Step 1: Identify the operation. The problem asks for the cube root of 216, which is written as $\sqrt[3]{216}$. This means we need to find a number that, when multiplied by itself three times, equals 216. Step 2: Find the prime factorization of 216. $$216 = 2 \times 108$$ $$108 = 2 \times 54$$ $$54 = 2 \times 27$$ $$27 = 3 \times 9$$ $$9 = 3 \times 3$$ So, the prime factorization of 216 is $2 \times 2 \times 2 \times 3 \times 3 \times 3$. Step 3: Group the prime factors into sets of three. $$216 = (2 \times 2 \times 2) \times (3 \times 3 \times 3)$$ $$216 = 2^3 \times 3^3$$ Step 4: Calculate the cube root. $$\sqrt[3]{216} = \sqrt[3]{2^3 \times 3^3}$$ Using the property $\sqrt[n]{a^n} = a$: $$\sqrt[3]{2^3 \times 3^3} = \sqrt[3]{2^3} \times \sqrt[3]{3^3}$$ $$= 2 \times 3$$ $$= 6$$ The cube root of 216 is 6. $$\boxed{6}$$