1. **State the problem:** Find the cube root of 81, which is written as $\sqrt[3]{81}$. 2. **Recall the definition:** The cube root of a number $a$ is a number $b$ such that $b^3 = a$. 3. **Express 81 in terms of prime factors:** $$81 = 3^4$$ 4. **Rewrite the cube root using exponents:** $$\sqrt[3]{81} = \sqrt[3]{3^4} = 3^{\frac{4}{3}}$$ 5. **Simplify the exponent:** $$3^{\frac{4}{3}} = 3^{1 + \frac{1}{3}} = 3^1 \times 3^{\frac{1}{3}} = 3 \times \sqrt[3]{3}$$ 6. **Final answer:** $$\boxed{3 \sqrt[3]{3}}$$