1. The problem is to find the value of $\left(8^{\frac{1}{2}}\right)^4$. 2. Use the power of a power rule: $\left(a^m\right)^n = a^{m \times n}$. 3. Apply the rule: $$\left(8^{\frac{1}{2}}\right)^4 = 8^{\frac{1}{2} \times 4}$$ 4. Simplify the exponent multiplication: $$\frac{1}{2} \times 4 = 2$$ 5. So, $$8^{\frac{1}{2} \times 4} = 8^2$$ 6. Calculate $8^2$: $$8^2 = 64$$ 7. Therefore, the value of $\left(8^{\frac{1}{2}}\right)^4$ is $64$.