Question

Conservamos o índice comum e multiplicação ou dividimos os radicais

1) efetue as operações:

a)
$\sqrt{3 \times \sqrt{4} }$
b)
$\sqrt{6 \times \sqrt{2} }$
c)
$\sqrt{12 \times \sqrt{12} }$
d)
$\sqrt{3 \div \sqrt{16} }$
e)
$\sqrt{8 \times \sqrt{9} }$
f)
$\sqrt{12 \div \sqrt{3} }$
g)
$\sqrt{51 \div \sqrt{2} }$
h)
$\sqrt{600 \times \sqrt{10} }$
i)
$\sqrt{200 \times \sqrt{6} }$
Após encontrar os valores acima, realizar a simplificação por MMC

​

Answer 1

Resposta:

mds isso tá mto dificil sei não desculpa

Answer 2

Explicação passo-a-passo:

a) $\sf \sqrt{3}\cdot\sqrt{4}=\sqrt{3\cdot4}=\red{\sqrt{12}}$

=> Simplificação:

$\sf 12~~~~|~~2$

$\sf ~~6~~~~|~~2$

$\sf ~~3~~~~|~~3$

$\sf ~~1$

$\sf 12=2^2\cdot3$

Assim:

$\sf \sqrt{12}=\sqrt{2^2\cdot3}$

$\sf \red{\sqrt{12}=2\sqrt{3}}$

b) $\sf \sqrt{6}\cdot\sqrt{2}=\sqrt{6\cdot2}=\red{\sqrt{12}}$

=> Simplificação:

$\sf 12~~~~|~~2$

$\sf ~~6~~~~|~~2$

$\sf ~~3~~~~|~~3$

$\sf ~~1$

$\sf 12=2^2\cdot3$

Assim:

$\sf \sqrt{12}=\sqrt{2^2\cdot3}$

$\sf \red{\sqrt{12}=2\sqrt{3}}$

c) $\sf \sqrt{12}\cdot\sqrt{12}=\sqrt{12\cdot12}=\red{\sqrt{144}}$

=> Simplificação:

$\sf 144~~~~|~~2$

$\sf ~~72~~~~|~~2$

$\sf ~~36~~~~|~~2$

$\sf ~~18~~~~|~~2$

$\sf ~~~9~~~~~|~~3$

$\sf ~~~3~~~~~|~~3$

$\sf ~~1$

$\sf 144=2^4\cdot3^2$

Assim:

$\sf \sqrt{144}=\sqrt{2^4\cdot3^2}$

$\sf \sqrt{144}=\sqrt{(2^2)^2\cdot3^2}$

$\sf \sqrt{144}=2^2\cdot3$

$\sf \sqrt{144}=4\cdot3$

$\sf \red{\sqrt{144}=12}$

d) $\sf \sqrt{3}\div\sqrt{16}=\sqrt{3\div16}=\sqrt{0,1875}$

$\sf =\sqrt{\dfrac{3}{16}}$

$\sf =\red{\dfrac{\sqrt{3}}{4}}$

e) $\sf \sqrt{8}\cdot\sqrt{9}=\sqrt{8\cdot9}=\red{\sqrt{72}}$

=> Simplificação:

$\sf 72~~~~|~~2$

$\sf 36~~~~|~~2$

$\sf 18~~~~|~~2$

$\sf ~~9~~~~|~~3$

$\sf ~~3~~~~|~~3$

$\sf ~~1$

$\sf 72=2^3\cdot3^2$

Assim:

$\sf \sqrt{72}=\sqrt{2^3\cdot3^2}$

$\sf \sqrt{72}=\sqrt{2\cdot2^2\cdot3^2}$

$\sf \sqrt{72}=2\cdot3\sqrt{2}$

$\sf \red{\sqrt{72}=6\sqrt{2}}$

f) $\sf \sqrt{12}\div\sqrt{3}=\sqrt{12\div3}=\red{\sqrt{4}}$

=> Simplificação:

$\sf 4~~~~|~~2$

$\sf 2~~~~|~~2$

$\sf 1$

$\sf 4=2^2$

Assim:

$\sf \sqrt{4}=\sqrt{2^2}$

$\sf \red{\sqrt{4}=2}$

g) $\sf \sqrt{51}\div\sqrt{2}=\sqrt{51\div2}=\red{\sqrt{25,5}}$

$\sf =\sqrt{\dfrac{255}{10}}$

$\sf =\sqrt{\dfrac{2550}{100}}$

$\sf =\dfrac{\sqrt{2550}}{\sqrt{100}}$

=> Simplificação:

$\sf 2550~~~~~|~~2$

$\sf 1275~~~~~|~~3$

$\sf ~~425~~~~~|~~5$

$\sf ~~~85~~~~~~|~~5$

$\sf ~~~17~~~~~~|~~17$

$\sf ~~~~1$

$\sf 2550=2\cdot3\cdot5^2\cdot17$

$\sf 100~~~~~~|~~2$

$\sf ~~50~~~~~~|~~2$

$\sf ~~25~~~~~~|~~5$

$\sf ~~~~5~~~~~~|~~5$

$\sf ~~~~~1$

$\sf 100=2^2\cdot5^2$

Assim:

$\sf \dfrac{\sqrt{2550}}{\sqrt{100}}=\dfrac{\sqrt{2\cdot3\cdot5^2\cdot17}}{\sqrt{2^2\cdot5^2}}$

$\sf \dfrac{\sqrt{2550}}{\sqrt{100}}=\dfrac{5\sqrt{2\cdot3\cdot17}}{2\cdot5}$

$\sf \dfrac{\sqrt{2550}}{\sqrt{100}}=\dfrac{5\sqrt{102}}{10}$

$\sf \dfrac{\sqrt{2550}}{\sqrt{100}}=\red{\dfrac{\sqrt{102}}{2}}$

h) $\sf \sqrt{600}\cdot\sqrt{10}=\sqrt{600\cdot10}=\red{\sqrt{6000}}$

=> Simplificação:

$\sf 6000~~~~~|~~2$

$\sf 3000~~~~~|~~2$

$\sf 1500~~~~~|~~2$

$\sf ~~750~~~~~|~~2$

$\sf ~~375~~~~~|~~3$

$\sf ~~125~~~~~~|~~5$

$\sf ~~~25~~~~~~|~~5$

$\sf ~~~~~5~~~~~~|~~5$

$\sf ~~~~~1$

$\sf 6000=2^4\cdot3\cdot5^3$

Assim:

$\sf \sqrt{6000}=\sqrt{2^4\cdot3\cdot5^3}$

$\sf \sqrt{6000}=\sqrt{(2^2)^2\cdot5^2\cdot5\cdot3}$

$\sf \sqrt{6000}=2^2\cdot5\sqrt{5\cdot3}$

$\sf \sqrt{6000}=4\cdot5\sqrt{15}$

$\sf \red{\sqrt{6000}=20\sqrt{15}}$

i) $\sf \sqrt{200}\cdot\sqrt{6}=\sqrt{200\cdot6}=\red{\sqrt{1200}}$

=> Simplificação:

$\sf 1200~~~~~~~|~~2$

$\sf ~~600~~~~~~~|~~2$

$\sf ~~300~~~~~~~|~~2$

$\sf ~~150~~~~~~~|~~2$

$\sf ~~~75~~~~~~~~|~~3$

$\sf ~~~25~~~~~~~~|~~5$

$\sf ~~~~5~~~~~~~~~|~~5$

$\sf ~~~~~1$

$\sf 1200=2^4\cdot3\cdot5^2$

Assim:

$\sf \sqrt{1200}=\sqrt{2^4\cdot3\cdot5^2}$

$\sf \sqrt{1200}=\sqrt{(2^2)^2\cdot5^2\cdot3}$

$\sf \sqrt{1200}=2^2\cdot5\sqrt{3}$

$\sf \sqrt{1200}=4\cdot5\sqrt{3}$

$\sf \red{\sqrt{1200}=20\sqrt{3}}$