Simplify. \[ w^{\frac{4}{5}} \cdot w^{\frac{2}{3}} \] Assume that the variable represents a positive real number.

To simplify the expression \( w^{\frac{4}{5}} \cdot w^{\frac{2}{3}} \), you can use the property of exponents that states \( a^m \cdot a^n = a^{m+n} \). Here, we will first find a common denominator for the exponents \( \frac{4}{5} \) and \( \frac{2}{3} \). The least common multiple of 5 and 3 is 15. We can rewrite the exponents: \[ \frac{4}{5} = \frac{4 \cdot 3}{5 \cdot 3} = \frac{12}{15} \] \[ \frac{2}{3} = \frac{2 \cdot 5}{3 \cdot 5} = \frac{10}{15} \] Now we can add the exponents: \[ w^{\frac{4}{5}} \cdot w^{\frac{2}{3}} = w^{\frac{12}{15} + \frac{10}{15}} = w^{\frac{22}{15}} \] Therefore, the simplified expression is: \[ w^{\frac{22}{15}} \]