Упрощение алгебраических выражений. Задача 16math100admin44242025-03-26T20:51:28+03:00
Задача 16. Упростите выражение \(\dfrac{{4x}}{{b + x}} + \left( {\dfrac{{2y}}{{{{\left( {x-b} \right)}^2}}}-\dfrac{{2y}}{{{x^2}-{b^2}}}} \right)\,{\left( {\dfrac{y}{{{{\left( {x-b} \right)}^2}}}} \right)^{-1}}\)
Решение
\(\dfrac{{4x}}{{b + x}} + \left( {\dfrac{{2y}}{{{{\left( {x-b} \right)}^2}}}-\dfrac{{2y}}{{{x^2}-{b^2}}}} \right)\,{\left( {\dfrac{y}{{{{\left( {x-b} \right)}^2}}}} \right)^{-1}} = \)
\( = \dfrac{{4x}}{{b + x}} + \left( {\dfrac{{2y}}{{{{\left( {x-b} \right)}^2}}}-\dfrac{{2y}}{{\left( {x-b} \right)\left( {x + b} \right)}}} \right) \cdot \dfrac{{{{\left( {x-b} \right)}^2}}}{y} = \)
\( = \dfrac{{4x}}{{b + x}} + \dfrac{{2yx + 2yb-2yx + 2yb}}{{{{\left( {x-b} \right)}^2}\left( {x + b} \right)}} \cdot \dfrac{{{{\left( {x-b} \right)}^2}}}{y} = \)
\( = \dfrac{{4x}}{{b + x}} + \dfrac{{4yb}}{{\left( {x + b} \right)y}} = \dfrac{{4x}}{{b + x}} + \dfrac{{4b}}{{x + b}} = \dfrac{{4x + 4b}}{{x + b}} = \dfrac{{4\left( {x + b} \right)}}{{x + b}} = 4.\)
Ответ: 4.