If $3 x - 2 y = 11$ and $x y = 12$ , find the value of $27 x^{3} - 8 y^{3}$ .

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Solution

$3 x - 2 y = 11$ $,$ $x y = 12$
$\Rightarrow 27 x^{3} - 8 y^{3} = {(3 x)}^{3} - {(2 y)}^{3} = (3 x - 2 y) (9 x^{2} + 4 y^{2} + 6 x y)$
Using ${(a - b)}^{2} = a^{2} + b^{2} - 2 a b$
$\Rightarrow {(3 x - 2 y)}^{2} = 9 x^{2} + 4 y^{2} - 12 x y$
$\Rightarrow 9 x^{2} + 4 y^{2} = {(11)}^{2} + 12 (12)$
$= 11^{2} + 12^{2}$
$= 265$
$\Rightarrow 27 x^{3} - 8 y^{3} = 11 (265 + 6 (12))$
$= 11 (265 + 72)$
$= 11 (337)$
$= 3707.$
Hence, the answer is $3707.$

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