Given \(2^x \cdot 3^y \cdot 5^z = 2160\), find \(x, y, z\) and evaluate \(2^x \cdot 2^{-y} \cdot 5^{-z}\)
Solution
\[ 2160 = 2^4 \cdot 3^3 \cdot 5^1 \]
\[ \Rightarrow x = 4,\; y = 3,\; z = 1 \]
\[ 2^x \cdot 2^{-y} \cdot 5^{-z} \]
\[ = 2^{x-y} \cdot 5^{-z} \]
\[ = 2^{4-3} \cdot 5^{-1} \]
\[ = 2^1 \cdot \frac{1}{5} \]
\[ = \frac{2}{5} \]
Final Answer:
\[ \boxed{x = 4,\; y = 3,\; z = 1,\quad \frac{2}{5}} \]