Question

Quais igualdades são verdadeiras?​

Answer 1

Utilizando a propriedade:

$\rightarrow~\sqrt[b]{a^c}~=~a^{\frac{a}{c}}$

$a)\\\\\sqrt[3]{\sqrt{18}}~=~\sqrt[3]{18^{\frac{1}{2}}}~=~\left(18^{\frac{1}{2}}\right)^{\frac{1}{3}}~=~18^{\frac{1}{2}.\frac{1}{3}}~=~18^{\frac{1}{6}}~=~\boxed{\sqrt[6]{18}}\\\\\\b)\\\\\sqrt[5]{\sqrt[3]{42}}~=~\sqrt[5]{42^{\frac{1}{3}}}~=~\left(42^{\frac{1}{3}}\right)^{\frac{1}{5}}~=~42^{\frac{1}{3}.\frac{1}{5}}~=~42^{\frac{1}{15}}~=~\boxed{\sqrt[15]{42}}$

$c)\\\\\sqrt[6]{\sqrt{22}}~=~\sqrt[6]{22^{\frac{1}{2}}}~=~\left(22^{\frac{1}{2}}\right)^{\frac{1}{6}}~=~22^{\frac{1}{2}.\frac{1}{6}}~=~22^{\frac{1}{12}}~=~\boxed{\sqrt[12]{22}}$

$d)\\\\\sqrt[3]{\sqrt[3]{\sqrt[4]{74}}}~=~\sqrt[3]{\sqrt[3]{74^{\frac{1}{4}}}}~=~\sqrt[3]{\left(74^{\frac{1}{4}}\right)^{\frac{1}{3}}}~=~\left(\left(74^{\frac{1}{4}}\right)^{\frac{1}{3}}\right)^{\frac{1}{3}}~=~74^{\frac{1}{3}.\frac{1}{3}.\frac{1}{4}}~=~74^{\frac{1}{36}}~=~\boxed{\sqrt[36]{74}}\\\\\\\\\sqrt{\sqrt[12]{74}}~=\left(74^{\frac{1}{12}}\right)^{\frac{1}{2}}~=~74^{\frac{1}{12}.\frac{1}{2}}~=~74^{\frac{1}{24}}~=~\boxed{\sqrt[24]{74}}$

Resposta: São verdadeiras as igualdades A e B.

Answer 2

a e b são verdadeiras,as restantes falsas