(a) Simplify \((\frac{4}{25})^{-\frac{1}{2}} \times 2^{4} \div (\frac{15}{2})^{-2}\)
(b) Evaluate \(\log_{5} (\frac{3}{5}) + 3 \log_{5} (\frac{5}{2}) - \log_{5} (\frac{81}{8})\).
(a) \((\frac{4}{25})^{-\frac{1}{2}} \times 2^{4} \div (\frac{15}{2})^{-2}\)
= \((\frac{25}{4})^{\frac{1}{2}} \times 2^{4} \div (\frac{2}{15})^{2}\)
= \(\frac{5}{2} \times 16 \times \frac{225}{4} = 2250\)
(b) \(\log_{5} (\frac{3}{5}) + 3\log_{5} (\frac{15}{2}) - \log_{5} (\frac{81}{8})\)
= \(\log_{5} 3 - \log_{5} 5 + 3\log_{5} 15 - 3\log_{5} 2 - \log_{5} 81 + \log_{5} 8\)
= \(\log_{5} 3 - \log_{5} 5 + 3\log_{5} (3 \times 5) - 3\log_{5} 2 - \log_{5} 3^{4} + \log_{5} 2^{3}\)
= \(\log_{5} 3 - \log_{5} 5 + 3\log_{5} 3 + 3\log_{5} 5 - 3\log_{5} 2 - 4\log_{5} 3 + 3\log_{5} 2\)
= \(2\log_{5} 5 \)
= \(2\)
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