Step-by-step Solution:

The number \[ 9^{\frac{1}{3}} \cdot 9^{\frac{1}{9}} \cdot 9^{\frac{1}{27}} \cdot \ldots = 9^{\frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \ldots \infty} \] The series \( \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \ldots \) is a geometric series with first term \( \frac{1}{3} \) and common ratio \( \frac{1}{3} \). Sum of the infinite series: \[ S = \frac{\frac{1}{3}}{1 - \frac{1}{3}} = \frac{\frac{1}{3}}{\frac{2}{3}} = \frac{1}{2} \] Thus, \[ 9^{S} = 9^{\frac{1}{2}} = \sqrt{9} = 3 \]