Преобразование выражений, содержащих радикалы. Задача 93math100admin44242024-05-14T20:54:22+03:00
Задача 93. Вычислите \(\sqrt {\,\left| {\,12\sqrt 5 -29\,} \right|} -\sqrt {12\sqrt 5 + 29} \)
Решение
\(\sqrt {\left| {12\sqrt 5 -29} \right|} -\sqrt {12\sqrt 5 + 29} .\)
Так как \(12\sqrt 5 = \sqrt {720} < \sqrt {841} = 29\), то \(12\sqrt 5 -29 < 0\) и \(\left| {12\sqrt 5 -29} \right| = 29-12\sqrt 5 .\)
\(\sqrt {\left| {12\sqrt 5 -29} \right|} -\sqrt {12\sqrt 5 + 29} = \sqrt {29-12\sqrt 5 } -\sqrt {12\sqrt 5 + 29} = \)
\( = \sqrt {20-12\sqrt 5 + 9} -\sqrt {20 + 12\sqrt 5 + 9} = \)
\( = \sqrt {{{\left( {2\sqrt 5 } \right)}^2}-2 \cdot 2\sqrt 5 \cdot 3 + {3^2}} -\sqrt {{{\left( {2\sqrt 5 } \right)}^2} + 2 \cdot 2\sqrt 5 \cdot 3 + {3^2}} = \)
\( = \sqrt {{{\left( {2\sqrt 5 -3} \right)}^2}} -\sqrt {{{\left( {2\sqrt 5 + 3} \right)}^2}} = \left| {2\sqrt 5 -3} \right|-\left| {2\sqrt 5 + 3} \right|.\)
Так как \(2\sqrt 5 = \sqrt {20} > \sqrt 9 = 3\), то \(2\sqrt 5 -3 > 0\) и \(\left| {2\sqrt 5 -3} \right| = 2\sqrt 5 -3.\)
\(\left| {2\sqrt 5 -3} \right|-\left| {2\sqrt 5 + 3} \right| = 2\sqrt 5 -3-2\sqrt 5 -3 = -6.\)
Ответ: \(-6.\)