Упрощение алгебраических выражений. Задача 21math100admin44242025-03-26T20:58:57+03:00
Задача 21. Упростите выражение \(\left( {\dfrac{n}{{m-n}} + \dfrac{m}{{m + n}}} \right)\,\left( {\dfrac{{{m^2}}}{{{n^2}}} + \dfrac{{{n^2}}}{{{m^2}}}-2} \right)\,{\left( {\dfrac{{{m^4}-{n^4}}}{{{m^2}{n^2}}}} \right)^{-1}}\)
Решение
\(\left( {\dfrac{n}{{m-n}} + \dfrac{m}{{m + n}}} \right)\left( {\dfrac{{{m^2}}}{{{n^2}}} + \dfrac{{{n^2}}}{{{m^2}}}-2} \right) \cdot {\left( {\dfrac{{{m^4}-{n^4}}}{{{m^2}{n^2}}}} \right)^{-1}} = \)
\( = \dfrac{{nm + {n^2} + {m^2}-mn}}{{{m^2}-{n^2}}} \cdot \dfrac{{{m^4} + {n^4}-2{n^2}{m^2}}}{{{n^2}{m^2}}} \cdot \dfrac{{{m^2}{n^2}}}{{{m^4}-{n^4}}} = \)
\( = \dfrac{{{n^2} + {m^2}}}{{{m^2}-{n^2}}} \cdot \dfrac{{{{\left( {{m^2}-{n^2}} \right)}^2}}}{{{n^2}{m^2}}} \cdot \dfrac{{{m^2}{n^2}}}{{\left( {{m^2}-{n^2}} \right)\left( {{m^2} + {n^2}} \right)}} = 1.\)
Ответ: 1.