Answer
$$ - 128 + 128i$$
Work Step by Step
$$\eqalign{
& {\left( {2 - 2i} \right)^5} \cr
& {\text{Write in the polar form}} \cr
& r = \sqrt {{{\left( 2 \right)}^2} + {{\left( { - 2} \right)}^2}} = 2\sqrt 2 = \cr
& \theta = {\tan ^{ - 1}}\left( {\frac{{ - 2}}{2}} \right) = - {45^ \circ } + {360^ \circ } \cr
& \theta = {315^ \circ } \cr
& {\left( {2 - 2i} \right)^5} = {\left( {2\sqrt 2 \angle {{315}^ \circ }} \right)^5} \cr
& {\text{Solve the power}} \cr
& = {\left( {2\sqrt 2 } \right)^5}\angle 5 \cdot {315^ \circ } \cr
& = {2^7}\sqrt 2 \angle {1575^ \circ } \cr
& = 128\sqrt 2 \angle {135^ \circ } \cr
& {\text{Write in the rectangular form}} \cr
& = 128\sqrt 2 \left( {\cos {{135}^ \circ } + i\sin {{135}^ \circ }} \right) \cr
& = 128\sqrt 2 \left( { - \frac{{\sqrt 2 }}{2} + i\frac{{\sqrt 2 }}{2}} \right) \cr
& = - 128 + 128i \cr} $$
