2025-04-15

Simplify the following expression completely. Express your answer using positive exponents only.

$$\begin{array}{rl}& \frac{(8^{2n+1}\cdot x^{-4}{)}^{-3}}{(\sqrt[3]{64\cdot x^9})\cdot (2^n{)}^{-5}}\\ & \\ & =\frac{(2^{3^{2n+1}}\cdot x^{-4}{)}^{-3}}{(\sqrt[3]{2^6\cdot x^9})\cdot (2^n{)}^{-5}}\\ & \\ & =\frac{(2^{6n+3}\cdot x^{-4}{)}^{-3}}{(\sqrt[3]{2^6\cdot x^9})\cdot (2^n{)}^{-5}}\\ & \\ & =\frac{(2^{6n+3}\cdot x^{-4}{)}^{-3}}{({2^6\cdot x^9)}^{\frac{1}{3}}\cdot (2^n{)}^{-5}}\\ & \\ & =\frac{(2^{6n+3}\cdot x^{-4}{)}^{-3}}{2^2\cdot x^3\cdot (2^n{)}^{-5}}\\ & \\ & =\frac{2^{-18n-9}\cdot x^{12}}{2^2\cdot x^3\cdot (2^n{)}^{-5}}\\ & \\ & =\frac{2^{-18n-9}\cdot x^{12}}{2^2\cdot x^3\cdot 2^{-5n}}\\ & \\ & =\frac{2^{-18n-9}\cdot x^{12}}{x^3\cdot 2^{2-5n}}\\ & \\ & =\frac{x^{12}}{x^3\cdot 2^{18n+9}\cdot 2^{2-5n}}\\ & \\ & =\frac{x^{12}}{x^3\cdot 2^{11+13n}}\\ & \\ & =\frac{x^9}{2^{11+13n}}\end{array}$$