Question

Сравните числа 12^67 и 8^77

Answer 1

a > b  a,b > 0

a/b > 1

(a^m)^n = a^(mn)

a^m * a^n = a^(m + n)

a^m/a^n = a^(m - n)

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12^67 / 8^77 = (3*4)^67 / (2^3)^77 = 3^67*(2^2)^67 / 2^231 = 3^67 * 2^134 / 2^231 = 3^67 / 2^97 = 3^3*3^64 / 2*2^96 = 27/2 * (3^2)^32/(2^3)^32 = 27/2 * 9^32/8^32 = 27/2 * (9/8)^32

27/2 > 1

(9/8)^32 > 1

27/2 * (9/8)^32 > 1

12^67  > 8^77