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 to each factor:** $$ (2a^2bc^3)^4 = 2^4 \cdot (a^2)^4 \cdot b^4 \cdot (c^3)^4 $$ 4. **Simplify each term:** - $2^4 = 16$ - $(a^2)^4 = a^{2 \times 4} = a^8$ - $b^4$ stays as is - $(c^3)^4 = c^{3 \times 4} = c^{12}$ 5. **Combine all factors:** $$ 16 a^8 b^4 c^{12} $$ **Final answer:** $$16 a^8 b^4 c^{12}$$