Question

Efetuar as operações:
a) 12√30 : 6√15 = √12².30 : √6².15 = √4320:540 = 18 (Está correto?)

b) 10√5 . 2√3 = √10².5 . √2².3 = √100.5 . √4.3 = √500.12 = 6000 (
Está correto?)

c) 3∛2 . 3∛5 = ∛3².2 . ∛3².5 = ∛9.2 . ∛9.5 = ∛18.45 = 810
(é isso?) 

d) 20 raiz quadrupla√8 / 5 raiz quadrupla√2 = raiz quadrupla√20².8 / raiz quadrupla√5².2 = raiz quadrupla√400.8 / raiz quadrupla√25.2 = raiz quadrupla√3200 / raiz quadrupla√50 (é isso?)

Answer 1

$\sqrt[n]{a}*\sqrt[n]{b}=\sqrt[n]{a*b}\\\sqrt[n]{a}/\sqrt[n]{b}=\sqrt[n]{a/b}$

Só multiplicar/dividir número com número e raiz com raiz
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a)

$\frac{12\sqrt{30}}{6\sqrt{15}}=\frac{12}{6}*\frac{\sqrt{30}}{\sqrt{15}}\\\\\frac{12\sqrt{30}}{6\sqrt{15}}=2*\sqrt{\frac{30}{15}}\\\\\boxed{\frac{12\sqrt{30}}{6\sqrt{15}}=2\sqrt{2}}$

b)

$10\sqrt{5}*2\sqrt{3}=(10*2)*(\sqrt{5}*\sqrt{3})\\10\sqrt{5}*2\sqrt{3}=20*\sqrt{5*3}\\10\sqrt{5}*2\sqrt{3}=20\sqrt{15}$

c)

$3\sqrt[3]{2}*3\sqrt[3]{5}=(3*3)*(\sqrt[3]{2}*\sqrt[3]{5})\\3\sqrt[3]{2}*3\sqrt[3]{5}=9*\sqrt[3]{2*5}\\3\sqrt[3]{2}*3\sqrt[3]{5}=9\sqrt[3]{10}$

d)

$\frac{20\sqrt[4]{8}}{5\sqrt[4]{2}}=\frac{20}{5}*\frac{\sqrt[4]{8}}{\sqrt[4]{2}}\\\\\frac{20\sqrt[4]{8}}{5\sqrt[4]{2}}=4*\sqrt[4]{\frac{8}{2}}\\\\\boxed{\frac{20\sqrt[4]{8}}{5\sqrt[4]{2}}=4\sqrt[4]{4}}$

Se quiser simplificar mais ainda:

$\sqrt[4]{4}=\sqrt[4]{2^{2}}\\\sqrt[4]{4}=2^{(2/4)}\\\sqrt[4]{4}=2^{(1/2)}\\\sqrt[4]{4}=\sqrt[2]{2^{1}}\\\sqrt[4]{4}=\sqrt{2}$

$\boxed{\boxed{\frac{20\sqrt[4]{8}}{5\sqrt[4]{2}}=4\sqrt{2}}}$