Упрощение алгебраических выражений. Задача 34math100admin44242025-03-26T21:23:27+03:00
Задача 34. Упростите выражение \(\dfrac{{5b}}{{\sqrt {a-b} }}\,\left( {\dfrac{1}{{\sqrt a -\sqrt {a-b} }} + \dfrac{1}{{\sqrt a + \sqrt {a + b} }}} \right)\,{\left( {1 + \sqrt {\dfrac{{a + b}}{{a-b}}} } \right)^{-1}}\)
Решение
\(\dfrac{{5b}}{{\sqrt {a-b} }}\,\left( {\dfrac{1}{{\sqrt a -\sqrt {a-b} }} + \dfrac{1}{{\sqrt a + \sqrt {a + b} }}} \right)\,{\left( {1 + \sqrt {\dfrac{{a + b}}{{a-b}}} } \right)^{-1}} = \)
\( = \dfrac{{5b}}{{\sqrt {a-b} }} \cdot \left( {\dfrac{{\sqrt a + \sqrt {a-b} }}{{\left( {\sqrt a -\sqrt {a-b} } \right)\left( {\sqrt a + \sqrt {a-b} } \right)}} + \dfrac{{\sqrt a -\sqrt {a + b} }}{{\left( {\sqrt a + \sqrt {a + b} } \right)\left( {\sqrt a -\sqrt {a + b} } \right)}}} \right) \cdot {\left( {\dfrac{{\sqrt {a-b} + \sqrt {a + b} }}{{\sqrt {a-b} }}} \right)^{-1}} = \)
\( = \dfrac{{5b}}{{\sqrt {a-b} }} \cdot \left( {\dfrac{{\sqrt a + \sqrt {a-b} }}{{a-a + b}} + \dfrac{{\sqrt a -\sqrt {a + b} }}{{a-a-b}}} \right) \cdot \dfrac{{\sqrt {a-b} }}{{\sqrt {a-b} + \sqrt {a + b} }} = \)
\( = \dfrac{{5b}}{{\sqrt {a-b} }} \cdot \dfrac{{\sqrt a + \sqrt {a-b} -\sqrt a + \sqrt {a + b} }}{b} \cdot \dfrac{{\sqrt {a-b} }}{{\sqrt {a-b} + \sqrt {a + b} }} = \)
\( = \dfrac{{5\left( {\sqrt {a-b} + \sqrt {a + b} } \right)}}{{\sqrt {a-b} + \sqrt {a + b} }} = 5.\)
Ответ: 5.