А4. Найдите значение выражения: a) 0,064.0,16; (0.2168) б) 0,090,027 B) (363) (1252)

a) Вычислим $$0,064^{-\frac{1}{3}} \cdot 0,16^{-\frac{1}{2}}$$.

1. $$0,064^{-\frac{1}{3}} = (\frac{64}{1000})^{-\frac{1}{3}} = (\frac{4^3}{10^3})^{-\frac{1}{3}} = (\frac{4}{10})^{-1} = \frac{10}{4} = \frac{5}{2} = 2.5$$
2. $$0,16^{-\frac{1}{2}} = (\frac{16}{100})^{-\frac{1}{2}} = (\frac{4^2}{10^2})^{-\frac{1}{2}} = (\frac{4}{10})^{-1} = \frac{10}{4} = \frac{5}{2} = 2.5$$
3. $$0,064^{-\frac{1}{3}} \cdot 0,16^{-\frac{1}{2}} = 2.5 \cdot 2.5 = 6.25$$

Ответ: 6.25

б) Вычислим $$ \frac{(0.216)^{\frac{2}{3}}}{0,09^{-\frac{3}{2}} \cdot 0,027^{\frac{1}{3}}}$$.

1. $$ (0.216)^{\frac{2}{3}} = (\frac{216}{1000})^{\frac{2}{3}} = (\frac{6^3}{10^3})^{\frac{2}{3}} = (\frac{6}{10})^2 = \frac{36}{100} = 0.36 $$
2. $$ 0,09^{-\frac{3}{2}} = (\frac{9}{100})^{-\frac{3}{2}} = (\frac{3^2}{10^2})^{-\frac{3}{2}} = (\frac{3}{10})^{-3} = (\frac{10}{3})^3 = \frac{1000}{27} $$
3. $$ 0,027^{\frac{1}{3}} = (\frac{27}{1000})^{\frac{1}{3}} = \frac{3}{10} $$
4. $$ \frac{(0.216)^{\frac{2}{3}}}{0,09^{-\frac{3}{2}} \cdot 0,027^{\frac{1}{3}}} = \frac{0.36}{\frac{1000}{27} \cdot \frac{3}{10}} = \frac{0.36}{\frac{100}{9} \cdot 1} = \frac{0.36}{\frac{100}{9}} = 0.36 \cdot \frac{9}{100} = \frac{36}{100} \cdot \frac{9}{100} = \frac{324}{10000} = 0.0324 $$

Ответ: 0.0324

в) Вычислим $$\left(\frac{36^3}{125^2}\right)^{-\frac{1}{6}}$$.

1. $$ \left(\frac{36^3}{125^2}\right)^{-\frac{1}{6}} = \left(\frac{(6^2)^3}{(5^3)^2}\right)^{-\frac{1}{6}} = \left(\frac{6^6}{5^6}\right)^{-\frac{1}{6}} = (\frac{6}{5})^{-1} = \frac{5}{6} $$

Ответ: $$\frac{5}{6}$$