465. Найдите значение выражения:

Решение:

Используем свойства степеней: \( (a^m)^n = a^{m \cdot n} \), \( a^m \cdot a^n = a^{m+n} \), \( a^m : a^n = a^{m-n} \) и \( a^0 = 1 \).

• а) \( \frac{2^5 \cdot (2^3)^4}{2^{13}} \): \( \frac{2^5 \cdot 2^{3 \cdot 4}}{2^{13}} = \frac{2^5 \cdot 2^{12}}{2^{13}} = \frac{2^{5+12}}{2^{13}} = \frac{2^{17}}{2^{13}} = 2^{17-13} = 2^4 = 16 \)
• б) \( \frac{(5^8)^2 \cdot 5^7}{5^{22}} \): \( \frac{5^{8 \cdot 2} \cdot 5^7}{5^{22}} = \frac{5^{16} \cdot 5^7}{5^{22}} = \frac{5^{16+7}}{5^{22}} = \frac{5^{23}}{5^{22}} = 5^{23-22} = 5^1 = 5 \)
• в) \( \frac{(2^5)^2}{2^6 \cdot 4^7} \): \( \frac{2^{5 \cdot 2}}{2^6 \cdot (2^2)^7} = \frac{2^{10}}{2^6 \cdot 2^{2 \cdot 7}} = \frac{2^{10}}{2^6 \cdot 2^{14}} = \frac{2^{10}}{2^{6+14}} = \frac{2^{10}}{2^{20}} = 2^{10-20} = 2^{-10} = \frac{1}{2^{10}} = \frac{1}{1024} \)
• г) \( \frac{(5^2)^4 \cdot 25}{5^9} \): \( \frac{5^{2 \cdot 4} \cdot 5^2}{5^9} = \frac{5^8 \cdot 5^2}{5^9} = \frac{5^{8+2}}{5^9} = \frac{5^{10}}{5^9} = 5^{10-9} = 5^1 = 5 \)
• д) \( \frac{(7^3)^3 \cdot 7^2}{(7^5)^2} \): \( \frac{7^{3 \cdot 3} \cdot 7^2}{7^{5 \cdot 2}} = \frac{7^9 \cdot 7^2}{7^{10}} = \frac{7^{9+2}}{7^{10}} = \frac{7^{11}}{7^{10}} = 7^{11-10} = 7^1 = 7 \)
• ж) \( \frac{3^{11} \cdot 27}{(3^4)^3 \cdot 9} \): \( \frac{3^{11} \cdot 3^3}{(3^{4 \cdot 3}) \cdot 3^2} = \frac{3^{11} \cdot 3^3}{3^{12} \cdot 3^2} = \frac{3^{11+3}}{3^{12+2}} = \frac{3^{14}}{3^{14}} = 3^{14-14} = 3^0 = 1 \)
• з) \( \frac{(11^2)^3}{11^2 \cdot 11^3} \): \( \frac{11^{2 \cdot 3}}{11^2 \cdot 11^3} = \frac{11^6}{11^{2+3}} = \frac{11^6}{11^5} = 11^{6-5} = 11^1 = 11 \)

Ответ: а) 16; б) 5; в) \( \frac{1}{1024} \); г) 5; д) 7; ж) 1; з) 11.