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

To simplify \( w^{\frac{1}{2}} \cdot w^{\frac{2}{3}} \), we can use the property of exponents that states \( a^m \cdot a^n = a^{m+n} \). Here, we add the exponents: \[ \frac{1}{2} + \frac{2}{3} \] To add these fractions, we need a common denominator. The least common multiple of 2 and 3 is 6. We can rewrite the fractions: \[ \frac{1}{2} = \frac{3}{6} \quad \text{and} \quad \frac{2}{3} = \frac{4}{6} \] Now, we can add them: \[ \frac{3}{6} + \frac{4}{6} = \frac{7}{6} \] Therefore, the exponents add up to \( \frac{7}{6} \), and we can write: \[ w^{\frac{1}{2}} \cdot w^{\frac{2}{3}} = w^{\frac{7}{6}} \] So, the simplified expression is: \[ \boxed{w^{\frac{7}{6}}} \]