284. Вычислите: А) 5^10 * (5^3)^4 / 5^18; Б) 3^34 / (3^17 * (3^5)^2); В) 7^11 / (7^2 * (7^4)^3); Г) 2^21 * (2^3)^5 / 2^40; Д) (0,1)^25 * 10^23; Е) 2,5^43 * (2/5)^43; Ё) (2 2/3)^11 * 0,375^8; Ж) 1,4^21 * (5/7)^19.

Для решения этих задач вспомним свойства степеней: $a^n \cdot a^m = a^{n+m}$, $\frac{a^n}{a^m} = a^{n-m}$, $(a^n)^m = a^{n \cdot m}$, $(a \cdot b)^n = a^n \cdot b^n$. А) $\frac{5^{10} \cdot (5^3)^4}{5^{18}} = \frac{5^{10} \cdot 5^{12}}{5^{18}} = \frac{5^{22}}{5^{18}} = 5^{22-18} = 5^4 = 625$ Б) $\frac{3^{34}}{3^{17} \cdot (3^5)^2} = \frac{3^{34}}{3^{17} \cdot 3^{10}} = \frac{3^{34}}{3^{27}} = 3^{34-27} = 3^7 = 2187$ В) $\frac{7^{11}}{7^2 \cdot (7^4)^3} = \frac{7^{11}}{7^2 \cdot 7^{12}} = \frac{7^{11}}{7^{14}} = 7^{11-14} = 7^{-3} = \frac{1}{7^3} = \frac{1}{343}$ Г) $\frac{2^{21} \cdot (2^3)^5}{2^{40}} = \frac{2^{21} \cdot 2^{15}}{2^{40}} = \frac{2^{36}}{2^{40}} = 2^{36-40} = 2^{-4} = \frac{1}{2^4} = \frac{1}{16} = 0,0625$ Д) $(0,1)^{25} \cdot 10^{23} = (10^{-1})^{25} \cdot 10^{23} = 10^{-25} \cdot 10^{23} = 10^{-2} = 0,01$ Е) $2,5^{43} \cdot (\frac{2}{5})^{43} = (2,5 \cdot \frac{2}{5})^{43} = (2,5 \cdot 0,4)^{43} = 1^{43} = 1$ Ё) $(2\frac{2}{3})^{11} \cdot 0,375^8 = (\frac{8}{3})^{11} \cdot (\frac{3}{8})^8 = \frac{8^{11}}{3^{11}} \cdot \frac{3^8}{8^8} = \frac{8^3}{3^3} = \frac{512}{27} = 18\frac{26}{27}$ Ж) $1,4^{21} \cdot (\frac{5}{7})^{19} = (\frac{14}{10})^{21} \cdot (\frac{5}{7})^{19} = (\frac{7}{5})^{21} \cdot (\frac{5}{7})^{19} = (\frac{7}{5})^{21-19} = (\frac{7}{5})^2 = \frac{49}{25} = 1,96$