Question

$\sqrt{\frac{x}{y}\sqrt[3]{\frac{y}{x}}} =$

Answer 1

Olá.

Nessa expressão, devemos usar as seguintes propriedades de potências:

$\left\{\begin{array}{l}\mathsf{\sqrt[n]{\dfrac{a}{b}}=\dfrac{\sqrt[n]{a}}{\sqrt[n]{b}}}\\\\\mathsf{\sqrt[s]{a^r}=a^{\frac{r}{s}}}\\\\\mathsf{a^m\div a^n=a^{m-n}}\\\\\mathsf{(a^m\cdot b^m)^n=a^{mn}\cdot b^{mn}}\end{array}\right.$

Desenvolvendo, teremos:

$\mathsf{\sqrt{\dfrac{x}{y}\cdot\sqrt[3]{\dfrac{y}{x}}}}\\\\\\\mathsf{\sqrt{\dfrac{x}{y}\cdot\dfrac{\sqrt[3]{y}}{\sqrt[3]{x}}}=}\\\\\\\mathsf{\sqrt{\dfrac{x}{y}\cdot\dfrac{y^{\frac{1}{3}}}{x^{\frac{1}{3}}}}=}\\\\\\\mathsf{\sqrt{\dfrac{x\cdot y^{\frac{1}{3}}}{y\cdot x^{\frac{1}{3}}}}=}\\\\\\\mathsf{\sqrt{\dfrac{y^{\frac{1}{3}}}{y}\cdot\dfrac{x}{x^{\frac{1}{3}}}}}\\\\\\\mathsf{\sqrt{x^{1-\frac{1}{3}}\cdot y^{\frac{1}{3}-1}}}$

Continuemos...

$\mathsf{\sqrt{x^{1-\frac{1}{3}}\cdot y^{\frac{1}{3}-1}}}\\\\\mathsf{\sqrt{x^{\frac{3-1}{3}}\cdot y^{\frac{1-3}{3}}}}\\\\\mathsf{\sqrt{x^{\frac{2}{3}}\cdot y^{\frac{-2}{3}}}}\\\\\mathsf{\left(x^{\frac{2}{3}}\cdot y^{\frac{-2}{3}}\right)^{\frac{1}{2}}}\\\\\mathsf{x^{\frac{2}{3}\cdot\frac{1}{2}}\cdot y^{\frac{-2}{3}\cdot\frac{1}{2}}}\\\\\mathsf{x^{\frac{2}{6}}\cdot y^{\frac{-2}{6}}}$

Essa já é uma forma final válida. Todavia, é possível simplificar.

$\mathsf{x^{\frac{2}{6}}\cdot y^{\frac{-2}{6}}}\\\\\mathsf{x^{\frac{1}{3}}\cdot y^{\frac{-1}{3}}}\\\\\mathsf{x^{\frac{1}{3}}\cdot\dfrac{1}{y^{\frac{1}{3}}}}\\\\\mathsf{\dfrac{x^{\frac{1}{3}}}{y^{\frac{1}{3}}}}\\\\\\\mathsf{\left(\dfrac{x}{y}\right)^{\frac{1}{3}}}\\\\\\\underline{\mathsf{\sqrt[3]{\dfrac{x}{y}}}}$

Qualquer dúvida, deixe nos comentários.
Bons estudos.