Найдите значение выражения:

а) 125^-1 ⋅ 25²

б) 0,0001 ⋅ (10³)² ⋅ (0,1)^-2

в) 16^-3 ⋅ 4⁵ / 8

г) 9⁴ ⋅ (1/27)^-3 ⋅ 81^-4

Решение:

а) 125^-1 ⋅ 25² = (5³ )^-1 ⋅ (5² )² = 5^{3 ⋅(-1)} ⋅ 5 ^2⋅2 = 5^-3 ⋅ 5⁴ = 5^-3+4 = 5

б) 0,0001 ⋅ (10³)² ⋅ (0,1)^-2 = 1/10⁴ ⋅ 10^3⋅2 ⋅ (1/10)^-2 = 10^-4 ⋅ 10⁶ ⋅ 10² = 10^-4+6+2 = 10⁴ = 10000

в) 16^-3 ⋅ 4⁵ / 8 = (2⁴)^-3 ⋅ (2²)⁵ = 2^4⋅(-3) ⋅ 2 ^2⋅5 / 2³ = 2^-12 ⋅ 2¹⁰ / 2³ = 2^-12+10 / 2³ = 2^-2/2³ = 2^-2-3 = 2^-5 = 1/2⁵ = 1/32

г) 9⁴ ⋅ (1/27)^-3 ⋅ 81^-4 = (3²)⁴ ⋅ (3^-3) ⋅ (3⁴)^-4 = 3^2⋅4 ⋅ 3^-3⋅(^-3) ⋅ 3^4⋅(-4) = 3⁸ ⋅ 3⁹ ⋅ 3^-16 = 3^8+9-16 = 3¹ = 3

Источник : Учебник по Алгебре 9 класса авторов Ю.Н. Макарычев, Н.Г. Миндюк, К.И. Нешков, С.Б. Суворова, 2014г.