#### Answer

$2a^{2}\sqrt[4] 3$

#### Work Step by Step

$\frac{\sqrt[4] (48a^{9}b^{3})}{\sqrt[4] (ab^{3})}=\sqrt[4] \frac {48a^{9}b^{3}}{ab^{3}}=\sqrt[4] (48a^{8})=\sqrt[4] (16\times a^{8}\times 3)=\sqrt[4] 16\times \sqrt[4] (a^{8})\times \sqrt[4] 3=2a^{2}\sqrt[4] 3$
We know that $\sqrt[4] 16=2$, because $2^{4}=16$.
We know that $\sqrt[4] (a^{8})=a^{2}$, because $(a^{2})^{4}=a^{2\times4}=a^{8}$