Задание 35. Представьте выражение в виде степени и найдите его значение:

Решение:

1. \( 2^3 \cdot 2^2 \cdot 2 = 2^{3+2+1} = 2^6 = 64 \)
2. \( 10^2 \cdot 10^2 \cdot 10^0 = 10^{2+2+0} = 10^4 = 10000 \)
3. \( 3^4 \cdot 3^5 : 3^7 = 3^{4+5-7} = 3^2 = 9 \)
4. \( 4^7 : 4^5 \cdot 4^0 = 4^{7-5+0} = 4^2 = 16 \)
5. \( 0.6^5 \cdot 0.6^{10} : 0.6^{13} = 0.6^{5+10-13} = 0.6^2 = 0.36 \)
6. \( \frac{9^{18}}{9^5 \cdot 9^{11}} = \frac{9^{18}}{9^{5+11}} = \frac{9^{18}}{9^{16}} = 9^{18-16} = 9^2 = 81 \)
7. \( \frac{14^8 \cdot 14^{13}}{14^{11} \cdot 14^9} = \frac{14^{8+13}}{14^{11+9}} = \frac{14^{21}}{14^{20}} = 14^{21-20} = 14^1 = 14 \)
8. \( \frac{6^0 \cdot 6^5}{6^3} = \frac{1 \cdot 6^5}{6^3} = 6^{5-3} = 6^2 = 36 \)
9. \( \left(\frac{1}{3}\right)^9 : \left(\left(\frac{1}{3}\right)^2 \cdot \left(\frac{1}{3}\right)^3\right) = \left(\frac{1}{3}\right)^9 : \left(\frac{1}{3}\right)^{2+3} = \left(\frac{1}{3}\right)^9 : \left(\frac{1}{3}\right)^5 = \left(\frac{1}{3}\right)^{9-5} = \left(\frac{1}{3}\right)^4 = \frac{1}{81} \)
10. \( \left(\frac{1}{2}\right)^8 \cdot \left(\frac{1}{2}\right)^7 : \left(\frac{1}{2}\right)^4 \cdot \left(\frac{1}{2}\right)^6 = \left(\frac{1}{2}\right)^{8+7-4+6} = \left(\frac{1}{2}\right)^{17} = \frac{1}{2^{17}} \)
11. \( \left(\frac{1}{3}\right)^7 : \left(\frac{1}{3}\right)^5 \cdot \frac{1}{3} = \left(\frac{1}{3}\right)^{7-5} \cdot \frac{1}{3} = \left(\frac{1}{3}\right)^2 \cdot \frac{1}{3} = \frac{1}{9} \cdot \frac{1}{3} = \frac{1}{27} \)
12. \( 2^5 \cdot 2^9 \cdot 2^n = 2^{5+9+n} = 2^{14+n} \)
13. \( 7^2 \cdot 7^4 \cdot 7^m = 7^{2+4+m} = 7^{6+m} \)
14. \( 15^6 \cdot 15^3 : 15^{2k} = 15^{6+3-2k} = 15^{9-2k} \)
15. \( 0.4^{2k} \cdot 0.4^{3n} : 0.4^m = 0.4^{2k+3n-m} \)
16. \( 4 \cdot 2^5 = 2^2 \cdot 2^5 = 2^{2+5} = 2^7 = 128 \)
17. \( 5^5 : 25 = 5^5 : 5^2 = 5^{5-2} = 5^3 = 125 \)
18. \( 81 \cdot 3^2 = 3^4 \cdot 3^2 = 3^{4+2} = 3^6 = 729 \)
19. \( 10^5 : 100 = 10^5 : 10^2 = 10^{5-2} = 10^3 = 1000 \)
20. \( 125 \cdot 25 : 5^4 = 5^3 \cdot 5^2 : 5^4 = 5^{3+2-4} = 5^1 = 5 \)
21. \( 6^5 : 36 = 6^5 : 6^2 = 6^{5-2} = 6^3 = 216 \)
22. \( \frac{3^{10}}{27 \cdot 9} = \frac{3^{10}}{3^3 \cdot 3^2} = \frac{3^{10}}{3^{3+2}} = \frac{3^{10}}{3^5} = 3^{10-5} = 3^5 = 243 \)
23. \( \frac{128}{2^2 \cdot 8} = \frac{2^7}{2^2 \cdot 2^3} = \frac{2^7}{2^{2+3}} = \frac{2^7}{2^5} = 2^{7-5} = 2^2 = 4 \)
24. \( 10000 : 10^3 \cdot 100 = 10^4 : 10^3 \cdot 10^2 = 10^{4-3+2} = 10^3 = 1000 \)
25. \( 5^{10} : 625 : 25 = 5^{10} : 5^4 : 5^2 = 5^{10-4-2} = 5^4 = 625 \)
26. \( 0.25 \cdot \left(\frac{1}{4}\right)^2 = \frac{1}{4} \cdot \left(\frac{1}{4}\right)^2 = \left(\frac{1}{4}\right)^{1+2} = \left(\frac{1}{4}\right)^3 = \frac{1}{64} \)
27. \( \left(\frac{1}{5}\right)^4 : 0.2 = \left(\frac{1}{5}\right)^4 : \frac{1}{5} = \left(\frac{1}{5}\right)^{4-1} = \left(\frac{1}{5}\right)^3 = \frac{1}{125} \)
28. \( \frac{1}{8} \cdot \left(\frac{1}{2}\right)^4 = \frac{1}{2^3} \cdot \frac{1}{2^4} = \frac{1}{2^{3+4}} = \frac{1}{2^7} = \frac{1}{128} \)
29. \( (216 : 6^2) \cdot (343 : 7^2) = (6^3 : 6^2) \cdot (7^3 : 7^2) = 6^1 \cdot 7^1 = 6 \cdot 7 = 42 \)
30. \( 512 : 8 : 16 = 2^9 : 2^3 : 2^4 = 2^{9-3-4} = 2^2 = 4 \)