Determine: \((8x)^x\), if \(9^{x+2} = 240 + 9^x\)
Solution
\[ 9^{x+2} = 240 + 9^x \]
\[ \Rightarrow 9^x \cdot 9^2 = 240 + 9^x \]
\[ \Rightarrow 81 \cdot 9^x = 240 + 9^x \]
\[ \Rightarrow 81 \cdot 9^x – 9^x = 240 \]
\[ \Rightarrow 80 \cdot 9^x = 240 \]
\[ \Rightarrow 9^x = 3 \]
\[ \Rightarrow (3^2)^x = 3 \]
\[ \Rightarrow 3^{2x} = 3^1 \]
\[ \Rightarrow 2x = 1 \]
\[ \Rightarrow x = \frac{1}{2} \]
\[ (8x)^x = (8 \cdot \tfrac{1}{2})^{1/2} \]
\[ = 4^{1/2} \]
\[ = 2 \]
Final Answer:
\[ \boxed{2} \]