1201. Вычислите: a) 8⁻² · 4³; б) 10⁰ : 10⁻³; в) 100 : 10⁻³; г) \(\frac{2^{-21}}{4^{-5} \cdot 8^{-6}}\); д) \(\frac{4^{-2} \cdot 8^{-6}}{2^{-22}}\); е) \(\frac{3^{-10} \cdot 9^{8}}{(-3)^{2}}\); ж) \(\frac{5^{-5} \cdot 25^{10}}{125^{3}}\).

Решение:

• а) 8⁻² · 4³ = (2³)⁻² · (2²)³ = 2⁻⁶ · 2⁶ = 2⁰ = 1
• б) 10⁰ : 10⁻³ = 1 : 10⁻³ = 10³ = 1000
• в) 100 : 10⁻³ = 10² : 10⁻³ = 10²⁻⁽⁻³⁾ = 10²⁺³ = 10⁵ = 100000
• г) \(\frac{2^{-21}}{4^{-5} \cdot 8^{-6}}\) = \(\frac{2^{-21}}{(2^2)^{-5} \cdot (2^3)^{-6}}\) = \(\frac{2^{-21}}{2^{-10} \cdot 2^{-18}}\) = \(\frac{2^{-21}}{2^{-10-18}}\) = \(\frac{2^{-21}}{2^{-28}}\) = 2⁻²¹⁻⁽⁻²⁸⁾ = 2⁻²¹⁺²⁸ = 2⁷ = 128
• д) \(\frac{4^{-2} \cdot 8^{-6}}{2^{-22}}\) = \(\frac{(2^2)^{-2} \cdot (2^3)^{-6}}{2^{-22}}\) = \(\frac{2^{-4} \cdot 2^{-18}}{2^{-22}}\) = \(\frac{2^{-4-18}}{2^{-22}}\) = \(\frac{2^{-22}}{2^{-22}}\) = 1
• е) \(\frac{3^{-10} \cdot 9^{8}}{(-3)^{2}}\) = \(\frac{3^{-10} \cdot (3^2)^{8}}{3^{2}}\) = \(\frac{3^{-10} \cdot 3^{16}}{3^{2}}\) = \(\frac{3^{-10+16}}{3^{2}}\) = \(\frac{3^{6}}{3^{2}}\) = 3⁶⁻² = 3⁴ = 81
• ж) \(\frac{5^{-5} \cdot 25^{10}}{125^{3}}\) = \(\frac{5^{-5} \cdot (5^2)^{10}}{(5^3)^{3}}\) = \(\frac{5^{-5} \cdot 5^{20}}{5^{9}}\) = \(\frac{5^{-5+20}}{5^{9}}\) = \(\frac{5^{15}}{5^{9}}\) = 5¹⁵⁻⁹ = 5⁶ = 15625

Ответ: а) 1; б) 1000; в) 100000; г) 128; д) 1; е) 81; ж) 15625.