a) \( \frac{x^{4} y^{-2} \cdot \sqrt{x^{3} y^{5}}}{\sqrt[3]{x^{3} y^{2}} \cdot x^{\frac{5}{3}} \cdot y^{\frac{2}{3}}} \)

Let’s break it down step by step. First, simplify each part of the expression. The top part is \( x^{4} y^{-2} \cdot \sqrt{x^{3} y^{5}} \). Recall that \( \sqrt{x^{3} y^{5}} = x^{\frac{3}{2}} y^{\frac{5}{2}} \). So, we multiply: \[ x^{4} y^{-2} \cdot x^{\frac{3}{2}} y^{\frac{5}{2}} = x^{4 + \frac{3}{2}} y^{-2 + \frac{5}{2}} = x^{\frac{8}{2} + \frac{3}{2}} y^{-\frac{4}{2} + \frac{5}{2}} = x^{\frac{11}{2}} y^{\frac{1}{2}} \] Now, let's simplify the denominator \( \sqrt[3]{x^{3} y^{2}} \cdot x^{\frac{5}{3}} \cdot y^{\frac{2}{3}} \). The cube root gives us \( \sqrt[3]{x^{3}} = x^{1} \) and \( \sqrt[3]{y^{2}} = y^{\frac{2}{3}} \): \[ \sqrt[3]{x^{3} y^{2}} = x^{1} y^{\frac{2}{3}} \Rightarrow x^{1} y^{\frac{2}{3}} \cdot x^{\frac{5}{3}} \cdot y^{\frac{2}{3}} = x^{1 + \frac{5}{3}} y^{\frac{2}{3} + \frac{2}{3}} = x^{\frac{3}{3} + \frac{5}{3}} y^{\frac{4}{3}} = x^{\frac{8}{3}} y^{\frac{4}{3}} \] Now, putting it all together, we have: \[ \frac{x^{\frac{11}{2}} y^{\frac{1}{2}}}{x^{\frac{8}{3}} y^{\frac{4}{3}}} = x^{\frac{11}{2} - \frac{8}{3}} y^{\frac{1}{2} - \frac{4}{3}} \] To simplify \( \frac{11}{2} - \frac{8}{3} \) and \( \frac{1}{2} - \frac{4}{3} \), we need a common denominator. 1. For \( \frac{11}{2} - \frac{8}{3} \): - Common Denominator: 6 - \( \frac{11}{2} = \frac{33}{6}, \frac{8}{3} = \frac{16}{6} \rightarrow \frac{33}{6} - \frac{16}{6} = \frac{17}{6} \) 2. For \( \frac{1}{2} - \frac{4}{3} \): - Common Denominator: 6 - \( \frac{1}{2} = \frac{3}{6}, \frac{4}{3} = \frac{8}{6} \rightarrow \frac{3}{6} - \frac{8}{6} = -\frac{5}{6} \) Thus, we have: \[ = x^{\frac{17}{6}} y^{-\frac{5}{6}} = \frac{x^{\frac{17}{6}}}{y^{\frac{5}{6}}} \] The final simplified expression is: \[ \frac{x^{\frac{17}{6}}}{y^{\frac{5}{6}}} \]