Сократите дробь (1-10):

Решение:

1. \(\(9x)\)^2 / \(x^{15} \cdot 5x\)\) = \(81x^2 / 5x^{16}\) = \(27 / (5x^{14})\)
2. \((3x)^3 \cdot x^{-3}) / (x^{-10} \cdot 2x^6)\) = \(27x^3 \cdot x^{-3} / (2x^{-4})\) = \(27 / (2x^{-4})\) = \(27x^4 / 2\)
3. \(18^2 / 3^{2 \cdot 2})\) = \(18^2 / 3^4\) = \((2 \cdot 3^2)^2 / 3^4\) = \(2^2 \cdot 3^4 / 3^4\) = 4
4. \(36^5 / (3^5 \cdot 4^5)\) = \((3 \cdot 12)^5 / (3^5 \cdot 4^5)\) = \(3^5 \cdot 12^5 / (3^5 \cdot 4^5)\) = \(12^5 / 4^5\) = \((12/4)^5\) = \(3^5\) = 243
5. \((5^x - 5^{x-1})) / (5^x - 5^{x-2})\) = \(5^{x-1}(5-1) / 5^{x-2}(5^2-1)\) = \(5^{x-1} \cdot 4 / (5^{x-2} \cdot 24)\) = \(5 \cdot 4 / 24\) = \(20 / 24\) = \(5/6\)
6. \((2^{x+2} + 2^{x+1})) / (2^x + 2^{x-1})\) = \(2^{x+1}(2+1) / 2^{x-1}(2^2+1)\) = \(2^{x+1} \cdot 3 / (2^{x-1} \cdot 5)\) = \(2^2 \cdot 3 / 5\) = \(4 \cdot 3 / 5\) = \(12/5\)
7. \((2^{x+1} + 2^{x+2})) / (2^{x-1} + 2^{x-2})\) = \(2^{x+1}(1+2) / 2^{x-2}(2+1)\) = \(2^{x+1} \cdot 3 / (2^{x-2} \cdot 3)\) = \(2^{x+1} / 2^{x-2}\) = \(2^{(x+1) - (x-2)}\) = \(2^3\) = 8
8. \((3^{x+2} + 3^{x+1})) / (3^{x-1} + 3^{x-2})\) = \(3^{x+1}(3+1) / 3^{x-2}(3+1)\) = \(3^{x+1}(4) / 3^{x-2}(4)\) = \(3^{x+1} / 3^{x-2}\) = \(3^{(x+1) - (x-2)}\) = \(3^3\) = 27
9. \((8 \cdot 100^n) / (4 \cdot 36^n))\) = \(2 \cdot (100/36)^n\) = \(2 \cdot (25/9)^n\)
10. \((4 \cdot 36^n) / (3^{2n} \cdot 2^{2n}))\) = \((4 \cdot 36^n) / (3^2)^n \cdot 2^{2n})\) = \((4 \cdot 36^n) / (9^n \cdot 4^n))\) = \((4 \cdot (9 \cdot 4)^n) / (9^n \cdot 4^n))\) = \((4 \cdot 9^n \cdot 4^n) / (9^n \cdot 4^n))\) = 4