The value of $${\left( {1 + 2\omega + {\omega ^2}} \right)^{3n}} - {\left( {1 + \omega + 2{\omega ^2}} \right)^{3n}}$$       is :

Solution :
We have,
$${\left( {1 + {\omega ^2} + 2\omega} \right)^{3n}} - {\left( {1 + \omega + 2{\omega ^2}} \right)^{3n}}$$
We know that, $$1 + \omega + {\omega ^2} = 0\,\,{\text{and}}\,\,{\omega ^3} = 1$$
∴ given expression is equal to
$$\eqalign{ & {\left( {2\omega - \omega } \right)^{3n}} - {\left( {2{\omega ^2} - {\omega ^2}} \right)^{3n}} \cr & = \,{\left( \omega \right)^{3n}} - {\left( {{\omega ^2}} \right)^{3n}} \cr & = {\left( {{\omega ^3}} \right)^n} - {\left( {{\omega ^3}} \right)^{2n}} \cr & = 1 - 1 \cr & = 0 \cr} $$