Выполни действия: 1) -54a^6b^9/c^12 * (-c^20/12a^4b^15)

1. 1) $$-\frac{54a^6b^9}{c^{12}} \cdot \left(-\frac{c^{20}}{12a^4b^{15}}\right) = \frac{54}{12} \cdot \frac{a^6}{a^4} \cdot \frac{b^9}{b^{15}} \cdot \frac{c^{20}}{c^{12}} = \frac{9}{2}a^{6-4}b^{9-15}c^{20-12} = \frac{9}{2}a^2b^{-6}c^8 = \frac{9a^2c^8}{2b^6}$$ 2) $$\frac{98m^8}{p^{17}}:(49m^5p^2) = \frac{98m^8}{p^{17}} \cdot \frac{1}{49m^5p^2} = \frac{98}{49} \cdot \frac{m^8}{m^5} \cdot \frac{1}{p^{17}p^2} = 2m^{8-5}p^{-17-2} = 2m^3p^{-19} = \frac{2m^3}{p^{19}}$$ 3) $$\frac{5a+5b}{b} \cdot \frac{6b^2}{a^2-b^2} = \frac{5(a+b)}{b} \cdot \frac{6b^2}{(a-b)(a+b)} = \frac{5 \cdot 6 \cdot (a+b) \cdot b^2}{b \cdot (a-b) \cdot (a+b)} = \frac{30b}{a-b}$$ 4) $$\frac{x^2-49}{3x-24}:\frac{5x+35}{x-8} = \frac{(x-7)(x+7)}{3(x-8)} \cdot \frac{x-8}{5(x+7)} = \frac{(x-7)(x+7)(x-8)}{3(x-8)5(x+7)} = \frac{x-7}{15}$$ 2. 1) $$\frac{3a}{a-3} + \frac{a+5}{6-2a} \cdot \frac{54}{5a+a^2} = \frac{3a}{a-3} + \frac{a+5}{-2(a-3)} \cdot \frac{54}{a(5+a)} = \frac{3a}{a-3} - \frac{1}{2(a-3)} \cdot \frac{54}{a} = \frac{3a}{a-3} - \frac{27}{a(a-3)} = \frac{3a^2-27}{a(a-3)} = \frac{3(a^2-9)}{a(a-3)} = \frac{3(a-3)(a+3)}{a(a-3)} = \frac{3(a+3)}{a}$$ 2) $$\left(\frac{a+4}{a-4} - \frac{a-4}{a+4}\right):\frac{48a}{16-a^2} = \frac{(a+4)^2 - (a-4)^2}{(a-4)(a+4)} : \frac{48a}{16-a^2} = \frac{a^2+8a+16 - (a^2-8a+16)}{(a-4)(a+4)} : \frac{48a}{16-a^2} = \frac{16a}{(a-4)(a+4)} \cdot \frac{16-a^2}{48a} = \frac{16a \cdot (4-a)(4+a)}{(a-4)(a+4) \cdot 48a} = -\frac{16a(a-4)(a+4)}{(a-4)(a+4)48a} = -\frac{16}{48} = -\frac{1}{3}$$ 3. $$\left(\frac{a^2}{a+5} - \frac{a^3}{a^2+10a+25}\right):\left(\frac{a}{a+5} - \frac{a^2}{a^2-25}\right) = \frac{a^2(a+5) - a^3}{(a+5)^2} : \frac{a(a-5) - a^2}{(a+5)(a-5)} = \frac{a^3 + 5a^2 - a^3}{(a+5)^2} : \frac{a^2 - 5a - a^2}{(a+5)(a-5)} = \frac{5a^2}{(a+5)^2} : \frac{-5a}{(a+5)(a-5)} = \frac{5a^2}{(a+5)^2} \cdot \frac{(a+5)(a-5)}{-5a} = -\frac{5a^2(a+5)(a-5)}{5a(a+5)^2} = -\frac{a(a-5)}{a+5} = \frac{5a-a^2}{a+5}$$ 4. Дано: $x^2 + \frac{49}{x^2} = 50$. Нужно найти значение выражения $x - \frac{7}{x}$. Заметим, что $\left(x - \frac{7}{x}\right)^2 = x^2 - 2 \cdot x \cdot \frac{7}{x} + \frac{49}{x^2} = x^2 - 14 + \frac{49}{x^2} = x^2 + \frac{49}{x^2} - 14$. Тогда $\left(x - \frac{7}{x}\right)^2 = 50 - 14 = 36$. Значит, $x - \frac{7}{x} = \pm \sqrt{36} = \pm 6$. **Ответ:** $x - \frac{7}{x} = \pm 6$