1. **State the problem:** Solve the equation $$\sqrt[3]{x} - 9 = 2$$ for $x$. 2. **Isolate the cube root:** Add 9 to both sides to isolate the cube root term. $$\sqrt[3]{x} - 9 + 9 = 2 + 9$$ $$\sqrt[3]{x} = 11$$ 3. **Remove the cube root:** Cube both sides to eliminate the cube root. The formula used is $\left(\sqrt[3]{x}\right)^3 = x$. $$\left(\sqrt[3]{x}\right)^3 = 11^3$$ $$x = 11^3$$ 4. **Calculate the cube:** $$x = 11 \times 11 \times 11 = 1331$$ 5. **Final answer:** $$\boxed{1331}$$ This means the value of $x$ that satisfies the equation is 1331.