Resolução de questões do livro Matemática Para Concurso - Hilder Góes e Ubaldo Tonar. 6º Edição.

1. 15
RADICIAÇÃO
01
3
=
3
. 5 = 3  5 = 3  5
5 5 5  52 5
7 7 7
02
2
=
2
.
 35 = 2  35 =
2  35
7
 32 7
 32 7 7
 35  37 7
5 2
= 2 2 =
2 . 3 . 5 2
= 
6 6 6 .5 2 3 2
03 = .
5 2 5 2 5 2 5 2 5.5 . 2 5
04 a)
15
=
15
. 5 = 15  5 = 15  5 = 3  5
5  5 5  52 5
b)
2
=
2
. 3 = 2 3 = 2 3
3 3  3 3 2 3
2 7
= 2 2 =
7 . 2 7
= 
7 7 7. 2 7 7
c) = .
2 7 2 7 2 7 2 7 2. 2 .7 2
3 3 3 3 3 3
d) 2 2
= 3 .  42 =
24
=  =  =
2
2 4 4
2 2
2 22 =  24 =
3
22
=
3
2
3 3 3 3
4 4  42 4 4 2 2 2 2
e)
5
=
5
.
 20 = 5  20 = 5  22 . 5 = 2 . 5  5 =  5
 20  20  20  20 2 20 2 . 2. 5 2
3 5
= 2  =
5 . 3 5
=
5 5 5. 3 5 5
f) = .
3 5 3 5 3 5 3 5
2 3.3 .5 3
5 3
= 2  =
2 . 3 . 5 3
= 
6 6 6. 5 3 2 3
g) = .
53 5 3 5 3 5 3 2 5. 5.3 5
7 7 7 7
h) 10 10 5  53 10 . 5  53 2 . 5 . 5  54 2  54
= . = = =
7
5  54 5  54
7 7
5  53
7
52  5 7 5. 5. 5 5
05 3 3 5 −  2 = 3  5 −  2 = 3 5 −  2 =  5 −  2
= .
5   2 5  2 5 −  2  52 −  2 2 5−2
diferença de quadrados: o quadrado do primeiro menos o quadrado do segundo

2. 06
2
=
2
.  7   3 = 2  7   3 = 2 7  3 =  7   3
7 − 3 7 − 3 7  3  72 −  32 7−3 2
2 2  2   3 2 2  2   3
. 
2 2 2 2  3
= = 2 2 =
2 2 − 3 2 2 − 3 2 2  3 2  2 −  32  22 . 2 − 3
07
2 2  2   3 2 2  2   3
=
4 . 2 − 3 5
08 1 1 5   2 = 1  5   2 =  5   2 = 5   2
= .
a) 5 −  2 5 − 2 5   2  52 −  2 2 5 − 2 3
b)
5
=
5
.  8   5 = 5  8   5 = 5  8  5 = 5  8   5
8 − 5 8 −  5 8  5  82 −  52 8 − 5 3
c)
4
=
4
.  7 −  3 = 4  7 −  3 = 4   7 −  3 =  7 −  3
7  3 7  3 7 − 3  72 −  32 7 − 3
3 3 3  2 3 3   2 3 3   2 3 3   2
d) = . = 2 = =
3 − 2 3 −  2 3  2 3 −  22 3 . 3 − 2 7
e)
4
=
4
. 
2 3  2 = 4 2  3   2 = 42  3   2 =
2 3 − 2 2 3 − 2 2 3   2 2 2  32 −  22 22 . 3 − 2
4 2  3   2 2 2  3   2
=
10 5
5 4  2 − 2 3 5 4  2 − 2  3
. 
5 5 4 2 − 2 3
f) = = 2 2 = 2
4 2  2 3 4  2  2 3 4 2 − 2 3 4  2 − 22  32 4 . 2 − 22 . 3
5 4  2 − 2  3 4 2 − 2 3 2 2  2 − 3 2 2 − 3
= = =
20 4 4 2
x x x
3 4 6
 a2 ,  a3 ,  a2 Redução de
÷ = ÷ = ÷ = Radical ao
Mesmo Índice
12 12 12
m.m.c.
a 8
, a 9
, a4

3. EXERCÍCIOS DE APLICAÇÃO
01 a) 6
 23 ,
6
3 2 d) 8
 a4 ,
8
 a 4 b6 , 8
 ab
12 12 12 12 12 12
b)  24 ,  33 ,  42 e)  x8 ,  x9 ,  x4
10 10 10 12 12
c)  25 ,  22 ,  21 f) a 6 ,  a8
02 a) 9 3 9
 27 =  33 =  3 d)  a2 b = a  b
8 8
b)  34 =  3 e)  a5 b4
4 5 5
c)  34 = 3 f)  a5 . a . b5 . b2 = ab  ab 2
03 a) 2 . 3 2 =  18 d) 3
3 . 33 =  81
3
3 4
b) 5 . 23 =  40
5
e) 2 . 24 =  32
4
6
c) 2 . 26 =  128
6
f)  c . a 2 2 . b2 =  a 4 b2 c
04 a) 3 . 52 = 5  3 d)  25 =  2 4 . 2 = 22  2 = 4  2
b)  22 . 32 . 5 = 2 . 3 5 = 6  5 e)  a6 . a . b8 = a 3 b 4  a
3 3
c) 2 . 72 = 7 2 f)  a3 . a 2 . b6 . b = ab 2  a2 b
05 a) 9  2
b) 8  5
Mesmo índice e radical (para poder somar e/ou subtrair)
3 Soma-se os números fora do radical
c) 6  5
d) 2  2
e)  25 −  2 . 5 2   2 . 72 −  2 . 3 2 = 22  2 − 5  2  7  2 − 3  2 = 3  2
3 3 3
f)  33 . 5 −  23 . 5   26 . 5 −  5 = 3  5 − 2  5  22 5 −  5 = 4  5
3 3 3 3 3 3
3 3 3
e)  23 . 6 −  3 4   26 . 3   3 = 2  3 − 3  3  22  3   3 = 4  3
3 3 3 3 3 3
f)  2 2 . 3  2  33  5  3 .
52 − 2  2 4 . 3 = 2  3  3 . 2  3  5 . 5  3 − 22 . 2  3 =
2  3  6  3  25  3 − 8  3 = 25  3

4. 06 a)  15
3
Mesmo índice (para poder multiplicar e/ou dividir)
b)  16 Multiplica-se os números de fora do radical e os r
2 radicandos (o de dentro da raiz)
c) 4  24
d) 24  60
6 6 6 6 6
e) 5  a 4 b 2 . 3  a 3 b 3 . 4  a 3 b4 = 5 . 3 . 4  a 4  3  3 . b 2  3  4 = 60  a 10 b9
f) 4 10 x 8 y 6 . 5 10 x 3 y 7 . 2 10 x5 y 5 = 4 . 5 . 2 10 x 8  3  5 . y 6  7  5 = 40 10 x 16 y 18
    
07 a) 4 = 2
3
b) 3
5
c) 4
d) 6

16
 4 2 ÷ 63 = 6 216 =

6 2
27
10
e)  a5 b5 ÷ a 4 b4 = 10 a 5 − 4
 . b5 − 4 = 10
a . b
f) 18 ÷ 6 12 a 8 y 4 ÷ x6 y3 = 3 12 a 8 y 4 − 3 ÷ x 6 = 3 12 a 8 y ÷ x 6
  
08 2  3 −  33  3  3 . 5 2 = 2  3 − 3  3  3 . 5  3 = 14  3
09  2   3 =  2 .  3  3 .  2 =  6   6 = 6   6 =
3  2  3  3  2  2  3 2  22 3 2
2 6
  = 
3 6 5 6
6 6 6
2
10
 625
10000
−
1
8  3
=
25
100
−

3 1
3 2
2 
1 1
= − 2=
4 2
1−1
4
=
1
2
4 4 4
11  12   12  64 = 12   12  4 =  12   16 =  12  4 = 16 = 2
3 4 4
6 3 6 6 6 2
12  215 .  26 = 22  23 . 22 = 4 . 4  2 3 = 16  23 = 16  21 = 16  2
1 3
13
1
2
3 . 3 =3 ⇒
1
2
3 2
=
3
3
2
=3
3
2
−
1
2
=3 =3
2
2
1 1
2 2
3 3

5. 14
  218 =  218 = 218
6 12 1
12
1
÷ 124 12 =  218 ÷ 124 = 1
0
4 3 12
 124 124
15
 2  2   2 . 2 =  2  2  2 . 2 =  2  2 . 2
2 4 2 4 2
 8
. 2 4 = 2  2 . 2 2 . 24 =
15
 2 . 28 2
. 2 4 . 28 =
16
2 . 22 . 2 4 . 28 =
16
 215 = 2 16
16 3  2 − 2  2 . 32  3  23 . 32 = 3  2 − 2 . 3  2  3 . 2 . 3  2 =
3  2 − 6  2  18  2 = 15  2
17 17 − 15 =  2 =  2 .  32 =  64 = 4 = 1
17  15  32  32  32  32 2 32 4
18 a 2
b
. 4
b =  a 2 b .  b = 12 a 6 b3 . 12 b 3 = 12 a 6 b6
4
6
4 
12 2 2  
12 2 2
=
12
a 4 b4 = ab
3
  ab
3
 ab a b a b
19 x = 2
y= 2 . 72 −  25 −  2 2 . 2 ⇒ 7  2 − 2 2  2 − 2  2 ⇒ 2
x=y
20 x 3 6 12
 x =  x2 = 12 x−7 = 1
= 4 3 
12 9 12 7
x . x 2
x x x
6 2 6 6 6
21  32 .   4 ⇒  32 .  16 ⇒  323 .  16 ⇒ 6 32 3 . 6 8 ⇒
A= 6 6 6  
2 2 2
6 6 6
 32 3 . 8 ⇒  253 . 23 ⇒  215 . 23 ⇒  2 18 ⇒ 23 ⇒ 8
22 4
aa 3
= a . a
4 3 3
=
12
a 4 = 12 a 4
4

12 3
= 12
a = a 12
1
 a a 1 a
23  3a 2

 6a −  5 a = 3a   6a − 5a = 3a  a =
4 2 8
 2 4 4 2 2
 4a 2 =  22 a 2 = 2a
24  31   10 −  83 − 2 =  31   10 −  81 =  31   10 − 9 =  31   1 =
5 6 5 6 5 6 5 6
5
 32 = 2
25  1   6   5  4 = 1   6   9 = 1   6  3 = 1   9 =
1  3 =  4 = 2

6. 26  9  16 . 16 =  9 16 2
=  9 . 16 =  32 . 4 2 = 3 . 4 = 12
27 5
6 4 3 48
=
24
516 = 52 = 25
28 3  33  5  2 2 . 3 =  3  3  3  5 . 2  3 = 3  3  3  10  3 =
83 8 3 8 3
14  3 14 7
= =
8 3 8 4
12 12 12
29  2  2   2
3 2
. 28  = 2 2  2 
3 4 10
= 2  2
3 4 10
. 24  = 2 2 
3 8 14
12
= 2
3 8 14
12
. 28  =
12 12
 24 222 
 =   2 11 = 211 = 2048
12
30 2 . 4 . 1  2 . 2 . 32 8  2 2 . 25 8  27 8 . 23  2 8  27 8 . 23  2
= = = = 5
= = 3 2
=
2  8 . 128 2  23 . 27 2  210 2. 2 2  210 2 .2 . 2
8 2
= 2
2. 4
2 . 5 2 10  2
31 2 . 3 3 . 3  = 6 9  = 6 . 3  10 = 18  10 = 28
2 2
32 a)  2 4 . 5   22 . 5 = 22  5  2  5 = 6  5
b) 3  5   32 . 5 − 2  22 . 5 = 3  5  3  5 − 2 . 2  5 = 2  5
c) 2  6 . 52 − 4  6 . 32  6  6 . 22 = 2 . 5  6 − 4 . 3 6  6 . 2  6 =
10  6 − 12  6  12 6 = 10  6
3 3
33  23 . 3 −  34
=
3 3
2 3 − 3  3
=
3
−1  3
= 12
3
− 3
=
3
− 3
=
12
22 3 5
. 92 226 5
. 92   25 . 9 2 . 26  25 . 322 . 26
6
 211 . 3 4
12 12 12 12
−  34 − 3 4 2 . 38 −  2 . 312 −3 12 2
 = −12 2

= . = =
12
 211 . 3 4 12
 2 11 . 34 12
2 . 38
12
 212 . 312 2. 3 2