Answer
$\dfrac{1-i\sqrt{3}}{2}$
Work Step by Step
Using $i=\sqrt{-1}$ and the properties of radicals, the given expression, $
\dfrac{5-\sqrt{-75}}{10}
,$ is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{5-\sqrt{-1}\cdot\sqrt{75}}{10}
\\\\=
\dfrac{5-i\cdot\sqrt{75}}{10}
\\\\=
\dfrac{5-i\cdot\sqrt{25\cdot3}}{10}
\\\\=
\dfrac{5-i\cdot\sqrt{(5)^2\cdot3}}{10}
\\\\=
\dfrac{5-i\cdot5\sqrt{3}}{10}
\\\\=
\dfrac{5-5i\sqrt{3}}{10}
\\\\=
\dfrac{5(1-i\sqrt{3})}{10}
\\\\=
\dfrac{\cancel{5}(1-i\sqrt{3})}{\cancel{5}\cdot2}
\\\\=
\dfrac{1-i\sqrt{3}}{2}
.\end{array}