(0) \( \frac{\left(x^{\frac{4}{3}} y^{-\frac{2}{3}}\right)^{-\frac{1}{2}} \cdot\left(x^{-\frac{2}{3}} y^{\frac{1}{3}}\right)^{2}}{\left(x^{\frac{1}{2}}\right)^{-8} \cdot\left(y^{\frac{1}{2}}\right)^{6}} \)
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Let's simplify the expression step by step. 1. Start with the expression: \[ \frac{\left(x^{\frac{4}{3}} y^{-\frac{2}{3}}\right)^{-\frac{1}{2}} \cdot\left(x^{-\frac{2}{3}} y^{\frac{1}{3}}\right)^{2}}{\left(x^{\frac{1}{2}}\right)^{-8} \cdot\left(y^{\frac{1}{2}}\right)^{6}} \] 2. Simplify the numerator: - The first part of the numerator: \[ \left(x^{\frac{4}{3}} y^{-\frac{2}{3}}\right)^{-\frac{1}{2}} = x^{-\frac{4}{6}} y^{\frac{1}{3}} = x^{-\frac{2}{3}} y^{\frac{1}{3}} \] - The second part of the numerator: \[ \left(x^{-\frac{2}{3}} y^{\frac{1}{3}}\right)^{2} = x^{-\frac{4}{3}} y^{\frac{2}{3}} \] Combining both parts in the numerator: \[ x^{-\frac{2}{3}} y^{\frac{1}{3}} \cdot x^{-\frac{4}{3}} y^{\frac{2}{3}} = x^{-\frac{2}{3} - \frac{4}{3}} y^{\frac{1}{3} + \frac{2}{3}} = x^{-\frac{6}{3}} y^{\frac{3}{3}} = x^{-2} y^{1} = \frac{y}{x^2} \] 3. Now simplify the denominator: - The first part: \[ \left(x^{\frac{1}{2}}\right)^{-8} = x^{-\frac{8}{2}} = x^{-4} \] - The second part: \[ \left(y^{\frac{1}{2}}\right)^{6} = y^{\frac{6}{2}} = y^{3} \] Therefore combining them: \[ x^{-4} \cdot y^{3} = \frac{y^3}{x^4} \] 4. Now put it all together: \[ \frac{\frac{y}{x^2}}{\frac{y^3}{x^4}} = \frac{y \cdot x^4}{y^3 \cdot x^2} = \frac{x^{4}y}{y^{3}x^{2}} = \frac{x^{4}}{x^{2}} \cdot \frac{1}{y^{2}} = x^{2} \cdot \frac{1}{y^{2}} = \frac{x^2}{y^2} \] Thus, the simplified expression is: \[ \frac{x^2}{y^2} \]
