Simplify: \(\frac{3^n \times 9^{n+1}}{3^{n-1} \times 9^{n-1}}\)
Solution
\[ = \frac{3^n \cdot (3^2)^{n+1}}{3^{n-1} \cdot (3^2)^{n-1}} \]
\[ = \frac{3^n \cdot 3^{2n+2}}{3^{n-1} \cdot 3^{2n-2}} \]
\[ = 3^{\,n+2n+2-(n-1+2n-2)} \]
\[ = 3^{\,3n+2-(3n-3)} = 3^5 \]
\[ = 243 \]
Final Answer:
\[ \boxed{243} \]