Factorize
i) $27 m^{3} - 216 n^{3}$ ii) $343 a^{3} - 512 b^{3}$

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Solution

i) $27 (m^{3} - 8 n^{3})$
...[Taking out the common factor 27]
$= 27 [m^{3} - (2 n)^{3}]$
Here, a = m and b = 2n
$∴ 27 m^{3} - 216 n^{3}$
$= 27 {\begin{matrix} (m - 2 n) [m^{2} + m (2 n) + (2 n)^{2}] \end{matrix}}$
.... $[∵ a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})]$
$= 27 (m - 2 n) (m^{2} + 2 m n + 4 n^{2})$

ii) $343 a^{3} - 512 b^{3}$
$= (7 a)^{3} - (8 b)^{3}$
Here, A = 7a and B = 8b
$∴ 343 a^{3} - 512 b^{3}$
$= (7 a - 8 b) [(7 a)^{2} + (7 a) (8 b) + (8 b)^{2}]$
... $[∵ a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})]$
$= (7 a - 8 b) (49 a^{2} + 56 a b + 64 b^{2}$

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