(2a+3b)^3 (2a 3b)^3 ∴ (a+b)^3 = a^3+6^3+3a^2b+3ab^2 ∴ (a b)^3 = a^3 b^3 3a^2b+3ab^3 ⇒ (2a)^3+(3b)^3+3x(2a)^2× 3b+3× 2a ×(3b)^2 [(2a)^3 (3b)^ ...

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Algebraic Identities

Solve: 2a+3b...

Question

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

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Solution

$(2 a + 3 b)^{3} - (2 a - 3 b)^{3}$
$∴ (a + b)^{3} = a^{3} + b^{3} + 3 a^{2} b + 3 a b^{2}$
$∴ (a - b)^{3} = a^{3} - b^{3} - 3 a^{2} b + 3 a b^{3}$
$\Rightarrow (2 a)^{3} + (3 b)^{3} + 3 \times (2 a)^{2} \times 3 b + 3 \times 2 a \times (3 b)^{2} - [(2 a)^{3} - (3 b)^{3} - 3 \times (2 a)^{2} \times 3 b + 3 \times 2 a \times (3 b)^{2}]$
$\Rightarrow 8 a^{3} + 276^{3} + 36 a^{2} b + 54 a b^{2} - [8 a^{3} - 27 b^{3} - 36 a^{2} b + 54 a b^{2}]$
$\Rightarrow 8 a^{3} + 27 b^{3} + 36 a^{2} b + 54 a b^{2} - 8 a^{3} + 27 b^{3} + 36 a^{2} b - 54 a b^{2}$
$\Rightarrow 72 a^{2} b + 54 b^{3}$
$\Rightarrow b (72 a^{2} + 54 b^{2})$ Ans

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