1. **State the problem:** Simplify the expression $\sqrt{64} - \sqrt[3]{125}$.\n\n2. **Recall the formulas and rules:**\n- The square root $\sqrt{a}$ is the number that when squared gives $a$.\n- The cube root $\sqrt[3]{a}$ is the number that when cubed gives $a$.\n\n3. **Calculate each root:**\n- $\sqrt{64} = 8$ because $8^2 = 64$.\n- $\sqrt[3]{125} = 5$ because $5^3 = 125$.\n\n4. **Substitute the values back into the expression:**\n$$8 - 5$$\n\n5. **Perform the subtraction:**\n$$8 - 5 = 3$$\n\n**Final answer:** $3$