Найдите произведение дробей и сократите получившуюся дробь:
Получаем:
\(\displaystyle \frac{p^{\frac{3}{7}}q^{\frac{2}{5}}+1}{12p^{\frac{6}{7}}-3q^{\frac{4}{5}}}:\frac{2p^{\frac{3}{7}}q^{\frac{2}{5}}+2}{8p^{\frac{6}{7}}-4p^{\frac{3}{7}}q^{\frac{2}{5}}}=\frac{p^{\frac{3}{7}}q^{\frac{2}{5}}+1}{12p^{\frac{6}{7}}-3q^{\frac{4}{5}}}\cdot \frac{8p^{\frac{6}{7}}-4p^{\frac{3}{7}}q^{\frac{2}{5}}}{2p^{\frac{3}{7}}q^{\frac{2}{5}}+2}=\)
\(\displaystyle =\frac{(p^{\frac{3}{7}}q^{\frac{2}{5}}+1)(8p^{\frac{6}{7}}-4p^{\frac{3}{7}}q^{\frac{2}{5}})}{(12p^{\frac{6}{7}}-3q^{\frac{4}{5}}) (2p^{\frac{3}{7}}q^{\frac{2}{5}}+2)}{\small .}\)
Чтобы сократить дробь, разложим выражения в числителе и знаменателе на множители:
- \(\displaystyle \color{green}{8p^{\frac{6}{7}}-4p^{\frac{3}{7}}q^{\frac{2}{5}}=4p^{\frac{3}{7}}(2p^{\frac{3}{7}}-q^{\frac{2}{5}})}\small,\)
- \(\displaystyle \color{blue}{12p^{\frac{6}{7}}-3q^{\frac{4}{5}}}=3(4p^{\frac{6}{7}}-q^{\frac{4}{5}})=3 ((2p^{\frac{3}{7}})^2-(q^{\frac{2}{5}})^2)=\color{blue}{3(2p^{\frac{3}{7}}-q^{\frac{2}{5}})(2p^{\frac{3}{7}}+q^{\frac{2}{5}})}\small,\)
- \(\displaystyle \color{purple}{2p^{\frac{3}{7}}q^{\frac{2}{5}}+2=2(p^{\frac{3}{7}}q^{\frac{2}{5}}+1)}{\small .}\)
Подставляя, получаем:
\(\displaystyle \frac{(p^{\frac{3}{7}}q^{\frac{2}{5}}+1)\color{green}{(8p^{\frac{6}{7}}-4p^{\frac{3}{7}}q^{\frac{2}{5}})}}{\color{blue}{(12p^{\frac{6}{7}}-3q^{\frac{4}{5}})}\color{purple}{(2p^{\frac{3}{7}}q^{\frac{2}{5}}+2)}}=\frac{(p^{\frac{3}{7}}q^{\frac{2}{5}}+1)\cdot\color{green}{4p^{\frac{3}{7}}(2p^{\frac{3}{7}}-q^{\frac{2}{5}})}}{{\color{blue}{3(2p^{\frac{3}{7}}-q^{\frac{2}{5}})(2p^{\frac{3}{7}}+q^{\frac{2}{5}})}}\cdot \color{purple}{2(p^{\frac{3}{7}}q^{\frac{2}{5}}+1)}}=\)
\(\displaystyle =\frac{4p^{\frac{3}{7}}(p^{\frac{3}{7}}q^{\frac{2}{5}}+1)(2p^{\frac{3}{7}}-q^{\frac{2}{5}})}{6(2p^{\frac{3}{7}}-q^{\frac{2}{5}})(2p^{\frac{3}{7}}+q^{\frac{2}{5}})(p^{\frac{3}{7}}q^{\frac{2}{5}}+1)} {\small .}\)
Сократим полученную дробь:
\(\displaystyle \begin{aligned}\frac{4p^{\frac{3}{7}}{\cancel{\color{green}{(p^{\frac{3}{7}}q^{\frac{2}{5}}+1)}}}{\cancel{\color{blue}{(2p^{\frac{3}{7}}-q^{\frac{2}{5}})}}}}{{6{\cancel{\color{blue}{(2p^{\frac{3}{7}}-q^{\frac{2}{5}})}}(2p^{\frac{3}{7}}+q^{\frac{2}{5}})}}\cancel{\color{green}{(p^{\frac{3}{7}}q^{\frac{2}{5}}+1)}}}=\frac{2p^{\frac{3}{7}}}{3(2p^{\frac{3}{7}}+q^{\frac{2}{5}})} {\small .}\end{aligned}\)
Ответ: \(\displaystyle \frac{2p^{\frac{3}{7}}}{3(2p^{\frac{3}{7}}+q^{\frac{2}{5}})} {\small .}\)