1. **State the problem:** Find the cube root of 64, which means finding a number $x$ such that $x^3 = 64$. 2. **Formula and rule:** The cube root of a number $a$ is written as $\sqrt[3]{a}$ and it satisfies $x = \sqrt[3]{a}$ if and only if $x^3 = a$. 3. **Evaluate the cube root:** We want to find $x$ such that $x^3 = 64$. 4. **Recall perfect cubes:** $4^3 = 4 \times 4 \times 4 = 64$. 5. **Conclusion:** Therefore, $\sqrt[3]{64} = 4$. **Final answer:** $$\boxed{4}$$