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Теория: 15 Умножение и деление буквенных выражений, содержащих степень с рациональным показателем

Задание

Найдите произведение дробей и сократите получившуюся дробь:
 

\(\displaystyle \frac{8x^{\frac{1}{5}}-16y^{\frac{1}{3}}}{72x^{\frac{2}{5}}-2y^{\frac{2}{3}}}\cdot \frac{6x^{\frac{1}{5}}+y^{\frac{1}{3}}}{4x^{\frac{1}{5}}-8y^{\frac{1}{3}}}=\)
\frac{1}{6x^{\frac{1}{5}}-y^{\frac{1}{3}}}
Решение

Получаем:

\(\displaystyle \frac{8x^{\frac{1}{5}}-16y^{\frac{1}{3}}}{72x^{\frac{2}{5}}-2y^{\frac{2}{3}}}\cdot \frac{6x^{\frac{1}{5}}+y^{\frac{1}{3}}}{4x^{\frac{1}{5}}-8y^{\frac{1}{3}}}=\frac{\left(8x^{\frac{1}{5}}-16y^{\frac{1}{3}}\right)\cdot \left(6x^{\frac{1}{5}}+y^{\frac{1}{3}}\right)}{\left(72x^{\frac{2}{5}}-2y^{\frac{2}{3}}\right)\cdot \left(4x^{\frac{1}{5}}-8y^{\frac{1}{3}}\right)}{\small .}\)


Разложим выражения в числителе и знаменателе на множители:

  • \(\displaystyle 8x^{\frac{1}{5}}-16y^{\frac{1}{3}}=8\left(x^{\frac{1}{5}}-2y^{\frac{1}{3}}\right)\small,\)
  • \(\displaystyle 72x^{\frac{2}{5}}-2y^{\frac{2}{3}}=2\left(36x^{\frac{2}{5}}-y^{\frac{2}{3}}\right)=2\left(\left(6x^{\frac{1}{5}}\right)^2-\left(y^{\frac{1}{3}}\right)^2\right)=\)
    \(\displaystyle =2\left(6x^{\frac{1}{5}}-y^{\frac{1}{3}}\right)\left(6x^{\frac{1}{5}}+y^{\frac{1}{3}}\right)\small,\)
  • \(\displaystyle 4x^{\frac{1}{5}}-8y^{\frac{1}{3}}=4\left(x^{\frac{1}{5}}-2y^{\frac{1}{3}}\right){\small .}\)


Подставляя, получаем:

\(\displaystyle \frac{\left(8x^{\frac{1}{5}}-16y^{\frac{1}{3}}\right)\cdot \left(6x^{\frac{1}{5}}+y^{\frac{1}{3}}\right)}{\left(72x^{\frac{2}{5}}-2y^{\frac{2}{3}}\right)\cdot \left(4x^{\frac{1}{5}}-8y^{\frac{1}{3}}\right)}=\frac{8\left(x^{\frac{1}{5}}-2y^{\frac{1}{3}}\right)\left(6x^{\frac{1}{5}}+y^{\frac{1}{3}}\right)}{8\left(6x^{\frac{1}{5}}-y^{\frac{1}{3}}\right)\left(6x^{\frac{1}{5}}+y^{\frac{1}{3}}\right)\left(x^{\frac{1}{5}}-2y^{\frac{1}{3}}\right)}{\small .}\)

Сократим полученную дробь: 

\(\displaystyle \begin{aligned}\frac{\cancel{\color{orange}{{8}}}^{\backslash1}{\cancel{\color{blue}{\left(x^{\frac{1}{5}}-2y^{\frac{1}{3}}\right)}}}\cancel{\color{green}{\left(6x^{\frac{1}{5}}+y^{\frac{1}{3}}\right)}}}{{\cancel{\color{orange}{{8}}}^{\backslash1}}{\left(6x^{\frac{1}{5}}-y^{\frac{1}{3}}\right)\cancel{\color{green}{\left(6x^{\frac{1}{5}}+y^{\frac{1}{3}}\right)}}}\cancel{\color{blue}{\left(x^{\frac{1}{5}}-2y^{\frac{1}{3}}\right)}}}=\frac{1}{6x^{\frac{1}{5}}-y^{\frac{1}{3}}} {\small .}\end{aligned}\)

Ответ: \(\displaystyle \frac{1}{6x^{\frac{1}{5}}-y^{\frac{1}{3}}} {\small .}\)