1. **State the problem:** Solve the proportion equation $4 \times x = 9 \times 12$ for $x$. 2. **Rewrite the equation:** We have $$4x = 9 \times 12$$ 3. **Calculate the right side:** $$9 \times 12 = 108$$ 4. **Isolate $x$ by dividing both sides by 4:** $$x = \frac{108}{4} = 27$$ 5. **Interpretation:** The value of $x$ that satisfies the equation is $27$. --- 6. **Next problem:** Find $x$ in the proportion $16 : 36 :: 36 : x$. 7. **Write the proportion as an equation:** $$16 \times x = 36 \times 36$$ 8. **Calculate the right side:** $$36 \times 36 = 1296$$ 9. **Solve for $x$:** $$x = \frac{1296}{16} = 81$$ 10. **Interpretation:** The third proportional to 16 and 36 is $81$. --- 11. **Next problem:** Simplify $$\sqrt{(6 + 3\sqrt{3})(8 - 4\sqrt{3})}$$. 12. **Multiply inside the square root:** $$(6)(8) + (6)(-4\sqrt{3}) + (3\sqrt{3})(8) + (3\sqrt{3})(-4\sqrt{3})$$ $$= 48 - 24\sqrt{3} + 24\sqrt{3} - 36$$ 13. **Simplify terms:** The $-24\sqrt{3}$ and $+24\sqrt{3}$ cancel out, so $$48 - 36 = 12$$ 14. **Take the square root:** $$\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}$$ --- 15. **Next problem:** Calculate $$0.08 \times 0.18$$ using square roots. 16. **Rewrite as square roots:** $$0.08 = \frac{8}{100}, \quad 0.18 = \frac{18}{100}$$ 17. **Multiply inside the square root:** $$\sqrt{\frac{8}{100} \times \frac{18}{100}} = \sqrt{\frac{144}{10000}} = \frac{12}{100} = 0.12$$ --- 18. **Next problem:** Find mean proportionals $x$ and $y$ such that $$xy = 36$$ $$y^3 = 729$$ 19. **Solve for $y$:** $$y^3 = 729 = 9^3 \implies y = 9$$ 20. **Find $x$ using $xy=36$:** $$x = \frac{36}{y} = \frac{36}{9} = 4$$ --- **Final answers:** - $x = 27$ from the first proportion. - $x = 81$ from the second proportion. - $\sqrt{(6 + 3\sqrt{3})(8 - 4\sqrt{3})} = 2\sqrt{3}$. - $0.08 \times 0.18 = 0.12$. - Mean proportionals: $x = 4$, $y = 9$.