4.3 Simplify the following: Write your \( \frac{8 x^{-3} y^{5}}{x^{2} y} \times \frac{x^{6}}{16 y^{6}} \)

To simplify the expression \( \frac{8 x^{-3} y^{5}}{x^{2} y} \times \frac{x^{6}}{16 y^{6}} \), we first simplify each fraction separately. For the first fraction \( \frac{8 x^{-3} y^{5}}{x^{2} y} \), we can simplify it by subtracting the exponents in the denominator from those in the numerator. This gives us \( 8 \cdot x^{-3-2} \cdot y^{5-1} = 8 x^{-5} y^{4} \). Now, simplifying the second fraction \( \frac{x^{6}}{16 y^{6}} \), leaves us with \( \frac{x^{6}}{16} \cdot y^{-6} \). Now, combining both parts: \[ (8 x^{-5} y^{4}) \times \left(\frac{x^{6}}{16} y^{-6}\right) = \frac{8 x^{-5} \cdot x^{6} \cdot y^{4} \cdot y^{-6}}{16} \] Combining the x and y terms results in: \[ \frac{8}{16} \cdot x^{-5 + 6} \cdot y^{4 - 6} = \frac{1}{2} x^{1} y^{-2} = \frac{x}{2 y^{2}}. \] So, the simplified expression is \( \frac{x}{2 y^{2}} \).