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

To simplify \( w^{\frac{2}{3}} \cdot w^{\frac{4}{9}} \), we can use the property of exponents that states: \( a^m \cdot a^n = a^{m+n} \). First, we need to add the exponents: \[ \frac{2}{3} + \frac{4}{9} \] To add these fractions, we need a common denominator. The least common multiple of 3 and 9 is 9. Now, we convert \( \frac{2}{3} \) into ninths: \[ \frac{2}{3} = \frac{2 \times 3}{3 \times 3} = \frac{6}{9} \] Now we can add the fractions together: \[ \frac{6}{9} + \frac{4}{9} = \frac{6 + 4}{9} = \frac{10}{9} \] Therefore, we can express the original multiplication as: \[ w^{\frac{2}{3}} \cdot w^{\frac{4}{9}} = w^{\frac{10}{9}} \] So, the simplified form is: \[ \boxed{w^{\frac{10}{9}}} \]