Find the value


Find the value

$(a-2 b)^{3}-512 b^{3}$


$=(a-2 b)^{3}-(8 b)^{3}$

$=(a-2 b-8 b)\left((a-2 b)^{2}+(a-2 b) 8 b+(8 b)^{2}\right)$

$\therefore\left[a^{3}-b^{3}=(a-b)\left(a^{2}+a b+b^{2}\right)\right]$

$=(a-10 b)\left(a^{2}+4 b^{2}-4 a b+8 a b-16 b^{2}+64 b^{2}\right)$

$=(a-10 b)\left(a^{2}+52 b^{2}+4 a b\right)$

$\therefore(a-2 b)^{3}-512 b^{3}=(a-10 b)\left(a^{2}+52 b^{2}+4 a b\right)$

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