Efectua as operações
8.
\sqrt[5]{\frac{x y}{x-y}} \cdot \sqrt{\frac{x-y}{x y}} \cdot \sqrt[10]{x^{3} y^{3}}

Passo 1

O enunciado pede para simplificar o produto de radicais
\sqrt[5]{\frac{x y}{x-y}} \;\cdot\; \sqrt{\frac{x-y}{x y}} \;\cdot\; \sqrt[10]{x^{3} y^{3}}
O conteúdo aqui é manipular radicais usando expoentes racionais. Vamos reescrever tudo em potências a^{p/q}.

Passo 2

Reescrevendo cada termo com expoente racional:
\sqrt[5]{\frac{x y}{x-y}} =\Bigl(\frac{x y}{x-y}\Bigr)^{\tfrac{1}{5}}, \quad \sqrt{\frac{x-y}{x y}} =\Bigl(\frac{x-y}{x y}\Bigr)^{\tfrac{1}{2}}, \quad \sqrt[10]{x^{3} y^{3}} =(x^{3}y^{3})^{\tfrac{1}{10}}.

Passo 3

Observe que \frac{x-y}{xy} = \bigl(\frac{xy}{x-y}\bigr)^{-1}. Logo
\Bigl(\tfrac{xy}{x-y}\Bigr)^{\tfrac{1}{5}} \cdot \Bigl(\tfrac{x-y}{xy}\Bigr)^{\tfrac{1}{2}} = \Bigl(\tfrac{xy}{x-y}\Bigr)^{\tfrac{1}{5} - \tfrac{1}{2}} = \Bigl(\tfrac{xy}{x-y}\Bigr)^{-\tfrac{3}{10}}.

Passo 4

Multiplicando pelo terceiro fator temos
\Bigl(\tfrac{xy}{x-y}\Bigr)^{-\tfrac{3}{10}} \cdot (x^{3}y^{3})^{\tfrac{1}{10}} = \frac{(x^{3}y^{3})^{\tfrac{1}{10}}} {\bigl(\tfrac{xy}{x-y}\bigr)^{\tfrac{3}{10}}} = \Bigl(\frac{x^{3}y^{3}}{(\tfrac{xy}{x-y})^{3}}\Bigr)^{\tfrac{1}{10}}.

Passo 5

Dentro do parêntesis:
\frac{x^{3}y^{3}}{\bigl(\tfrac{xy}{x-y}\bigr)^{3}} = \frac{x^{3}y^{3}}{\tfrac{x^{3}y^{3}}{(x-y)^{3}}} = (x-y)^{3}.
Portanto o resultado final é
\bigl((x-y)^{3}\bigr)^{\tfrac{1}{10}} =(x-y)^{\tfrac{3}{10}}.

Resposta

\sqrt[10]{(x-y)^{3}}

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