1. **State the problem:** Simplify the expression $ (2a^2bc^3)^4 $. 2. **Recall the power of a product rule:** When raising a product to a power, raise each factor to that power: $$ (xyz)^n = x^n y^n z^n $$ 3. **Apply the rule:** $$ (2a^2bc^3)^4 = 2^4 (a^2)^4 b^4 (c^3)^4 $$ 4. **Simplify each term:** $$ 2^4 = 16 $$ $$ (a^2)^4 = a^{2 \times 4} = a^8 $$ $$ b^4 = b^4 $$ $$ (c^3)^4 = c^{3 \times 4} = c^{12} $$ 5. **Combine all:** $$ 16 a^8 b^4 c^{12} $$ **Final answer:** $16 a^8 b^4 c^{12}$