Answer
$$8\left( {\cos {{30}^ \circ } + i\sin {{30}^ \circ }} \right)$$
Work Step by Step
$$\eqalign{
& 4\sqrt 3 + 4i \cr
& {\text{Use }}r = \sqrt {{a^2} + {b^2}} {\text{ and }}\theta = {\tan ^{ - 1}}\left( {\frac{b}{a}} \right),{\text{ so}} \cr
& r = \sqrt {{{\left( {4\sqrt 3 } \right)}^2} + {{\left( 4 \right)}^2}} = 8 \cr
& \theta = {\tan ^{ - 1}}\left( {\frac{{\sqrt 3 }}{1}} \right) = {30^ \circ } \cr
& {\text{write the vector in the trigonometric form }}r\left( {\cos \theta + i\sin \theta } \right) \cr
& = 8\left( {\cos {{30}^ \circ } + i\sin {{30}^ \circ }} \right) \cr} $$
