Factorise each of the following:

(i) $27 y^{3}+125 z^{3}$

(ii) $64 m^{3}-343 n^{3}$
Solution:

(i) $27 y^{3}+125 z^{3}$

$=(3 y)^{3}+(5 z)^{3}$

$=(3 y+5 z)\left[(3 y)^{2}+(5 z)^{2}-(3 y)(5 z)\right]$

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

$=(3 y+5 z)\left[9 y^{2}+25 z^{2}-15 y z\right]$

(ii) $64 m^{3}-343 n^{3}$

$=(4 m)^{3}-(7 n)^{3}$

$=(4 m-7 n)\left[(4 m)^{2}+(7 n)^{2}+(4 m)(7 n)\right]$

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

$=(4 m-7 n)\left[16 m^{2}+49 n^{2}+28 m n\right]$
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