Упрощение алгебраических выражений. Задача 25math100admin44242025-03-26T21:05:35+03:00
Задача 25. Упростите выражение \(\left( {\dfrac{{{a^2}-ba}}{{{b^2} + ab}}-\dfrac{{{a^2}-2ab + {b^2}}}{{{a^2} + ab}}} \right)\,{\left( {\dfrac{{{b^2}}}{{{a^3}-a{b^2}}} + \dfrac{1}{{a + b}}} \right)^{-1}}\)
Ответ
ОТВЕТ: \(\dfrac{{{{\left( {a-b} \right)}^2}}}{b}\).
Решение
\(\left( {\dfrac{{{a^2}-ba}}{{{b^2} + ab}}-\dfrac{{{a^2}-2ab + {b^2}}}{{{a^2} + ab}}} \right)\,{\left( {\dfrac{{{b^2}}}{{{a^3}-a{b^2}}} + \dfrac{1}{{a + b}}} \right)^{-1}} = \)
\( = \left( {\dfrac{{{a^2}-ba}}{{b\left( {a + b} \right)}}-\dfrac{{{a^2}-2ab + {b^2}}}{{a\left( {a + b} \right)}}} \right) \cdot {\left( {\dfrac{{{b^2}}}{{a\left( {a-b} \right)\left( {a + b} \right)}} + \dfrac{1}{{a + b}}} \right)^1} = \)
\( = \dfrac{{{a^3}-b{a^2}-{a^2}b + 2a{b^2}-{b^3}}}{{ab\left( {a + b} \right)}} \cdot {\left( {\dfrac{{{b^2} + {a^2}-ab}}{{a\left( {a-b} \right)\left( {a + b} \right)}}} \right)^{-1}} = \)
\( = \dfrac{{{a^3}-{b^3} + 2a{b^2}-2{a^2}b}}{{ab\left( {a + b} \right)}} \cdot \dfrac{{a\left( {a-b} \right)\left( {a + b} \right)}}{{{a^2}-ab + {b^2}}} = \)
\( = \dfrac{{\left( {\left( {a-b} \right)\left( {{a^2} + ab + {b^2}} \right)-2ab\left( {a-b} \right)} \right)\left( {a-b} \right)}}{{b\left( {{a^2}-ab + {b^2}} \right)}} = \)
\( = \dfrac{{\left( {a-b} \right)\left( {{a^2} + ab + {b^2}-2ab} \right)\left( {a-b} \right)}}{{b\left( {{a^2}-ab + {b^2}} \right)}} = \dfrac{{{{\left( {a-b} \right)}^2}\left( {{a^2}-ab + {b^2}} \right)}}{{b\left( {{a^2}-ab + {b^2}} \right)}} = \dfrac{{{{\left( {a-b} \right)}^2}}}{b}.\)
Ответ: \(\dfrac{{{{\left( {a-b} \right)}^2}}}{b}.\)