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Консультация № 191811
20.11.2017, 18:07
0.00 руб.
0
2
1
Здравствуйте, уважаемые эксперты! Прошу вас ответить на следующий вопрос:
Даны натуральные числа a и b
Число (а×3^1/2+b)÷(b×3^1/2+с) рационально
Доказать, что (а^2+b^2+c^2)÷(a+b+c) целое число
Задания данного типа регулярно не получаются, поэтому прошу объяснить подробно
Даны натуральные числа a и b
Число (а×3^1/2+b)÷(b×3^1/2+с) рационально
Доказать, что (а^2+b^2+c^2)÷(a+b+c) целое число
Задания данного типа регулярно не получаются, поэтому прошу объяснить подробно
Обсуждение
21.11.2017, 06:10
общий
это ответ
Здравствуйте, Dipper!
Чтобы избавиться от иррациональности в знаменателе, умножим числитель и знаменатель дроби
(a[$8730$]3 + b)/(b[$8730$]3 + c) на b[$8730$]3 - c. Получим ((b2 - ac)[$8730$]3 + 3ab - bc)/(3b2 - c2).
Это число рационально, только если b2 = ac. С учетом полученного равенства, преобразуем
(a2+b2+c2)/(a+b+c) = ((a+c)2 - 2ac+b2)/(a+b+c)=
((a+c)2 - b2)/(a+b+c)) = ((a+c+b)(a+c-b))/(a+b+c) = a+c-b, целое.
Чтобы избавиться от иррациональности в знаменателе, умножим числитель и знаменатель дроби
(a[$8730$]3 + b)/(b[$8730$]3 + c) на b[$8730$]3 - c. Получим ((b2 - ac)[$8730$]3 + 3ab - bc)/(3b2 - c2).
Это число рационально, только если b2 = ac. С учетом полученного равенства, преобразуем
(a2+b2+c2)/(a+b+c) = ((a+c)2 - 2ac+b2)/(a+b+c)=
((a+c)2 - b2)/(a+b+c)) = ((a+c+b)(a+c-b))/(a+b+c) = a+c-b, целое.
5
Спасибо, все понятно<br>
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