Упрощение алгебраических выражений. Задача 19math100admin44242025-03-26T20:56:01+03:00
Задача 19. Упростите выражение \(\left( {\dfrac{a}{{a + b}} + \dfrac{b}{{a-b}} + \dfrac{{2ab}}{{{b^2}-{a^2}}}} \right)\dfrac{a}{{a-b}} + \left( {\dfrac{b}{{b-a}} + \dfrac{{2ab}}{{{a^2}-{b^2}}}} \right)\)
Решение
\(\left( {\dfrac{a}{{a + b}} + \dfrac{b}{{a-b}} + \dfrac{{2ab}}{{{b^2}-{a^2}}}} \right)\dfrac{a}{{a-b}} + \left( {\dfrac{b}{{b-a}} + \dfrac{{2ab}}{{{a^2}-{b^2}}}} \right) = \)
\( = \left( {\dfrac{a}{{a + b}} + \dfrac{b}{{a-b}}-\dfrac{{2ab}}{{\left( {a-b} \right)\left( {a + b} \right)}}} \right) \cdot \dfrac{a}{{a-b}} + \left( {\dfrac{{2ab}}{{\left( {a-b} \right)\left( {a + b} \right)}}-\dfrac{b}{{a-b}}} \right) = \)
\( = \dfrac{{{a^2}-ab + ab + {b^2}-2ab}}{{\left( {a + b} \right)\left( {a-b} \right)}} \cdot \dfrac{a}{{a-b}} + \dfrac{{2ab-ab-{b^2}}}{{\left( {a-b} \right)\left( {a + b} \right)}} = \)
\( = \dfrac{{{a^2}-2ab + {b^2}}}{{\left( {a + b} \right)\left( {a-b} \right)}} \cdot \dfrac{a}{{a-b}} + \dfrac{{ab-{b^2}}}{{\left( {a-b} \right)\left( {a + b} \right)}} = \)
\( = \dfrac{{{{\left( {a-b} \right)}^2} \cdot a}}{{\left( {a + b} \right){{\left( {a-b} \right)}^2}}} + \dfrac{{b\left( {a-b} \right)}}{{\left( {a-b} \right)\left( {a + b} \right)}} = \dfrac{a}{{a + b}} + \dfrac{b}{{a + b}} = \dfrac{{a + b}}{{a + b}} = 1.\)
Ответ: 1.