If $$z{\left( {2 - i2\sqrt 3 } \right)^2} = i{\left( {\sqrt 3 + i} \right)^4}$$     then amplitude of $$z$$ is

Solution :
$$\eqalign{ & z = \frac{{i{{\left( {\sqrt 3 + i} \right)}^4}}}{{4{{\left( {1 - \sqrt 3 i} \right)}^2}}} = \frac{i}{4} \cdot \frac{{{{\left( {2 + 2\sqrt {3}i } \right)}^2}}}{{ - 2 - 2\sqrt {3}i }} = \frac{{i\left( { - 2 + 2\sqrt {3}i } \right)}}{{2\left( { - 1 - \sqrt {3}i } \right)}} = \frac{{\sqrt 3 + i}}{{1 + \sqrt {3}i }} \cr & z = \frac{{\left( {\sqrt 3 + i} \right)\left( {1 - \sqrt {3}i } \right)}}{{1 + 3}} = \frac{{2\sqrt 3 - 2i}}{4} = \frac{{\sqrt 3 }}{2} - \frac{i}{2} \cr & \therefore \,\,{\text{amp}}\,z = - {\tan ^{ - 1}}\frac{1}{{\sqrt 3 }}. \cr} $$