1. **State the problem:** Solve the equation $$27x^3 = (x + 5)^3$$ for $x$. 2. **Recall the formula:** The equation is of the form $a^3 = b^3$, which implies $a = b$ or the cube roots are equal. 3. **Apply the cube root property:** Since $$27x^3 = (x + 5)^3$$, we can write $$\sqrt[3]{27x^3} = \sqrt[3]{(x + 5)^3}$$ which simplifies to $$3x = x + 5$$ because $\sqrt[3]{27} = 3$ and $\sqrt[3]{x^3} = x$. 4. **Solve the linear equation:** $$3x = x + 5$$ Subtract $x$ from both sides: $$3x - \cancel{x} = \cancel{x} + 5$$ $$2x = 5$$ 5. **Divide both sides by 2:** $$x = \frac{5}{2}$$ 6. **Final answer:** $$x = \frac{5}{2}$$