С.2.40. Вычислите: 1) \(10^{3-5\lg2}\); 2) \(\left(\frac13\right)^{3+2\log_{1/3}6}\); 3) \(\left(\frac{25}{7}\right)^{-2}\cdot25^{2-\log_5 7}\); 4) \(49^{0,5\log_7 4+1}\).

1. \(10^{3-5\lg2}=\frac{10^3}{10^{5\lg2}}=\frac{1000}{2^5}=\frac{125}{4}\).
2. \(\left(\frac13\right)^{3+2\log_{1/3}6}=\frac1{27}\cdot\left(\left(\frac13\right)^{\log_{1/3}6}\right)^2=\frac{36}{27}=\frac43\).
3. \(\left(\frac{25}{7}\right)^{-2}=\frac{49}{625}\), а \(25^{2-\log_5 7}=\frac{625}{49}\). Произведение равно \(1\).
4. \(49^{0,5\log_7 4+1}=7^{\log_7 4+2}=4\cdot49=196\).

Ответ: 1) \(\frac{125}{4}\); 2) \(\frac43\); 3) 1; 4) 196.