1. Problem: Calculate the cube root of 360, denoted as $\sqrt[3]{360}$.\n\n2. Formula and rules: The cube root of a number $x$ is a number $y$ such that $y^3 = x$. We can simplify roots by factoring the number into prime factors and extracting perfect cubes.\n\n3. Factorize 360: $360 = 2^3 \times 3^2 \times 5$.\n\n4. Extract perfect cubes: $\sqrt[3]{360} = \sqrt[3]{2^3 \times 3^2 \times 5} = 2 \times \sqrt[3]{3^2 \times 5} = 2 \times \sqrt[3]{45}$.\n\n5. Final answer: $\sqrt[3]{360} = 2\sqrt[3]{45}$.