Given \(1176 = 2^a \cdot 3^b \cdot 7^c\), find \(a, b, c\) and evaluate \(2^a \cdot 3^b \cdot 7^{-c}\)
Solution
\[ 1176 = 2^3 \cdot 3^1 \cdot 7^2 \]
\[ \Rightarrow a = 3,\; b = 1,\; c = 2 \]
\[ 2^a \cdot 3^b \cdot 7^{-c} \]
\[ = 2^3 \cdot 3^1 \cdot 7^{-2} \]
\[ = 8 \cdot 3 \cdot \frac{1}{49} \]
\[ = \frac{24}{49} \]
Final Answer:
\[ \boxed{a = 3,\; b = 1,\; c = 2,\quad \frac{24}{49}} \]