1.49. Найдите значение выражения: а) \(\frac{8 \cdot (2^3)^2}{16^2}\); б) \(\frac{9^6}{(3^3)^3 \cdot 27}\); в) \(\frac{100^3 \cdot 1000^2}{(10^5)^2}\); г) \(\frac{(5^2)^4 \cdot 625^2}{125^5}\); д) \(\frac{85}{4^5 \cdot 24}\); е) \(\frac{93 \cdot 27^4}{81^5}\); ж) \(\frac{625^4}{25^3 \cdot 5^9}\); з) \(\frac{1000^4}{100^2 \cdot 10^5}\); и) \(\frac{125^4 \cdot 25^5}{625^2 \cdot 5^{12}}\).

1.49. Нахождение значения выражений:

1. а) \[ \frac{8 \cdot (2^3)^2}{16^2} = \frac{2^3 \cdot 2^{3 \times 2}}{(2^4)^2} = \frac{2^3 \cdot 2^6}{2^{4 \times 2}} = \frac{2^{3+6}}{2^8} = \frac{2^9}{2^8} = 2^{9-8} = 2^1 = 2 \]
2. б) \[ \frac{9^6}{(3^3)^3 \cdot 27} = \frac{(3^2)^6}{3^{3 \times 3} \cdot 3^3} = \frac{3^{2 \times 6}}{3^9 \cdot 3^3} = \frac{3^{12}}{3^{9+3}} = \frac{3^{12}}{3^{12}} = 1 \]
3. в) \[ \frac{100^3 \cdot 1000^2}{(10^5)^2} = \frac{(10^2)^3 \cdot (10^3)^2}{10^{5 \times 2}} = \frac{10^{2 \times 3} \cdot 10^{3 \times 2}}{10^{10}} = \frac{10^6 \cdot 10^6}{10^{10}} = \frac{10^{6+6}}{10^{10}} = \frac{10^{12}}{10^{10}} = 10^{12-10} = 10^2 = 100 \]
4. г) \[ \frac{(5^2)^4 \cdot 625^2}{125^5} = \frac{5^{2 \times 4} \cdot (5^4)^2}{(5^3)^5} = \frac{5^8 \cdot 5^{4 \times 2}}{5^{3 \times 5}} = \frac{5^8 \cdot 5^8}{5^{15}} = \frac{5^{8+8}}{5^{15}} = \frac{5^{16}}{5^{15}} = 5^{16-15} = 5^1 = 5 \]
5. д) \[ \frac{8^5}{4^5 \cdot 24} = \frac{(2^3)^5}{(2^2)^5 \cdot (8 \cdot 3)} = \frac{2^{15}}{2^{10} \cdot 2^3 \cdot 3} = \frac{2^{15}}{2^{10+3} \cdot 3} = \frac{2^{15}}{2^{13} \cdot 3} = \frac{2^{15-13}}{3} = \frac{2^2}{3} = \frac{4}{3} \]
6. е) \[ \frac{9^3 \cdot 27^4}{81^5} = \frac{(3^2)^3 \cdot (3^3)^4}{(3^4)^5} = \frac{3^{2 \times 3} \cdot 3^{3 \times 4}}{3^{4 \times 5}} = \frac{3^6 \cdot 3^{12}}{3^{20}} = \frac{3^{6+12}}{3^{20}} = \frac{3^{18}}{3^{20}} = 3^{18-20} = 3^{-2} = \frac{1}{3^2} = \frac{1}{9} \]
7. ж) \[ \frac{625^4}{25^3 \cdot 5^9} = \frac{(5^4)^4}{(5^2)^3 \cdot 5^9} = \frac{5^{4 \times 4}}{5^{2 \times 3} \cdot 5^9} = \frac{5^{16}}{5^6 \cdot 5^9} = \frac{5^{16}}{5^{6+9}} = \frac{5^{16}}{5^{15}} = 5^{16-15} = 5^1 = 5 \]
8. з) \[ \frac{1000^4}{100^2 \cdot 10^5} = \frac{(10^3)^4}{(10^2)^2 \cdot 10^5} = \frac{10^{3 \times 4}}{10^{2 \times 2} \cdot 10^5} = \frac{10^{12}}{10^4 \cdot 10^5} = \frac{10^{12}}{10^{4+5}} = \frac{10^{12}}{10^9} = 10^{12-9} = 10^3 = 1000 \]
9. и) \[ \frac{125^4 \cdot 25^5}{625^2 \cdot 5^{12}} = \frac{(5^3)^4 \cdot (5^2)^5}{(5^4)^2 \cdot 5^{12}} = \frac{5^{3 \times 4} \cdot 5^{2 \times 5}}{5^{4 \times 2} \cdot 5^{12}} = \frac{5^{12} \cdot 5^{10}}{5^8 \cdot 5^{12}} = \frac{5^{12+10}}{5^{8+12}} = \frac{5^{22}}{5^{20}} = 5^{22-20} = 5^2 = 25 \]