7°série - Matemática3 - Calculando Radicais
Autor: Resuminho | Publicado: 2012-04-21T10:43:00-07:00




Exercício P.55

1.&nbsp;&nbsp;&nbsp;&nbsp;
Nas
expressões seguintes, identifique: &nbsp;&nbsp;3√-8 , 10√1,
4√-16, 5√32, √49, 3√-125, √-1, 7√-1,
8√256



a)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
As
que são definidas no conjunto R;



Resposta: 3√-8, 5√32, 10√1, √49, 7√-1,
3√-125, 8√256



b)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
As
que não são definidas no conjunto R.



Resposta: √-1,&nbsp; 4√-16



2.&nbsp;&nbsp;&nbsp;&nbsp;
Diga se é definida ou não no conjunto R a raiz
quadrada de :



a)&nbsp;&nbsp;&nbsp;&nbsp;
49 = 7

b)&nbsp;&nbsp;&nbsp;
121 = 11

c)&nbsp;&nbsp;&nbsp;&nbsp;
-25 = não é raiz quadrado

d)&nbsp;&nbsp;&nbsp;
64 = 8

e)&nbsp;&nbsp;&nbsp;&nbsp;
10 = não é raiz quadrado 

f)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
-9 = não é raiz quadrado



3.&nbsp;&nbsp;&nbsp;&nbsp;
Verifique se a expressão √(b²-4ac) representa
um numero real quando a=10, b=-1 e c=-3.



√(b²-4ac) = √((-1)²-4(10)(-3)) =
√(1+120) = √121 = 11



4.&nbsp;&nbsp;&nbsp;&nbsp;
Sendo x=5 e y=4, verifique se a expressão
√(x²-y²) é definida no conjunto R



√(x²-y²) = √(5²-4²) = √(25-16) = √9
= 3



5.&nbsp;&nbsp;&nbsp;&nbsp;
Toda s as expressões a seguir são definidas no
conjunto R. Então, calcule o valor de:












a)&nbsp;&nbsp;&nbsp;&nbsp;
&nbsp;&nbsp;√25 

= 5



b)&nbsp;&nbsp;&nbsp;
&nbsp;&nbsp;√(-6)² 

= √36 

= √6² 

= 6



c)&nbsp;&nbsp;&nbsp;&nbsp;
&nbsp;&nbsp;&nbsp;&nbsp;5√-32 

= √(-2)5 

= -2



d)&nbsp;&nbsp;&nbsp;
&nbsp;&nbsp;&nbsp;√0,001


= √0,1² 

= 0,1



e)&nbsp;&nbsp;&nbsp;&nbsp;
&nbsp;&nbsp;&nbsp;-4√81


= -4√34 

= -3

f)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
&nbsp;&nbsp;&nbsp;&nbsp;-3√-8


= -3√(-2)³ 

= -2



g)&nbsp;&nbsp;&nbsp;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;6√64 

= 6√26 

= 6



h)&nbsp;&nbsp;&nbsp;
&nbsp;&nbsp;&nbsp;-√(-2)²


= -√4 

= -√2² 



i)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
&nbsp;&nbsp;√121 

= 11



j)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
&nbsp;&nbsp;&nbsp;-3√-125


= -³√(-5)³ 

= 5&nbsp;







6.&nbsp;&nbsp;&nbsp;&nbsp;
Determine o valor das expressões numéricas:









a)&nbsp;&nbsp;&nbsp;&nbsp;
&nbsp;&nbsp;&nbsp;4√16 - 3√-8

=4√24 - 3√(-2)³

=2&nbsp; – (-2)

=2+2

=4



b)&nbsp;&nbsp;&nbsp;
&nbsp;&nbsp;3√-125-4√1+√(-3)²

=3√(-5)³-4(1)+√9

=-5-4+3

=-9+3

=-6

c)&nbsp;&nbsp;&nbsp;&nbsp;
&nbsp;&nbsp;5√32 - 3√-27 + 6√1

=5√25 - 3√(-3)³ + 1

=2 – (-3) + 1

=2+3+1

=6



d)&nbsp;&nbsp;&nbsp;
&nbsp;&nbsp;&nbsp;7√-1-√16-3√-64

=7√(-1)7-√4²-3√(-4)³

=-1-4-(-4)

=-5+4

=-1

e)&nbsp;&nbsp;&nbsp;&nbsp;
&nbsp;&nbsp;(53√2³-3²)
/ (√6²+8²)

=(53√8-9) / (√36+64)

=(53√-1) / (√100)

=(53√(-1)-3) / (√10²)

=(5(-1)) / (10)

=(-5) / (10)

=-1/2




f)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;√((-4)²+(-3)²) - 3√(-5²+17)

=√(16 + 9) - 3√(-25+17)

=√25 - 3√-8

=√5² - 3√(-2)³

=5-(-2)

=5+2

=7


1.&nbsp;&nbsp;&nbsp;&nbsp;
Use o sinal = ou ≠ para comparar os números
reais a e b. &nbsp;A=√36 + √64&nbsp; e B=√(36+64)









A=√36 + √64


A=√6² + √8² 

A=6+8 

A=14 

B=√(36+64)

B=√100

B=√10²

B=10







A(14) &gt; B(10)


2.&nbsp;&nbsp;&nbsp;&nbsp;
Determine x, sabendo que

X=(-(-2)² - 3√-27) / (20-2)

X=(-4 - 3√(-3)³) / (1-2)

X=(-4-(-3)) / (-1)

X=(-4+3) / (-1)

X=(-1) / (-1)

X = 1


3.&nbsp;&nbsp;&nbsp;&nbsp;
Sendo x um número real positivo e y um número
real positivo, simplifique a expressão:

&nbsp; √x² * √y²

=√x²y²


Exercicio P.60

1.&nbsp;&nbsp;&nbsp;&nbsp;
Dê o valor de cada uma das expressões:







a)&nbsp;&nbsp;&nbsp;&nbsp; √10²

=10


b)&nbsp;&nbsp;&nbsp; 5√35

=5


c)&nbsp;&nbsp;&nbsp;&nbsp; 9√29

=2


d)&nbsp;&nbsp;&nbsp; 3√73

=7


e)&nbsp;&nbsp;&nbsp;&nbsp; 6√(2x)6

=2x


f)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;7√(2 * 5)7

=7√(10)7

=10


g)&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;√(5²)²

=√(25)²

=25


h)&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;4√(x²y)4

= x²y




2.&nbsp;&nbsp;&nbsp;&nbsp;
Decomponha o radicando em fatores primos e , em
seguida, use a propriedade dos radicais aritméticos para encontrar o valor das
expressões:



a)&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;√49 

= 7


b)&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;6√729

=6√36

=3


c)&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;4√625

=4√54

=5


d)&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;10√1024

=10√210

=2


e)&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;4√81

=4√34

=3


f)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;³√343

=³√7³

=7


1.&nbsp;&nbsp;&nbsp;&nbsp;
&nbsp;Dividindo o índice do radical e o expoente do
radicando por um mesmo numero, diferente de zero, simplifique os radicais:




a)&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;15√25

=3√2


b)&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;14√37

=2√3


c)&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;16√104

=4√10


d)&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;9√x6

=3√x2


e)&nbsp;&nbsp;&nbsp;&nbsp; 10√58

=5√54


f)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;20√a12

=5√a³


g)&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;8√y4

=2√y1


h)&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;21√614

=3√62



2.&nbsp;&nbsp;&nbsp;&nbsp;
Determine o valor do numero x em cada uma das
igualdades:




a)&nbsp;&nbsp;&nbsp;&nbsp; 14√28 = x√24

7√24 = x√24

x=7


b)&nbsp;&nbsp;&nbsp; 15√105 = 3√10x

3√10 = 3√10x

x=1

c)&nbsp;&nbsp;&nbsp;&nbsp; 8√54 = √5x

2√5 = √5x

x=1


d)&nbsp;&nbsp;&nbsp; 10√6x = 5√6

10√6x = 10√6²

x=2




3.&nbsp;&nbsp;&nbsp;&nbsp;
Decomponha o radicando em fatores primos e, em
seguida, simplifique cada um dos radicais:




a)&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;10√32

=10√25

=2√2


b)&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;9√27

=9√3³

=3√3


c)&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;16√81

=16√34

=4√3


d)&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;6√16

=6√24

=3√2² ou 6√4


e)&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;8√64

=8√26

=4√2³&nbsp; ou 4√8


f)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;12√1024

=12√210

=6√25




4.&nbsp;&nbsp;&nbsp;&nbsp;
Escreva sob a forma de um único radical:



a)&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;√5√x

=10√x


b)&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;√√6

=4√6


c)&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;4√3√a

=12√a


d)&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;3√3√2

=9√2


e)&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;8√√10

=16√10


f)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;√√√2

=8√2




5.&nbsp;&nbsp;&nbsp;&nbsp;
Escreva na forma mais simples possível:




a)&nbsp;&nbsp;&nbsp;&nbsp; 4√3√64

12√26

2√2


b)&nbsp;&nbsp;&nbsp; √5√243

10√35

2√3



6.&nbsp;&nbsp;&nbsp;&nbsp;
Sendo x um número real positivo, transforme em
um único radical:





a)&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;√4√x

=8√x


b)&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;6√√2x

=12√2x


c)&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;√3√√x

=6√√x

=12√x


d)&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;7√3√x5

=21√x5




7.&nbsp;&nbsp;&nbsp;&nbsp;
Determine o número real x das igualdades:







a)&nbsp;&nbsp;&nbsp;&nbsp; x√6√10=24√10

6x√10=24√10

6x=24

X=4



b)&nbsp;&nbsp;&nbsp; 5√x√3 = 15√3

5x√3 = 15√3

5x=15

X=3




8.&nbsp;&nbsp;&nbsp;&nbsp;
Escreva como um produto de radicais:







a)&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;√(5 * 7)

=√5 *√7


b)&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;³√ax

=³√a * ³√x


c)&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;7√(3² * 11)

=7√3²&nbsp; * 7√11


d)&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;6√(x * y)

=6√x * &nbsp;6√y


e)&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;√2ab

=√2 * √a * √b


f)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;³√x²y

=³√x² * ³√y




9.&nbsp;&nbsp;&nbsp;&nbsp;
Decomponha o radicando em fatores primos e
escreva cada expressão na forma de um produto de radicais:



a)&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;√10

=√(2 *5)

=√2 * √5


b)&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;6√21

=6√(3*7)

=6√3 * √7


c)&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;9√35

=9√(5 * 7)

=9√5 * 9√7


d)&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;7√30

=7√(2*3*5)

=7√2 * 7√3 * 7√5


e)&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;10√15

=10√3*5

=10√3 * 10√5


f)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;³√154

=³√(2*7*11)

=³√2 * ³√7 * ³√11





10.&nbsp; Transforme
em um único radical positivo, transforme em um único radical cada um dos produtosm
simplificando o radical obtido:




a)&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;√3 * √5

=√15


b)&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;³√2 * ³√7

=³√14


c)&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;6√3 * 6√13

=6√39


d)&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;√2*√5 * =√7

=√70




11.&nbsp; Sendo x e
y dois números reais positivos, transforme em um único radical cada um dos
produtos, simplificando o radical obtido:




a)&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;12√x5 &nbsp;* 12√x

=12√x6

=2√x


b)&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;20√y3 *
20√y

=20√y4

=5√y

c)&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;15√x4y²
* 15√xy³

=15√x5y5

=15√(xy)5

=3√xy

d)&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;14√y³ * 14√y³
* 14√y

=14√y7

=2√y




12.&nbsp; Transforme
em um quociente de radicais cada uma das expressões:




a)&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;√(11/6)

=√11 / √6


b)&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;³√(7/5)

=³√7 ³ / √5


c)&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;8√(3/11)

=8√3 / 8√11


d)&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;√(13/2)

=√13 / √2 

e)&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;6√(2/13)

=6√2 6 / √13


f)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;7√(4/5)

=7√4 / 7√5



13.&nbsp; Os
números a e b são números reais positivos. Nessas condições, simplifique os
radicais 6√a³ e 12√b6, calculando em seguida a
expressão que representa o produto dos radicais obtidos.



&nbsp;&nbsp;




6√a³

=²√a



&nbsp;&nbsp; 12√b6

=²√a






&nbsp;&nbsp;²√a * ²√a

=²√a²

=a






Exercicios P.64

1.&nbsp;&nbsp;&nbsp;&nbsp;
Simplifique os radicais retirando fatores do
radicando.&nbsp;







a)&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;√(2*72)

&nbsp;=7√2



b)&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;5√(35*11)

=35√11&nbsp;



c)&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;3√(2³*3*5³)

=2*5 3√3

=103√3



d)&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;√10³

=10√10&nbsp;



e)&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;√27

=2³√2

=8√2



f)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;³√(2³*54)

=2*5³√5

=10³√5







2.&nbsp;&nbsp;&nbsp;&nbsp;
Os números x e y são numero reais positivos.
Simplifique, então, os radicais, retirando fatores do radicando.&nbsp;



a)&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;√x5

=x²√x



b)&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;³√y4

=y&nbsp; ³√y



c)&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;√x9

=x4 √x



d)&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;5√y12

=y² 5√y2



e)&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;√x²y³

=xy√y



f)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;5√x5y7

=xy 5√y2



g)&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;9√y10

=y 9√y



h)&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;10√x13

=x 10√x3





3.&nbsp;&nbsp;&nbsp;&nbsp;
Você deve fatorar o numero que aparece no
radicando. Em seguida, retirando fatores do radicando, simplifique os radicais.&nbsp;







a)&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;√75

=√3*5²

=5√3



b)&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;√700

=√2² * 5² * 7

=10√7



c)&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;3√250

=3√2*5³

=5 3√2



d)&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;5√192

=5√26 * 3

=2 5√3*2

=2 5√6



e)&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;4√176

=4√24 * 11

=2 4√11&nbsp;



f)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
&nbsp;&nbsp;√800

=√25 * 5²

=2² * 5 √2

=20√2



g)&nbsp;&nbsp;&nbsp;
&nbsp;&nbsp;&nbsp;√1800

=√(2³ * 3² * 5²)

=2 * 3 * 5√2

=30√2



h)&nbsp;&nbsp;&nbsp;
&nbsp;&nbsp;&nbsp;3√375

=3√(3*5³)

=5 3√3



i)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;√2700

=√(2² * 3³ * 5²)

=10√3



j)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;6√640

=6√27 * 5

=2 6√10





4.&nbsp;&nbsp;&nbsp;&nbsp;
Se x=√5184, qual é o valor de x?



x=√5184

x=√(26 * 34)

x= 2³ * 3²

x=8 * 9

x=72



5.&nbsp;&nbsp;&nbsp;&nbsp;
Considerando que √2=1,14; √3=1,73; √5=2,23 e
√6=2,44, simplifique cada um dos radicais e determine, na forma decimal, o
valor de:&nbsp;







a)&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;√50

=√(5² * 2)

=5√2

=5 * 1.41

=7,05



b)&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;√27

=√3³

=3√3

=3 * 1.73

=5,19



c)&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;√80

=√(24 * 5)

=2√5

=2 * 2,23

=8,92



d)&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;√150

=√(2 * 3 * 5²)

=5√6

=5 * 2,44

=12,20&nbsp;



e)&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;√200

=√(2³ * 3²)

=10√2

=10 * 1,41

=14,1



f)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;√500

=√(2² * 5³)

=2 * 5√5

=10√5

=10 * 2,23

=22,3



g)&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;√294

=√(2 * 3 * 7²)

=7√2 * 3

=7√6

=7 * 2,44

=17,08



h)&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;√675

=√5² * 3³

=5 * 3√3

=15√3

=15 * 1,73

=25,95





6.&nbsp;&nbsp;&nbsp;&nbsp;
Considere que a&nbsp;
e b são números reais positivos. Nessas condições, simplifique os
radicais:&nbsp;







a)&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;√(9a³)

=√(3²a³)

=3a√a



b)&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;b√(20b²)

=b√(2² * 5 *b²)

=2b²√5



c)&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;ab³√(27a4)

=ab³√(3³a4)

=3a²b3√a



d)&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;ab√(a²b5)

=a²b³√(b)



e)&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;1/2√(176a4)

=1/2√(24 * 11 * a4)

=½
* 2² * a²√11

=2a²√11



f)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;1/(ab)√(12a4b3)

=1/(ab)√(2² * 3 * a4b3)

=1/(ab)
* 2 * a² * b√(3b)

=2a√3b



g)&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;1/a2√(50a7)

=1/a2√(2 * 5² * a7)

= (1/a2) * 5 * a³ √2a

=5a√2a



h)&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;1/4a²b√(48a²b4)

=1/4a²b√(24 * 3 * a²b4)

= (¼) * 2² a²b * ab² √3

=a³b³√3





7.&nbsp;&nbsp;&nbsp;&nbsp;
Sabe-se que a é a raiz quadrada exata de 4096,
b é a raiz quarta exata de 1296 e c é a raiz cubica exata de 3375. Qual é o
valor da expressão a+b+c?&nbsp;







a²=4096

a=√4096

a=√212

a=26

a=64



b4=1296

b=4√1296

b=4√(24 * 34)

b=2 * 3

b=6



c³=3375

c=³√3375

c=√(3³ * 5³)

c=3*5

c=15





&nbsp;&nbsp; a+b+c

=64+6+15

=85



8.&nbsp;&nbsp;&nbsp;&nbsp;
Simplificando os radicais, calcule o valor da
expressão 6√729+5√1024-³√125

&nbsp; &nbsp; 6√729+5√1024-³√125

=6√36 + 5√210 -
³√5³

=3+2²-5

=3+4-5

=2



9.&nbsp;&nbsp;&nbsp;&nbsp;
Se A=³√1728 e B=6√64, qual é o valor
da razão A/B?&nbsp;







A=³√1728

A=³√26 * 3³

A= 2² * 3

A=4*3

A=12



B=6√64

B=6√26

B=2

&nbsp;&nbsp; 

A/B

=12/2

=6&nbsp;







10.&nbsp; Considerando
√3=1,73 e √10=3,16, qual é o valor, na forma decimal, da expressão √1000-√27?

&nbsp; √10³-√3³

=10√10 - 3√3

=10 * 3,16 * 3 * 1,73

=31.16 – 5.19

=26.41



11.&nbsp; Simplificando
o radical e, em seguida, colocando o fator comum em evidencia, fatore as
expressões.







a)&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;5+√50

=5+√(5² * 2)

=5+5√2

=5(1+√2)



b)&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;3-√18

=3-√(3² * 2)

=3-3√2

=3(1-√2)



c)&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;10-√8

=10-√2³

=10-2√2

=2(5-√2)



d)&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;10+√200

=10+√(2³ * 5²)

=10+10√2

=10(1+√2)







12.&nbsp; Qual p
valor que você vai encontrar ao simplificar a expressão 6√1728 / 6√64?

&nbsp;&nbsp; 6√1728 / 6√64

=6√(26 * 3³) / 6√26

=2 6√3³ / 2

=2 √3 / 2

=√3



13.&nbsp; Um
terreno quadrado tem 9800m² de área. Sendo l a medida do lado desse terreno e
considerando √2=1,41, calcule o valor de l.


l²=9800

l=√9800

l=√(2³ * 5² * 7²)

l=2*5*7√2

l=70√2

l=70*1,41

l=98,7



14.&nbsp; Considere
a expressão E=√(ab+c).Qual é o valor de E quando a=40, b=25 e c=200?

E=√(ab+c)

E=√((40)(25)+200)

E=√(1000+200)

E=√1200

E=√(24 * 3 * 5²)

E=2² * 5√3

E=20√3

15.&nbsp; Qual é o
numero que se obtém simplificando a expressão √(32*(0,0004)*25000) ,
considerando que √5=2,23?

&nbsp; &nbsp;√(32*(0,0004)*25000)

=√(0.0128 * 25000)

=√320

=√26 * 5

=2³√5

=8√5

=8 * 2,23

=17,84









<!--[if !supportLists]-->1.&nbsp;&nbsp;&nbsp;&nbsp;
<!--[endif]-->Transforme em um só radical e determine o valor
das expressões a seguir.










<!--[if !supportLists]-->a)&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;√³√√4096

=6√√212

=12√212

=2

<!--[if !supportLists]-->b)&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;&nbsp;√√10000

=4√104

=10







Exercicio P.69

<!--[if !supportLists]-->1.&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->Copie no
caderno apenas os itens com igualdades verdadeiras.&nbsp;








<!--[if !supportLists]-->a)&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->√7+√7 = 2√7

Verdadeiro



<!--[if !supportLists]-->b)&nbsp;&nbsp;&nbsp; <!--[endif]-->√6+√5 = √11

Falso



<!--[if !supportLists]-->c)&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->1+√2 = √3

Falso



<!--[if !supportLists]-->d)&nbsp;&nbsp;&nbsp; <!--[endif]-->√10+√10+√10
= 3√10 

Verdadeiro








<!--[if !supportLists]-->2.&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->Escreva
na forma mais simples possível as expressões seguintes.&nbsp;








<!--[if !supportLists]-->a)&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;&nbsp;9√10-5√10

= 4√10



<!--[if !supportLists]-->b)&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;&nbsp;&nbsp;2√5+7√5-16√5

= 2+7-16√5

= -7√5



<!--[if !supportLists]-->c)&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;√6+√6+√6

= 3√6



<!--[if !supportLists]-->d)&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;&nbsp;6 ³√2 - 10 ³√2

= 6-10√2

= -4 ³√2



<!--[if !supportLists]-->e)&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;&nbsp;2√x+2√x+2√x+2√x

= (2+2+2+2)√x

= 8√x



<!--[if !supportLists]-->f)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;4√3 + 7 4√3
- 11 4√3 + 2 4√3

= (1+7-11+2) 4√3

= (-1) 4√3

= - 4√3&nbsp; àdeu diferente



<!--[if !supportLists]-->g)&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;&nbsp;5a√10+7a√10-9a√10

= (5a+7a-9ª)√10

= (12a-9ª)√10

= 3a√10



<!--[if !supportLists]-->h)&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;&nbsp;(2+7)√6+2√6-1

= 1+9√6



<!--[if !supportLists]-->i)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;&nbsp;2√7+2+√7-3-√7

= (2+1-1)√7+2-3

= 2√7-1



<!--[if !supportLists]-->j)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;&nbsp;&nbsp;2√5+8√2-6√2+8√5-2√2

= (2+8)√5+(8-6-2)√2

= 10√5+0√2

= 10√5+0

= 10√5&nbsp;





<!--[if !supportLists]-->3.&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->Reduza
cada expressão à sua forma mais simples.&nbsp;








<!--[if !supportLists]-->a)&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;√12+√75-9√3+√27+√48

=√(2² * 3) + √(3*5²) - 9√3+√3³+√(24 * 3)

=2√3+5√3-9√3+3√3+2²√3

= (2+5-9+3+2²)√3

=5√3



<!--[if !supportLists]-->b)&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;4√125+3√45-30√5

=4√5³+3√(3²*5)-30√5

=4*5√5+3*3√5-30√5

=20√5+9√5-30√5

= (20+9-30)√5

=-√5



<!--[if !supportLists]-->c)&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;&nbsp;√54+√6-√150+2√24

=√(2*3³) +√6-√(2*3*5²)+2√(2³ * 3)

=3√(2*3)+ √6-5√(2*3)+2*2√(2*3)

=3√6+√6-5√6+4√6

= (3+1-5+4)√6

=3√6&nbsp;&nbsp;





<!--[if !supportLists]-->4.&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->Observe o
triangulo seguinte, em que aparecem as medidas dos lados, em unidades de
comprimento. Qual é o perímetro desse triangulo?



&nbsp;&nbsp; √28+√112+√175

=√(2²*7)+√(24*7)+ √(5²*7)

=2√7+2²√7+5√7

=2√7+4√7+5√7

= (2+4+5) √7

=11√7



<!--[if !supportLists]-->5.&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->Um número
A é tal que:

A=√18+3√50+√98

Fazendo
√2=1,41,qual é a forma decimal do numero A?



A=√18+3√50+√98

A=√(2*3²)+3√(5²*2)+ √(2*7²)

A=3√2+3*5√2+7√2

A=3√2+15√2+7√2

A=(3+15+7)√2

A=25√2

A=25*1,41

A=35,25



<!--[if !supportLists]-->6.&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->Calcule:








<!--[if !supportLists]-->a)&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;&nbsp;√(16x)+√(9x)√-√(36x)

=√(24x)+ √(3²x)- √(2² *3²x)

=2²√x+3√-2*3√x

=4√x+3√x-6x

= (4+3-6)√x

=√x



<!--[if !supportLists]-->b)&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;&nbsp;√(8a³)+√(72a³)-√(18a³)

=√(2³a³)+√(2³ * 3²a³)-√(2 * 3²a)

=2a√2a+2*3a√2a-3a√2

=2a√2a+6a√2a-3a√2

= (2a+6a-3a)√2a

=5a√2a



<!--[if !supportLists]-->c)&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;3x√(x²y)-2√(x4y)+6x²√y

=3x*x√y-2*x²√y+6x²√y

=3x²√y+2x²√y+6x²√y

= (3x²-2x²+6x²)√y

=7x²√y



<!--[if !supportLists]-->d)&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;&nbsp;3a√(ab³)-7b√(a³b)+ab√(4ab)

=3ª*b√ab-7b*a√ab+ab√2²ab

=3ab√ab-7ab√ab+2ab√ab

= (3ab-7ab+2ab)√ab

=-2ab√ab







<!--[if !supportLists]-->7.&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->Qual é o
valor de x na expressão a seguir

x=1/4√48+1/2√243-1/6√12

x=1/4√(24*3)+1/2√35-1/6√2²*3

x=1/4*2²√3+1/2*3²√3-1/6*2√3

x=1/4*4√3+1/2*9√3-1/3√3

x=√3+9/2√3-1/3√3

x=(1 + 9/2 - 1/3)√3

x=(6+27-2)/6√3

x=(31√3) /6



<!--[if !supportLists]-->8.&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->Qual [e o
perímetro de uma região triangular, cujos lados medem 4√486 cm, 4√96 cm e 5√216
cm



&nbsp;&nbsp; 4√486+4√96+5√216

=4√(2*35)+4√(25*3)+5√(2³*3³)

=4*3²√(2*3)+4*2²√(2*3)+5*2*3√(2*3)

=4*9√9+4*4√6+30√6

=36√6+16√6.30√6

= (36+16+30)√6

=82√6



<!--[if !supportLists]-->9.&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->Qual é a
forma simplificada da fração a seguir?

&nbsp;&nbsp; (√28+√175)/ √63

= (√ (2²*7)+ √(5²*7))/ √(3²*7)

= (2√7+5√7) / 3√7

= (7√7)/(3√7)

=7/3



<!--[if !supportLists]-->10.&nbsp; <!--[endif]-->Veja a
região retangular a seguir, em que as medidas dos lados são expressas em cm.
Qual é o perímetro dessa região retangular? Lado=√40, comprimento=√250.



&nbsp; √250+√250+√40+√40

=2√250+2√40

=2√(5³*2)+2√(2³*5)

=2*5√(2*5)+2*2√(2*5)

=10√10+4√10

= (10+4) √10

=14√10



<!--[if !supportLists]-->11.&nbsp; <!--[endif]-->Um numero
real x é expresso por ³√250-³√16+³√54-³√2. Qual é o numero x?



&nbsp;&nbsp; ³√250-³√16+³√54 - ³√2

=³√(5³*2) - ³√24 + ³√(2*3³) - ³√2

=5³√2- 2³√2 + 3³√2 - ³√2

= (5-2+3-1)³√2

=5 ³√2



<!--[if !supportLists]-->12.&nbsp; <!--[endif]-->Qual é a
forma mais simples de escrever a fração a seguir?



&nbsp;&nbsp; (√50-√18)/ √200

= (√(5²*2)-√(2*3²))/ √(2³*5²)

= (5√2-3√2)/ 2*5√5

= (2√2) / (10√2)

=1 / 5



<!--[if !supportLists]-->13.&nbsp; <!--[endif]-->Simplificando
o numerador, escreva na forma irredutível a expressão.

&nbsp;&nbsp; (3√20+√80-2√45)/8.

= (3√(2²*5)+√(24*5)-2√(3²*5)) / 8

= (3*2√5+2√5-2*3√5) / 8

= (6√5+4√5-6√5) / 8

= ((6+4-6)√5) / 8

= (4√5) / 8

=√5 / 2



<!--[if !supportLists]-->14.&nbsp; <!--[endif]-->Se x=√2 e
y=√98-√32-√8, qual é o valor de x+y?



y=√98-√32-√8

y=√(7²*2)-√25-√2³

y=7√2-2²√2-2√2

y=7√2 - 4√2 - 2√2

y=(7-4-2)√2

y= √2



&nbsp;&nbsp;&nbsp; x+y

=√2+√2

=2√2



<!--[if !supportLists]-->15.&nbsp; <!--[endif]-->São dados
os números reais a, b e c tais que a=1-√27, b=1+√75 e c=2-√108. Determine o valor
de:&nbsp;

<!--[if !supportLists]-->a)&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;a+b+c

=1-√27+1+√75+2-√108

=1+1+2-√3³+√(3*5²)-√(2²*3³)

=4-3√3+5√3-3*2√3

=4+(-3+5-6) √6

=4-4√3

<!--[if !supportLists]-->b)&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;&nbsp;a-b-c

=1-√27-(1+√75)-(2-√108)

=1-√3³-1-√(3*5²)-2+√(2²*3³)

=1-1-2-3√3-5√3+2*3√3

=-2+(-3-5+6)√3

=-2-2√3



<!--[if !supportLists]-->16.&nbsp; <!--[endif]-->Fazendo
√52,23 e √2=1,41, qual é o valor do numero real x na forma decimal sabendo que



x=√5000+√500+√50+√5.

x=√2*54+√2*5³+√2*5²+√5.

x=2*5²√2+2*5√5+5√2+√5.

x=2*25√2+10√5+5√2+√5.

x=50√2 + 10√5 + 5√2 + √5

x=(50+5) √2+(10+1) √5

x=55√2+11√5

x=55*1,41+11*2,23

x=77,55+24,53

x=102,08



<!--[if !supportLists]-->17.&nbsp; <!--[endif]-->Sabe-se
que A=√243-√162 e B=√300-√50. Qual é o valor de A+B?








A=√243-√162

A=√35-√2*34

A=3²√3-3²√2

A=9√3-9√2



B=√300-√50

B=√(32²*5²*3)-√(2*5²)

B=2*5√3-5√2

B=10√3-5√2



&nbsp;&nbsp; A+B

=9√3-9√2+10√3-5√2

= (9+10) √3+(-9-5)√2

=19√3-14√2




<!--[if !supportLists]-->18.&nbsp; <!--[endif]-->Considerando
√3=1,7 e √2=1,4 dê, na forma decimal, o valor da expressão 4√3-7√18+5√48+√200.

4√3-7√18+5√48+√200

=4√3-7√2*3²+5√24*3+√2*5²

=4√3 - 7*3√2 + 5*2²√3 +
2*5√2

=4√3-21√2+20√3+10√2

=4*1,7 - 21*1,4 + 20*1,7
+ 10*1,4

=6,8 - 29,4 + 34 + 14

=25,4



19.&nbsp;&nbsp;&nbsp;&nbsp;
Considerando √3=1,7 e √2=1,4 dê, na forma
decimal, o valor da expressão 4√3-7√18+5√48+√200.



4√3-7√18+5√48+√200

=4√3-7√2*3²+5√24*3+√2*5²

=4√3 - 7*3√2 + 5*2²√3 +
2*5√2

=4√3-21√2+20√3+10√2

=4*1,7 - 21*1,4 + 20*1,7
+ 10*1,4

=6,8 - 29,4 + 34 + 14

=25,4



Exercício P.72

1.&nbsp;&nbsp;&nbsp;&nbsp; Efetue as
multiplicações:



a)&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;√5*√7

=√35



b)&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;5√2a*5√7a

=5√14a²



c)&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;3*√2*9√3

=27√6



d)&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;³√(xy) * ³√(xy)

=³√(x²y²)



2.&nbsp;&nbsp;&nbsp;&nbsp; Escreva
as multiplicações na forma mais simples:



a)&nbsp;&nbsp;&nbsp;&nbsp;√6*√3

&nbsp; &nbsp; =√18

&nbsp; &nbsp; =√(2*3²)

&nbsp; &nbsp; =3√2



b)&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;√28 * √21

&nbsp; &nbsp; &nbsp; &nbsp; =√588

&nbsp; &nbsp; &nbsp; &nbsp; =√(2² * 7² * 3)

&nbsp; &nbsp; &nbsp; &nbsp; =2*7√3

&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;=14√3&nbsp;



c)&nbsp;&nbsp;&nbsp;&nbsp;√10 * √20

&nbsp; &nbsp; &nbsp; =√200

&nbsp; &nbsp; &nbsp; =√(2³ * 5²)

&nbsp; &nbsp; &nbsp; =2*5√2

&nbsp; &nbsp; &nbsp; =10√2



d)&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;2√21*5√2*√7&nbsp;

&nbsp; &nbsp; &nbsp; =10√294

&nbsp; &nbsp; &nbsp; =10√(7² * 2*3)

&nbsp; &nbsp; &nbsp; =10*7√6

&nbsp; &nbsp; &nbsp; =70√6




<!--[if !supportLists]-->3.&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->No
retângulo seguinte, as medidas indicadas são dadas em centímetros. Determine:
lado=5√2, e comprimento=9√2.

a)&nbsp;&nbsp;&nbsp;&nbsp; O
perímetro do retângulo;

&nbsp;9√2+9√2+5√2+5√2

= (9+9+5+5)√2

=28√2



b)&nbsp;&nbsp;&nbsp; A área do
retângulo.

&nbsp;9√2*5√2

=45√2²

=45*2

=90cm²



<!--[if !supportLists]-->4.&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->Qual é a
área de um quadrado cujo lado mede 9√5 unidades de comprimento?

(Ao
multiplicarmos a medida do lado de um quadrado por ela mesma, obtemos a área do
quadrado.)

&nbsp;9√5*9√5

=81√5

=81*5

=405&nbsp;

<!--[if !supportLists]-->5.&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->Calcule,
na forma decimal, a área do triangulo da figura, adotando&nbsp; √3=1,73.

Triangulo
tem 5√6cm de comprimento e 3√2cm de altura.

(A área
de um triangulo é dada pela metade do produto da medida da base pela medida da
altura.)



&nbsp; (5√6*3√2) / 2

=15√12 /2

=(15√(2²*3)) /2

=(15*2√3) / 2

=(30√3) /2

=(30*1,75) /2

=51,9/2

=25,95 cm²



<!--[if !supportLists]-->6.&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->A área de
um trapézio é dada pela fórmula A=((B+b)h)/2, em que B representa a medida da
vase maior, b representa a medida da vase menor e h representa a medida da
altura, Calcule a área do trapézio a seguir: comprimento1=2√5, comprimento2=3√5
e √5 de altura.

A=((B+b)h)/2

A=((3√5 + 2√5)*√5)/2

A=((5√5)*√5)/2

A=(5√5) /2

A=(5*5) / 2

A=25/2

A=12,5



<!--[if !supportLists]-->7.&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->Um bloco
tem √12 cm de comprimento, √6 cm de largura e √3 cm de altura. Fazendo √6=2,45,
calcule o volume desse bloco.

(O volume
de um bloco retangular(paralelepípedo retângulo) á obtido multiplicando entre
si as dimensões do bloco)



&nbsp;&nbsp; √12*√6*√3

=√72*√3

=√216

=√(2³*3³)

=2*3√2*3

=6√6

=6*2,45

=14,7cm³



<!--[if !supportLists]-->8.&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->Considerando
que todas as letras representam números reais positivos, efetue as multiplicações:&nbsp;

a)&nbsp;&nbsp;³√x² * ³√x²

&nbsp; &nbsp; =³√x4

&nbsp; &nbsp; &nbsp;=x ³√x



b)&nbsp;&nbsp;6√a5 *
6√x3 * 6√a5

&nbsp; &nbsp; =6√a13

&nbsp; &nbsp; &nbsp;=a2 6√a



c)&nbsp;&nbsp;&nbsp;&nbsp;4√st³ * 4√t

&nbsp; &nbsp; &nbsp; =4√st4

&nbsp; &nbsp; &nbsp;=t 4√s

&nbsp; &nbsp; &nbsp;

d)&nbsp;&nbsp;&nbsp;5√(a4b)
* 5√(ab6)

=5√(a5 b7)

=ab 5√b²



e)&nbsp;&nbsp;&nbsp;√(a/x) * √((ab)/x)

=√((a²b)/x²)

= (a√b) /x&nbsp; ou a/b√b



f)&nbsp;8√(x³y7)
* 8√(2x5y)

=8√(2x8y8)

=xy 8√2



g)√((2a)/n) * √((2x)/3) * √((2ax)/n)

= √(2³a²x²) / √(3n²)

= (2ax √2)
/ n √3

=√(2ax/n) &nbsp;√(2/3)



Exercicios P.74

1.&nbsp;&nbsp;&nbsp;&nbsp; Efetue as
multiplicações?

a)&nbsp;√2*(√6-√3)

=√12-√6

=√(2²*3)- √6

=2√3-√6

b)&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;√7*(√7+√2)

=√7²+√14

c)&nbsp;&nbsp;&nbsp;&nbsp; √10*(5√2-3√10)

=5√20-3√10²

=5√(2² * 5)-3*10

=5*2√5-30

=10√5-30

d)&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;√5*(7+√5)

=7√5+√5²

=7√5+5&nbsp;

e)&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;√15*(√3+√5)

=√45+√75

=√(3²*5)+ √(5²*3)

=3√5+5√3

f)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;√8*(2-√6)

=2√8 - √48

=2√2³ - √(243)

=2*2√2 - 2²√3

=4√2 - 4√3

2.&nbsp;&nbsp;&nbsp;&nbsp; Efetue as
multiplicações:

a)&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;(√2-√6)*( √2+2√6)

=√2²+2√12-√12-2√6²

=2+√12-2*6

=2+√(2²*3)-12

=10+2√3

b)&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;(5-√7)*(5+√7)

=25+5√7-5√7-√7

=25-7

=18

c)&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;(3√5-2)*( √5+3)

=3√5²+9√5-2√5-6

=3*5+7√5-6

=15-6+7√5

=9+7√5

d)&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;(4+√13)*(4-√13)

=16-4√13+4√13-√13²

=16-13

=3

3.&nbsp;&nbsp;&nbsp;&nbsp; Qual é o
valor do numero real A?

A=√6(√2+1)-
√2(√3-√6)

A=√12+√6-√6+√12

A= 2√12

A=2√(2²*3)

A=2*2√3

A=4√3

4.&nbsp;&nbsp;&nbsp;&nbsp; Encontre
o valor do numero real x, tal que x=(3-2√3)*(1+√3)

x=(3-2√3)*(1+√3)

X=3+3√3-2√3-2√3²

X=3+√3-2*3

X=3+√3-6

X=-3+√3

5.&nbsp;&nbsp;&nbsp;&nbsp; Dada a
região retangular da figura seguinte, determine:

Lado =
(6-√5)cm,&nbsp; comprimento =(5+√5)cm

a)&nbsp;&nbsp;&nbsp;&nbsp; O
perímetro do retângulo;

&nbsp;(5+√5)2+(6-√5)2

=10+2√5+12-2√5

=22

b)&nbsp;&nbsp;&nbsp; A área do
retângulo.

(5+√5)(6-√5)

=30-5√5+6√5-√5²

=30+√5-5

=25+√5

6.&nbsp;&nbsp;&nbsp;&nbsp; Usando a
multiplicação, calcule:

a)&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;(1+√5)²

= (1+√5)(1+√5)

=1+√5+√5+√5²

=1+2√5+5

=6+2√5

b)&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;(2-√3)²

= (2-√3)(2-√3)

=4-2√3-2√3+√3²

=4-4√3+3

=7-4√3

c)&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;(√5+√3)²

= (√5+√3)( √5+√3)

=√5²+√15+√15+√3²

=5+2√15+3

=8+2√15

d)&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;(√7-√2)²

= (√7-√2)( √7-√2)

=√7²-√14-√14+√2²

=7-2√14+2

=9-2√14

7.&nbsp;&nbsp;&nbsp;&nbsp; Qual é o
resultado da multiplicação?

√(7+√5) *
√(7-√5)

=√((7+√5)(7-√5))

=√(49-√5²)

=√(49-5)

=√44

=√(2²*11)

=2√11

8.&nbsp;&nbsp;&nbsp;&nbsp; Qual é o
numero real x expresso por √(10+√10)* √(10-√10)?

&nbsp; &nbsp;√(10+√10)* √(10-√10)

=√((10+√10)(10-√10))

=√(100-√10²)

=√(100-10)

=√90

=√(2*3²*5)

=3√(2*5)

=3√10

9.&nbsp;&nbsp;&nbsp;&nbsp; A
expressão (-5+2√7)*(4+√7)-3√7 pode ser representada por um numero real inteiro.
Qual é esse número?

&nbsp;(-5+2√7)*(4+√7)-3√7

=-20-5√7+8√7+2√7²-3√7

=-20-5√7+8√7+2*7-3√7

=-20+14+(-5+8-3) √7

=-6+0√7

=-6

10.&nbsp; Considerando
√2=1,41, qual é o numero decimal que representa o resultado da multiplicação
(4√2+3)*(√2-1)?

&nbsp;(4√2+3)*(√2-1)

=4√2²-4√2+3√2-3

=4*2+√2-3

=8-3-√2

=5-√2

=5&nbsp; * 1,41

=3,59

11.&nbsp; Qual é a
forma mais simples de se escrever a fração a seguir?

((4+√2)*(4-√2))/((3+√3)*(3-√3))

= (16-4√2+4√2-√2²) / (9-3√3+3√3-√3²)

= (16-2) / (9-3)

=14/6

=7/3

12.&nbsp; A
propriedade fundamental das proporções nos diz que o “produto dos extremos é
igual ao produto dos meios”. Usando essa propriedade, determine o valor de x
nas proporções:

a)&nbsp;&nbsp;&nbsp;&nbsp; x/√6=√24/2

2x=√144

2x=12

X=6

b)&nbsp;&nbsp;&nbsp; 2/(√13+√10)=(
√13-√10)/9

9x=(√13+√10)( √13-√10)

9x=√13²-√10²

9x=13-10

9x=3

X=3/9

X=1/3

13.&nbsp; O volume
de um bloco retangular é calculado multiplicando-se as dimensões do bloco. Um
bloco retangular tem 9cm de comprimento, (6-√2)cm de largura e (6+√2)cm de
altura. Qual é o volume desse bloco, em cm³?

&nbsp;(6-√2)(6+√2)9

= (36-√2²)9

= (36-2)9

=34*9

=306








Exercicios P.75

<!--[if !supportLists]-->1.&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->Efetue as
divisões:










<!--[if !supportLists]-->a)&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;&nbsp;√15/√3

=√5



<!--[if !supportLists]-->b)&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;&nbsp;4√21/4√7

=4√3



<!--[if !supportLists]-->c)&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;&nbsp;√162/√3

=√54

=√(2*3³)

=3√6



<!--[if !supportLists]-->d)&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;&nbsp;√240/√6

=√40

=√(2³ * 5)

=2√10



<!--[if !supportLists]-->e)&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;&nbsp;√90/√5

=√18

=√2*3²

=3√2





<!--[if !supportLists]-->f)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;&nbsp;5√x9/5√x³

=5√x6

=X 5√x



<!--[if !supportLists]-->g)&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;³√a8/³√a³

=³√a5

=a ³√a2



<!--[if !supportLists]-->h)&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;&nbsp;4√a5b²/4√ab

=4√a4b

=a4√b










<!--[if !supportLists]-->2.&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->Qual é a
forma mais simples de escrever a expressão (√8/*√20)/2?



&nbsp; √80 / √(24*5)

=2²√5

=4√5



<!--[if !supportLists]-->3.&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->Você já
sabe que n√a/n√b=n√(a/b)= n√a:b.
Simplifique, então, as expressões:










<!--[if !supportLists]-->a)&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;&nbsp;√40/√5

=√8

=√2³

=2√2



<!--[if !supportLists]-->b)&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;&nbsp;√54/√3

=√18

=√(3²*2)

=3√2



<!--[if !supportLists]-->c)&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;&nbsp;√486/√3

=√(2*35)/ √3

= (3²√(2*3)) / √3

= (9√6)/√3

=9√2



<!--[if !supportLists]-->d)&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;√150/√3

=√50

=√(2*5²)

=5√2



<!--[if !supportLists]-->e)&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;&nbsp;7√x11/7x³

=7√x8

=x 7√x



<!--[if !supportLists]-->f)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;&nbsp;√(972x6)/√(3x³)

=√(2² * 35 * x6) / √(3x³)

= (2*3²x³√3) / (x√(3x))

= (18x²) / √x

= (18x²√x) / √x²

= (18x²√x) / x

=18x√x



<!--[if !supportLists]-->g)&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;√(225a³)/√(5a)

=√(5²*3²*a³) / √(5a)

= (5*3a√a) / √5ª

= (15a) / √5

= (15a√5) / √5²

= (15a√5) / 5

=3a√5



<!--[if !supportLists]-->h)&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;&nbsp;&nbsp;5√(192b7)/5√(3b²)

=5√(26*3*b7) / 5√(3b²)

= (2b5√2*3b²) / 5√(3b²)

=2b5√2














<!--[if !supportLists]-->4.&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->Simplifique
a expressão ((2+√3)/(2+√6))/((√6-2)/ √3)



&nbsp;&nbsp; ((2+√3)/(2+√6))*(√3/(√6-2))

= (2√3+√3²) / (2√6-4+√6²-2√6)

= (2√3+3) / (-4+6)

= (2√3+3) / 2

Exercicios P77

<!--[if !supportLists]-->1.&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->Reduza ao
mesmo índice os radicais a seguir.










<!--[if !supportLists]-->a)&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;³√2, √3

=6√2² , 6√3³



<!--[if !supportLists]-->b)&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;&nbsp;&nbsp;7√a³, ³√b²

=21√a9, 21√b14



<!--[if !supportLists]-->c)&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;&nbsp;&nbsp;4√3*√3

=20√38 ,&nbsp; 20√315



<!--[if !supportLists]-->d)&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;&nbsp;14√25,
21√29

=42√215 , 42√218



<!--[if !supportLists]-->e)&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;&nbsp;14√25,
21√29

=30√36 , 30√25, 30√28



<!--[if !supportLists]-->f)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;&nbsp;5√34,
10√6, √2

=10√38 , 10√6 , 10√25

















<!--[if !supportLists]-->2.&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->Reduza os
radicais ao mesmo índice; em seguida compare cada par de radicais usando os
sinais &gt; ou &lt;.










<!--[if !supportLists]-->a)&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->10√2 e15√2²

30√2³&nbsp; &lt; 30√24



<!--[if !supportLists]-->b)&nbsp;&nbsp;&nbsp; <!--[endif]-->12√310 e 18√311

36√330&nbsp; &gt; &nbsp;36√322



<!--[if !supportLists]-->c)&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->6√25 e 9√27

18√215 &gt; 18√214



<!--[if !supportLists]-->d)&nbsp;&nbsp;&nbsp; <!--[endif]-->8√2³ e 6√2³

24√29 &lt; 24√212








&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Exercicios
P.77

<!--[if !supportLists]-->1.&nbsp;&nbsp;&nbsp;&nbsp;
<!--[endif]-->Efetue as operações indicadas. Simplifique o
resultado quando possível.










<!--[if !supportLists]-->a)&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;&nbsp;³√10*5√10

=15√105 * 15√10³

=15√108



<!--[if !supportLists]-->b)&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;√7 / 5√7

=10√75 / 10√7²

=10√73



<!--[if !supportLists]-->c)&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;&nbsp;4√3 *√3

=4√3 * 4√3²

=4√3³



<!--[if !supportLists]-->d)&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;√2 / 20√27

=20√210 / 20√27

=20√23



<!--[if !supportLists]-->e)&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;&nbsp;&nbsp;6√5² / 10√5³

=30√510 / 30√59

=30√5



<!--[if !supportLists]-->f)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;&nbsp;6√75 /
³√7²

=6√75 / 6√74

=6√7



<!--[if !supportLists]-->g)&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;&nbsp;&nbsp;4√2³ * 5√24
* 10√27

=20√215 * 20√216 * 20√214

=20√245

=2 20√25

=4 4√2



<!--[if !supportLists]-->h)&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;&nbsp;8√65
/ 12√6²

=8√65 / 6√6

=24√615 / 24√64

=24√611








<!--[if !supportLists]-->2.&nbsp;&nbsp;&nbsp;&nbsp;
<!--[endif]-->Sabendo que a e b são dois números reais
positivos, escreva a expressão algébrica que representa o resultado de:










<!--[if !supportLists]-->a)&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;&nbsp;&nbsp;8√(a5b³)
/ 6√(ab²)

=24√(a15b9) / 24√(a4b8)

=24√(a11b)



<!--[if !supportLists]-->b)&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;&nbsp;&nbsp;9√(a7b6)
/ 6√(a³b²)

=18√(a14b12) / 18√(a9b6)

=18√(a5b6) 








&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
Exercicios P.79

<!--[if !supportLists]-->1.&nbsp;&nbsp;&nbsp;&nbsp;
<!--[endif]-->Calcule:








<!--[if !supportLists]-->a)&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;(√17)²

=√17²

=17



<!--[if !supportLists]-->b)&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;(³√2)4

=³√24

=2 ³√2

<!--[if !supportLists]-->c)&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;(6√2)²

=6²√2²

=36 * 2

=72







<!--[if !supportLists]-->d)&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;&nbsp;(1/2√10)²

= (½)²√10²

=¼ *10

=10/4

=5/2














<!--[if !supportLists]-->2.&nbsp;&nbsp;&nbsp;&nbsp;
<!--[endif]-->Sabendo que a e b são números reais positivos,
escreva a expressão algébrica mais simples que representa a expressão:










<!--[if !supportLists]-->a)&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;(a√b)²

=a²√b²

=a²b



<!--[if !supportLists]-->b)&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;(b ³√a)4

=b4 ³√a4

=ab4 3√a



<!--[if !supportLists]-->c)&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;&nbsp;(ab ³√b)4

=a4b4 ³√b4

=a4b5 ³√b



<!--[if !supportLists]-->d)&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;&nbsp;(a/b√(ab))²

=a2/b2√(a2b2))

= (a2/b2)ab

=a3/b








<!--[if !supportLists]-->3.&nbsp;&nbsp;&nbsp;&nbsp;
<!--[endif]-->Determine&nbsp;
o valor da expressão 4√8-(√2)³



&nbsp; 4√8-(√2)³

=4√2³-√2³

=4*2√2-2√2

=8√2-2√2

= (8-2)√2

=6√2

<!--[if !supportLists]-->4.&nbsp;&nbsp;&nbsp;&nbsp;
<!--[endif]-->Sendo x=2√3 e y=3√2, calcule o valor de x² *
y².

&nbsp;&nbsp; x² * y²

= (2√3)² * (3√2)²

=2²√3² * 3²√2²

=2 * 3 * 9 * 2

=6*18

=108

<!--[if !supportLists]-->5.&nbsp;&nbsp;&nbsp;&nbsp;
<!--[endif]-->Sabendo que a =√10 e b=2√5, calcule o valor de
a²-b²+10.

&nbsp;&nbsp; (√10)²-(2√5)²+10

=√10²-2²√5²+10

=10-4*5+10

=10-20+10

=0



<!--[if !supportLists]-->6.&nbsp;&nbsp;&nbsp;&nbsp;
<!--[endif]-->Aplicando a regra dos produtos notáveis,
calcule:










<!--[if !supportLists]-->a)&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->(√3+√2)²

=√3²+2√(3*2)+√2²

=3+2√6+2

=5+2√6



<!--[if !supportLists]-->b)&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;(1-√7)²

=1²-2√7+√7²

=1-27+7

=8-2√7



<!--[if !supportLists]-->c)&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;(4√2+5)(4√2-5)

=4²√2²-5²

=16*2-25

=32-25

=7





<!--[if !supportLists]-->d)&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;(2+√10)²

=2²+4√10+√10²

=4+4√10+10

=14+4√10



<!--[if !supportLists]-->e)&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;(√11+√7)( √11-√7)

=√11²-√7²

=11-7

=4



<!--[if !supportLists]-->f)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;(3√3+√2)²

=3²√3²+2*3√3*2+√2²

=9*3+6√6+2

=27+6√+2

=29+6√6





<!--[if !supportLists]-->g)&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;(7+√19)(7-√19)

=7²-√19²

=49-19

=30



<!--[if !supportLists]-->h)&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;&nbsp;&nbsp;(-3√5+1)(-3√5-1)

= (-3)² √5²-1²

=9*5-1

=45-1

=44



<!--[if !supportLists]-->i)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;(2√7+3√5)²

=2²√7²+2*2*3√(7*5)+3²√5²

=4*7+12√35+9*5

=28+12√35+45

=73+12√35








<!--[if !supportLists]-->7.&nbsp;&nbsp;&nbsp;&nbsp;
<!--[endif]-->Calcule a área dos seguintes quadrados:










<!--[if !supportLists]-->a)&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->Lado:
3+√3 por 3+√3

&nbsp; (3+√3)²

=3²+2*3√3+√3²

=9+6√3+3

=12+6√3



<!--[if !supportLists]-->b)&nbsp;&nbsp;&nbsp; <!--[endif]-->Lado: 5 -√7
por 5-√7

&nbsp; (5 -√7)²

=5²-2*5√7+√7²

=25-10√7+7

=32-10√7








<!--[if !supportLists]-->8.&nbsp;&nbsp;&nbsp;&nbsp;
<!--[endif]-->Sabe-se que a=√5+√3. Qual é a expressão que
representa a²?

&nbsp;&nbsp;&nbsp; A²

= (√5+√3)²

=√5²+2√15 +√3²

=5+2√15+3

=8+2√15



<!--[if !supportLists]-->9.&nbsp;&nbsp;&nbsp;&nbsp;
<!--[endif]-->Quando x=3+√2, qual é o valor numérico das
expressões?










<!--[if !supportLists]-->a)&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;&nbsp;&nbsp;X²-6√2

= (3+√2)²-6√2

=3²+2*3√2+√2²-6√2

=9+6√2+2-6√2

=11



<!--[if !supportLists]-->b)&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;&nbsp;X²-4x-4

= (3+√2)²-4(3+√2)-4

=3²+2*3√2+√2²-12-4√2-4

=9+6√2+2-12-4√2-4

=-5+2√2








<!--[if !supportLists]-->10.&nbsp; <!--[endif]-->Dada a
igualdade x²-4x+2=0, verifique se ela é verdadeira para x=2+√2.



&nbsp;&nbsp;&nbsp;&nbsp; x²-4x+2=0

à (2+√2)²-4(2+√2)+2=0

à2² + 2*2√2 + √2² - 8 - 4√2 + 2=0

à4 + 4√2 + 2 - 8 - 4√2 + 2=0

à4 + 2 + 2 + 4√2 - 4√2 =0

à0=0

Verdadeiro



<!--[if !supportLists]-->11.&nbsp; <!--[endif]-->Sabendo
que a=8+√6 e b=8-√6, calcule o valor de a²+b².



&nbsp;&nbsp; a²+b².

= (8+√6)²+(8-√6)²

=8²+8*2√6+√6²+8²-8*2√6+√6²

=64+16√6+6+64-16√6+6

=140



<!--[if !supportLists]-->12.&nbsp; <!--[endif]-->Determine
o numero real inteiro que representa a expressão:

&nbsp; (√21+√13)*( √21-√13)-( √10+√7)*(
√10-√7)

=√21²-√13²-(√10²-√7²)

=21-13-(10-7)

=8-3

=5



<!--[if !supportLists]-->13.&nbsp; <!--[endif]-->Escreva
na forma mais simples possível: ((3+2√5)²-√720-18)



&nbsp;&nbsp; ((3+2√5)²-√720-18)

=3²+3*2*2√5+2²√5²-√(24*3²*5)-18

=9+12√5+4*5-2²*3√5-18

=9+12√5+20-12√5-18

=11



<!--[if !supportLists]-->14.&nbsp; <!--[endif]-->Qual é a
forma mais simples de escrever a expressão a seguir?

&nbsp; (√7+√5)²-( √7+√5)*( √7-√5)

=√7²+2√(7*5)+
√5²-(√7²-√5²)

=7+2√35+5-(7-5)

=7+2√35+5-2

=10+2√35



<!--[if !supportLists]-->15.&nbsp; <!--[endif]-->Simplifique
a fração a seguir. (√6+√2)²/&nbsp; ((√7+√3)*(
√7-√3))



&nbsp;&nbsp;&nbsp; (√6+√2)²/&nbsp; ((√7+√3)*( √7-√3))

= (√6²+√12+√2²) /&nbsp;
(√7²-√3²)

= (6+2√(2³*3)+2) / (7-3)

= (8+2*2√3) / 4 

=(8+4√3) / 4

=(4(2+√3)) / 4

=2+√3









Exercicios P.81

<!--[if !supportLists]-->1.&nbsp;&nbsp;&nbsp;&nbsp;
<!--[endif]-->Resolva as equações, não esquecendo de fazer a
verificação dos valores encontrados.










<!--[if !supportLists]-->a)&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->√(3x)=6

√(3x)²=6²

3x=36

X=36/3

X=12



<!--[if !supportLists]-->b)&nbsp;&nbsp;&nbsp; <!--[endif]-->√(3x-2)=5

√(3x-2)²=5²

3x-2=25

3x=25+2

3x=27

X=9



<!--[if !supportLists]-->c)&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->√(2x+1)=-3

√(2x+1)²=(-3)²

2x+1=9

2x=8

X=8/2

X=4 à Não deu certo



<!--[if !supportLists]-->d)&nbsp;&nbsp;&nbsp; <!--[endif]-->2√x=4

(2√x)²=4²

2²√x²=16

4x=16

X=16/4

X=4





<!--[if !supportLists]-->e)&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->3√x=12

(3√x)²=12²

3²√x²=144

9x=144

X=144/9

X=16



<!--[if !supportLists]-->f)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->√(x²+3x-9)=x

√(x²+3x-9)²=x²

x²+3x-9=x²

x²-x²+3x=9

3x=9

X=9/3

X=3





<!--[if !supportLists]-->g)&nbsp;&nbsp;&nbsp; <!--[endif]-->√(2x+5)=
√(x+8)

√(2x+5)²= √(x+8)²

2x+5= x+8

2x-x=8-5

X=3



<!--[if !supportLists]-->h)&nbsp;&nbsp;&nbsp; <!--[endif]-->3√2x=6

(3√2x)²=6²

3²√2²x²=36

9*2x=36

18x=36

X=2










<!--[if !supportLists]-->2.&nbsp;&nbsp;&nbsp;&nbsp;
<!--[endif]-->Determine o conjunto solução da equação
√(x²+2)=x+1.



√(x²+2)=x+1

√(x²+2)²=(x+1)²

x²+2=x²+2x+1

x²-x²-2x=1-2

-2x=-1 

-x=-1/2&nbsp;&nbsp; (-1)

X=1/2



<!--[if !supportLists]-->3.&nbsp;&nbsp;&nbsp;&nbsp;
<!--[endif]-->Qual é o valor real de x na equação?



√(5x+2)=√(-6+9x)

√(5x+2)²=√(-6+9x)²

5x+2=-6+9x

5x-9x=-6-2

-4x=-8

-X=-8/4

-x=-2&nbsp; (-1)

X=2











<!--[if !supportLists]-->4.&nbsp;&nbsp;&nbsp;&nbsp;
<!--[endif]-->Determine o numero x para que se tenha
√(x-5)=3/√(x-5), com x≠5.










√(x-5)²=(3/√(x-5))²

x-5=3²/√(x-5)²

x-5=9/(x-5)

(x-5)²=9

x²-10x+25=9

x²-10x+16=0





Δ=b²-4ac

Δ=10²-4(1)(16)

Δ=100-64

Δ=36



X=(-b±√Δ)/2a

X=(-(-10)± √36)/2(1)

X=(10±6)/2

X’=(10+6)/2

X’=16/2

X’=8



X”=(10-6)/2

X”=4/2

X”=2










S={8 e 2}



Exercicios P.85



<!--[if !supportLists]-->1.&nbsp;&nbsp;&nbsp;&nbsp;
<!--[endif]-->Racionalize o denominador das seguintes
expressões:










<!--[if !supportLists]-->a)&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;&nbsp;2 / √10

=2√10 / √10²

=2 √10/ 10

=√10/ 5



<!--[if !supportLists]-->b)&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;&nbsp;6 / √6

=6√6 / √6²

=6√6 / 6

=√6



<!--[if !supportLists]-->c)&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;9 / √3

=9√3 / √3²

=9√3 / 3

=3√3



<!--[if !supportLists]-->d)&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;√5 / √2

= (√5 * √2) / √2²

=√10 / 2



<!--[if !supportLists]-->e)&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;&nbsp;20 / (2√5)

=20√5 / (2√5²)

=20√5 / (2*5)

=20√5 / 10

=2√5



<!--[if !supportLists]-->f)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;3 / √6

=3√6 / √6²

=3√6 / 6

=√6 / 2



<!--[if !supportLists]-->g)&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;&nbsp;20 / (3√10)

= (20√10) / (3√10²)

= (20√10) / (3*10)

= (20√10) / 30

= (2√10) / 3







<!--[if !supportLists]-->h)&nbsp;&nbsp;&nbsp; <!--[endif]-->&nbsp;&nbsp;&nbsp;1 / √7

=√7 / √7²

=√7 / 7



<!--[if !supportLists]-->i)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<!--[endif]-->(2√3) / (5√2)

= (2√3) / (5√2)

= (2√3*√2) / (5√2²)

= (2√6) / (5*2)

= (2√6) / 10

=√6 / 5



<!--[if !supportLists]-->j)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<!--[endif]-->&nbsp;&nbsp;&nbsp;(7√3)
/ (2√7)

= (7√3*√7) / (2√7²)

= (7√21) / (2*7)

= (7√21) / 14

=√21 / 2
















<!--[if !supportLists]-->2.&nbsp;&nbsp;&nbsp;&nbsp;
<!--[endif]-->Racionalize o denominador das expressões
seguintes:










<!--[if !supportLists]-->a)&nbsp;&nbsp;&nbsp;&nbsp;
<!--[endif]-->&nbsp;&nbsp;&nbsp;(1-√3)
/ √3

= (1-√3) √3 / √3²

= (√3-√3²) / 3

= (√3-3) / 3



<!--[if !supportLists]-->b)&nbsp;&nbsp;&nbsp;
<!--[endif]-->&nbsp;&nbsp;&nbsp;(3-√2)
/ √2

= (3-√2) √2/ √2²

= (3√2-√2²) / 2 

= (3√2-2) / 2



<!--[if !supportLists]-->c)&nbsp;&nbsp;&nbsp;&nbsp;
<!--[endif]-->&nbsp;&nbsp;&nbsp;&nbsp;(√5+√2)
/ √5

= (√5+√2) √5 / √5²

= (√5²+√2*√5) / 5

= (5+√10) / 5










<!--[if !supportLists]-->3.&nbsp;&nbsp;&nbsp;&nbsp;
<!--[endif]-->Sabendo que x e y são números reais positivos,
racionalize o denominador das seguintes expressões:










<!--[if !supportLists]-->a)&nbsp;&nbsp;&nbsp;&nbsp;
<!--[endif]-->&nbsp;&nbsp;&nbsp;x /√x

=x√x /√x²

=x √x /x

=√x 



<!--[if !supportLists]-->b)&nbsp;&nbsp;&nbsp;
<!--[endif]-->&nbsp;&nbsp;x
/(2√y)

=x√y /(2√y²)

=x√y /(2y)



<!--[if !supportLists]-->c)&nbsp;&nbsp;&nbsp;&nbsp;
<!--[endif]-->&nbsp;&nbsp;(xy) /
(5√x)

=(xy√x) / (5√x²)

=y√x / 5



<!--[if !supportLists]-->d)&nbsp;&nbsp;&nbsp;
<!--[endif]-->&nbsp;&nbsp;&nbsp;(x√y)
/ (y√x)

= (x√y*√x) / (y√x²)

= (x√(xy)) / (yx)

= (√xy) / y










<!--[if !supportLists]-->4.&nbsp;&nbsp;&nbsp;&nbsp;
<!--[endif]-->Você sabe que n√(a/b)= n√a
/ n√b. Racionalize o denominador das expressões:










<!--[if !supportLists]-->a)&nbsp;&nbsp;&nbsp;&nbsp;
<!--[endif]-->&nbsp;&nbsp;√(3/10)

=√3/√10

= (√3*√10)/√10²

=√30/10



<!--[if !supportLists]-->b)&nbsp;&nbsp;&nbsp;
<!--[endif]-->&nbsp;&nbsp;√(5/3)

=√5/√3

= (√5*√3)/√3²

=√15/3















<!--[if !supportLists]-->c)&nbsp;&nbsp;&nbsp;&nbsp;
<!--[endif]-->&nbsp;&nbsp;√(1/2)

=√1/√2

=√2/√2²

=√2/2



<!--[if !supportLists]-->d)&nbsp;&nbsp;&nbsp;
<!--[endif]-->&nbsp;&nbsp;&nbsp;√(1/8)


=√(1/√8)

=√(2³/√8²)

=2√(2/8)

=√2/4













<!--[if !supportLists]-->e)&nbsp;&nbsp;&nbsp;&nbsp;
<!--[endif]-->&nbsp;&nbsp;&nbsp;√0,9

=√(9/10)

=√9/√10

= (√9*√10) / √10²

=√90/10

=√(2*3²*5)/10

=3√10/10



<!--[if !supportLists]-->f)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<!--[endif]-->&nbsp;&nbsp;&nbsp;√(5/8)

=√5/√8

= (√5*√8) / √8²

=√40/8

=√(2³*5)/8

=2√(2*5)/8

=√10/4










<!--[if !supportLists]-->5.&nbsp;&nbsp;&nbsp;&nbsp;
<!--[endif]-->Considerando √6=2,449, √2=1,414 e √10=3,162,
calcule o valor na forma decimal das expressões:










<!--[if !supportLists]-->a)&nbsp;&nbsp;&nbsp;&nbsp;
<!--[endif]-->&nbsp;&nbsp;√(3/2)

=√3/√2

= (√3*√2)/ √2²

=√6/2

=2,449/2

=1,2245



<!--[if !supportLists]-->b)&nbsp;&nbsp;&nbsp;
<!--[endif]-->&nbsp;&nbsp;&nbsp;√(2/5)

=√2/√5

= (√2*√5)/ √5²

=√10/5

=3,162/5

=0,6324



<!--[if !supportLists]-->c)&nbsp;&nbsp;&nbsp;&nbsp;
<!--[endif]-->&nbsp;&nbsp;&nbsp;√(1/2)

=√1/√2

=√2/√2²

=√2/2

=1,414/2

=0,707










<!--[if !supportLists]-->6.&nbsp;&nbsp;&nbsp;&nbsp;
<!--[endif]-->Racionalize o denominador das expressões a
segui:










<!--[if !supportLists]-->a)&nbsp;&nbsp;&nbsp;&nbsp;
<!--[endif]-->&nbsp;&nbsp;&nbsp;1 / 5√6³

=5√6² / 5√6³

=5√6² / 5√6³ * 5√6²

=5√6²/ 5√65

=5√6²/6



<!--[if !supportLists]-->b)&nbsp;&nbsp;&nbsp;
<!--[endif]-->&nbsp;&nbsp;&nbsp;15 / ³√5

=15 ³√5² / ³√5 *³√5²

=15 ³√5² / ³√5³

=15 ³√5² / 5

=3 ³√5² 



<!--[if !supportLists]-->c)&nbsp;&nbsp;&nbsp;&nbsp;
<!--[endif]-->&nbsp;&nbsp;2 / 9√27

=2 9√22/ 9√27*9√22

=2 9√22/ 9√29

=2 9√22/ 2

=9√22



<!--[if !supportLists]-->d)&nbsp;&nbsp;&nbsp;
<!--[endif]-->&nbsp;&nbsp;6 / 10√35

=6 10√35/ 10√35
* 10√35

=6 10√35/ 10√310

=6 10√35/ 3

=210√35



<!--[if !supportLists]-->e)&nbsp;&nbsp;&nbsp;&nbsp;
<!--[endif]-->&nbsp;&nbsp;4/4√8³

=44√8/4√8³ *4√8

=44√8/4√84

=44√84/8

=4√8/2



<!--[if !supportLists]-->f)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<!--[endif]-->&nbsp;&nbsp;20 11√103/
11√108 *11√103

=20 11√103/ 11√1011

=20 11√103/ 10

=2 11√103










<!--[if !supportLists]-->7.&nbsp;&nbsp;&nbsp;&nbsp;
<!--[endif]-->Racionalize o denominador das seguintes
expressões:



a)&nbsp;&nbsp;&nbsp;&nbsp;
&nbsp;&nbsp;&nbsp;1/(3-√6)

=1(3+√6)/(3-√6) (3+√6)

= (3+√6) / (3²-√6²)

= (3+√6) / (9-6)

= (3+√6) / 3



b)&nbsp;&nbsp;&nbsp;
&nbsp;&nbsp;2/(√5+√3)

=2(√5-√3) / (√5+√3) (√5-√3)

= (2√5-2√3)/(5-3)

= (2√5-2√3)/2

=√5-√3



c)&nbsp;&nbsp;&nbsp;&nbsp;
&nbsp;&nbsp;1/(4-√5)

=1(4+√5)/(4-√5)(4+√5)

= (4+√5) / (4²-√5²)

= (4+√5)/(16-5)

= (4+√5)/ 11



d)&nbsp;&nbsp;&nbsp;
&nbsp;&nbsp;&nbsp;11/(2√3-1)

= (22√3+11) / (4√3²-1²)

= (22√3+11) / (4*3-1)

= (11+22√3) / (12-1)

= (11(1-2√3)) / 11

=1+2√3



e)&nbsp;&nbsp;&nbsp;&nbsp;
&nbsp;&nbsp;&nbsp;(2-√2)
/ (3+√2)

= ((2-√2)(3-√2)) / ((3+√2)(3-√2))

= (2*3-2√2-3√2+√2²) / (3²-√2²)

= (6-5√2+2) / (9-2)

= (8-5√2) /7



f)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
&nbsp;&nbsp;&nbsp;&nbsp;(2-2√2)
/ (2-√2)

= ((2-2√2)(2+√2)) / ((2-√2)(2+√2))

= (4+2√2-4√2-2√2²) / (2²-√2²)

= (4-2√2-2*2) / (4-2)

= (4-2√2-4) / 2

= (-2√2)/2

=-√2



g)&nbsp;&nbsp;&nbsp;
&nbsp;&nbsp;&nbsp;&nbsp;(1+√5) / (√3+√5)

= (1+√5)( √3-√5) / ((√3+√5)(√3-√5))

= (√3-√5+√15-√5²) / (√3²-√5²)

= (√3-√5+√15-5) / (3-5)

= (√3-√5+√15-5) / -2

= (-√3+√5-√15+5) / 2



h)&nbsp;&nbsp;&nbsp;
&nbsp;&nbsp;&nbsp;(√3-√2)
/ (√3+√2)

= (√3-√2)(√3-√2)&nbsp;
/ (√3+√2)(√3-√2)

= (√3²-2√6+√2²)&nbsp;
/ (3-2)

= (3-2√6+2) &nbsp;/ 1

=5-2√6