14. Найдите значение выражения: 1) (1 1/4)^(-5) * (1 1/2)^10 * (1 2/3)^5 2) (0,04)^(-3) * (1 1/5)^(-4) * (30)^(-2) 3) (0,25)^(-4) * (2 1/2)^8 * 125^-2 4) (0,2)^(-5) * (12 1/2)^(-1) * (2 1/2)^(-3)

Решение:

1) \( \left(1\frac{1}{4}\right)^{-5} \cdot \left(1\frac{1}{2}\right)^{10} \cdot \left(1\frac{2}{3}\right)^5 = \left(\frac{5}{4}\right)^{-5} \cdot \left(\frac{3}{2}\right)^{10} \cdot \left(\frac{5}{3}\right)^5 = \left(\frac{4}{5}\right)^{5} \cdot \left(\frac{3^2}{2^2}\right)^{5} \cdot \left(\frac{5}{3}\right)^5 = \left(\frac{4}{5} \cdot \frac{9}{4} \cdot \frac{5}{3}\right)^5 = \left(\frac{4 \cdot 9 \cdot 5}{5 \cdot 4 \cdot 3}\right)^5 = \left(\frac{180}{60}\right)^5 = 3^5 = 243 \)

2) \( (0.04)^{-3} \cdot (1 \frac{1}{5})^{-4} \cdot (30)^{-2} = \left(\frac{4}{100}\right)^{-3} \cdot \left(\frac{6}{5}\right)^{-4} \cdot \left(\frac{1}{30}\right)^2 = \left(\frac{1}{25}\right)^{-3} \cdot \left(\frac{5}{6}\right)^4 \cdot \left(\frac{1}{30}\right)^2 = (25)^3 \cdot \frac{5^4}{6^4} \cdot \frac{1}{30^2} = (5^2)^3 \cdot \frac{5^4}{(2 \cdot 3)^4} \cdot \frac{1}{(2 \cdot 3 \cdot 5)^2} = 5^6 \cdot \frac{5^4}{2^4 \cdot 3^4} \cdot \frac{1}{2^2 \cdot 3^2 \cdot 5^2} = \frac{5^{6+4-2}}{2^{4+2} \cdot 3^{4+2}} = \frac{5^8}{2^6 \cdot 3^6} \)

3) \( (0.25)^{-4} \cdot (2 \frac{1}{2})^8 \cdot 125^{-2} = \left(\frac{1}{4}\right)^{-4} \cdot \left(\frac{5}{2}\right)^8 \cdot (5^3)^{-2} = 4^4 \cdot \left(\frac{5}{2}\right)^8 \cdot 5^{-6} = (2^2)^4 \cdot \frac{5^8}{2^8} \cdot 5^{-6} = 2^8 \cdot \frac{5^8}{2^8} \cdot 5^{-6} = 5^{8-6} = 5^2 = 25 \)

4) \( (0.2)^{-5} \cdot (12 \frac{1}{2})^{-1} \cdot (2 \frac{1}{2})^{-3} = \left(\frac{1}{5}\right)^{-5} \cdot \left(\frac{25}{2}\right)^{-1} \cdot \left(\frac{5}{2}\right)^{-3} = 5^5 \cdot \left(\frac{2}{25}\right) \cdot \left(\frac{2}{5}\right)^3 = 5^5 \cdot \frac{2}{5^2} \cdot \frac{2^3}{5^3} = \frac{5^5 \cdot 2 \cdot 2^3}{5^2 \cdot 5^3} = \frac{5^5 \cdot 2^4}{5^5} = 2^4 = 16 \)

Ответ: 1) \( 243 \) 2) \( \frac{5^8}{2^6 \cdot 3^6} \) 3) \( 25 \) 4) \( 16 \)