Answer
$3a^5b^6\sqrt{2ab}$
Work Step by Step
RECALL:
(1) The quotient rule:
$\dfrac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\dfrac{a}{b}}$
where
$\sqrt[n]{a}$ and $\sqrt[n]{b}$ are real numbers and $b\ne0$
(2) $\dfrac{a^m}{a^n} = a^{m-n}, a \ne =0$
Use the quotient rule above to obtain:
$\require{cancel}=\sqrt{\dfrac{54a^7b^{11}}{3a^{-4}b^{-2}}}
\\=\sqrt{\dfrac{18\cancel{54}a^7b^{11}}{\cancel{3}a^{-4}b^{-2}}}
\\=\sqrt{\dfrac{18a^7b^{11}}{a^{-4}b^{-2}}}$
Use rule (2) above to obtain:
$=\\=\sqrt{18a^{7-(-4)}b^{11-(-2)}}
\\=\sqrt{18a^{7+4}b^{11+2}}
\\=\sqrt{18a^{11}b^{13}}$
Factor the radicand so that at least one factor is a perfect square to obtain:
$=\sqrt{(9a^{10}b^{12})(2ab)}
\\=\sqrt{(3a^5b^6)^2(2ab)}$
Simplify to obtain:
$=3a^5b^6\sqrt{2ab}$
