Simplify: \[ \left[\frac{5^{-1}\cdot7^2}{5^2\cdot7^{-4}}\right]^{7/2} \times \left[\frac{5^{-2}\cdot7^3}{5^3\cdot7^{-5}}\right]^{-5/2} \]
Solution
\[ \left[\frac{5^{-1}}{5^2}\cdot\frac{7^2}{7^{-4}}\right]^{7/2} = \left[5^{-3}\cdot7^{6}\right]^{7/2} \]
\[ = 5^{-21/2}\cdot7^{21} \]
\[ \left[\frac{5^{-2}}{5^3}\cdot\frac{7^3}{7^{-5}}\right]^{-5/2} = \left[5^{-5}\cdot7^{8}\right]^{-5/2} \]
\[ = 5^{25/2}\cdot7^{-20} \]
\[ \text{Multiplying: } = 5^{-21/2+25/2}\cdot7^{21-20} \]
\[ = 5^{2}\cdot7 \]
\[ = 25\cdot7 = 175 \]
Final Answer:
\[ \boxed{175} \]