1. **State the problem:** Simplify or expand the expression $$(4^a - b)^3$$. 2. **Formula used:** We use the binomial expansion formula for cubes: $$(x - y)^3 = x^3 - 3x^2y + 3xy^2 - y^3$$. 3. **Identify terms:** Here, $x = 4^a$ and $y = b$. 4. **Apply the formula:** $$ (4^a - b)^3 = (4^a)^3 - 3(4^a)^2 b + 3(4^a) b^2 - b^3 $$ 5. **Simplify powers:** $$ (4^a)^3 = 4^{3a}, \quad (4^a)^2 = 4^{2a} $$ 6. **Write the expanded form:** $$ 4^{3a} - 3 \cdot 4^{2a} b + 3 \cdot 4^a b^2 - b^3 $$ 7. **Explanation:** We expanded the cube of a binomial by applying the binomial theorem, carefully substituting and simplifying powers of 4 raised to powers involving $a$. **Final answer:** $$ 4^{3a} - 3 \cdot 4^{2a} b + 3 \cdot 4^a b^2 - b^3 $$