#### Answer

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

#### Work Step by Step

$\frac{\sqrt [4] {48a^{9}b^{3}}}{\sqrt [4] {ab^{3}}}$
=$\sqrt [4] \frac{48a^{9}b^{3}}{ ab^{3}}$
=$\sqrt [4] {48}\times \sqrt [4]\frac{a^{9}}{a}\times\sqrt [4]\frac{b^{3}}{b^{3}}$
=$\sqrt [4] {16\times3}\times \sqrt [4] {a^{9-1}}\times\sqrt [4]1$
=$\sqrt [4] {16}\times \sqrt [4] {3}\times \sqrt [4] {a^{8}}\times 1$
=$\sqrt [4] {2^{4}}\times \sqrt [4] {3}\times \sqrt [4] {a^{8}}\times 1$
=$2\times \sqrt [4] {3}\times a^{2}\times 1$
=$2a^{2}\sqrt [4] {3}$