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Expansion of (a ± b)³

If 3x+2y=9 ...

Question

If $3 x + 2 y = 9$ and $x y = 3$ , then find $27 x^{3} + 8 y^{3}$ .

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Solution

$3 x + 2 y = 9$
Cubing both sides,
$(3 x + 2 y)^{3} = 9^{3}$
Applying the formula,
${(a + b)}^{3} = a^{3} + b^{3} + 3 a b (a + b)$
$∴ (3 x)^{3} + (2 y)^{3} + 3 \times 3 x \times 2 y \times (3 x + 2 y)) = 729$
$(3 x)^{3} + (2 y)^{3} + 18 \times x y \times (3 x + 2 y) = 729$
Substituting, $3 x + 2 y = 9; x y = 3$
$∴ 27 x^{3} + 8 y^{3} + 18 \times 3 \times 9 = 729$
$27 x^{3} + 8 y^{3} = 729 - 486 = 243$

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