A 26 24.25 26 128.24 Б 23.22 34.33 27 16.22 32 37 38 81.35 36 B 34.32 65 38 9.27 64.65 36 6.216 Γ 63.64 53.55 69 60.62 63 59 55.125 625 д 56 46 53-54 42.45 59 25-52 49 16.256 E 42.42 104.102 47 64.42 47 104 104. 1000 1000 105 103.102 107 103.100

Решение заданий с упрощением выражений

Давай разберем по порядку каждый пример и упростим его, используя свойства степеней.

A

• \(\frac{2^6}{2^3 \cdot 2^2} = \frac{2^6}{2^{3+2}} = \frac{2^6}{2^5} = 2^{6-5} = 2^1 = 2\)
• \(\frac{2^4 \cdot 2^5}{2^7} = \frac{2^{4+5}}{2^7} = \frac{2^9}{2^7} = 2^{9-7} = 2^2 = 4\)
• \(\frac{2^6}{16 \cdot 2^2} = \frac{2^6}{2^4 \cdot 2^2} = \frac{2^6}{2^{4+2}} = \frac{2^6}{2^6} = 1\)
• \(\frac{128 \cdot 2^4}{32} = \frac{2^7 \cdot 2^4}{2^5} = \frac{2^{7+4}}{2^5} = \frac{2^{11}}{2^5} = 2^{11-5} = 2^6 = 64\)

Б

• \(\frac{3^4 \cdot 3^3}{3^7} = \frac{3^{4+3}}{3^7} = \frac{3^7}{3^7} = 1\)
• \(\frac{3^8}{3^4 \cdot 3^2} = \frac{3^8}{3^{4+2}} = \frac{3^8}{3^6} = 3^{8-6} = 3^2 = 9\)
• \(\frac{81 \cdot 3^5}{3^8} = \frac{3^4 \cdot 3^5}{3^8} = \frac{3^{4+5}}{3^8} = \frac{3^9}{3^8} = 3^{9-8} = 3^1 = 3\)
• \(\frac{3^6}{9 \cdot 27} = \frac{3^6}{3^2 \cdot 3^3} = \frac{3^6}{3^{2+3}} = \frac{3^6}{3^5} = 3^{6-5} = 3^1 = 3\)

В

• \(\frac{6^5}{6^3 \cdot 6^4} = \frac{6^5}{6^{3+4}} = \frac{6^5}{6^7} = 6^{5-7} = 6^{-2} = \frac{1}{6^2} = \frac{1}{36}\)
• \(\frac{6^4 \cdot 6^5}{6^9} = \frac{6^{4+5}}{6^9} = \frac{6^9}{6^9} = 1\)
• \(\frac{36}{6^0 \cdot 6^2} = \frac{6^2}{1 \cdot 6^2} = \frac{6^2}{6^2} = 1\)
• \(\frac{6 \cdot 216}{6^3} = \frac{6 \cdot 6^3}{6^3} = \frac{6^4}{6^3} = 6^{4-3} = 6^1 = 6\)

Г

• \(\frac{5^3 \cdot 5^5}{5^6} = \frac{5^{3+5}}{5^6} = \frac{5^8}{5^6} = 5^{8-6} = 5^2 = 25\)
• \(\frac{5^9}{5^3 \cdot 5^4} = \frac{5^9}{5^{3+4}} = \frac{5^9}{5^7} = 5^{9-7} = 5^2 = 25\)
• \(\frac{5^5 \cdot 125}{5^9} = \frac{5^5 \cdot 5^3}{5^9} = \frac{5^{5+3}}{5^9} = \frac{5^8}{5^9} = 5^{8-9} = 5^{-1} = \frac{1}{5}\)
• \(\frac{625}{25 \cdot 5^2} = \frac{5^4}{5^2 \cdot 5^2} = \frac{5^4}{5^{2+2}} = \frac{5^4}{5^4} = 1\)

Д

• \(\frac{4^6}{4^2 \cdot 4^2} = \frac{4^6}{4^{2+2}} = \frac{4^6}{4^4} = 4^{6-4} = 4^2 = 16\)
• \(\frac{4^2 \cdot 4^5}{4^7} = \frac{4^{2+5}}{4^7} = \frac{4^7}{4^7} = 1\)
• \(\frac{4^9}{64 \cdot 4^2} = \frac{4^9}{4^3 \cdot 4^2} = \frac{4^9}{4^{3+2}} = \frac{4^9}{4^5} = 4^{9-5} = 4^4 = 256\)
• \(\frac{16 \cdot 256}{4^7} = \frac{4^2 \cdot 4^4}{4^7} = \frac{4^{2+4}}{4^7} = \frac{4^6}{4^7} = 4^{6-7} = 4^{-1} = \frac{1}{4}\)

Е

• \(\frac{10^4 \cdot 10^2}{10^5} = \frac{10^{4+2}}{10^5} = \frac{10^6}{10^5} = 10^{6-5} = 10^1 = 10\)
• \(\frac{10^4}{10^3 \cdot 10^2} = \frac{10^4}{10^{3+2}} = \frac{10^4}{10^5} = 10^{4-5} = 10^{-1} = \frac{1}{10} = 0.1\)
• \(\frac{10^4 \cdot 1000}{10^7} = \frac{10^4 \cdot 10^3}{10^7} = \frac{10^{4+3}}{10^7} = \frac{10^7}{10^7} = 1\)
• \(\frac{1000}{10^3 \cdot 100} = \frac{10^3}{10^3 \cdot 10^2} = \frac{10^3}{10^{3+2}} = \frac{10^3}{10^5} = 10^{3-5} = 10^{-2} = \frac{1}{10^2} = \frac{1}{100} = 0.01\)

Ответ: Все выражения упрощены.