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Rational Algebraic Expression

The value of ...

Question

The value of $(2 x + 3 y)^{3} - (2 x - 3 y)^{3}$ is _____.

A

72 x^{2} y + 54 y^{3}

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B

54 x^{2} y + 72 y^{3}

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C

72 x^{2} y

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D

54 x^{2} y

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Solution

The correct option is A $72 x^{2} y + 54 y^{3}$
Given,
$(2 x + 3 y)^{3} - (2 x - 3 y)^{3}$ .
This is of the form $(a^{3} - b^{3}) = (a - b) (a^{2} + a b + b^{2})$ , [Here a = $(2 x + 3 y)^{3}$ , b = $(2 x - 3 y)^{3}$ ]

Applying the identity,
$(2 x + 3 y)^{3} - (2 x - 3 y)^{3}$
$= [(2 x + 3 y) - (2 x - 3 y)]$ $[(2 x + 3 y)^{2} + (2 x + 3 y) (2 x - 3 y) + (2 x - 3 y)^{2}]$
$= [2 x + 3 y - 2 x + 3 y] [4 x^{2} + 12 x y + 9 y^{2} + 4 x^{2} - 9 y^{2} + 4 x^{2} - 12 x y + 9 y^{2}]$
$= 6 y (12 x^{2} + 9 y^{2})$
$= 72 x^{2} y + 54 y^{3}$

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