1. **State the problem:** Simplify the expression $\frac{1}{2}x^2 \cdot 32$. 2. **Recall the multiplication rule:** When multiplying a constant by a variable expression, multiply the constants and keep the variable part as is. 3. **Multiply the constants:** $\frac{1}{2} \times 32 = \frac{1 \times 32}{2} = \frac{32}{2}$. 4. **Simplify the fraction:** $\frac{32}{2} = \cancel{\frac{32}{2}} = 16$. 5. **Combine with the variable part:** $16 \times x^2 = 16x^2$. **Final answer:** $$16x^2$$