Try not to bash this out.

De Moivre's Theorem states that $(a+bi )^{ n } = r^{ n }*(\cos (\theta n)+i\sin ((\theta n))$ where $r$ is equal to $\sqrt { { a }^{ 2 }+{ b }^{ 2 } }$ . To find $\theta$ for this expression, graph it on the imaginary plane to get $\theta = \frac { 5\pi }{ 4 }$ For this problem you get $( -2 -2i)^{ 7 }= (2\sqrt { 2 } )^{ 7 }*(\cos { \frac { 35\pi }{ 4 } } +i\sin { \frac { 35\pi }{ 4 } ) } = 1024\sqrt { 2 } *(\frac { -\sqrt { 2 } }{ 2 } +i\frac { \sqrt { 2 } }{ 2 } ) = -1024+1024i,$ where $a=-1024$ and $b = 1024$ giving $a +b = \boxed { 0 } .$ Whew...that took a while to type up!