Найдите значение выражения
\(\displaystyle \frac{b^{\frac{2}{9}}}{x^{\frac{2}{7}}y^{\frac{2}{5}}}+\frac{(b^{\frac{1}{9}}+x^{\frac{2}{7}})^2}{x^{\frac{4}{7}}-x^{\frac{2}{7}}y^{\frac{2}{5}}}-\frac{(b^{\frac{1}{9}}+y^{\frac{2}{5}})^2}{x^{\frac{2}{7}}y^{\frac{2}{5}}-y^{\frac{4}{5}}}\)
при \(\displaystyle b=5\small,\) \(\displaystyle x=\frac{1}{2}\small,\) \(\displaystyle c=3\small.\)
Сначала упростим выражение.
Разложим знаменатели дробей на множители:
\(\displaystyle \frac{b^{\frac{2}{9}}}{x^{\frac{2}{7}}y^{\frac{2}{5}}}+\frac{(b^{\frac{1}{9}}+x^{\frac{2}{7}})^2}{x^{\frac{2}{7}}(x^{\frac{2}{7}}-y^{\frac{2}{5}})}+\frac{(b^{\frac{1}{9}}+y^{\frac{2}{5}})^2}{y^{\frac{2}{5}}(x^{\frac{2}{7}}-y^{\frac{2}{5}})}{\small .}\)
Ни одна из дробей не сокращается.
- \(\displaystyle \frac{b^{\frac{2}{9}}}{x^{\frac{2}{7}}y^{\frac{2}{5}}}=\frac{b^{\frac{2}{9}}(x^{\frac{2}{7}}-y^{\frac{2}{5}})}{x^{\frac{2}{7}}y^{\frac{2}{5}}(x^{\frac{2}{7}}-y^{\frac{2}{5}})}{ \small ,}\)
- \(\displaystyle \frac{(b^{\frac{1}{9}}+x^{\frac{2}{7}})^2}{x^{\frac{2}{7}}(x^{\frac{2}{7}}-y^{\frac{2}{5}})}=\frac{(b^{\frac{1}{9}}+x^{\frac{2}{7}})^2y^{\frac{2}{5}}}{x^{\frac{2}{7}}y^{\frac{2}{5}}(x^{\frac{2}{7}}-y^{\frac{2}{5}})}{ \small ,}\)
- \(\displaystyle \frac{(b^{\frac{1}{9}}+y^{\frac{2}{5}})^2}{y^{\frac{2}{5}}(x^{\frac{2}{7}}-y^{\frac{2}{5}})}=\frac{(b^{\frac{1}{9}}+y^{\frac{2}{5}})^2x^{\frac{2}{7}}}{x^{\frac{2}{7}}y^{\frac{2}{5}}(x^{\frac{2}{7}}-y^{\frac{2}{5}})}{\small .}\)
Записывая под одним знаменателем, получаем:
\(\displaystyle \begin{aligned}\frac{b^{\frac{2}{9}}}{x^{\frac{2}{7}}y^{\frac{2}{5}}}+\frac{(b^{\frac{1}{9}}+x^{\frac{2}{7}})^2}{x^{\frac{4}{7}}-x^{\frac{2}{7}}y^{\frac{2}{5}}}-\frac{(b^{\frac{1}{9}}+y^{\frac{2}{5}})^2}{x^{\frac{2}{7}}y^{\frac{2}{5}}-y^{\frac{4}{5}}}=\\[10px^{\frac{2}{7}}]&=\frac{b^{\frac{2}{9}}(x^{\frac{2}{7}}-y^{\frac{2}{5}})+(b^{\frac{1}{9}}+x^{\frac{2}{7}})^2y^{\frac{2}{5}}-(b^{\frac{1}{9}}+y^{\frac{2}{5}})^2x^{\frac{2}{7}}}{x^{\frac{2}{7}}y^{\frac{2}{5}}(x^{\frac{2}{7}}-y^{\frac{2}{5}})}{\small .}\end{aligned}\)
\(\displaystyle \frac{b^{\frac{2}{9}}(x^{\frac{2}{7}}-y^{\frac{2}{5}})+(b^{\frac{1}{9}}+x^{\frac{2}{7}})^2y^{\frac{2}{5}}+(b^{\frac{1}{9}}+y^{\frac{2}{5}})^2x^{\frac{2}{7}}}{x^{\frac{2}{7}}y^{\frac{2}{5}}(x^{\frac{2}{7}}-y^{\frac{2}{5}})}\)
\(\displaystyle =\frac{ b^{\frac{2}{9}}x^{\frac{2}{7}}-b^{\frac{2}{9}}y^{\frac{2}{5}}+(b^{\frac{2}{9}}+2b^{\frac{1}{9}}x^{\frac{2}{7}}+x^{\frac{4}{7}})y^{\frac{2}{5}}-(b^{\frac{2}{9}}+2b^{\frac{1}{9}}y^{\frac{2}{5}}+y^{\frac{4}{5}})x^{\frac{2}{7}}}{ x^{\frac{2}{7}}y^{\frac{2}{5}}(x^{\frac{2}{7}}-y^{\frac{2}{5}}) }=\)
\(\displaystyle =\frac{ b^{\frac{2}{9}}x^{\frac{2}{7}}-b^{\frac{2}{9}}y^{\frac{2}{5}}+b^{\frac{2}{9}}y^{\frac{2}{5}}+2b^{\frac{1}{9}}x^{\frac{2}{7}}y^{\frac{2}{5}}+x^{\frac{4}{7}}y^{\frac{2}{5}}-b^{\frac{2}{9}}x^{\frac{2}{7}}-2b^{\frac{1}{9}}x^{\frac{2}{7}}y^{\frac{2}{5}}-x^{\frac{2}{7}}y^{\frac{4}{5}}}{ x^{\frac{2}{7}}y^{\frac{2}{5}}(x^{\frac{2}{7}}-y^{\frac{2}{5}}) }{\small .} \)
\(\displaystyle \frac{ \cancel{\color{red}{ b^{\frac{2}{9}}x^{\frac{2}{7}}}}-\cancel{\color{green}{ b^{\frac{2}{9}}y^{\frac{2}{5}}}}+\cancel{\color{green}{ b^{\frac{2}{9}}y^{\frac{2}{5}}}}+\cancel{\color{blue}{ 2b^{\frac{1}{9}}x^{\frac{2}{7}}y^{\frac{2}{5}}}}+x^{\frac{4}{7}}y^{\frac{2}{5}}-\cancel{\color{red}{ b^{\frac{2}{9}}x^{\frac{2}{7}}}}-\cancel{\color{blue}{ 2b^{\frac{1}{9}}x^{\frac{2}{7}}y^{\frac{2}{5}}}}-x^{\frac{2}{7}}y^{\frac{4}{5}}}{ x^{\frac{2}{7}}y^{\frac{2}{5}}(x^{\frac{2}{7}}-y^{\frac{2}{5}}) }=\frac{ x^{\frac{4}{7}}y^{\frac{2}{5}}-x^{\frac{2}{7}}y^{\frac{4}{5}}}{x^{\frac{2}{7}}y^{\frac{2}{5}}(x^{\frac{2}{7}}-y^{\frac{2}{5}}) }{\small .} \)
Разложим числитель на множители. Получаем:
\(\displaystyle \frac{ x^{\frac{4}{7}}y^{\frac{2}{5}}-x^{\frac{2}{7}}y^{\frac{4}{5}}}{x^{\frac{2}{7}}y^{\frac{2}{5}}(x^{\frac{2}{7}}-y^{\frac{2}{5}}) }=\frac{ x^{\frac{2}{7}}y^{\frac{2}{5}}(x^{\frac{2}{7}}-y^{\frac{2}{5}})}{x^{\frac{2}{7}}y^{\frac{2}{5}}(x^{\frac{2}{7}}-y^{\frac{2}{5}}) }=1{\small .} \)
При всех допустимых значениях переменных значение выражения равно \(\displaystyle 1\small.\)
В частности, при \(\displaystyle b=5\small,\) \(\displaystyle x=\frac{1}{2}\small,\) \(\displaystyle c=3\small\) значение выражения равно \(\displaystyle 1\small.\)
Ответ: \(\displaystyle 1{\small .} \)