\frac{18^{n+2}}{3^{2n+2} \cdot 9^n \cdot 2^n}

\frac{18^{n+2}}{3^{2n+2} \cdot 9^n \cdot 2^n} = \frac{(2 \cdot 3^2)^{n+2}}{3^{2n+2} \cdot (3^2)^n \cdot 2^n} = \frac{2^{n+2} \cdot 3^{2n+4}}{3^{2n+2} \cdot 3^{2n} \cdot 2^n} = \frac{2^{n+2} \cdot 3^{2n+4}}{3^{4n+2} \cdot 2^n} = 2^{(n+2)-n} \cdot 3^{(2n+4)-(4n+2)} = 2^2 \cdot 3^{-2n+2} = 4 \cdot \frac{3^2}{3^{2n}} = \frac{36}{9^n}