Question:
Find the value
$3 \sqrt{3} a^{3}-b^{3}-5 \sqrt{5} c^{3}-3 \sqrt{15} a b c$
Solution:
$=(\sqrt{3} a)^{3}+(-b)^{3}-(\sqrt{5} c)^{3}-3(\sqrt{3} a)(-b)(-\sqrt{5} c)$
$=(\sqrt{3} a+(-b)+(-\sqrt{5} c))$
$\left((\sqrt{3} a)^{2}+(-b)^{2}+(-\sqrt{5} c)^{2}-\sqrt{3} a(-b)-(-b)(-\sqrt{5} c)-(-\sqrt{5} c) \sqrt{3} a\right)$
$=(\sqrt{3} a-b-\sqrt{5} c)\left(3 a^{2}+b^{2}+5 c^{2}+\sqrt{3} a b-\sqrt{5} b c+\sqrt{15} a c\right)$
$\therefore 3 \sqrt{3} a^{3}-b^{3}-5 \sqrt{5} c^{3}-3 \sqrt{15} a b c$
$=(\sqrt{3} a-b-\sqrt{5} c)\left(3 a^{2}+b^{2}+5 c^{2}+\sqrt{3} a b-\sqrt{5} b c+\sqrt{15} a c\right)$