11. \( \frac{\left(27 x^{9} y^{-3}\right)^{-\frac{2}{3}}}{\sqrt{9^{-1} x^{-8}} y^{2}} \)

To simplify the expression: \[ \frac{\left(27 x^{9} y^{-3}\right)^{-\frac{2}{3}}}{\sqrt{9^{-1} x^{-8}} y^{2}} \] Let's break it down step by step. ### **Step 1: Simplify the Numerator** \[ \left(27 x^{9} y^{-3}\right)^{-\frac{2}{3}} \] - **Break down 27:** \[ 27 = 3^3 \] - **Apply the exponent:** \[ \left(3^3 x^{9} y^{-3}\right)^{-\frac{2}{3}} = 3^{3 \times -\frac{2}{3}} \times x^{9 \times -\frac{2}{3}} \times y^{-3 \times -\frac{2}{3}} \] - **Simplify each part:** \[ 3^{-2} \times x^{-6} \times y^{2} = \frac{y^2}{9x^6} \] ### **Step 2: Simplify the Denominator** \[ \sqrt{9^{-1} x^{-8}} \times y^{2} \] - **Express the square root as an exponent:** \[ \left(9^{-1} x^{-8}\right)^{\frac{1}{2}} \times y^{2} \] - **Simplify each component:** \[ 9^{-\frac{1}{2}} \times x^{-4} \times y^2 = \frac{1}{3} \times \frac{y^2}{x^4} \] ### **Step 3: Combine Numerator and Denominator** \[ \frac{\frac{y^2}{9x^6}}{\frac{y^2}{3x^4}} = \frac{y^2}{9x^6} \times \frac{3x^4}{y^2} = \frac{3x^4}{9x^6} = \frac{1}{3x^2} \] ### **Final Simplified Expression** \[ \frac{1}{3x^{2}} \] **Answer:** \(\boxed{\dfrac{1}{3\,x^{2}}}\)