1. **Problem:** Find $x$ such that $$(x-1)^3 = 4^3$$ 2. **Formula and rules:** For equations of the form $a^3 = b^3$, if $a$ and $b$ are real numbers, then $a = b$. 3. **Step-by-step solution:** 1) Given $$(x-1)^3 = 4^3$$ 2) Apply the cube root to both sides: $$\sqrt[3]{(x-1)^3} = \sqrt[3]{4^3}$$ 3) Simplify using the property $\sqrt[3]{a^3} = a$: $$x-1 = 4$$ 4) Solve for $x$: $$x = 4 + 1$$ $$x = 5$$ 4. **Answer:** $$\boxed{5}$$