
1
362
2
c) ( )( )
− =
−
=−6
1
6
1
362
2
d) ( , )0 222 2
9
9
2
729
83
3 3
−−
=
= =
x
x
x
x
x
= ⋅
=
− =
=
=
0 222 10
10 2 2 22
0 222
9 2
2
9
,
,
,
e) − −
= − −
= − +
= −
−2
3
3
2
81
16
81
16
4 4
II. a)1
44
3
3=
− d) 19
10
110
19
1
10
19
10
191
1
= =
=
−
b)1
6
1
66
1
1= =
− e)1
77
9
9=
−
c)1
1010
5
5=
− f) 3
4
1
4
3
4
3
7
7
7 =
=
−
04 I. a) (–2)–3 – (–2)–2 – (–2)–5
−
− −
− −
−
− +
− −
1
2
1
2
1
2
1
8
1
4
1
32
3 2 5
− − + = − − +
= −1
8
1
4
1
32
4 8 1
32
11
32
b) (3 · 4)–1 · (3–1 + 4–1)–1
1
12
1
3
1
4
1
12
4 3
12
1
12
7
121
12
12
7
1
1
1
1
⋅ +
⋅ +
⋅
⋅ =
−
−
−
77
Testando seus Conhecimentos – pág. 4
Potenciação Aula 1Capítulo 1
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2 | 9o Ano – Ensino Fundamental II
LIVRO 1 | ÁLGEBRA
01 a) 28.000 = 2,8000 · 10.000 = 2,8 · 104
b) 360 = 3,60 · 100 = 3,6 · 102
c) 0 03 3
100
3
103 10
2
2, = = = ⋅−
d) 0 00128 1 28
1 000
1 28
10 128 10
3
3, ,
.
,,= = = ⋅
−
e) 23,04 = 2,304 · 10
f) 128,03 = 1,2803 · 100 = 1,2803 · 102
g) 346,15 = 3,4615 · 100 = 3,4615 · 102
h)0 006937
6 937
1 000
6 937
10 6 937 103
3
,
,
.
,
,= = = ⋅
−
02 a) 1,5 · 100.000.000 = 1,5 · 108
b) 3,84 · 100.000 = 3,84 · 105
01 I. a) ( ) ( ) ( )
( )( )
− ⋅ −
−( )
= −
− = −
5 5
5
5
55
3 38
2 20
41
40
1
b) ( ) ( ) ( ) ( )
( )
( )
( )
− ⋅ −
⋅
=
− ⋅ −[ ]⋅
=+
+
2 10
4 5
2 10
4 5
20
20
50 50
37 37
50
37
50
37 == +( )20 13
c) ( ) ( )− − − +
− −
− +
− −
− − − − −
−
2 2 2 2
1
2
1
4
1
2
1
4
2 1
4
1 2 2 1 2 2
2 2
−
+
−
−
+ − = + −
−
−
2 2
2 2
2 1
4
3
4
3
4
16
9
9
16
256 81
144 ==
+175
144
d)
3
2
2
3
2
3 3 2
2
3
2
3
5 4
10
1 3
5
÷
÷ ⋅
=
÷
− −
−( )
÷ ⋅
=
÷
∴
∴
4
10 3
9
10 32
3
1
3 2
2
3
2
3
2
3
2
33
2
3
2
3
4
9
9
7
2
=
=
II. a) 1,32 = (1,3) · (1,3) = 1,69 b) 0,42 = (0,4) · (0,4) = 0,16 c) –0,26 = –(0,2) · (0,2) · (0,2) · (0,2) · (0,2) · (0,2) =
– 0,000064 d) (–0,08)3 = (–0,08) · (–0,08) · (–0,08) = – 0,000512
05 2 2 2 1 2 3
16
1
2 010
1
2
1 1 2 2
1 1
− − −− −
+ − − − ÷ − +
+ −
( ) ( ) ( ).
−− −
÷ −
+ −
−[ ] ÷ −
+ −
1
2 4 1
1
4
16
3 2 010
4
4 1
4
16
3 2 010
.
.
−−[ ] ÷ +
+ −
−[ ] ÷ +
+ −
− + −
4 3
4
16
3 2 010
4 4
3
16
3 2 010
16
3
16
3
.
.
22 010 2 010. .⇒ −
Atividades Propostas – pág. 5
c) 9 11
100 009 11 10
31
31,,
…
zeros
= ⋅−
d) 5 974 1 00 0 5 974 1024
24, ,⋅ = ⋅…
zeros
e) 1 66
100 0
166 10
23
23,,
…
zeros
= ⋅−
f) 3 · 100.000.000 = 3 · 108
03 x y
xy
x y
xy
y x
xy
xy
x y
xy
xyy x
− −
−
+=
+
=
+
=+
⋅ = +
1 1
1
1 1
1 1 1( )
04 a) − − − −
− − − −
− − − − − − +[
2 3 2 3
8 9 8 9
8 9 1
8 9 1
3 2 3 2 51
51
51
( )
( )
( )
]]
− − = −8 10 18
b) 2
5
5
42 3
5
2
5
4
1
49
25
4
2
2 2
2
÷ − − − −
÷ − +
− +
+
−−( ) ( )
( )
⋅ +
− −
+ − −
− − = − −
= −
4
5
1
49
5
1
4 9
1
4
4
1
1 16
4
17
4
05 E x y x y
x y x y
x y x
x y y
x
y
E
= −
+ =
⋅ −⋅ +
= −
+
=−
3 3 4 3
3 3 3 4
3 3
3 3
1
1
1
1
1 1
2
( )
( )
+ − =
−−
= −−
= +
−1
1 3
1 2
1 3
1
2
1
2( )
06 C, F
Escrevendo o valor em função da potência de base dez,tem-se o seguinte resultado: 3,75 · 10–8, que, por sua vez,pode ser representado como 0,375 · 10–7 (item C) ou375 · 10–10 (item F), os quais determinam o mesmo resul-tado da expressão dada na questão.
Testando seus Conhecimentos – pág. 7
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9o Ano – Ensino Fundamental II | 3
LIVRO 1 | ÁLGEBRA
c) ( , )
( )
0 2
5
2
10
1
5
1
5
1
5
21
1 15
21
15
21
− − − −=
=
115
36
361
55=
= −
d) ( )x xn n3 2 6− −
=
II. a b
c
c
a b
a b
c
c
a
b b
c
2 3
3
2 2
6 3
3
2
6
2 5
1
1
3 1
⋅
⋅
⋅⋅ ⋅ = =
⋅
−
( 00
2 10
3 10
2 10
3
2 10
243 100
224 300
2 12 15
8 5
38
5 40
38
52
)
..
⋅ =
= ⋅
⋅ = ⋅ =
⋅=
= = 00
02 a) 2 10 7 3 2 1 0 7
10 7 2 10 7
8 10 7
3 10 7
8
3
⋅ −
+ ⋅ ⋅ −
− + ⋅ −
=⋅
−
⋅ −
=
b)( ) ( )
( ) ( )
10 3 3 10 3 2 3
10 2 3 10 4 4
10 9 1018
10 6 10 16
− ⋅ −
−
− ⋅ − =
− ⋅− ⋅ −
==−
=109
10 22 1031
03 ( ) ( )
( ) ( )
( ) (10 2 5 10
2 10 5
10 2 55 4 2 3 2 3
3 2 2
5 4 2− − − −
−
− −÷ ÷ ÷− ⋅ − ÷
= ⋅ ÷ 33 2 3
10 8
9 6
10
1
8 100 25
10 2
5 10
100
8
⋅
−
⋅ +
÷
∴
⋅⋅
−
−
−
)
( )
÷
=
⋅ ⋅⋅ ⋅
− ⋅⇒
⋅⋅
−=
− −
−
− −
−
25
1
2 5 2
5 2 5
100
8
1
25
2 5
2 5
1
2
10 10 8
9 6 6
2 10
6 3 22 5
2
2 5 2 5 10
8 7
1
7 7 7 7
− −
−
− − − −
⋅−
∴
− ⋅ = − ⋅ = −( )
04 a) (V)
b) (F) (+3 + 4)2 = (+7)2 = 49, logo, falso 32 + 42 = 9 + 16 = 25
c) (V)
d) (F) 4 3 1 024
27
4
3
16
9
1 024
27
16
9
5 3
2
÷ =
=
≠
.
.
e) (F) (a – b)2
= a2
– 2ab + b2
≠ a2
– b2
f) (F) a
b
a
ba b
= ≠ −
2 2
2
2 2
05 − ⋅ − ⋅ − −( )
− −( ) ⋅
∴
∴
−
− −
− −
0 25 0 125 0 0625
0 5 0 03125
4 2 3 2 3
3 5
, ( , ) [ , ]
, ,
[(00 5 0 5 0 5
0 5 0 5
2 4 3 2 3 4 2 3
3 5
, ) ] { [( , ) ] } { [ , ] }
( , ) [( , ) ]
⋅ − ⋅ − −( )− − ⋅
− −
− −55
8 18 24
3 25
34
0 5 0 5 0 5
0 5 0 5
0 5
∴
∴− ⋅ ⋅ −
+ ⋅
∴
∴−
− −
− −
−
( , ) ( , ) ( , )
( , ) ( , )
( , )(( , )
( , )0 5
0 528
6
−
−
= −
01 a) 33p · 3–2p = 33p + (–2p) = 3p
b) ( ) ( ) ( ) ( ) ( ) ( ) ( )− ⋅ − ⋅
= − ⋅ − ⋅ + = − ⋅ + = −−2 2
1
2 2 2 2 2 2 24
6
4 1 6 5 6 11
c) 52x + 1 ÷ 5x + 1 = 52x + 1 – (x + 1) = 52x – x + 1 – 1 = 5x
d) [ ( ) ] ( )
[ ( ) ] (
4 8 125 10 2 25
2 2 5 10 2
5 2 14 5 11 1 4
10 6 3 14 20 44
⋅ ÷ ⋅ ⋅ ∴
∴ ⋅ ⋅ ÷ ⋅ ⋅
−
55
2 2 5 10 2 5 2 5
2 5 2 5
8
10 84 4 2 20 44 894 42
20 20 44 8
−
−
−
∴
∴ ⋅ ⋅ ÷ ⋅ ⋅ ⇒⋅
⋅ ⋅ ⋅
)
[ ] ( ) ∴∴
∴⋅
⋅
= ⋅ = ⋅ =2 5
2 5 2 5 2 5 10
94 42
64 12
30 30 30 30( )
e)[( ) ]
[ ] [ ]( )a
a a
a
a
a
aa a
p p pp p
−
−
−
−
−
−
− + − ⋅ −
⋅
= = = =
3 2
4 10 5
6
6 5
6
30
6 30 6 5
02 a) −( ) +( ) = −
−( ) +( )
4 4 47 7 14
.
b) − − ⋅ + = −
− +
( ) ( )( ) ( )
3 3 310 13 23
c)− − ⋅ − ⋅ − = +
− + −
( ) ( ) ( )( ) ( ) ( )
2 2 2 218 20 21 59
d) − −
÷ − +
− +
1
5
1
5
3 5 5 4
( ) ( )
= −
÷
= −
−1
5
1
5
1
5
15 20 5
03 m m=
⋅ ⋅
⋅
= = ∴ = =
− −
− −
−
−
10 10 10
10 10
10
10
1 1 15 6 5
3 1 3
6
6
2 2
( )
( )
04 D
a2 = 996, b3 = 997 e c4 = 998
(abc)12 = a12 · b12 · c12
= (a2)6 · (b3)4 · (c4)3
= (996)6 · (997)4 · (998)3
= 9936 · 9928 · 9924 = 9988
01 P = 3 · (–1)2n + 5 · (–1)2n + 1 – (–1)2n + 6 + 4 · (–1)3n
a) n e ímpar: P = 3 · (+1) + 5 · (–1) – (+1) + 4 · (–1) + 3 – 5 – 1 – 4 = – 7 b) n é par: P = 3 · (+1) + 5 · (–1) – (+1) + 4 · (+1) + 3 – 5 – 1 + 4 = +1
02 x = (–1)8n – (–1)4n + 7 + (–1)7n – 2 · (–1)n Considerando n par:
x = (+1) – (–1) + (+1) – 2 · (+1) + 1 + 1 + 1 – 2 = + 1 Considerando n ímpar: X = (+1) – (–1) + (–1) – 2 · (–1) + 1 + 1 – 1 + 2 = + 3
03 a)2 2 2 2
2 2 2
2 2 2
2 2 1
8 2
1
2 1
43 1
1
3 1
1
n n
n n
n
n
⋅ − ⋅
⋅ +
=
−( )+( )
=−
+
=− −
Atividades Propostas – pág. 9
Testando seus Conhecimentos – pág. 11
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4 | 9o Ano – Ensino Fundamental II
LIVRO 1 | ÁLGEBRA
01 a)
( ) [( ) ( ) ]
( )
( ) (10 10 10 10 10
10 10
10 10 102 2
2
2x y x x y y
x y
x x y+ ⋅ − +
+=
− ++ + yy
x y
)2
10 10+
b) 2 1 2 5 2
2 1 2 17
3 20
1 8 17
17
24
17
24
2
3
x
x
⋅ + − ⋅
⋅ − −
=−
− −
=−
−
=+( )
( )
02 a) 10 10
10
10 10 1
10 10
1000 1
100
999
100 9
2 3 2
2 2
2 3
2 2
a a
a
a
a
+
+
−=
⋅ −
⋅
=−
= =( )
,999
b) a b
a ba b
−( )
−( )= −
3
2
03 E a ba b
a b=−
−
= − = ⋅ − ⋅ = ⋅ ⋅ − ⋅ =
⋅ − ⋅
( )( )
3
2
8 6 2 6 6
6
5 10 2 10 5 10 10 2 10
500 10 2 100 500 2 10 498 10 4 98 106 6 6 8= − ⋅ = ⋅ = ⋅( ) ,
04 2 2 2 2
2 2 2
20 1
21
4
1
2
40 1
21 2
4
41
23
4
41
2
4 2 1
2 1
n
n
⋅ + +
⋅ +
=
+
+
=
+
+= = ⋅
−
− −
( )
( )
44
3
82
3
2
=
05 E
1
2
1
2
1
22 2
1
2
1
2
1
2
4
4
4
− ⋅
⋅
=
⋅
−
+
− −
x x
x
x
⋅
1
2
1
2
4x
07 Basta contar os fatores 5.
⇒ 5, 10, 15, ... , 1.000⇒ Tem 200 fatores. ⇒ 25, 50, 75, ... , 1.000 ⇒ Tem 40 fatores.
⇒ 125, 250, ... , 1.000⇒ Tem 8 fatores.
⇒ 625⇒ 1
249
fator
fatores
Logo⇒ 249 zeros.
08 ⇒ 5, 10, 15, ... , 100 ⇒ Tem 20 fatores. ⇒ 25, ... , 100 ⇒ Tem 4 fatores.
Logo, 24 zeros, portanto a maior potência de 10 quedivide P é 1024.
09 1 + 2 + 1 = 4 = 22
1 + 2 + 3 + 2 + 1 = 9 = 32
1 + 2 + 3 + 4 + 3 + 2 + 1 = 16 = 42
A = 1 + 2 + 3 + ... + 2.008 + 2.009 + 2.008 + ... + 1 = 2.0092
A
287
2 009
287
2 009
287 7 49
2
2
2
2
2= =
= =
( . )
( )
.
10 Basta contar os fatores 5. 5, 10, 15, ... , 200⇒ Tem 40 fatores. 25, 50, ... , 200⇒ Tem 8 fatores. 125⇒ 1 fator.
Logo, 49 fatores, portanto, 49 zeros.
06 C 2x = b
2 2 2
1
42
1
4
1
4 42 2 2 2 2 2 2
2− + −
= ⋅ = ⋅ = ⋅ = ⋅ =x x x b b
b( ) ( )
b) 5 5 5 5 5 5 5
5 5 5
5 5 5 5
5 1 5
95
6
2 2 2 2 2 3
2 2
2 2 3
2
n n n n
n n
n
n
⋅ + ⋅ − − ⋅
+ ⋅
=
+ −( )+( )
= −
c) 2 2 2 2 2 2
13 2 2
2 2 2 2
2 13 2
7
13
3 3 3 3 2
3 1
3 3 2
3
n n n
n
n
n
⋅ + ⋅ + ⋅
⋅( ) =
+ +( )⋅
=
04 a) 3 1
81
3 1
33
4
4
4
x
x
x
=
= =
= −
−
d) 10x = 0,00000001 10x = 10–8
x = – 8
b) 10x = 1.000.000 e) (0,001)x =1
1093
10x =106 (10–3)x = 10–93
x = 6 10–3x = 10–93
– 3x = –93 c) 5x = 0,2 =
2
10
1
5= x = 31
5x = 5–1⇒ x = –1
05 E x y x yx y x y
x y x
x y y
E x
y
= −+ = ⋅ −⋅ +
= −
+ =
−
5 5 6 5
5 5 5 6
5 5
5 52 3
1
2 3
1
2 3
1 3
5
( )
( )
+ ⋅ − =
−
− =
−
− =
−
− =
−⋅
−
=
+
−1
2 3 3
1 5
32 9
3 5
37
2
37
2
3
1
7
2
21
( )
Atividades Propostas – pág. 12
− −
−=
−
=
−
= −
1
16
1
21
16
1 8
161
16
7
161
16
77
16
16
1 7⋅ = −
Mergulhando Fundo
01 E
x = 0,00375 · 10–6
e y = 22,5 · 10–8
x = 3,75 · 10–9 e y = 225 · 10–9
y
x=
⋅
⋅
=
−
−
225 10
3 75 1060
9
9,, logo, y = 60x
02 2x = 23y + 3 32y = 3x – 9
x = 3y + 3 2y = x –9
x y
y x
= +
= −
3 3
2 9
Utilizando o método da substituição, tem-se:
2y = 3y + 3 – 9 x = 3 · 6 + 3 –y = –6 · (–1) x = 21 y = 6 x + y = 21 + 6 = 27
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9o Ano – Ensino Fundamental II | 5
LIVRO 1 | ÁLGEBRA
03 a) − = − ⋅ = − ⋅ = − ⋅ = − = −2 744 2 7 2 7 2 7 14 143 3 33 33
3
3 1. ( ) ( ) ( )
b)343
1 000
7
10
7
10
7
10
7
10
7
10 03
3
33
3
3
3
31
. ,= =
=
=
= = 77
c) − − = − − = − − = − = +4 913 17 17 17 173 33
3
3 1. ( ) ( ) ( )
d) − =
−
=
− +
= −
∴
∴ −
0 000064 64
1 000 000
2
10
2
10
2
10
3 3
6
6
3
6
3,. .
( )
= −
= − = −
6
32
2
10
4
100 0 04,
01 M x x x
M x x x
M x x
= + + >
= ( ) + ⋅ ⋅ + = +( )
= + = +
4 4 0
2 2 2 2
2 2
22
2
,
02 B
x x x x
x x x
x x
x x
x x
x x
+ + − < <
+ ( ) − ⋅ ⋅ +
+ −( )+ −
+ − +
− + = +
1 2 0 1
2 1 1
1
1
1
1
22
2
,
*
( )
11
* Como 0 < x < 1
x x x− = − −( ) = − +1 1 1
03 a) 7 7 7 71 1= =, pois
b) − ∃ − ∉356 3564R R
+, pois
c) − = − = − − = −127 13 13 13 1273
3
3
3
, pois
d) 0 04 4
100
2
10
2
10
4
100
2
, ,= =
=pois
04 a b2 23 2 23 3 3 32 8 6 2 100 2 10 2 8 2+ − = + − = − = − = =
05 (F), pois x x2=
(V)(V)
(V)
(F), 49 7=
(V) (F) 8 2 23 33
= =
(F) − = − ∉3 92R
01 a) R +
b) R +
c) R *– , a ∉R
d) Existe
e) a se e a se3 30 0 0 0≥ ≥ < <, ,a a
f) a b e= ⇔ = ≥b a b2 0
g) a b3 3= ⇔ =b a
02 a) 1 764 2 3 7 2 3 7 2 3 7 422 2 22
2
2
2
2
2 1 1 1. = ⋅ ⋅ = ⋅ ⋅ = ⋅ ⋅ =
b) 15 876 2 3 7 2 3 7 2 3 7 1262 4 22
2
4
2
2
2 1 2 1. = ⋅ ⋅ = ⋅ ⋅ = ⋅ ⋅ =
c) 5 856 400 2 5 11 2 5 11 2 5 11 2 4204 2 44
2
2
2
4
2 2 1 2. . .= ⋅ ⋅ = ⋅ ⋅ = ⋅ ⋅ =
Atividades Propostas – pág. 18
03 −÷ +
÷
÷
− −
− − −( )
( )( ( ) ) ( ) (
10
1010 10 10
4 9
2 27
1 9 2 3 4 3 ))4 1
36
54
18 12 12
1
10
1010 10 10
10
−÷
÷
÷ ∴
∴ −
−
− −
− 88 18 12 12
12 12 12
10 10 10
1 10 10 10 10 112
÷ ÷{ } ÷ ∴
∴ − ÷{ } ÷ ⇒ − ÷ = −
− −
− −
04 I. P = 32 · 7 · 1050 · 231 · 540 · 26
P = 32 · 7 · 1050 · 237 · 537 · 53
P = 32 · 53 · 7 · 1050 · (2 · 5)37
P = 32 · 53 · 7 · 1050 · 1037
P = 32 · 53 · 7 · 1087
Logo, 87 zeros.
II. A2 – B2 = (A + B) · (A – B)
3 3
2
3 3
2
3 3
2
3 3
2
3 3 3 10
x x x x x x x x
x x
++
−
⋅
+−
+
⇒
⋅ = =
− − − −
−
III. a) 3n + 2 · 2n + 3 = 2.592 = 25 · 34
n + 3 = 5 ou n + 2 = 4, logo n = 2
b) 3n · (1 + 3 + 9 + 27) = 1.080
3n · 40 = 1.080 ⇒ 3n = 1 080
40
. ⇒ 3n = 27 = 33, n = 3
Radiciação Aula 1Capítulo 2
Testando seus Conhecimentos – pág. 16
04 a) 676 2 13 2 13 2 13 262 22
2
2
2 1 1= ⋅ = ⋅ = ⋅ =
b) 0 216 216
1 000
6
10
6
10
6
10
6
103 3
3
33
3
3
3
31
,.
= = =
=
=
∴∴
∴ =6
10 0 6,
c) 12 167 23 23 23 23
3 333
3 1
. = = = =
d) 3 136 2 2 2 7 56 3 136 566 7 3 2 2. ( ) .= ⋅ = ⋅ = ∴ =
e) − ∃625 R
f) − = −
= −
= −
= −
729
2 197
9
13
9
13
9
13
9
133
3
3
3
31
.
05 B
16 4 4 162= ⇔ =
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6 | 9o Ano – Ensino Fundamental II
LIVRO 1 | ÁLGEBRA
01 a) 8 82
3 23=
b) 6 61
2 1=
c) a a
3
5 35=
d) x x1
4 14=
e) 9 1
9
1
9
1
21
21
−
= =
f) 17 1
17
1
17
1
17
3
53
5
3
535
−
= = =
g) ( )
( ) ( )a b
a b a b
2 31
6
2 31
62 3 16
1 1−
= =
h) ( ) ( ) ( ) ( ),xy z xy z xy z xy z xy z2 0 4 2
4
10 2
2
5 2 252
2
5 1
1
−
− −
−= = = = ∴
∴xx y z2 4 2
5
i) a
a a
b
bb
−
= =
5
5 5
1 1
02 a) 4 0 5 0 25 8 4 0 0625 0 5 8
0 25 0 5 64 0 25 0 5
42
3 23
3
⋅ + + ⇒ ⋅ + +
+ + ⇒ + +
( , ) , ( , ) ,
, , , , 44 4 75⇒
,
b) 100 81 16 1
10081 16
1
1003 16
1
0 51
4 0 25
0 5
4
25
100
25
25
1
4
−
÷
÷
+ − ⇒ + −
+ − ⇒
, ,
,
1103 16
1
103 2
1
101
1 10
10
11
10
4+ −
+ − ⇒ + =+
=
c) ( )
( ) ( ) ( )
( )
, ,
, ,
− + −
− + −
− + −
− −
− −
−
1 36 256
1 6 2
1 6
3
7 0 5 0 25
3
7 2 0 5 8 0 25
3
7 1 221
1
1
6
1
4
12 2 3
12
13
12
2−
− + − =− + −
= −
03 K
K
= + −
+
−
= ( ) + ( )
− −
36 64 1
121
1
4625
6 4
1
2
1
3
0 5 1
0 25
21
2 3
,
,
11
3
2 0 5 1
4 0 25
1 1
1
11
1
45
6 4
1
11
−
+
− ( )
= + −
− −,
,
K
+
−= + − + −= −
− −1 1
11
4 56 4 11 4 5
2
K
K
01 C
8 2 2 83 3= =, pois
02 ( )
( ) ( )
,
2 5 5
2 5 2 5 2 5 5 2
2 5 5 5 2 5 2
2
2
− −
− = − = − − = −
− − ⇒ − − = −Logo
03 a) a ∈ R + e b ∈ R
+
b) (a, b) ∈ R
04 M x x x x
x x
= − + = − ⋅ ⋅ + ∴
∴ − = −
4 4 1 2 2 2 1 1
2 1 2 1
2 2 2
2
( ) ( )
( )
a) M x M= − = ⋅ − = − =
−
= − =2 1 2 1
81
1
41
1 4
4
3
4
3
4
b) M x= − = ⋅ − = − = =2 1 2 12
1 1 1 0 0
c) M x= − = ⋅ − = − = =2 1 2 19 1 38 1 37 37
05 K x x x x x= + + + = + = +3 23 333 3 1 1 1( )
∴ para x = 1,996, tem-se 1,996 + 1 = 2,996.
06 0 < b < a e ab > 2
a)a ab b
a ab b
a b
a b
a b
a b
2 2
2 2
2
2 2
2
2
+ +
+ +
=+
+
=
+
+
( )
( ) ( )
como 0 < b < a a b a b
a b
a b a b+ = + =
+
+
=
+
( )
( )2
1
04 a) 5 51
2=
b) 36 3651
5=
c) 4 4383
8=
d) x x373
7=
e)
1
2
1
2
9 9
2 =
f) 2
1
3
2
1
3
2 1
33
5
3
5
3
5
=
= ⋅
−
05 a) 49 49 7 751
5 21
5
2
5= = ( ) =
b) − = −7 731
3( )
c) 8 8 2 241
4 31
4
3
4= = ( ) =
d) a a a1520
15
20
5
53
4= =
÷
÷
Testando seus Conhecimentos – pág. 19
Atividades Propostas – pág. 20
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9o Ano – Ensino Fundamental II | 7
LIVRO 1 | ÁLGEBRA
b) a ab b
a a b ab b
a b
a b
a b
a b
a b
a b
2 2
3 2 2 33
2
33
2
3 3
− +
+ + +
=−
+
=
−
+
=−
+
( )
( )
Observação: |a – b| = a – b, pois 0 < b < a.
c) a b ab
a b
ab
ab ab
ab
ab ab
2 2
2 2
24 4
4
2
2 2
2
2 2
− +
−
=−
+ ⋅ −
=
−
+ ⋅ −
=
=
( )
( ) ( ) ( ) ( )
(aab
ab ab ab
−
+ ⋅ −
=
+
2
2 2
1
2
)
( ) ( )
Observação: |ab – 2| = ab – 2, pois ab > 2.
01 a) a a a a2020
2 10 5 2 23 9= = = = =( )
b) a a a15315
3 5 3= = =
c) a a a a75375
3 25 5 5 53 243= = = = =( )
02 a) ( ) ( )− = − = −5 5 51
5 15 5
b) ( , ) ( , )0 4 0 43
4 34=
c) −
= −
1
9
1
9
3
73
7
d) ( ) ( )abx abx abx1
2 1= =
03 a) 8 1
25 0 017
9
8 2 5 1
1
3
1
20 3
1
3 21
2+
⋅
⋅ ⇒ + ⋅
−− −
, ( ) ( )
⋅ ⇒
⇒ + ⋅ ⋅ ⇒ + ⋅ = ⋅ =
9
8
2 5 1 9
8 2 5 9
8 7 9
8
63
81 1
{ } { }
b) 2 2
50 3444 9
3
21
20 1 53
3
−− −
+
÷ ( ) +
⋅
⇒
⇒
, ... ,
11
2
25
41 9
8
27
2 25
41 9
8
2
1 5
3
1
2+
÷ +
⋅ ⇒
+
÷ +
⋅
,
77
27
41 3
8
27
27
4
1
4
8
27
2
4
1
2
2
2
⇒
⇒
÷ + ⋅ ⇒ ⋅ ⋅ ⇒ =
÷
÷[ ]
04 a) 10 79 6 4 4 489
10 79 6 2 67 10 79 8
34
34 2 34
⋅ + +
⋅ ⇒
⋅ + +( ) ⋅ ⇒ ⋅ +( ) ⋅
( . )
( ) 667
10 79 2 67 10 81 67 10 3 67 2 0104 4
⇒⋅ +( ) ⋅ ⇒ ⋅ ⋅ ⇒ ⋅ ⋅ = .
b)2 2
10
2 2 2
10
2 10
10
2 2 2 512
28 30
3
27 3
3
27
3
27327
3 9
+=
⋅ +=
⋅⇒
= = =
( )
01 a) a ∈ R +
b) n ∈ R e n ≥ 2
02 a) x x x x777
7 1= = =
b)
16
9
16
9
16
9
9
16
5
5
5
51
=
=
=
− −
−
c) m m m
m
−
−
−
= = =66
6
6 1 1
d)
m n
p
m n
p
m n
p
p
m m p mp
x
x
x
x+
=
+
=
+
∴
∴+ +
− −
−3 3
3
3
3 2 23 3 ++ p3
03 x y x y x y x ym mm mm mm
m
m
m
m3 4 3 43 4
3 4⋅ = ⋅ = ⋅ = ⋅
04 a) 2 187 32 2 187 32
3 3 2 3 9 2 12 9
25 5 25
5 25 105 5 2 5
. ( ) . ( )⋅ = ⋅ =
⋅ ⋅ = ⋅ ⋅ =
b) ( . ) ( )
.
343 2 197 7 3 7 3
7 3 3 7 3 3 3 969
23 3 7 23 6 143
6 12 23 2 4 23
⋅ = ⋅ = ⋅ =
⋅ ⋅ = ⋅ ⋅ = 993
05 a) 2 2 2 2 2 2 4 2204 208 16 48 2 4 48 4= = ⋅ = ⋅ = ⋅
÷÷
b) 4 096 2 2 845 12 440 4 310 10. = = =÷÷
c) 81 3 352 4 420 4 5= =
÷÷
d) 256 2 2 165 8 210 2 45 5= = =
÷÷
06 a)3 3
3
3
33 3 9
1421 1021
1021
2421
1021
14 721 7 23 3⋅
= = = =÷÷
b) x x y
xy
x y
x yx y y x
5 7
5
6 7
5 5
12 6⋅ ⋅
= = ⋅ = ⋅
− −
− −
( )
01 b
ab
a a b ab
ab
− ÷
÷
= = =
=
=
2
2
1
4 2
2
4
2 24
2 2
4 2
16
1
16
1
16
1
4
1
4
1
2 (( )ab − 1
02 a)x x y
y
x x y
y
x x x y x y
y
x x y
10 4
93
10 43
93
9 33
3
3
⋅ +=
⋅ +
=
⋅ ⋅ + ⋅ +
=⋅ +
( ) ( ) ( ) ( )
( )
yyx x y
33
⋅ ⋅ +( )
b)a
x
a
x
a a
x x
a a
x x
a
x
a
x
n
nn
nn
nn
n nn
n nn
nn
nn
3
52
32
52
22
42
2
2 2= =
⋅
⋅= =
n
n2
c) x x x x x x x x
x x
3 2 2 210 25 10 25 5
5
− + = ⋅ − + = ⋅ − ∴
∴ − ⋅
( ) ( )
( )
d) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
a b a b a b a b a b
a b a b a b a b
2 2
2
− ⋅ + = + ⋅ − ⋅ + ∴
∴ + ⋅ − = + ⋅ −
Mergulhando Fundo
Atividades Propostas – pág. 25
Radicais Aula 1Capítulo 3
Testando seus Conhecimentos – pág. 24
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8 | 9o Ano – Ensino Fundamental II
LIVRO 1 | ÁLGEBRA
01 a) 7 7 768 6 28 2 34= =
÷÷
b) a b a b a b ab3 69 3 39 3 6 39 3 3 23 23⋅ = ⋅ = ⋅ =
÷÷ ÷÷
c) x
a b
x
ab
x
ab
4
2 610
2
3
2 2
10 2
2
35=
=÷
÷
d) 32 2 2
2
1510 5 510 5 10 5 15 5 3
3
⋅ − = ⋅ − = ⋅ − ∴
∴ ⋅ −
÷÷÷
÷
( ) ( ) ( )
( )
m n m n m n
m n
e) ( ) ( )a b a b a bou
a b+=
+=
+ +2
612
212
612
6
6
2 2 2 8
f)
( )p q
a b
p q
a a b
+=
+
⋅
9
129
39
02 a) 12 2 3 2 32= ⋅ =
b) 75 3 5 5 32= ⋅ =
c) 48 2 3 2 34 44 4= ⋅ =
d) 180 2 3 5 2 3 5 6 52 2= ⋅ ⋅ = ⋅ ⋅ =
e) 243 3 3 3 3 34 54 44 4= = ⋅ =
f) 56 2 7 2 73 33 3= ⋅ =
03 a) 25 52 2x y x y= ⋅
b) 16 42mx x m= ⋅
c) 64 46 123 2 4a m a m=
d) 75 5 3 5 33 3 3 2 2 2 2a b c a a b b c c abc abc⋅ = ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ =
e)3
2
3
2 2
3
2 2
3
2
4 2 8
6 64
4 2 4 4
4 2 4 24
2 2
2 24
2
m n
p
m n n
p p
n
p
m
p
n
p
⋅
= ⋅ ⋅ ⋅
⋅ ⋅ ⋅
= ⋅
⋅ ⇒
⋅ mm
p
n
p
m
p2
3
2 2
2 2
4 2
2
= ⋅
÷
÷
f)250 5 2 5 10 5 10
3 4
3
2 4 2 2x y a x xy a xy xa xy xa=
⋅
⋅ ⋅
= =
04 a) a ab b a b a b2 2 22+ + = + = +( )
b) 81 18 9 92 2+ + = + = +x x x x( )
c) x xy y x y x y2 24 2 24 22− + = − = −÷÷ ( )
d) a a a a a a a a a a3 2 2 22 2 1 1 1− + = ⋅ − + = ⋅ − = − ⋅( ) ( ) ( )
05 a) 2 8 2 2 2 2 2 2
4 2
3 4 2 2 4 2
2 2
m m n m m m n m m n m
m n m
= ⋅ ⋅ ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅ ∴
∴
03 a) 5 3 5 3 753 2 6 2 6 2x y x y x y= ⋅ ⋅ =
b) 3 3 3 3 813 3 3 93 103a a a a a= ⋅ ⋅ =
c) 1 1
4 3
3
4 3
3
4 3
9
16 3
3
16
2 3
3−
⋅ = ⋅ =
⋅ = ⋅ =
÷
÷
y y y y y
d) 1
1
9 4
10
9 4
10
9 4
1 000
729 4
4 0
2 23 2 23
3
6 23
6 23
a a a a a a
a a
= =
⋅ ⇒
⋅ ⋅ =. . 000
729
8
3 a
04 Fx
a
x
x
aF
x
a
x
x x
a
Fa
a
= ⋅ ⋅ ⋅ ⇒ = ⋅ ⋅ ⋅⋅
⋅
= =
⋅ ⋅
= =
⋅
3 4
3 2
3 4
3 2
4
2
4
2 2 2
4
2 4
4
2 2
2 3
4 2
== =4
41
05 a) 5 2 3
5 2 3
23 53 55
6 15 25
,
,
ou
ou
m.m.c. (6, 15, 25) = 150
5 2 3
5 2 3 5
25150 10150 6150
25150 10150 6150 23
ou
o
⇒
> > log , . b) 3 2 5 4
6 20
6 20 216 400
5 4
3 3
3
36 26 6 6
3 3
⋅ ⋅
⇒
⋅
ou
ou
ou ou
Logo, .
c) 10 7
10 7
10
1 000 117 649
7
3
4
3
2
3
4
6
4
34
4 4
3
2
ou
ou
Logo
;
. .
,
06 E ab
c
ab
c
ab
c
n
n
n
n
=
=
=
=
⋅ −
−
3 3
35 4
1
2
( )
33 3
3
20
1
2
40 64 000
= −
−
∴
∴ =[ ] .
07 a) a b c a b b c a b b c
a b bc
n n n n n n
n
4 4 14 44 44 4 4 4
4
⋅ ⋅ = ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ ∴
∴ ⋅ ⋅
+
( ) ( )
b) 32 343 243 2 7 3 2 2 7 3 3
2 2 7 3 3 168
53 10 3 53 93 33 3 23
3 3 23
⋅ ⋅ = ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅
= ⋅ ⋅ ⋅ ⋅ = ⋅⋅ 183
08 a) ( ) ( )16 9
5
2 3
5
2 3
5
2 2 3 3
5 5
2 3
5
73
4 2 5
73
20 10
73
18 23 93
63
6 3
⋅
=
⋅
=
⋅
=
⋅ ⋅ ⋅
⋅
=
⋅ ⋅⋅
=
12
5 5
1 728
25
12
5
3
2 33
.
b) ( , )0 08
8
100
2
25
25
2
5
2
5
2 2
12
3
3 3 3 6
3
3
−− −
=
=
=
= ∴
∴ = 55
2 2
c) 32 2
2 2
7 165 55 5 25 15 15
3 2 153
2
5
x y x x y y
x y x y x
y
x
y
− − −
− −
= ⋅ ⋅ ⋅ ⋅
= ⋅ ⋅ ⋅ ⋅ =
d)2 3
52 3 5 360
3 2
5 3 25 5n n
n
n n n n⋅
= ⋅ ⋅ =−
( ) ( ) ( )
Não é possível extrair o radical.
Testando seus Conhecimentos – pág. 27
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9o Ano – Ensino Fundamental II | 9
LIVRO 1 | ÁLGEBRA
b) 2 32 2 2 2 22
3
12
4 55
2
3
5 10 2
4 55
2 2
3
2
45
a
b
b
a c
a
b
b b
a c
a b
b c
b
c= ⋅
⋅ ⋅
⋅
=
⋅ ⋅
⋅
⋅ ∴
∴
44 42 2
3
2
45
2 2
45
a b
b c
b
c
a
bc
b
c=
c)m n m n
m b
m n m n
m b
m n
m bm n
+
⋅
−( )
+( )=
+
⋅
−( )
+( )∴
∴+
⋅ +
⋅ −
2 2
2
2
63
23
2
2
23
( )( )
d) a b
a
a
a b
a b
a
a a
a b a b
a b
a
aa a
−
−( )=
− ⋅ ⋅
− ⋅ −
∴
∴
−
⋅
⋅
2
8
2
8
2
8
13
65
10 35
55
2
( ) ( )
335
5
3
5
2
8
( ) ( )a b a b
a a
a b− ⋅ −
=
−
01 E x x
x x
x x x x
x x
x
=+ ⋅ +
− +
=+ ⋅ − + ⋅ +
− +
∴
∴ + =
( ) ( ) ( ) ( ) ( )
( )
( )
3
2
2
2
2
1 1
1
1 1 1
1
1 xx + 1
a) |0,47 + 1| = 1,47b) |–349 + 1| = 348
c) 3 444 1 4 44 40
9, ... , ...+ = =
02 y y y y y y y y y y
1
4
3
4
1
2
1
4 2
1 1
4
3
4
1
2
1
4 21
⋅ + + +
⇒ ⋅ + + +
−
⇒
⇒ + + + ⇒ + + + ⇒
⇒ + + +
y y y y13
4
1
2
9
4 4 1 43
4 41
2 49
4
4 3 2
2 2 2 2
2 2 2 2
( ) ( ) ( ) ( )99 16 8 4 512 540⇒ + + + =
03 a b
ca
b
ca
b
ca
b
c
ac b
c
a b
cac b a b
ac a
+
=
⇒ + = ⇒
⇒ +
= ⇒ + =
⇒ =
2 2
2
22
222 1
b b c ab a
a− ⇒ =
⋅ −( )
04 3 3
3 3
3 3
3 3
3
3
3
33
3
220 415
330 6
10 415
10 6
415
6
830
530
330
3 3
⋅
⋅
=
⋅
⋅
= = = ∴
∴ ÷660 3 20 3÷
=
05 9 2
9 2
3 2
3 2
9
34 0 333 0 111
123
9
1
9
2121
27
6 27
ou
ou
ou
ou
, ... , ...
( )
334
06 M b
a
a b
b bM
b
a
a b
b
M b
=−
− +
⇒ =
−( )⋅
−( )∴
∴ = = =
1
3
9
2 1
1
3
3
1
676 26
2
2
01 a) 2 2 2 233 33 433 49⋅ = =
b) 5 5 5 5 5 5 54 6105 2010 6105 26105 26 250 2 1325⋅ = ⋅ = = =
÷÷
c) 2 2
2
2
22 2
4530 3430
20303
7930
20303 59303 5990⋅
= = =
02 E x x x x x= − + − = −( ) = −
− =
−
=
3 23 333 3 1 1 1
40
11 1 40 11
11
29
11
03 12 1
21
2 1
4
4 2 1
4
1
8 4 2
2 2
8 4
4
4 8 4
4
4 2
+ − +( )
( )= +
− +=
+ − +∴
∴ +( )
x x
x
x x
x
x x x
x
x
22
1
2
1
22 2
4
2
2 4
2x
x
x
x
x( )=
+
=
+
04 En n
nn n
n
nn
=
⋅ − ⋅
=
⋅ − ⋅
=
⋅ ⋅ −
=
600
25 25 5 5
600
5 5 5 5
600
5 5 5 1
6
2 2 2 2 4 2 2
2 2 2( )
000
5 600
1
5
1
5
1
252 2 2nn
nn
⋅
= = =
01 a) 8 18 32 50 2 4 2 9 16 2 25 2
2 2 3 2 4 2 5 2 14 2
+ + + = ⋅ + ⋅ + ⋅ + ⋅ ⇒
⇒ + + + =
b) 12 48 75 4 3 16 3 25 3
2 3 4 3 5 3 3
+ − = ⋅ + ⋅ − ⋅ ⇒
⇒ + − =
c) 8 3
42
3
168
3
42
3
16
8 3
2
2 3
4
7 3
22
− = ⋅ − ⋅ = − =
d) 75 48 3 108 25 3 48 3 36 3
5 3 48 3 6 3 37 3
− + = ⋅ − + ⋅ ⇒
⇒ − + = −
e) 2 16 54 128 2 8 2 27 2 64 2
2 2 2 3 2 4 2 10 2
3 3 3 3 3 3 3 3
3 3 3 3 3
+ + + = + ⋅ + ⋅ + ⋅ ⇒
⇒ + + + =
f) 162 1 250 2 592 81 2 625 2 1 296 2
3 2 5 2 6 2 8 2
4 4 4 4 4 4
4 4 4 4
− − = ⋅ − ⋅ − ⋅ ⇒
⇒ − − = −
. . .
Operações com radicais I
Testando seus Conhecimentos – pág. 31
Mergulhando Fundo
Aula 1Capítulo 4
Atividades Propostas – pág. 29
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9o Ano – Ensino Fundamental II | 11
LIVRO 1 | ÁLGEBRA
01 a) a b ab6 6 6⋅ =
b) 144 6 2 3 2 3 2 3 2 3 2 33 3 4 23 3 3 33 3÷ = ⋅ ÷ ⋅ = ⋅ = ⋅ =
c) 2 3 65 25 5 45x x x x⋅ ⋅ =
d) ( )8 4 2 32 2 16 234 34 4 64 4 54 4a a a a a a a a⋅ ÷ = ÷ = =
e) ( . ) .
.
162 16 000 12 15 216 000 15
14 400 120
⋅ ÷ ÷ ⇒ ÷ ∴
∴ =
f) 8 18 2 3 4 3 124 2 2a a a a a⋅ = ⋅ ⋅ = ⋅ ⋅ =
g) 2 14
22 7
xy
yx=
h) a a a a a a6 3 3 3 3 31 1 1 1 1 1− ÷ + = + ⋅ − ÷ + = −( ) ( )
i) 2 2 2 2 214 34 1 34 24x x x x− − − − +
÷ = = =
02 a) x x x x x215 310 430 930 1330⋅ = ⋅ =
b) a a a a a a4 12 312 112 212 6÷ = ÷ = =
c)8
4
8
4
64
64
2 524
216
2 5 248
2 348
448 10
348 6
448x y
xy
x y
xy
x y
x yxy= = =
( )
( )
d) m m
m
m m
m
m
mm
215 1720
1130
860 5160
2260
5960
2260
3760⋅
=
⋅
= =
e) ( )
(
9 32 6 8 27 16
2 2
4 28 4 5 312 5 3 26
15 12 6
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ÷ ⋅ ⋅ ⋅ ⋅
⋅ ⋅ ⋅
a b c a b c a b c
a b ⋅⋅ ⋅ ⋅ ⋅ ⋅ ÷ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ ÷
c a b c a b c
a b c
324 6 8 10 624 16 20 12 824
21 20 16 924
2 2
2 2
)
22 2 22 32
16 20 12 824 5 424
424
⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ ∴
∴
a b c b cb c
03 a) 2 4 3 6 4 6 6 9 2 4 7 6 6 93 3 3 3 3 3 3− − + ⇒ − +
b) 6 12 4 18 12 12 4 6 8 2
6 4 3 4 9 2 12 4 3 4 6 8 2
12 3 12 2 24 3 4 6
− − + + =
= ⋅ − ⋅ − ⋅ + + =
= − − + + 88 2
12 3 4 2 4 6
=
= − − +
04 A
B
= − = − =
= − = − =
+
=
9 2 10 81 40 41
7 2 3 49 12 37
41
37 4
4
2 2
2 2
2 010
( )
( ).
11
41 1 1
2 010
2 010
= =
.
.
05 a)ab b
a ab
a ab
ab b
b a b
a a b
a a b
b a b
a b
+
−
⋅+
−
=⋅ +
⋅ −
⋅ +
⋅ −
∴
∴+
2
2
2
2
2
( )
( )
( )
( )
( )
(( )a b
a b
a b−
=+
−2
b) x y x y x y
x y x y x y
x y x y
+ ⋅ + ÷ +
+ ⋅ + ÷ +
+ = +
3 4 12
412 312 12
612
( ) ( ) ( )
( )
06 A B
A B
A A B
= − ⋅ + ⋅ = ⋅ − ⋅
= − + = − =
=−
5 2 3 5 2 7 12 2 5 2 2 3 3 2
5 2 15 2 84 2 10 2 9 2 2
74 27
2
33
73 2
73 2= =
07 x x x= − ⇒ = − ⇒ = − =2 27
91
2 3
31 2 1 1
6
4
612
612
08 x x= − = = =36 27 9 9
09 a) 3 5 3 3 2 5 3 3 5− − + =
b) 7 15 2 4 8 5 3 4 5 5 20 2 20 5
105 2 32 5 12 5 100 2 20 5 5 2
⋅ − ⋅ + ⋅ − ⋅ + ⇒
⇒ − + − + =
c) 15 25 40 15 5 2 10− + = − +
d)3
25
2
53
1
310 6
150
56
5 6
56 0
⋅ ⋅ ⋅ ⋅ − ⇒ − ∴
∴ − =
10 a) ( ) ( )15 3 15 3 1− ÷ − =
b) x + y – y = x
c)( )
( ) ( )
a
a a a
−
+ ⋅ −
=
+
5
5 5
1
5
d) 3
48
3
42
3
233 m m m= ⋅ ⋅ =
Mergulhando Fundo
01 A = 49 – 63 = –14B = 25 – 48 = –23
− −− −
=
−
= −14 23
23 14
37
37 1
2 010 2 010
2 010
( ) ( )
. .
. == 1
02 2 243 1 2 27 32 9 3 2 2 3 3 6
18 3 2 6 3 6
2 24 3 4
⋅ − + ⋅ + =
= ⋅ − + ⋅ + =
= − + + ⇒
⇒ = +
( ) ( )
p
A
A
A
= − ⋅ + = + − −
= + − −
= +
( ) ( )3 1 3 3 3 3 3 3 3
81 27 3 3 3 3
24 3 78
5 3 8 5 3
03 266
1914= cm
04 a) 18 9
12 9
18
12
9
9
3
29
2
3
3 36
26
36a a
a
a a
aa a
−
−
= ⋅
⋅ −
−
= ⋅ ⋅ −
( )
( )( )
Atividades Propostas – pág. 35
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9o Ano – Ensino Fundamental II | 13
LIVRO 1 | ÁLGEBRA
01 Substituindo x por –1, tem-se:
−( ) + − −( )
−( ) + + −( )=
+
−
1 2 2 1
1 2 2 1
3 2
3 2
2
2
Racionalizando o denominador, tem-se:
=
+ +( )− +( )
=+ + +
+ − −
∴
∴+ +
−
=+
−
3 2 3 2
3 2 3 2
3 2 3 2 3 2
3 2 3 2 3 2
3 4 3 4
3 4
7 4 3
1
2 2
2 2
== − −
7 4 3
02 a)x y
x y
x y
x y
x y x y
x yx y
−
−
⋅
+
+
=
− ⋅ +
−
= +
( )
( )
( ) ( )
( )
b)a a
a a
a a
a a
a a a
a a
a a
+ − −
+ + −
⋅+ − −
+ − −
=
=+ − − + −
+ − +
=−
1 1
1 1
1 1
1 1
1 2 1 1
1 1
2 22 2−−
= − −1
212a a
c) a mm a
m am a
a m m am a
m a m a
2 2
2
1
−
−
⋅+
+
=− ⋅ +
−
∴
∴ − ⋅ + = − −
( )( )
( )
( )
d) a a
a a
a a
a a
a a a a
a a
a
+ −
− −
⋅+ −
+ −
=+ − + −
− −
∴
∴
2
2
2
2
2 2 2
2 2
2
1
1
1
1
2 1 1
1
2
( )
( ) ( )
++ − −
+ +
=+ − −
∴
∴ + − = + − −
2 1 1
2 2 1
2 2 1 1
1
2 2 1 2 2 1 1
2
2
2 2
2 2 2 2
a a
a a
a a a
a a a a a a
03 18 7 18 7 8 7 8 7 2 8 4 2+
+
−
= − + + = =
04 am an
m n mn
a m n m n
m mn n mn
a mn m n
m n
2 2
2 2 2 2
−
− +
=⋅ + ⋅ −
− + +
⋅ ⋅ −
+
( )
( ) ( )
( ) ( )
( )) ( ) . ( ) . .= ⋅ − = + − = ⋅ =a m n n n2 011 1 2 011 1 2 011
05 E
E
E
E
=
+
+
−
− ∴
∴ =−
−
++
−
− ∴
∴ = − + − − − ∴
∴ = −
1
2 3
1
2 32 2
2 3
1
2 3
1
2 2
2 3 2 3 2 2
4 2
f)10
2 3
10
4 3
10
12
10 12
12 12
10 2 3
12
10 2 2
4
34
4 34
2 34
44
4 2
=
⋅
= =
⋅
⋅
∴
∴
⋅ ⋅
=
⋅ ⋅( ) ⋅⋅
∴
∴
⋅ ⋅ ⋅
=
⋅
=
⋅
3
12
10 2 4 27
12
20 108
12
5 108
3
34
4 4 4
g)a
a b
a ab
a b ab
a ab
a b
a ab
ab
ab
b34
34
34 34
34
4 44
34 34
=
⋅
⋅
=
⋅
= =
h) 5
5
5
5
5 5
5 5
5 5
5
5 5
53 125
3 6
56
6 56
56
66
566
= =
⋅
⋅
= = = .
02 a) 6
2 2
2 2
2 2
6 2 2
4 2
6 2 2
23 2 2
+
⋅−
−
=⋅ −
−
=⋅ −
= ⋅ −( )
( )
( ) ( )( )
b)3
3 6
3 6
3 6
3 3 6
9 6
3 3 6
3 3 6
−
⋅+
+
=⋅ +
−
=⋅ −
= +( )
( )
( ) ( )
c)3 2 2
3 2 2
3 2 2
3 2 2
3 2 2
1
2+
−
⋅
+
−
=+( )
( )
( )
d) 3 2
1 2
1 2
1 2
3 3 2 1 2 2
1 2
5 4 2
15 4 2
−
+
⋅−
−
=− − +
−
∴
∴−
−
= − +
( )
( )
e)3 2
7 2 6 3
7 2 6 3
7 2 6 3
42 18 6
98 108
42 18 6
10
42 18 6
10
−
⋅+
+
=+
−
∴
∴+
−
=− −
( )
( )
f)17
3 5 2 7
3 5 2 7
3 5 2 7
17 3 5 2 7
45 28
17 3 5 2 7
173 5 2 7
+
⋅−
−
=⋅ −
−
∴
∴⋅ −
= −
( )
( )
( )
( )
03 a)2
2
3
3
2 2
2
3 3
32 3+ = + = +
b)2
3
3
2
2 3
3
6
2
4 3 3 6
6+ = + =
+
c)1
2
2
4
3
8
4
16
1
21
3
2 21
2
2
3 2
4
2
1
2 2 3 2 8
4
5 2 8
4
+ + + = + + + =
= + + =+ +
=+
04 A =
+
⋅−
−
=⋅ −
−
= −3
5 2
5 2
5 2
3 5 2
5 25 2
( )
( )
( )
B =
+
⋅−
−
= ⋅ −
−
= −13
3 5 4 2
3 5 4 2
3 5 4 2
13 3 5 4 2
45 323 5 4 2
( )
( )
( )
C = − − − = ⋅ − − + ∴
∴
4 5 2 3 5 4 2 4 5 4 2 3 5 4 25
( ) ( )
05 4 3
4 3
4 3
4 3
4 3 4 3
4 34 3
+
+
⋅+
+
=+ ⋅ +
+
= +( ) ( )
Atividades Propostas – pág. 48
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14 | 9o Ano – Ensino Fundamental II
LIVRO 1 | ÁLGEBRA
01 a)1
3 2 6
3 2 6
3 2 6
3 2 6
3 2 2 6 6
3 2 6
2 6 1
2 6 1
2 6
( )
( )
( )
( )
(
+ −
+ +
+ +
=+
+ + −
∴+ +
−
+
⋅+
∴
⋅
++
∴
∴+ + + + +
−
+ + + + +∴
+ + +
∴
∴
∴
1
2 18 3 2 12 2 12 6
24 1
6 2 3 4 3 2 12 6
23
7 2 5 3 6 12
23
)
06 a) 6 2 5 1 5 1 5 5 12− = − = − = −( )
b) a b a b a b a b
a a b b
a b b
2 2 2 2 2 2 2
2 2 2 2
2 2
2 4
2
2
2
2
− − = − ⋅ − ∴
∴+ −
− ∴
∴ − −
( )
c) 7 48 2 3 2 3 2 32+ = + = + = +( )
d) 23 45 23 22
2
23 22
2
45
2
1
2+ =
++
−= +
07 a) 3 2
3 2
3 2
3 2
3 2
1
3 2 3 2
2−
+
⋅−
−
=−
∴
∴ − = −
( )
b) 7 2 6 6 2 5 1 6 1 5
1 6 1 5 1 6 5 1 6 5
2 2
+ + − ⇒ + + − ⇒
⇒ + + − ⇒ + + − ⇒ +
( ) ( )
08 a) 10 96 10 2
2
10 2
26 2± =
+±
−= ±
b) 57 12 15 57 2 160 57 33
20
57 33
2
45 12
− = − =+
−−
∴
∴ −
.
c) 3 1
210
7
210
7
2
3
22
2
51− = − =
−
⇒ −
d) 17 12 2 17 288 17 1
2
17 1
2
9 8 3 8
− = − ⇒+
−−
∴
∴ − ⇒ −
09 m m m m m m
m
+ + =+ + −
++ − +
∴
∴ +
3 12 3 3
2
3 3
2
3
10 a
ba b=
+
+
⋅−
−
=−
−
= = =5 1
3 1
5 1
3 1
5 1
3 1
4
22 2
b) 11
6 3 2
11
6 3 2
6 3 2
6 3 2
11 6 5 6 3 2
19
4 4
4
4
4
− −
=
− +
⋅+ +
+ +
=
=⋅ − ⋅ + +
( )
( )
( )
( ) ( )
02 a) 3
4 2 3
4 2 3
4 2 3
3 4 2 3
4
6 2 3
43 2 3
2
20
2 3
2 3
2 3
−
⋅+
+
=⋅ +
=+
=
=⋅ +
=
+
⋅−
−
( ) ( )
( ) ( )
( ))( )= ⋅ −20 2 3
Logo, 20
2 3+
b)5 2
7 3
5 2
7 3
3
41
−
+
⋅+
−
= <
a < b. Logo,7 3
5 2
−
+
03 I. 5 2
5 2
5 2
5 25 2 5 2
5 2 5 2
5 2 5 2
5 2 5 2
+
−
+−
+
∴
+ ⋅ +
− ⋅ +
+− ⋅ −
+ ⋅ −
( ) ( )
( ) ( )
( ) ( )
( ) ( ))
( ) ( )
∴
∴+
−
+−
−
∴
∴ + + − =
5 2
5 4
5 2
5 4
5 2 5 2 2 5
2 2
II. E x
x
x
xE
E E
E
=
−
−−
⇒ =
−
−
−
⇒
⇒ = − ⇒ = ⋅ − = ⇒
⇒ =
3
1
1
3
3
1 1
3
1 1
3
3
32
3
2
33
3 32
23
13
3
2
2
22
2
3
3 2
2
2
3
9 2 2 2
6
7 2
6− ⇒ = − ⇒ =
−=E E
III. a) 1
9 2
9 2
9 2
9 2 9 2
9 2 9 2
9 2 9 2
7
4 4
4 4
4 4
4 4
4 4
+
⋅−
−
=− ⋅ +
− + +
∴
∴− ⋅ +
( ) ( ) ( )
( ) ( )
b) 35 3
5 35 3
3 5 35 3
5 35 3
3 5 3
4 4
4 4
4 4
4 4
4 4
−
⋅ +
+
= ⋅ +
−
⋅ +
+
∴
∴⋅ + ⋅
( )( )
( ) ( )( )
( ) ( 55 3
2
+ )
04 I. a)1
1 2
1 2 4
1 2 4
1 2 4
11 2 4
3
3 3
3 3
3 33 3
−
⋅+ +
+ +
=+ +
−
= − − −( )
( )
b) 14
25 10 4
14
5 5 2
5 2
5 2
14 5 2
7
2
3 3 3 3 3 3 2
3 3
3 3
3 3
− +
=
− ⋅
⋅+
+
=
=+
= ⋅
( ) ( )
( )
( )
( )(( )5 23 3
+
c)4 1
3 1
9 3 1
9 3 1
4 1 9 3 1
2
3
3
3 3
3 3
3 3 3−
−
⋅+ +
+ +
=− ⋅ + +( )
( )
( ) ( )
Mergulhando Fundo
7/23/2019 Resolucao 2014 9ºano Algebra l1
http://slidepdf.com/reader/full/resolucao-2014-9oano-algebra-l1 15/15
LIVRO 1 | ÁLGEBRA
II. 1
4 4 3 3
4 3
4 3
4 3
2 3
2 3
2 3
4 3 2
6 2 6 6 6 2
6 6
6 6
6 6
6 6
( ) ( )
( ) (
− ⋅ +
⋅+
+
=+
+
⋅−
−
=
= + ⋅ − 33 )
III.1
729 32 3 4 256
1
3 32 3 4 21
1 32 3 2
1
1 2 2 3 2
1
1
6 4 8 8 4 8
4 4 4 4
− + −
=
− + −
=
=
− +
=
− +
=
+ 22
1 2
1 2
1 2
1 2
1 2
1 2
1 2 1 2
1
4
4
4
4 4
⋅−
−
=
=−
−
⋅+
+
=− ⋅ +
−
( )
( )
( )
( )
( ) ( )
IV. C
5
7 7 2 2
7 2
7 2
5 7 2
7 2
7 2
3 2 3 3 3 2
3 3
3 3
3 33 3
( ) ( )
( )
+ ⋅ +
⋅−
−
=
⋅ −
−
= −
d)4 1
2 1
4 2 1
4 2 14 1 4 2 1
3
3
3 3
3 3
3 3 3−
−
⋅+ +
+ +
= − ⋅ + +( )
( )( ) ( )
e)3 2
2 18 9
3 2
2 2 9 9
2 9
2 9
3 2 2
3 6 3 6 2 6 6 6 2
6 6
6 6
+
− +
=+
− ⋅ +
⋅+
+
=
=+ ⋅
( ) ( )
( )
( )
( ) ( 66 66 69
2 3
2 9+
+
= +)
( )
f)1
2 2 3
4 4 2 6 9
4 4 2 6 9
4 2 9
16 3
4 4 2 6 9
19
33
3 6 3
3 6 3
3 3
3 3 3
+
⋅− +
− +
⋅−
+
∴
∴− +
( )
( )