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Expansion of (x-y)^3

Simplify: 2...

Question

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

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Solution

$(2 a + 3 b)^{3} = (2 a)^{3} + (3 b)^{3} + 3 (2 a) (3 b) (2 a + 3 b)$
Using, $(a + b)^{3} = a^{3} + b^{3} + 3 a b (a + b)$
$= 8 a^{3} + 27 b^{3} + 18 a b (2 a + 3 b)$
$= 8 a^{3} + 27 b^{3} + 36 a^{2} b + 54 a b^{2}$ .........(i)
$(2 a - 3 b)^{3} = (2 a)^{3} - (3 b)^{3} - 3 (2 a) (3 b) (2 a - 3 b)$
Using, $(a - b)^{3} = a^{3} - b^{3} - 3 a b (a - b)$
$= 8 a^{3} - 27 b^{3} - 18 a b (2 a - 3 b)$
$= 8 a^{3} - 27 b^{3} - 36 a^{2} b + 54 a b^{2}$ ......(ii)
Subtracting (i) from (ii) we get,
$(2 a + 3 b)^{3} - (2 a - 3 b)^{3} = 8 a^{3} + 27 b^{3} + 36 a^{2} b + 54 a b^{2} - (8 a^{3} - 27 b^{3} - 36 a^{2} b + 54 a b^{2})$
$= 8 a^{3} + 27 b^{3} + 36 a^{2} b + 54 a b^{2} - 8 a^{3} + 27 b^{3} + 36 a^{2} b - 54 a b^{2}$
$= 54 b^{3} + 72 a^{2} b$

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