Simplify. \[ x^{\frac{4}{7}} \cdot x^{\frac{2}{5}} \] Assume that the variable represents a positive real number. \( \square \)

To simplify \( x^{\frac{4}{7}} \cdot x^{\frac{2}{5}} \), you can use the property of exponents which states that \( a^m \cdot a^n = a^{m+n} \). Therefore, \[ x^{\frac{4}{7}} \cdot x^{\frac{2}{5}} = x^{\left(\frac{4}{7} + \frac{2}{5}\right)}. \] Next, you need a common denominator to add the fractions \( \frac{4}{7} \) and \( \frac{2}{5} \). The least common multiple of 7 and 5 is 35. Rewriting the fractions: \[ \frac{4}{7} = \frac{20}{35}, \quad \text{and} \quad \frac{2}{5} = \frac{14}{35}. \] Now, add the fractions: \[ \frac{20}{35} + \frac{14}{35} = \frac{34}{35}. \] Thus, you can simplify the expression as follows: \[ x^{\frac{4}{7}} \cdot x^{\frac{2}{5}} = x^{\frac{34}{35}}. \] So, the simplified expression is: \[ \boxed{x^{\frac{34}{35}}}. \]