Решите уравнение 3 * (16^x) + 2 * (81^x) = 5 * (36^x)

3 * (16^x) + 2 * (81^x) = 5 * (36^x).

3 * (4^ (2 * x)) + 2 * (9^ (2 * x)) = 5 * (6^ (2 * x)).

Разделим обе части на 5 * (6^ (2 * x)), тогда:

3/5 * ((2/3) ^ (2 * x)) + 2/5 * ((3/2) ^ (2 * x)) = 1.

3/5 * ((2/3) ^ (2 * x)) + 2/5 * ((3/2) ^ ( - 2 * x)) = 1.

3/5 * ((2/3) ^ (4 * x)) + 2/5 = (2/3) ^ (2 * x).

Обозначим ((2/3) ^ (2 * x)) = t.

Следовательно,

3/5 * t^2 + 2/5 - t = 0.

3 * t^2 - 5 * t + 2 = 0.

D = 25 - 4 * 3 * 2 = 25 - 24 = 1.

t1,2 = (5 ± 1) / 6.

1) ((2/3) ^ (2 * x)) = 1.

((2/3) ^ (2 * x)) = ((2/3) ^0).

x = 0.

Проверка: 3 * 1 + 2 * 1 = 5 * 1, 5 = 5.

2) ((2/3) ^ (2 * x)) = 2/3.

2 * x = 1.

x = ½.

Проверка:

3 * (16^ (1/2)) + 2 * (81^ (1/2)) = 5 * (36^ (1/2));

3 * 4 + 2 * 9 = 5 * 6;

30 = 30.

Ответ: х1 = 0, х2 = ½.