Simplify. \[ v^{\frac{4}{7}} \cdot v^{\frac{1}{2}} \] Assume that the variable represents a positive real number.

To simplify the expression \( v^{\frac{4}{7}} \cdot v^{\frac{1}{2}} \), we can use the property of exponents that states \( a^m \cdot a^n = a^{m+n} \). First, we need to add the exponents together: \[ \frac{4}{7} + \frac{1}{2} \] To add these fractions, we need a common denominator. The least common multiple of 7 and 2 is 14. We can rewrite each fraction: \[ \frac{4}{7} = \frac{4 \cdot 2}{7 \cdot 2} = \frac{8}{14} \] \[ \frac{1}{2} = \frac{1 \cdot 7}{2 \cdot 7} = \frac{7}{14} \] Now, we can add the two fractions: \[ \frac{8}{14} + \frac{7}{14} = \frac{8 + 7}{14} = \frac{15}{14} \] Now we can rewrite the original expression using the simplified exponent: \[ v^{\frac{4}{7}} \cdot v^{\frac{1}{2}} = v^{\frac{15}{14}} \] So, the simplified expression is \[ \boxed{v^{\frac{15}{14}}} \]