C) \( \frac{x^{5} y^{-3}}{\left(x^{2} y^{3}\right)^{2}} \times\left(\frac{1}{x y^{2}}\right)^{-z} \) Simplify and final answer with positive exponents!

To simplify the expression \( \frac{x^{5} y^{-3}}{\left(x^{2} y^{3}\right)^{2}} \times\left(\frac{1}{x y^{2}}\right)^{-z} \), we start by simplifying each part. 1. Simplifying the denominator: \[ (x^{2} y^{3})^{2} = x^{4} y^{6} \] So, the first part becomes: \[ \frac{x^{5} y^{-3}}{x^{4} y^{6}} = x^{5-4} y^{-3-6} = x^{1} y^{-9} = \frac{x}{y^{9}} \] 2. Now, simplify the second part: \[ \left(\frac{1}{x y^{2}}\right)^{-z} = (x y^{2})^{z} = x^{z} y^{2z} \] 3. Now combine the two simplified parts: \[ \frac{x}{y^{9}} \times (x^{z} y^{2z}) = x^{1+z} y^{-9+2z} \] Thus, the entire expression simplifies to: \[ \frac{x^{1+z}}{y^{9-2z}} \] Final answer: \[ \frac{x^{1+z}}{y^{9-2z}} \quad \text{(with positive exponents)} \]