Find the value of $a^{3} - 8 b^{3} - 36 a b - 216$ when $a = 2 b + 6$

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Solution

The correct option is A $0$
Let us directly substitute $a = 2 b + 6$ in the expression $a^{3} - 8 b^{3} - 36 a b - 216$ as shown below:

$(2 b + 6)^{3} - 8 b^{3} - 36 b (2 b + 6) - 216$
$= [(2 b)^{3} + 6^{3} + (3 \times (2 b)^{2} \times 6) + (3 \times 2 b \times 6^{2})] - 8 b^{3} - 72 b^{2} - 216 b - 216$
$(∵ (x + y)^{3} = x^{3} + y^{3} + 3 x^{2} y + 3 x y^{2})$
$= 8 b^{3} + 216 + 72 b^{2} + 216 b - 8 b^{3} - 72 b^{2} - 216 b - 216$
$= (8 b^{3} - 8 b^{3}) + (72 b^{2} - 72 b^{2}) + (216 b - 216 b) + (216 - 216)$
$= 0$

Hence, $a^{3} - 8 b^{3} - 36 a b - 216 = 0$ when $a = 2 b + 6$ .

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