$(\frac{a + 4}{a - 4} - \frac{a - 4}{a + 4}) * \frac{16 - a^2}{32a^3} = \frac{(a + 4)^2 - (a - 4)^2}{(a - 4)(a + 4)} * \frac{16 - a^2}{32a^3} = \frac{a^2 + 8a + 16 - (a^2 - 8a + 16)}{a^2 - 16} * \frac{16 - a^2}{32a^3} = \frac{a^2 + 8a + 16 - a^2 + 8a - 16}{a^2 - 16} * (-\frac{a^2 - 16}{32a^3}) = \frac{16a}{1} * (-\frac{1}{32a^3}) = -\frac{1}{2a^2}$

$(7x - \frac{4x}{x - 3}) : \frac{14x - 50}{3x - 9} = \frac{7x(x - 3) - 4x}{x - 3} : \frac{2(7x - 25)}{3(x - 3)} = \frac{7x(x - 3) - 4x}{x - 3} * \frac{3(x - 3)}{2(7x - 25)} = \frac{7x^2 - 21x - 4x}{1} * \frac{3}{2(7x - 25)} = \frac{7x^2 - 25x}{1} * \frac{3}{2(7x - 25)} = \frac{x(7x - 25)}{1} * \frac{3}{2(7x - 25)} = \frac{3x}{2}$

$\frac{2a}{a - 2} + \frac{a + 7}{8 - 4a} * \frac{32}{7a + a^2} = \frac{2a}{a - 2} + \frac{a + 7}{4(2 - a)} * \frac{32}{a(7 + a)} = \frac{2a}{a - 2} + \frac{1}{2 - a} * \frac{8}{a} = \frac{2a}{a - 2} - \frac{8}{a(a - 2)} = \frac{2a^2 - 8}{a(a - 2)} = \frac{2(a^2 - 4)}{a(a - 2)} = \frac{2(a - 2)(a + 2)}{a(a - 2)} = \frac{2(a + 2)}{a}$

$(\frac{9c}{c - 8} + \frac{7c}{c^2 - 16c + 64}) : \frac{9c - 65}{c^2 - 64} - \frac{8c + 64}{c - 8} = (\frac{9c}{c - 8} + \frac{7c}{(c - 8)^2}) : \frac{9c - 65}{(c - 8)(c + 8)} - \frac{8(c + 8)}{c - 8} = \frac{9c(c - 8) + 7c}{(c - 8)^2} * \frac{(c - 8)(c + 8)}{9c - 65} - \frac{8(c + 8)}{c - 8} = \frac{9c^2 - 72c + 7c}{c - 8} * \frac{c + 8}{9c - 65} - \frac{8(c + 8)}{c - 8} = \frac{9c^2 - 65c}{c - 8} * \frac{c + 8}{9c - 65} - \frac{8(c + 8)}{c - 8} = \frac{c(9c - 65)}{c - 8} * \frac{c + 8}{9c - 65} - \frac{8(c + 8)}{c - 8} = \frac{c}{c - 8} * \frac{c + 8}{1} - \frac{8(c + 8)}{c - 8} = \frac{c(c + 8)}{c - 8} - \frac{8(c + 8)}{c - 8} = \frac{c(c + 8) - 8(c + 8)}{c - 8} = \frac{(c + 8)(c - 8)}{c - 8} = c + 8$

$(\frac{a^2}{a + b} - \frac{a^3}{a^2 + ab + b^2}) : (\frac{a}{a - b} - \frac{a^2}{a^2 - b^2}) = \frac{a^2(a^2 + ab + b^2) - a^3(a + b)}{(a + b)(a^2 + ab + b^2)} : (\frac{a}{a - b} - \frac{a^2}{(a - b)(a + b)}) = \frac{a^4 + a^3b + a^2b^2 - a^4 - a^3b}{(a + b)(a^2 + ab + b^2)} : \frac{a(a + b) - a^2}{(a - b)(a + b)} = \frac{a^2b^2}{(a + b)(a^2 + ab + b^2)} : \frac{a^2 + ab - a^2}{(a - b)(a + b)} = \frac{a^2b^2}{(a + b)(a^2 + ab + b^2)} * \frac{(a - b)(a + b)}{ab} = \frac{ab}{a^2 + ab + b^2} * \frac{a - b}{1} = \frac{ab(a - b)}{a^2 + ab + b^2}$

$(\frac{b}{b + 6} + \frac{36 + b^2}{36 - b^2} - \frac{b}{b - 6}) : \frac{6b + b^2}{(6 - b)^2} = (\frac{b}{b + 6} - \frac{36 + b^2}{b^2 - 36} - \frac{b}{b - 6}) : \frac{b(6 + b)}{(6 - b)^2} = (\frac{b}{b + 6} - \frac{36 + b^2}{(b - 6)(b + 6)} - \frac{b}{b - 6}) : \frac{b(6 + b)}{(6 - b)^2} = \frac{b(b - 6) - (36 + b^2) - b(b + 6)}{(b - 6)(b + 6)} * \frac{(6 - b)^2}{b(6 + b)} = \frac{b^2 - 6b - 36 - b^2 - b^2 - 6b}{b + 6} * \frac{6 - b}{b(6 + b)} = \frac{-b^2 - 12b - 36}{b + 6} * \frac{6 - b}{b(6 + b)} = \frac{-(b^2 + 12b + 36)}{b + 6} * \frac{6 - b}{b(6 + b)} = \frac{-(b + 6)^2}{b + 6} * \frac{6 - b}{b(6 + b)} = -\frac{6 - b}{b}$

$(\frac{2x}{x^3 + 1} : \frac{1 - x}{x^2 - x + 1} + \frac{2}{x - 1}) * \frac{x^2 - 2x + 1}{4} : \frac{x - 1}{x + 1} = (\frac{2x}{(x + 1)(x^2 - x + 1)} * \frac{x^2 - x + 1}{1 - x} + \frac{2}{x - 1}) * \frac{x^2 - 2x + 1}{4} * \frac{x + 1}{x - 1} = (\frac{2x}{x + 1} * \frac{1}{1 - x} - \frac{2}{1 - x}) * \frac{x^2 - 2x + 1}{4} * \frac{x + 1}{x - 1} = \frac{2x - 2(1 + x)}{(1 + x)(1 - x)} * \frac{x^2 - 2x + 1}{4} * \frac{x + 1}{x - 1} = \frac{2x - 2 - 2x}{(1 + x)(1 - x)} * \frac{(x - 1)^2}{4} * \frac{x + 1}{x - 1} = \frac{-2}{(1 + x)(1 - x)} * \frac{x - 1}{4} * \frac{x + 1}{1} = \frac{2}{(x + 1)(x - 1)} * \frac{x - 1}{4} * \frac{x + 1}{1} = \frac{2}{4} = \frac{1}{2}$