Answer
$$ - \frac{1}{2} - \frac{{\sqrt 3 }}{2}i$$
Work Step by Step
$$\eqalign{
& {\left( {\cos {{100}^ \circ } + i\sin {{100}^ \circ }} \right)^6} \cr
& {\text{Write in the polar form}} \cr
& \cos {100^ \circ } + i\sin {100^ \circ } = 1\angle {100^ \circ } \cr
& {\left( {\cos {{100}^ \circ } + i\sin {{100}^ \circ }} \right)^6} = {\left( {1\angle {{100}^ \circ }} \right)^6} \cr
& {\text{Solve the power}} \cr
& = {\left( 1 \right)^6}\angle 6 \cdot {100^ \circ } \cr
& = 1\angle \left( {{{600}^ \circ } - {{360}^ \circ }} \right) \cr
& = 1\angle {240^ \circ } \cr
& {\text{Write in the rectangular form}} \cr
& = \left( {\cos {{240}^ \circ } + i\sin {{240}^ \circ }} \right) \cr
& = - \frac{1}{2} + i\left( { - \frac{{\sqrt 3 }}{2}} \right) \cr
& = - \frac{1}{2} - \frac{{\sqrt 3 }}{2}i \cr} $$
