Преобразование выражений, содержащих степени. Задача 13math100admin44242025-03-26T20:16:44+03:00
Задача 13. Вычислите \({0,12^{\frac{1}{9}}} \cdot {5^{\frac{1}{3}}} \cdot {15^{\frac{8}{9}}} + {1,25^{\frac{1}{5}}} \cdot {2^{\frac{3}{5}}} \cdot {10^{\frac{4}{5}}}\)
Решение
\({0,12^{\frac{1}{9}}} \cdot {5^{\frac{1}{3}}} \cdot {15^{\frac{8}{9}}} + {1,25^{\frac{1}{5}}} \cdot {2^{\frac{3}{5}}} \cdot {10^{\frac{4}{5}}} = {\left( {\dfrac{3}{{25}}} \right)^{\frac{1}{9}}} \cdot {5^{\frac{1}{3}}} \cdot {\left( {5 \cdot 3} \right)^{\frac{8}{9}}} + {\left( {\dfrac{5}{4}} \right)^{\frac{1}{5}}} \cdot {2^{\frac{3}{5}}} \cdot {\left( {5 \cdot 2} \right)^{\frac{4}{5}}} = \)
\( = \dfrac{{{3^{\frac{1}{9}}} \cdot {5^{\frac{1}{3}}} \cdot {5^{\frac{8}{9}}} \cdot {3^{\frac{8}{9}}}}}{{{5^{\frac{2}{9}}}}} + \dfrac{{{5^{\frac{1}{5}}} \cdot {2^{\frac{3}{5}}} \cdot {5^{\frac{4}{5}}} \cdot {2^{\frac{4}{5}}}}}{{{2^{\frac{2}{5}}}}} = {3^{\frac{1}{9} + \frac{8}{9}}} \cdot {5^{\frac{3}{9} + \frac{8}{9}-\frac{2}{9}}} + {5^{\frac{1}{5} + \frac{4}{5}}} \cdot {2^{\frac{3}{5} + \frac{4}{5}-\frac{2}{5}}} = 3 \cdot 5 + 5 \cdot 2 = 15 + 10 = 25.\)
Ответ: 25.