Question

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Answer 1

$\sqrt{\frac{x}{y}.\sqrt[3]{\frac{y}{x}.\sqrt[4]{\frac{x}{y}}} }=(\frac{y}{x})^p\\\\\sqrt{\frac{x}{y}.\sqrt[3]{\frac{y}{x}.(\frac{x}{y})^{\frac{1}{4}}}} }=(\frac{y}{x})^p\\\\\sqrt{\frac{x}{y}.\sqrt[3]{\frac{y}{x}.(\frac{x^{\frac{1}{4}}}{y^{\frac{1}{4}}})} }=(\frac{y}{x})^p\\\\\sqrt{\frac{x}{y}.\sqrt[3]{{y^{\frac{3}{4}}x^{\frac{-3}{4}}}}}=(\frac{y}{x})^p\\\\\sqrt{\frac{x}{y}.(\frac{y^{\frac{3}{4}}}{x^{\frac{3}{4}}})^{\frac{1}{3}}} }=(\frac{y}{x})^p$

$\sqrt{\frac{x}{y}.(\frac{y^{\frac{3}{4}}}{x^{\frac{3}{4}}})^{\frac{1}{3}}} }=(\frac{y}{x})^p\\\\\sqrt{\frac{x}{y}.(\frac{y^{\frac{1}{4}}}{x^{\frac{1}{4}}})} }=(\frac{y}{x})^p\\\\\sqrt{x^{\frac{3}{4}}.y^{\frac{-3}{4}}}=(\frac{y}{x})^p\\\\\sqrt{\frac{y^{\frac{-3}{4}}}{x^{\frac{-3}{4}}}}=(\frac{y}{x})^p\\\\(\frac{y^{\frac{-3}{4}}}{x^{\frac{-3}{4}}})^{\frac{1}{2}}=(\frac{y}{x})^p\\\\(\frac{y^{\frac{-3}{8}}}{x^{\frac{-3}{8}}})=(\frac{y}{x})^p\\\\(\frac{y}{x})^{\frac{-3}{8}}}=(\frac{y}{x})^p$

$p=\frac{-3}{8}$