1. **Stating the problem:** Simplify the expression $$(-2a)^3 (-3a)(-b^2)^3.$$ 2. **Recall the rules:** - Power of a product: $ (xy)^n = x^n y^n $. - Power of a power: $ (x^m)^n = x^{mn} $. - Multiplying powers with the same base: $ x^m x^n = x^{m+n} $. - Negative signs multiply normally. 3. **Calculate each part:** - $$(-2a)^3 = (-2)^3 a^3 = -8 a^3$$ - $$(-b^2)^3 = (-1)^3 (b^2)^3 = - b^{6}$$ 4. **Substitute back:** $$(-8 a^3) \times (-3 a) \times (- b^{6})$$ 5. **Multiply coefficients:** $$-8 \times -3 \times -1 = (-8 \times -3) \times -1 = 24 \times -1 = -24$$ 6. **Multiply variables:** $$a^3 \times a = a^{3+1} = a^4$$ 7. **Combine all:** $$-24 a^4 b^6$$ **Final answer:** $$\boxed{-24 a^4 b^6}$$