Question

Aplique as propriedades dos radicais e transforme a expressão abaixo em um único radical:

Answer 1

$\frac{ \sqrt[4]{ \sqrt{16}* } \sqrt[8]{ \frac{5}{2} } }{ \sqrt[8]{15}* \sqrt[8]{ \frac{4}{3} } } \\ \\ \frac{ \sqrt[8]{16}* \sqrt[8]{ \frac{5}{2} } }{ \sqrt[8]{ \frac{60}{3} } } \\ \\ \frac{ \sqrt[8]{ \frac{80}{2} } }{ \sqrt[8]{ \frac{60}{3} } } \\ \\ \frac{ \sqrt[8]{40} }{ \sqrt[8]{20} } \\ \\ \sqrt[8]{1} =1$

Boa Noite e Bons Estudos !!

Answer 2

Propriedades utilizadas:

$\begin{array}{rclc} \sqrt[m]{\sqrt[n]{a}}&=&\sqrt[m \cdot n]{a}&\;\;\;\text{(i)}\\ \\ \sqrt[n]{a}\cdot \sqrt[n]{b}&=&\sqrt[n]{a \cdot b}&\;\;\;\text{(ii)}\\ \\ \dfrac{\sqrt[n]{a}}{\sqrt[n]{b}}&=&\sqrt[n]{\dfrac{a}{b}}&\;\;\;\text{(iii)} \end{array}$


$\dfrac{\sqrt[4]{\sqrt{16}}\cdot \sqrt[8]{\frac{5}{2}}}{\sqrt[8]{15}\cdot \sqrt[8]{\frac{4}{3}}}\\ \\ =\dfrac{\sqrt[4\cdot 2]{16}\cdot \sqrt[8]{\frac{5}{2}}}{\sqrt[8]{15\cdot \frac{4}{3}}}\\ \\ =\dfrac{\sqrt[8]{16}\cdot \sqrt[8]{\frac{5}{2}}}{\sqrt[8]{20}}\\ \\ =\dfrac{\sqrt[8]{16 \cdot \frac{5}{2}}}{\sqrt[8]{20}}\\ \\ =\dfrac{\sqrt[8]{40}}{\sqrt[8]{20}}\\ \\ =\sqrt[8]{\dfrac{40}{20}} \\ \\=\sqrt[8]{2}\\ \\ \\ \boxed{\dfrac{\sqrt[4]{\sqrt{16}}\cdot \sqrt[8]{\frac{5}{2}}}{\sqrt[8]{15}\cdot \sqrt[8]{\frac{4}{3}}}=\sqrt[8]{2}}$