Упрощение алгебраических выражений. Задача 30math100admin44242025-03-26T21:14:22+03:00
Задача 30. Упростите выражение \(\dfrac{{x + 7}}{{x + 9}} + \left( {\dfrac{{x + 7}}{{{x^2} + 81-18x}} + \dfrac{{x + 5}}{{{x^2}-81}}} \right)\,{\left( {\dfrac{{\,\,x + {3^{}}}}{{\,\,x-{9^{}}}}} \right)^{-2}}\)
Решение
\(\dfrac{{x + 7}}{{x + 9}} + \left( {\dfrac{{x + 7}}{{{x^2} + 81-18x}} + \dfrac{{x + 5}}{{{x^2}-81}}} \right)\,{\left( {\dfrac{{\,\,x + {3^{}}}}{{\,\,x-{9^{}}}}} \right)^{-2}} = \)
\( = \dfrac{{x + 7}}{{x + 9}} + \left( {\dfrac{{x + 7}}{{{{\left( {x-9} \right)}^2}}} + \dfrac{{x + 5}}{{\left( {x-9} \right)\left( {x + 9} \right)}}} \right) \cdot \dfrac{{{{\left( {x-9} \right)}^2}}}{{{{\left( {x + 3} \right)}^2}}} = \)
\( = \dfrac{{x + 7}}{{x + 9}} + \dfrac{{{x^2} + 7x + 9x + 63 + {x^2} + 5x-9x-45}}{{{{\left( {x-9} \right)}^2}\left( {x + 9} \right)}} \cdot \dfrac{{{{\left( {x-9} \right)}^2}}}{{{{\left( {x + 3} \right)}^2}}} = \)
\( = \dfrac{{x + 7}}{{x + 9}} + \dfrac{{2{x^2} + 12x + 18}}{{\left( {x + 9} \right){{\left( {x + 3} \right)}^2}}} = \dfrac{{x + 7}}{{x + 9}} + \dfrac{{2{{\left( {x + 3} \right)}^2}}}{{\left( {x + 9} \right){{\left( {x + 3} \right)}^2}}} = \dfrac{{x + 7}}{{x + 9}} + \dfrac{2}{{x + 9}} = \dfrac{{x + 9}}{{x + 9}} = 1.\)
Ответ: 1.