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44. $\frac{b}{a-b} \cdot (\frac{1}{a} - \frac{1}{b}) = \frac{b}{a-b} \cdot (\frac{b-a}{ab}) = \frac{b}{a-b} \cdot \frac{-(a-b)}{ab} = -\frac{1}{a}$ 45. $\frac{b}{2a-b} \cdot (\frac{1}{2} - \frac{a}{b}) = \frac{b}{2a-b} \cdot (\frac{b-2a}{2b}) = \frac{b}{2a-b} \cdot \frac{-(2a-b)}{2b} = -\frac{1}{2}$ 46. $\frac{a}{a-3b} \cdot (\frac{3}{a} - \frac{1}{b}) = \frac{a}{a-3b} \cdot (\frac{3b-a}{ab}) = \frac{a}{a-3b} \cdot \frac{-(a-3b)}{ab} = -\frac{1}{b}$ 47. $\frac{u}{v} + \frac{4v}{u} - 4 \cdot \frac{uv}{2v-u} = \frac{u^2 + 4v^2 - 4uv}{uv} \cdot \frac{uv}{2v-u} = \frac{(u-2v)^2}{uv} \cdot \frac{uv}{2v-u} = \frac{(u-2v)^2}{2v-u} = \frac{(2v-u)^2}{2v-u} = 2v-u$ 48. $\frac{4u}{v} + \frac{9v}{u} + 12 \cdot \frac{uv}{3u+2v} = \frac{4u^2 + 9v^2 + 12uv}{uv} \cdot \frac{uv}{3u+2v} = \frac{(2u+3v)^2}{uv} \cdot \frac{uv}{3u+2v} = 2u+3v$ 49. $\frac{36u}{v} + \frac{v}{u} - 12 \cdot \frac{uv}{v-6u} = \frac{36u^2 + v^2 - 12uv}{uv} \cdot \frac{uv}{v-6u} = \frac{(6u-v)^2}{uv} \cdot \frac{uv}{v-6u} = \frac{(v-6u)^2}{v-6u} = v-6u$ 50. $\frac{x^3 - 8}{x+2} \cdot \frac{x^2 + 4x + 4}{x^2 + 2x + 4} = \frac{(x-2)(x^2 + 2x + 4)}{x+2} \cdot \frac{(x+2)^2}{x^2 + 2x + 4} = (x-2)(x+2) = x^2 - 4$ 51. $\frac{x^3 + 216}{x-6} \cdot \frac{x^2 - 12x + 36}{x^2 - 6x + 36} = \frac{(x+6)(x^2 - 6x + 36)}{x-6} \cdot \frac{(x-6)^2}{x^2 - 6x + 36} = (x+6)(x-6) = x^2 - 36$