Add. \[ \frac{4}{9 u^{2} w}+\frac{5}{6 u w^{3}} \] Simplify your answer as much as possible.

To add the fractions \( \frac{4}{9 u^{2} w} + \frac{5}{6 u w^{3}} \), we first need to find a common denominator. The least common multiple (LCM) of \( 9 u^{2} w \) and \( 6 u w^{3} \) is \( 18 u^{2} w^{3} \). Now, we rewrite each fraction with the common denominator: 1. For the first fraction: \[ \frac{4}{9 u^{2} w} = \frac{4 \cdot 2 w^{2}}{9 u^{2} w \cdot 2 w^{2}} = \frac{8 w^{2}}{18 u^{2} w^{3}} \] 2. For the second fraction: \[ \frac{5}{6 u w^{3}} = \frac{5 \cdot 3 u}{6 u w^{3} \cdot 3 u} = \frac{15 u}{18 u^{2} w^{3}} \] Now we can add these two fractions: \[ \frac{8 w^{2}}{18 u^{2} w^{3}} + \frac{15 u}{18 u^{2} w^{3}} = \frac{8 w^{2} + 15 u}{18 u^{2} w^{3}} \] Thus, the final simplified result is: \[ \frac{8 w^{2} + 15 u}{18 u^{2} w^{3}} \]