Answer
$$8i$$
Work Step by Step
$$\eqalign{
& {\left( {\sqrt 3 + i} \right)^3} \cr
& {\text{Write in the polar form}} \cr
& r = \sqrt {{{\left( {\sqrt 3 } \right)}^2} + {{\left( 1 \right)}^2}} = 2 \cr
& \theta = {\tan ^{ - 1}}\left( {\frac{1}{{\sqrt 3 }}} \right) = {30^ \circ } \cr
& {\left( {\sqrt 3 + i} \right)^3} = {\left( {2\angle {{30}^ \circ }} \right)^3} \cr
& {\text{Solve the power}} \cr
& = {2^3}\angle 3 \cdot {30^ \circ } \cr
& = {2^3}\angle {90^ \circ } \cr
& {\text{Write in the rectangular form}} \cr
& = 8\left( {\cos {{90}^ \circ } + i\sin {{90}^ \circ }} \right) \cr
& = 8\left( {0 + i\left( 1 \right)} \right) \cr
& = 8i \cr} $$
