Answer
$$2\sqrt 3 - 2i$$
Work Step by Step
$$\eqalign{
& \frac{8}{{\sqrt 3 + i}} \cr
& {\text{Convert the numerator and the denominator to trigonometric form}} \cr
& 8 = 8\left( {\cos {0^ \circ } + i\sin {0^ \circ }} \right) \cr
& \sqrt 3 + i = 2\left( {\cos {{30}^ \circ } + \sin {{30}^ \circ }} \right) \cr
& {\text{Then,}} \cr
& \frac{8}{{\sqrt 3 + i}} = \frac{{8\left( {\cos {0^ \circ } + i\sin {0^ \circ }} \right)}}{{2\left( {\cos {{30}^ \circ } + \sin {{30}^ \circ }} \right)}} \cr
& {\text{Using the Quotient Theorem}} \cr
& = \frac{8}{2}\left[ {\cos \left( {{0^ \circ } - {{30}^ \circ }} \right) + i\sin \left( {{0^ \circ } - {{30}^ \circ }} \right)} \right] \cr
& {\text{Simplify}} \cr
& = 4\left[ {\cos \left( { - {{30}^ \circ }} \right) + i\sin \left( { - {{30}^ \circ }} \right)} \right] \cr
& = 4\left[ {\cos \left( {{{30}^ \circ }} \right) - i\sin \left( {{{30}^ \circ }} \right)} \right] \cr
& {\text{Write in Rectangular form}} \cr
& = 4\left[ {\frac{{\sqrt 3 }}{2} - i\left( {\frac{1}{2}} \right)} \right] \cr
& = 2\sqrt 3 - 2i \cr} $$
