Simplify: $(3 x y - 2 a b)^{3} - (3 x y + 2 a b)^{3}$

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Solution

$(3 x y - 2 a b)^{3} - (3 x y + 2 a b)^{3}$
Here, A = 3xy - 2ab and B = 3xy + 2ab
$∴ (3 x y - 2 a b)^{3} - (3 x y + 2 a b)^{3}$
$= [(3 x y - 2 a b) - (3 x y + 2 a b)] [(3 x y - 2 a b)^{2} + (3 x y - 2 a b) (3 x y + 2 a b) + (3 x y + 2 a b)^{2}]$
... $[∵ A^{3} - B^{3} = (A - B) (A^{2} + A B + B^{2})]$
$= (3 x y - 2 a b - 3 x y - 2 a b) [(9 x^{2} y^{2} - 12 x y a b + 4 a^{2} b^{2}) + (9 x^{2} y^{2} - 4 a^{2} b^{2}) + (9 x^{2} y^{2} + 12 x y a b + 4 a^{2} b^{2})]$
$= (- 4 a b) (9 x^{2} y^{2} + 9 x^{2} y^{2} + 9 x^{2} y^{2} - 12 x y a b + 12 x y a b + 4 a^{2} b^{2} - 4 a^{2} b^{2} + 4 a^{2} b^{2})$
$= (- 4 a b) (27 x y^{2} + 4 a^{2} b^{2})$
$= - 108 x^{2} y^{2} a b - 16 a^{3} b^{3}$

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