1. **State the problem:** Simplify the expression $\left(\frac{1}{2}x^2\right) \cdot 8x$. 2. **Recall the multiplication rule for exponents:** When multiplying terms with the same base, add the exponents: $x^a \cdot x^b = x^{a+b}$. 3. **Multiply the coefficients:** $\frac{1}{2} \times 8 = \cancel{\frac{1}{2}} \times \cancel{8} = 4$. 4. **Multiply the variables:** $x^2 \cdot x^1 = x^{2+1} = x^3$. 5. **Combine the results:** The simplified expression is $4x^3$. **Final answer:** $4x^3$