If $3 x - 4 y = 5 a n d x y = 3, f i n d : 27 x^{3} - 64 y^{3} .$

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Solution

$27 x^{3} - 64 x^{3} = (3 x)^{3} - (4 y)^{3} = (3 x - 4 y)^{3} (3 x - 4 y)^{3} + 3 (3 x) (4 y) (3 x - 4 y) [∵ a^{3} - b^{3} = (a - b)^{3} + 3 a b (a - b)] = (5)^{3} + 36 (x y) (3 x - 4 y) = 125 + 36 (3) (5) = 125 + 540 = 665$

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