1. **State the problem:** Simplify the expression $$\frac{8m^6n^2}{4m^3} \cdot \frac{12(mn)^3}{n^2}$$ and write an expression for the area of a rectangle with dimensions $$(2a^2 b)^3$$ and $$(4b^3)^2$$. 2. **Simplify problem 5:** Start with $$\frac{8m^6n^2}{4m^3} \cdot \frac{12(mn)^3}{n^2}$$. 3. **Simplify each fraction:** $$\frac{8m^6n^2}{4m^3} = \frac{\cancel{8}^2 m^6 n^2}{\cancel{4}^2 m^3} = 2 m^{6-3} n^2 = 2 m^3 n^2$$ 4. **Expand and simplify the second fraction:** $$(mn)^3 = m^3 n^3$$ So, $$\frac{12(mn)^3}{n^2} = \frac{12 m^3 n^3}{n^2} = 12 m^3 n^{3-2} = 12 m^3 n^1 = 12 m^3 n$$ 5. **Multiply the simplified parts:** $$2 m^3 n^2 \cdot 12 m^3 n = (2 \cdot 12)(m^{3+3})(n^{2+1}) = 24 m^6 n^3$$ --- 6. **Simplify problem 7:** Write an expression for the area of a rectangle with dimensions $$(2a^2 b)^3$$ and $$(4b^3)^2$$. 7. **Use the formula for area:** $$\text{Area} = \text{length} \times \text{width}$$ So, $$\text{Area} = (2a^2 b)^3 \times (4b^3)^2$$ 8. **Expand each term:** $$(2a^2 b)^3 = 2^3 (a^2)^3 b^3 = 8 a^{6} b^{3}$$ $$(4b^3)^2 = 4^2 (b^3)^2 = 16 b^{6}$$ 9. **Multiply the expanded terms:** $$8 a^{6} b^{3} \times 16 b^{6} = (8 \times 16) a^{6} b^{3+6} = 128 a^{6} b^{9}$$ **Final answers:** Problem 5: $$24 m^6 n^3$$ Problem 7: $$128 a^{6} b^{9}$$