(c) \( \left(\frac{16 x^{9} y^{3}}{x y^{11}}\right)^{-1 / 2} \)

To simplify the expression: \[ \left(\frac{16 x^{9} y^{3}}{x y^{11}}\right)^{-1 / 2} \] **Step 1: Simplify the Fraction Inside the Parentheses** \[ \frac{16 x^{9} y^{3}}{x y^{11}} = 16 \frac{x^{9}}{x} \frac{y^{3}}{y^{11}} = 16 x^{8} y^{-8} = \frac{16 x^{8}}{y^{8}} \] **Step 2: Apply the Exponent \(-\frac{1}{2}\) to the Simplified Fraction** \[ \left(\frac{16 x^{8}}{y^{8}}\right)^{-\frac{1}{2}} = \frac{y^{8}}{16 x^{8}}^{\frac{1}{2}} = \frac{\sqrt{y^{8}}}{\sqrt{16 x^{8}}} \] **Step 3: Simplify the Square Roots** \[ \sqrt{y^{8}} = y^{4} \quad \text{and} \quad \sqrt{16 x^{8}} = 4 x^{4} \] **Step 4: Combine the Results** \[ \frac{y^{4}}{4 x^{4}} \] **Final Simplified Expression:** \[ \boxed{\dfrac{y^{4}}{4\,x^{4}}} \]