Answer
$2-i\sqrt{2}$
Work Step by Step
Using $i=\sqrt{-1}$ and the properties of radicals, the given expression, $
\dfrac{4-\sqrt{-8}}{2}
,$ is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{4-\sqrt{-1}\cdot\sqrt{8}}{2}
\\\\=
\dfrac{4-i\cdot\sqrt{4\cdot2}}{2}
\\\\=
\dfrac{4-i\cdot\sqrt{(2)^2\cdot2}}{2}
\\\\=
\dfrac{4-2i\sqrt{2}}{2}
\\\\=
\dfrac{2(2-i\sqrt{2})}{2}
\\\\=
\dfrac{\cancel{2}(2-i\sqrt{2})}{\cancel{2}}
\\\\=
2-i\sqrt{2}
.\end{array}