Simplify. \[ w^{\frac{2}{3}} \cdot w^{\frac{4}{9}} \] Assume that the variable represents a positive real number.

To simplify the expression \( w^{\frac{2}{3}} \cdot w^{\frac{4}{9}} \), we can use the property of exponents that states \( a^m \cdot a^n = a^{m+n} \). First, we'll find a common denominator for the exponents \( \frac{2}{3} \) and \( \frac{4}{9} \). The least common multiple of the denominators \( 3 \) and \( 9 \) is \( 9 \). Now, express \( \frac{2}{3} \) with a denominator of \( 9 \): \[ \frac{2}{3} = \frac{2 \cdot 3}{3 \cdot 3} = \frac{6}{9} \] Now, we can add the exponents: \[ w^{\frac{2}{3}} \cdot w^{\frac{4}{9}} = w^{\frac{6}{9}} \cdot w^{\frac{4}{9}} = w^{\frac{6}{9} + \frac{4}{9}} = w^{\frac{10}{9}} \] Thus, the simplified expression is: \[ \boxed{w^{\frac{10}{9}}} \]