1. Problem: Find the area of a square with side length $5\sqrt{2}$ cm. 2. Formula: Area of a square = side length squared, i.e., $A = s^2$. 3. Calculation: $$A = (5\sqrt{2})^2 = 5^2 \times (\sqrt{2})^2 = 25 \times 2 = 50$$ 4. Explanation: Squaring the side length means multiplying it by itself. The square of $\sqrt{2}$ is 2, so the area is $50$ cm$^2$. --- 5. Problem: Find the perimeter and area of a rectangle with dimensions $(3\sqrt{3}+4)$ cm and $(3\sqrt{3}-4)$ cm. 6. Formulas: - Perimeter $P = 2(l + w)$ - Area $A = l \times w$ 7. Calculation of perimeter: $$P = 2\big((3\sqrt{3}+4) + (3\sqrt{3}-4)\big) = 2(3\sqrt{3} + 4 + 3\sqrt{3} - 4) = 2(6\sqrt{3}) = 12\sqrt{3}$$ 8. Calculation of area: $$A = (3\sqrt{3}+4)(3\sqrt{3}-4)$$ Use difference of squares formula: $(a+b)(a-b) = a^2 - b^2$ $$= (3\sqrt{3})^2 - 4^2 = 9 \times 3 - 16 = 27 - 16 = 11$$ 9. Explanation: The perimeter adds all sides, simplifying the sum inside the parentheses. The area uses the difference of squares to simplify multiplication. --- 10. Problem: Simplify $\sqrt{75} - 2\sqrt{27} + 3\sqrt{\frac{1}{3}}$. 11. Simplify each term: - $\sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3}$ - $2\sqrt{27} = 2 \times \sqrt{9 \times 3} = 2 \times 3\sqrt{3} = 6\sqrt{3}$ - $3\sqrt{\frac{1}{3}} = 3 \times \frac{1}{\sqrt{3}} = 3 \times \frac{\sqrt{3}}{3} = \sqrt{3}$ 12. Combine: $$5\sqrt{3} - 6\sqrt{3} + \sqrt{3} = (5 - 6 + 1)\sqrt{3} = 0$$ --- 13. Problem: Simplify $\sqrt[3]{81} + \sqrt[3]{24} - 3 \sqrt[3]{\frac{1}{9}}$. 14. Simplify each term: - $\sqrt[3]{81} = \sqrt[3]{3^4} = 3 \sqrt[3]{3}$ - $\sqrt[3]{24} = \sqrt[3]{8 \times 3} = 2 \sqrt[3]{3}$ - $3 \sqrt[3]{\frac{1}{9}} = 3 \times \sqrt[3]{3^{-2}} = 3 \times 3^{-\frac{2}{3}} = 3^{1 - \frac{2}{3}} = 3^{\frac{1}{3}} = \sqrt[3]{3}$ 15. Combine: $$3\sqrt[3]{3} + 2\sqrt[3]{3} - \sqrt[3]{3} = (3 + 2 - 1)\sqrt[3]{3} = 4\sqrt[3]{3}$$ --- 16. Problem: Simplify $\sqrt{12} + \sqrt[3]{54} - 2\sqrt{3} - \sqrt[3]{16}$. 17. Simplify each term: - $\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}$ - $\sqrt[3]{54} = \sqrt[3]{27 \times 2} = 3\sqrt[3]{2}$ - $2\sqrt{3}$ stays as is - $\sqrt[3]{16} = \sqrt[3]{8 \times 2} = 2\sqrt[3]{2}$ 18. Combine: $$2\sqrt{3} + 3\sqrt[3]{2} - 2\sqrt{3} - 2\sqrt[3]{2} = (2\sqrt{3} - 2\sqrt{3}) + (3\sqrt[3]{2} - 2\sqrt[3]{2}) = 0 + \sqrt[3]{2} = \sqrt[3]{2}$$ --- 19. Problem: Simplify $\sqrt[3]{24} + \sqrt[3]{15} \times \sqrt{25}$. 20. Simplify each term: - $\sqrt[3]{24} = \sqrt[3]{8 \times 3} = 2\sqrt[3]{3}$ - $\sqrt{25} = 5$ 21. Multiply: $$\sqrt[3]{15} \times 5 = 5\sqrt[3]{15}$$ 22. Combine: $$2\sqrt[3]{3} + 5\sqrt[3]{15}$$ This is the simplest form since the cube roots are different. --- 23. Problem: Simplify $\frac{\sqrt{72} \times \sqrt{75}}{\sqrt[3]{6}}$. 24. Simplify numerator: $$\sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2}$$ $$\sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3}$$ 25. Multiply numerator: $$6\sqrt{2} \times 5\sqrt{3} = 30 \times \sqrt{2 \times 3} = 30\sqrt{6}$$ 26. Expression: $$\frac{30\sqrt{6}}{\sqrt[3]{6}} = 30 \times \frac{\sqrt{6}}{\sqrt[3]{6}}$$ 27. Rewrite radicals with exponents: $$\sqrt{6} = 6^{\frac{1}{2}}, \quad \sqrt[3]{6} = 6^{\frac{1}{3}}$$ 28. Simplify: $$30 \times 6^{\frac{1}{2} - \frac{1}{3}} = 30 \times 6^{\frac{3}{6} - \frac{2}{6}} = 30 \times 6^{\frac{1}{6}} = 30 \sqrt[6]{6}$$ --- 29. Problem: Given $x = \sqrt{7} + \sqrt{5}$ and $y = \sqrt{7} - \sqrt{5}$, find: ① $x^2 - y^2$ ② $\frac{x + y}{x - y}$ 30. Calculate $x^2$ and $y^2$: $$x^2 = (\sqrt{7} + \sqrt{5})^2 = 7 + 2\sqrt{35} + 5 = 12 + 2\sqrt{35}$$ $$y^2 = (\sqrt{7} - \sqrt{5})^2 = 7 - 2\sqrt{35} + 5 = 12 - 2\sqrt{35}$$ 31. Calculate $x^2 - y^2$: $$x^2 - y^2 = (12 + 2\sqrt{35}) - (12 - 2\sqrt{35}) = 4\sqrt{35}$$ 32. Calculate $x + y$ and $x - y$: $$x + y = (\sqrt{7} + \sqrt{5}) + (\sqrt{7} - \sqrt{5}) = 2\sqrt{7}$$ $$x - y = (\sqrt{7} + \sqrt{5}) - (\sqrt{7} - \sqrt{5}) = 2\sqrt{5}$$ 33. Calculate $\frac{x + y}{x - y}$: $$\frac{2\sqrt{7}}{2\sqrt{5}} = \frac{\cancel{2}\sqrt{7}}{\cancel{2}\sqrt{5}} = \frac{\sqrt{7}}{\sqrt{5}} = \sqrt{\frac{7}{5}}$$ --- Final answers: - Area of square: $50$ cm$^2$ - Rectangle perimeter: $12\sqrt{3}$ cm - Rectangle area: $11$ cm$^2$ - Simplifications: ① $0$ ② $4\sqrt[3]{3}$ ③ $\sqrt[3]{2}$ ④ $2\sqrt[3]{3} + 5\sqrt[3]{15}$ ⑤ $30 \sqrt[6]{6}$ - Expressions with $x,y$: ① $4\sqrt{35}$ ② $\sqrt{\frac{7}{5}}$