1. **Problem statement:** Simplify each expression by applying the power of a product rule and the power of a power rule. 2. **Part a:** Simplify $ (3 m^4)^2 $. $$ (3 m^4)^2 = (3 m^4) \times (3 m^4) = 3 \times 3 \times m^4 \times m^4 = 9 m^{4+4} = 9 m^8 $$ 3. **Part b:** Simplify $ (2 n^5)^3 $. $$ (2 n^5)^3 = (2 n^5) \times (2 n^5) \times (2 n^5) = 2 \times 2 \times 2 \times n^5 \times n^5 \times n^5 = 8 n^{5+5+5} = 8 n^{15} $$ 4. **Part c:** Simplify $ (4 v^3)^2 $. $$ (4 v^3)^2 = (4 v^3) \times (4 v^3) = 4 \times 4 \times v^3 \times v^3 = 16 v^{3+3} = 16 v^6 $$ 5. **Part d:** Simplify $ -2 (a^2)^3 $. $$ -2 (a^2)^3 = -2 \times a^{2 \times 3} = -2 a^6 $$ 6. **Part e:** Simplify $ 4 (v^3)^2 $. $$ 4 (v^3)^2 = 4 \times v^{3 \times 2} = 4 v^6 $$ 7. **Part f:** Simplify $ (-2 b^2)^3 $. $$ (-2 b^2)^3 = (-2 b^2) \times (-2 b^2) \times (-2 b^2) = (-2)^3 \times b^{2 \times 3} = -8 b^6 $$ **Final answers:** - a) $9 m^8$ - b) $8 n^{15}$ - c) $16 v^6$ - d) $-2 a^6$ - e) $4 v^6$ - f) $-8 b^6$