1. **State the problem:** Calculate the cube root of 64, written as $\sqrt[3]{64}$. 2. **Formula and rule:** The cube root of a number $a$ is a number $b$ such that $b^3 = a$. So, $\sqrt[3]{a} = b$ where $b^3 = a$. 3. **Evaluate the cube root:** We want to find $b$ such that $b^3 = 64$. 4. **Factor 64:** $64 = 4 \times 4 \times 4 = 4^3$. 5. **Apply the cube root:** $\sqrt[3]{64} = \sqrt[3]{4^3}$. 6. **Simplify using the property $\sqrt[3]{x^3} = x$:** $$\sqrt[3]{4^3} = 4.$$ 7. **Final answer:** $$\boxed{4}.$$