1. The problem is to simplify the expression $\left(7^{\frac{1}{2}}\right)^4$. 2. Recall the power of a power rule: $\left(a^m\right)^n = a^{m \cdot n}$. 3. Apply this rule: $\left(7^{\frac{1}{2}}\right)^4 = 7^{\frac{1}{2} \times 4}$. 4. Multiply the exponents: $\frac{1}{2} \times 4 = 2$. 5. So, the expression simplifies to $7^2$. 6. Calculate $7^2 = 49$. Final answer: $49$.