1. Express \( \sqrt[3]{x y^{-2}} \) as a positive power of \( x y \).

To express \( \sqrt[3]{x y^{-2}} \) as a positive power of \( x y \), we start by rewriting the expression using exponent rules. The term \( y^{-2} \) can be rewritten as \( \frac{1}{y^2} \). Thus, we have: \[ \sqrt[3]{x y^{-2}} = \sqrt[3]{\frac{x}{y^2}} = \sqrt[3]{x} \cdot \sqrt[3]{y^{-2}} = \sqrt[3]{x} \cdot y^{-\frac{2}{3}}. \] Next, we can express \( \sqrt[3]{x} \) as \( x^{\frac{1}{3}} \) and \( y^{-\frac{2}{3}} \) remains the same: \[ x^{\frac{1}{3}} \cdot y^{-\frac{2}{3}} = x^{\frac{1}{3}} \cdot \frac{1}{y^{\frac{2}{3}}} = \frac{x^{\frac{1}{3}}}{y^{\frac{2}{3}}}. \] To convert this expression into a form involving \( xy \), we note that \( xy = x^{1}y^{1} \). Thus, we can rewrite the terms as a single fraction: \[ \frac{x^{\frac{1}{3}}}{y^{\frac{2}{3}}} = \frac{x^{\frac{1}{3}}y^{-\frac{2}{3}}}{1} = x^{\frac{1}{3}} \cdot (xy)^{-\frac{2}{3}}. \] Putting it all together gives us: \[ \sqrt[3]{x y^{-2}} = (xy)^{-\frac{2}{3}} \cdot x^{\frac{1}{3}} = (xy)^{\frac{1}{3}} \cdot (xy)^{-\frac{2}{3}} = (xy)^{\frac{1}{3} - \frac{2}{3}} = (xy)^{-\frac{1}{3}}. \] Therefore, the final result is: \[ \sqrt[3]{x y^{-2}} = \frac{x^{\frac{1}{3}}}{y^{\frac{2}{3}}} = (xy)^{-\frac{1}{3}}. \]