Найдите значение выражения: 31) 2⁵ · 5⁵ = 32) 14⁶ · (1/7)⁶ = 33) 0,25⁹ · 4⁹ = 34) (2²)³ : 2³ = 35) (2¹⁰)² : 2⁷ / 2⁷ = 36) 5¹³ · 5⁷ / (5⁴)⁵ = 37) 3¹⁰ · (3³)³ / 3¹⁶ = 38) (2⁴)³ · (2⁵)² / (2³)⁶ = 39) 2³ · 4³ = 40) 16² · 4³ · 8⁴ = 41) 625 · 5⁷ : 125³ = 42) 100³ · 10¹² : 1000⁴ = 43) (2²)⁹ · 8 / 2²⁰ = 44) 27 · (3⁴)⁶ / (3²⁰ · 3⁵) = 45) 5¹⁵ · 5¹² / (125 · (5⁸)³) =

Решение:

1. \( 2^5 \cdot 5^5 = (2 \cdot 5)^5 = 10^5 = 100000 \)
2. \( 14^6 \cdot (\frac{1}{7})^6 = (14 \cdot \frac{1}{7})^6 = 2^6 = 64 \)
3. \( 0.25^9 \cdot 4^9 = (0.25 \cdot 4)^9 = 1^9 = 1 \)
4. \( (2^2)^3 : 2^3 = 2^{2 \cdot 3} : 2^3 = 2^6 : 2^3 = 2^{6-3} = 2^3 = 8 \)
5. \( \frac{(2^{10})^2}{2^7} : 2^7 = \frac{2^{10 \cdot 2}}{2^7} : 2^7 = \frac{2^{20}}{2^7} : 2^7 = 2^{20-7} : 2^7 = 2^{13} : 2^7 = 2^{13-7} = 2^6 = 64 \)
6. \( \frac{5^{13} \cdot 5^7}{(5^4)^5} = \frac{5^{13+7}}{5^{4 \cdot 5}} = \frac{5^{20}}{5^{20}} = 5^{20-20} = 5^0 = 1 \)
7. \( \frac{3^{10} \cdot (3^3)^3}{3^{16}} = \frac{3^{10} \cdot 3^{3 \cdot 3}}{3^{16}} = \frac{3^{10} \cdot 3^9}{3^{16}} = \frac{3^{10+9}}{3^{16}} = \frac{3^{19}}{3^{16}} = 3^{19-16} = 3^3 = 27 \)
8. \( \frac{(2^4)^3 \cdot (2^5)^2}{(2^3)^6} = \frac{2^{4 \cdot 3} \cdot 2^{5 \cdot 2}}{2^{3 \cdot 6}} = \frac{2^{12} \cdot 2^{10}}{2^{18}} = \frac{2^{12+10}}{2^{18}} = \frac{2^{22}}{2^{18}} = 2^{22-18} = 2^4 = 16 \)
9. \( 2^3 \cdot 4^3 = 2^3 \cdot (2^2)^3 = 2^3 \cdot 2^{2 \cdot 3} = 2^3 \cdot 2^6 = 2^{3+6} = 2^9 = 512 \)
10. \( 16^2 \cdot 4^3 \cdot 8^4 = (2^4)^2 \cdot (2^2)^3 \cdot (2^3)^4 = 2^{4 \cdot 2} \cdot 2^{2 \cdot 3} \cdot 2^{3 \cdot 4} = 2^8 \cdot 2^6 \cdot 2^{12} = 2^{8+6+12} = 2^{26} \)
11. \( 625 \cdot 5^7 : 125^3 = 5^4 \cdot 5^7 : (5^3)^3 = 5^{4+7} : 5^{3 \cdot 3} = 5^{11} : 5^9 = 5^{11-9} = 5^2 = 25 \)
12. \( 100^3 \cdot 10^{12} : 1000^4 = (10^2)^3 \cdot 10^{12} : (10^3)^4 = 10^{2 \cdot 3} \cdot 10^{12} : 10^{3 \cdot 4} = 10^6 \cdot 10^{12} : 10^{12} = 10^{6+12-12} = 10^6 = 1000000 \)
13. \( \frac{(2^2)^9 \cdot 8}{2^{20}} = \frac{2^{2 \cdot 9} \cdot 2^3}{2^{20}} = \frac{2^{18} \cdot 2^3}{2^{20}} = \frac{2^{18+3}}{2^{20}} = \frac{2^{21}}{2^{20}} = 2^{21-20} = 2^1 = 2 \)
14. \( \frac{27 \cdot (3^4)^6}{3^{20} \cdot 3^5} = \frac{3^3 \cdot 3^{4 \cdot 6}}{3^{20+5}} = \frac{3^3 \cdot 3^{24}}{3^{25}} = \frac{3^{3+24}}{3^{25}} = \frac{3^{27}}{3^{25}} = 3^{27-25} = 3^2 = 9 \)
15. \( \frac{5^{15} \cdot 5^{12}}{125 \cdot (5^8)^3} = \frac{5^{15+12}}{5^3 \cdot 5^{8 \cdot 3}} = \frac{5^{27}}{5^3 \cdot 5^{24}} = \frac{5^{27}}{5^{3+24}} = \frac{5^{27}}{5^{27}} = 5^{27-27} = 5^0 = 1 \)

Ответ: 100000; 64; 1; 8; 64; 1; 27; 16; 512; 2²⁶; 25; 1000000; 2; 9; 1.