1. **Problem 11:** Solve the equation $$\frac{x+5}{2} - \frac{4-x}{3} = 1$$. 2. To solve, find a common denominator for the fractions, which is 6. 3. Multiply both sides by 6 to clear denominators: $$6 \times \left(\frac{x+5}{2} - \frac{4-x}{3}\right) = 6 \times 1$$ 4. Distribute: $$3(x+5) - 2(4-x) = 6$$ 5. Expand terms: $$3x + 15 - 8 + 2x = 6$$ 6. Combine like terms: $$5x + 7 = 6$$ 7. Subtract 7 from both sides: $$5x + \cancel{7} - \cancel{7} = 6 - 7$$ $$5x = -1$$ 8. Divide both sides by 5: $$\frac{5x}{\cancel{5}} = \frac{-1}{\cancel{5}}$$ $$x = -\frac{1}{5}$$ --- 9. **Problem 12a:** Simplify $$\frac{4^{n+3} \times 2^{4-n}}{8^{n+1}}$$. 10. Express all bases as powers of 2: $$4 = 2^2, \quad 8 = 2^3$$ 11. Rewrite: $$\frac{(2^2)^{n+3} \times 2^{4-n}}{(2^3)^{n+1}} = \frac{2^{2(n+3)} \times 2^{4-n}}{2^{3(n+1)}}$$ 12. Simplify exponents: $$\frac{2^{2n+6} \times 2^{4-n}}{2^{3n+3}} = \frac{2^{2n+6 + 4 - n}}{2^{3n+3}} = \frac{2^{n+10}}{2^{3n+3}}$$ 13. Subtract exponents in division: $$2^{n+10 - (3n+3)} = 2^{n+10 - 3n - 3} = 2^{-2n + 7}$$ --- 14. **Problem 12b:** Simplify $$2\sqrt{5}a^4 \times \sqrt{5} \times \sqrt[3]{a}$$. 15. Combine radicals: $$2 \times \sqrt{5} \times \sqrt{5} = 2 \times 5 = 10$$ 16. Combine powers of $$a$$: $$a^4 \times a^{\frac{1}{3}} = a^{4 + \frac{1}{3}} = a^{\frac{13}{3}}$$ 17. Final expression: $$10 a^{\frac{13}{3}}$$ --- 18. **Problem 13:** Simplify and express with positive indices: $$\frac{2w^6 r^7}{9h^2} \times \frac{4w^{-5} h^7}{\left(3w^{-2} h^3\right)^2}^{-6}$$ 19. Simplify the denominator inside the power: $$(3w^{-2} h^3)^2 = 3^2 w^{-4} h^6 = 9 w^{-4} h^6$$ 20. Raise to the power $$-6$$: $$\left(9 w^{-4} h^6\right)^{-6} = 9^{-6} w^{24} h^{-36}$$ 21. Rewrite the expression: $$\frac{2w^6 r^7}{9h^2} \times \frac{4w^{-5} h^7}{9^{-6} w^{24} h^{-36}}$$ 22. Multiply numerator and denominator: $$= \frac{2w^6 r^7}{9h^2} \times 4w^{-5} h^7 \times 9^{6} w^{-24} h^{36}$$ 23. Combine coefficients: $$2 \times 4 \times 9^{6} = 8 \times 9^{6}$$ 24. Combine powers of $$w$$: $$w^{6} \times w^{-5} \times w^{-24} = w^{6 - 5 - 24} = w^{-23}$$ 25. Combine powers of $$h$$: $$h^{-2} \times h^{7} \times h^{36} = h^{-2 + 7 + 36} = h^{41}$$ 26. Include $$r^7$$: $$r^7$$ 27. Final expression: $$8 \times 9^{6} w^{-23} h^{41} r^{7}$$ 28. Express with positive indices: $$\frac{8 \times 9^{6} h^{41} r^{7}}{w^{23}}$$