Задание 1. Сократите дробь: 1) 100" / 527-1.47-2 2) 50 / 52n-1.2n-3 3) 18 / 32n-1.2n-2 4) 48 / 42n-1.3n-3 5) 20 / 22n-2.5n-2 6) 12" / 22n-3.3n-1 7) 75 / 52n-1.3n-2 8) 36 / 32n-1.4n-2

Ответ: Решение смотри в пошаговом объяснении.

1) \(\frac{100^n}{5^{2n-1} \cdot 4^{n-2}}\) = \(\frac{(5^2 \cdot 4)^n}{5^{2n-1} \cdot 4^{n-2}}\) = \(\frac{5^{2n} \cdot 4^n}{5^{2n-1} \cdot 4^{n-2}}\) = \(5^{2n-(2n-1)} \cdot 4^{n-(n-2)}\) = \(5^{2n-2n+1} \cdot 4^{n-n+2}\) = \(5^1 \cdot 4^2\) = \(5 \cdot 16\) = 80

2) \(\frac{50^n}{5^{2n-1} \cdot 2^{n-3}}\) = \(\frac{(2 \cdot 5^2)^n}{5^{2n-1} \cdot 2^{n-3}}\) = \(\frac{2^n \cdot 5^{2n}}{5^{2n-1} \cdot 2^{n-3}}\) = \(2^{n-(n-3)} \cdot 5^{2n-(2n-1)}\) = \(2^{n-n+3} \cdot 5^{2n-2n+1}\) = \(2^3 \cdot 5^1\) = \(8 \cdot 5\) = 40

3) \(\frac{18^n}{3^{2n-1} \cdot 2^{n-2}}\) = \(\frac{(2 \cdot 3^2)^n}{3^{2n-1} \cdot 2^{n-2}}\) = \(\frac{2^n \cdot 3^{2n}}{3^{2n-1} \cdot 2^{n-2}}\) = \(2^{n-(n-2)} \cdot 3^{2n-(2n-1)}\) = \(2^{n-n+2} \cdot 3^{2n-2n+1}\) = \(2^2 \cdot 3^1\) = \(4 \cdot 3\) = 12

4) \(\frac{48^n}{4^{2n-1} \cdot 3^{n-3}}\) = \(\frac{(4 \cdot 3 \cdot 4)^n}{4^{2n-1} \cdot 3^{n-3}}\) = \(\frac{4^{2n} \cdot 3^n}{4^{2n-1} \cdot 3^{n-3}}\) = \(4^{2n-(2n-1)} \cdot 3^{n-(n-3)}\) = \(4^{2n-2n+1} \cdot 3^{n-n+3}\) = \(4^1 \cdot 3^3\) = \(4 \cdot 27\) = 108

5) \(\frac{20^n}{2^{2n-2} \cdot 5^{n-2}}\) = \(\frac{(2^2 \cdot 5)^n}{2^{2n-2} \cdot 5^{n-2}}\) = \(\frac{2^{2n} \cdot 5^n}{2^{2n-2} \cdot 5^{n-2}}\) = \(2^{2n-(2n-2)} \cdot 5^{n-(n-2)}\) = \(2^{2n-2n+2} \cdot 5^{n-n+2}\) = \(2^2 \cdot 5^2\) = \(4 \cdot 25\) = 100

6) \(\frac{12^n}{2^{2n-3} \cdot 3^{n-1}}\) = \(\frac{(2^2 \cdot 3)^n}{2^{2n-3} \cdot 3^{n-1}}\) = \(\frac{2^{2n} \cdot 3^n}{2^{2n-3} \cdot 3^{n-1}}\) = \(2^{2n-(2n-3)} \cdot 3^{n-(n-1)}\) = \(2^{2n-2n+3} \cdot 3^{n-n+1}\) = \(2^3 \cdot 3^1\) = \(8 \cdot 3\) = 24

7) \(\frac{75^n}{5^{2n-1} \cdot 3^{n-2}}\) = \(\frac{(5^2 \cdot 3)^n}{5^{2n-1} \cdot 3^{n-2}}\) = \(\frac{5^{2n} \cdot 3^n}{5^{2n-1} \cdot 3^{n-2}}\) = \(5^{2n-(2n-1)} \cdot 3^{n-(n-2)}\) = \(5^{2n-2n+1} \cdot 3^{n-n+2}\) = \(5^1 \cdot 3^2\) = \(5 \cdot 9\) = 45

8) \(\frac{36^n}{3^{2n-1} \cdot 4^{n-2}}\) = \(\frac{(3^2 \cdot 4)^n}{3^{2n-1} \cdot 4^{n-2}}\) = \(\frac{3^{2n} \cdot 4^n}{3^{2n-1} \cdot 4^{n-2}}\) = \(3^{2n-(2n-1)} \cdot 4^{n-(n-2)}\) = \(3^{2n-2n+1} \cdot 4^{n-n+2}\) = \(3^1 \cdot 4^2\) = \(3 \cdot 16\) = 48

Ответ: Решение смотри в пошаговом объяснении.