1. **State the problem:** Simplify the expression $$(-3v^3 u)^4$$. 2. **Recall the power of a product rule:** When raising a product to a power, raise each factor to that power: $$ (abc)^n = a^n b^n c^n $$ 3. **Apply the rule:** $$ (-3v^3 u)^4 = (-3)^4 (v^3)^4 (u)^4 $$ 4. **Calculate each part:** - $$(-3)^4 = (-3) \times (-3) \times (-3) \times (-3) = 81$$ - $$(v^3)^4 = v^{3 \times 4} = v^{12}$$ - $$u^4$$ remains as is. 5. **Combine all parts:** $$ 81 v^{12} u^4 $$ **Final answer:** $$81 v^{12} u^4$$