Simplify. \[ w^{\frac{1}{2}} \cdot w^{\frac{2}{3}} \] Assume that the variable represents a positive real number.

To simplify the expression \( w^{\frac{1}{2}} \cdot w^{\frac{2}{3}} \), we first use the property of exponents that states when multiplying like bases, we can add the exponents: \[ w^{\frac{1}{2}} \cdot w^{\frac{2}{3}} = w^{\frac{1}{2} + \frac{2}{3}}. \] Next, we need to add the fractions \( \frac{1}{2} \) and \( \frac{2}{3} \). To do this, we find a common denominator, which is 6. We can convert both fractions: \[ \frac{1}{2} = \frac{3}{6} \quad \text{and} \quad \frac{2}{3} = \frac{4}{6}. \] Now, adding these gives: \[ \frac{3}{6} + \frac{4}{6} = \frac{7}{6}. \] Thus, we can rewrite the expression as: \[ w^{\frac{1}{2}} \cdot w^{\frac{2}{3}} = w^{\frac{7}{6}}. \] In conclusion, the simplified expression is: \[ \boxed{w^{\frac{7}{6}}}. \]