$\frac{a - b}{3b} * \frac{3}{a - b} = \frac{1}{b} * \frac{1}{1} = \frac{1}{b}$

$\frac{2mn + n^2}{6m} * \frac{2m}{n} = \frac{n(2m + n)}{6m} * \frac{2m}{n} = \frac{2m + n}{3} * \frac{1}{1} = \frac{2m + n}{3}$

$\frac{7a + 7b}{b^6} * \frac{b^3}{a + b} = \frac{7(a + b)}{b^6} * \frac{b^3}{a + b} = \frac{7}{b^3} * \frac{1}{1} = \frac{7}{b^3}$

$\frac{32a}{a^2 - 9} * \frac{a - 3}{8a} = \frac{32a}{(a - 3)(a + 3)} * \frac{a - 3}{8a} = \frac{4}{a + 3} * \frac{1}{1} = \frac{4}{a + 3}$

$\frac{c - 1}{c + 6} * \frac{c + 6}{c^2 - 2c + 1} = \frac{c - 1}{c + 6} * \frac{c + 6}{(c - 1)^2} = \frac{1}{1} * \frac{1}{c - 1} = \frac{1}{c - 1}$

$\frac{m - 2}{m^2 - 49} * \frac{m + 7}{m - 2} = \frac{m - 2}{(m - 7)(m + 7)} * \frac{m + 7}{m - 2} = \frac{1}{m - 7} * \frac{1}{1} = \frac{1}{m - 7}$

$(a + 4) * \frac{a}{2a + 8} = (a + 4) * \frac{a}{2(a + 4)} = 1 * \frac{a}{2} = \frac{a}{2}$

$\frac{x - 9}{4x + 8} * \frac{x^2 + 2x}{x - 9} = \frac{x - 9}{4(x + 2)} * \frac{x(x + 2)}{x - 9} = \frac{1}{4} * \frac{x}{1} = \frac{x}{4}$

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

$\frac{a^2 - 25}{4a} * \frac{4a^2}{a^2 - 5a} = \frac{(a - 5)(a + 5)}{4a} * \frac{4a^2}{a(a - 5)} = \frac{(a - 5)(a + 5)}{4a} * \frac{4a}{a - 5} = \frac{a + 5}{1} * \frac{1}{1} = a + 5$