Преобразование выражений, содержащих степени. Задача 14math100admin44242025-03-26T20:21:15+03:00
Задача 14. Вычислите \({0,75^{\frac{1}{5}}} \cdot {4^{\frac{2}{5}}} \cdot {12^{\frac{4}{5}}}-{1,25^{\frac{1}{7}}} \cdot {2^{\frac{3}{7}}} \cdot {10^{\frac{6}{7}}}\)
Решение
\({0,75^{\frac{1}{5}}} \cdot {4^{\frac{2}{5}}} \cdot {12^{\frac{4}{5}}}-{1,25^{\frac{1}{7}}} \cdot {2^{\frac{3}{7}}} \cdot {10^{\frac{6}{7}}} = {\left( {\frac{3}{4}} \right)^{\frac{1}{5}}} \cdot {4^{\frac{2}{5}}} \cdot {\left( {4 \cdot 3} \right)^{\frac{4}{5}}}-{\left( {\frac{5}{4}} \right)^{\frac{1}{7}}} \cdot {2^{\frac{3}{7}}} \cdot {\left( {2 \cdot 5} \right)^{\frac{6}{7}}} = \)
\( = \dfrac{{{3^{\frac{1}{5}}} \cdot {4^{\frac{2}{5}}} \cdot {4^{\frac{4}{5}}} \cdot {3^{\frac{4}{5}}}}}{{{4^{\frac{1}{5}}}}}-\dfrac{{{5^{\frac{1}{7}}} \cdot {2^{\frac{3}{7}}} \cdot {2^{\frac{6}{7}}} \cdot {5^{\frac{6}{7}}}}}{{{2^{\frac{2}{7}}}}} = \)
\( = {3^{\frac{1}{5} + \frac{4}{5}}} \cdot {4^{\frac{2}{5} + \frac{4}{5}-\frac{1}{5}}}-{5^{\frac{1}{7} + \frac{6}{7}}} \cdot {2^{\frac{3}{7} + \frac{6}{7}-\frac{2}{7}}} = 3 \cdot 4-5 \cdot 2 = 12-10 = 2.\)
Ответ: 2.