#### Answer

$5a^2b^3c^{4}$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
To simplify the given expression, $
-\sqrt[3]{-125a^6b^9c^{12}}
,$ find a factor of the radicand that is a perfect power of the index. Then extract the root of that factor. Note that all variables are assumed to represent positive real numbers.
$\bf{\text{Solution Details:}}$
Expressing the radicand of the expression above with a factor that is a perfect power of the index results to
\begin{array}{l}\require{cancel}
-\sqrt[3]{(-5a^2b^3c^{4})^3}
\\\\=
-(-5a^2b^3c^{4})
\\\\=
5a^2b^3c^{4}
.\end{array}