1. **State the problem:** Simplify the expression $$(2a^2b)^3 (4b^5)^2$$. 2. **Recall the exponent rules:** - Power of a product: $$(xy)^n = x^n y^n$$ - Power of a power: $$(x^m)^n = x^{mn}$$ - When multiplying like bases, add exponents: $$x^a x^b = x^{a+b}$$ 3. **Apply the power to each factor inside the parentheses:** $$(2a^2b)^3 = 2^3 (a^2)^3 b^3 = 8 a^{2 \times 3} b^3 = 8 a^6 b^3$$ $$(4b^5)^2 = 4^2 (b^5)^2 = 16 b^{5 \times 2} = 16 b^{10}$$ 4. **Multiply the two results:** $$8 a^6 b^3 \times 16 b^{10} = (8 \times 16) a^6 b^{3+10} = 128 a^6 b^{13}$$ 5. **Final simplified expression:** $$128 a^6 b^{13}$$