237. Вычислите:

Решение:

1. \( 100^5 : 1000^2 = (10^2)^5 : (10^3)^2 = 10^{10} : 10^6 = 10^{10-6} = 10^4 = 10000 \)
2. \( \frac{3^{10} \cdot (3^3)^5}{(3^5)^4 \cdot 3} = \frac{3^{10} \cdot 3^{15}}{3^{20} \cdot 3^1} = \frac{3^{25}}{3^{21}} = 3^{25-21} = 3^4 = 81 \)
3. \( \frac{4^3 \cdot 16^2}{2^{12}} = \frac{(2^2)^3 \cdot (2^4)^2}{2^{12}} = \frac{2^6 \cdot 2^8}{2^{12}} = \frac{2^{14}}{2^{12}} = 2^{14-12} = 2^2 = 4 \)
4. \( \frac{3^{10} \cdot (35)^4}{45^{10}} = \frac{3^{10} \cdot (5 \cdot 7)^4}{(5 \cdot 3^2)^{10}} = \frac{3^{10} \cdot 5^4 \cdot 7^4}{5^{10} \cdot 3^{20}} = \frac{7^4}{5^6 \cdot 3^{10}} \)

Ответ: 1) 10000; 2) 81; 3) 4; 4) \(\frac{7^4}{5^6 \cdot 3^{10}}\).