If $5 x - 4 y = 7$ and $x y = 8$ , then find $125 x^{3} - 64 y^{3}$ .

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Solution

$5 x - 4 y = 7$
Cubing both sides,
$(5 x - 4 y)^{3} = 7^{3}$
Applying the formula, ${(a - b)}^{3} = a^{3} - b^{3} - 3 a b (a - b)$
$∴ (5 x)^{3} - (4 y)^{3} - 3 \times 5 x \times 4 y \times (5 x - 4 y)) = 343$
$(5 x)^{3} - (4 y)^{3} - 60 \times x y \times (5 x - 4 y) = 343$
Substituting, $5 x - 4 y = 7; x y = 8$
$∴ 125 x^{3} - 64 y^{3} - 60 \times 8 \times 7 = 343$
$125 x^{3} - 64 y^{3} = 343 + 3360 = 3703$

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