Answer
$-\dfrac{125y^{3}}{x^{30}}$
Work Step by Step
Using the laws of exponents, the given expression, $
\left( \dfrac{-3x^4y^6}{15x^{-6}y^7} \right)^{-3}
,$ is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{(-3)^{-3}x^{4(-3)}y^{6(-3)}}{15^{-3}x^{-6(-3)}y^{7(-3)}}
\\\\=
\dfrac{(-3)^{-3}x^{-12}y^{-18}}{(-3)^{-3}(-5)^{-3}x^{18}y^{-21}}
\\\\=
\dfrac{\cancel{(-3)^{-3}}x^{-12-18}y^{-18-(-21)}}{\cancel{(-3)^{-3}}(-5)^{-3}}
\\\\=
\dfrac{x^{-12-18}y^{-18+21}}{(-5)^{-3}}
\\\\=
\dfrac{x^{-30}y^{3}}{(-5)^{-3}}
\\\\=
\dfrac{(-5)^{3}y^{3}}{x^{30}}
\\\\=
\dfrac{-125y^{3}}{x^{30}}
\\\\=
-\dfrac{125y^{3}}{x^{30}}
.\end{array}