Solve: \(27^x = \frac{9}{3^x}\)
Solution
\[ 27 = 3^3,\quad 9 = 3^2 \]
\[ (3^3)^x = \frac{3^2}{3^x} \]
\[ 3^{3x} = 3^{2-x} \]
\[ \Rightarrow 3x = 2 – x \]
\[ \Rightarrow 4x = 2 \]
\[ \Rightarrow x = \frac{1}{2} \]
Final Answer:
\[ \boxed{x = \frac{1}{2}} \]