#### Answer

$6-2i\sqrt{21}$

#### Work Step by Step

Using $i=\sqrt{-1}$ and the properties of radicals, the given expression, $
6-\sqrt{-84}
,$ is equivalent to
\begin{array}{l}\require{cancel}
6-\sqrt{84}\cdot\sqrt{-1}
\\\\=
6-\sqrt{4\cdot21}\cdot i
\\\\=
6-2i\sqrt{21}
.\end{array}