Упрощение алгебраических выражений. Задача 47math100admin44242025-03-26T21:54:34+03:00
Задача 47. Упростите выражение \(\left( {\dfrac{{\sqrt {x-2} }}{{\sqrt {x + 2} + \sqrt {x-2} }} + \dfrac{{x-2}}{{\sqrt {{x^2}-4} -x + 2}}} \right)\,\sqrt {\,{{\left( {\dfrac{{{x^2}}}{4}-1} \right)}^{-1}}} \)
Решение
\(\left( {\dfrac{{\sqrt {x-2} }}{{\sqrt {x + 2} + \sqrt {x-2} }} + \dfrac{{x-2}}{{\sqrt {{x^2}-4} -x + 2}}} \right)\sqrt {{{\left( {\dfrac{{{x^2}}}{4}-1} \right)}^{-1}}} = \)
\( = \left( {\dfrac{{\sqrt {x-2} }}{{\sqrt {x + 2} + \sqrt {x-2} }} + \dfrac{{{{\left( {\sqrt {x-2} } \right)}^2}}}{{\sqrt {x-2} \sqrt {x + 2} -{{\left( {\sqrt {x-2} } \right)}^2}}}} \right)\sqrt {{{\left( {\dfrac{{{x^2}-4}}{4}} \right)}^{-1}}} = \)
\( = \left( {\dfrac{{\sqrt {x-2} }}{{\sqrt {x + 2} + \sqrt {x-2} }} + \dfrac{{{{\left( {\sqrt {x-2} } \right)}^2}}}{{\sqrt {x-2} \left( {\sqrt {x + 2} -\sqrt {x-2} } \right)}}} \right)\sqrt {\dfrac{4}{{{x^2}-4}}} = \)
\( = \left( {\dfrac{{\sqrt {x-2} }}{{\sqrt {x + 2} + \sqrt {x-2} }} + \dfrac{{\sqrt {x-2} }}{{\sqrt {x + 2} -\sqrt {x-2} }}} \right) \cdot \dfrac{2}{{\sqrt {x-2} \sqrt {x + 2} }} = \)
\( = \left( {\dfrac{1}{{\sqrt {x + 2} + \sqrt {x-2} }} + \dfrac{1}{{\sqrt {x + 2} -\sqrt {x-2} }}} \right) \cdot \dfrac{2}{{\sqrt {x + 2} }} = \)
\( = \dfrac{{\sqrt {x + 2} -\sqrt {x-2} + \sqrt {x + 2} + \sqrt {x-2} }}{{{{\left( {\sqrt {x + 2} } \right)}^2}-{{\left( {\sqrt {x-2} } \right)}^2}}} \cdot \dfrac{2}{{\sqrt {x + 2} }} = \dfrac{{2\sqrt {x + 2} }}{{x + 2-x + 2}} \cdot \dfrac{2}{{\sqrt {x + 2} }} = \frac{4}{4} = 1.\)
Ответ: 1.