Answer
$4a^5b^5$
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:
$\require{cancel}=\sqrt{\dfrac{48a^8b^7}{3a^{-2}b^{-3}}}
\\=\sqrt{\dfrac{16\cancel{48}a^8b^7}{\cancel{3}a^{-2}b^{-3}}}
\\=\sqrt{\dfrac{16a^8b^7}{a^{-2}b^{-3}}}$
Use rule (2) above to obtain:
$=\\=\sqrt{16a^{8-(-2)}b^{7-(-3)}}
\\=\sqrt{16a^{8+2}b^{7+3}}
\\=\sqrt{16a^{10}b^{10}}$
Factor the radicand so that at least one factor is a perfect square to obtain:
$=\sqrt{(4a^5b^5)^2}$
Simplify to obtain:
$=4a^5b^5$
