1. **Write 324 as a product of powers of its prime factors.** Step 1: Find prime factors of 324. 324 ÷ 2 = 162 162 ÷ 2 = 81 81 ÷ 3 = 27 27 ÷ 3 = 9 9 ÷ 3 = 3 3 ÷ 3 = 1 Step 2: Count the powers. Number of 2's: 2 Number of 3's: 4 Step 3: Write as product of prime powers: $$324 = 2^2 \times 3^4$$ --- 2. **Find the square root of** $$A = 2^4 \times 3^6 \times 5^2 \times 7^4$$ Step 1: Use the property $$\sqrt{a^m} = a^{m/2}$$. Step 2: Apply square root to each prime power: $$\sqrt{A} = 2^{4/2} \times 3^{6/2} \times 5^{2/2} \times 7^{4/2} = 2^2 \times 3^3 \times 5^1 \times 7^2$$ Step 3: Calculate numeric value: $$2^2 = 4, \quad 3^3 = 27, \quad 5^1 = 5, \quad 7^2 = 49$$ $$\sqrt{A} = 4 \times 27 \times 5 \times 49 = 26460$$ --- 3. **Calculate and simplify:** $$\frac{11}{45} \times \frac{15}{22}$$ Step 1: Multiply numerators and denominators: $$\frac{11 \times 15}{45 \times 22} = \frac{165}{990}$$ Step 2: Simplify fraction by dividing numerator and denominator by 15: $$\frac{165 \div 15}{990 \div 15} = \frac{11}{66}$$ Step 3: Simplify further by dividing numerator and denominator by 11: $$\frac{11 \div 11}{66 \div 11} = \frac{1}{6}$$ --- 4. **Calculate:** $$\frac{1}{2} \div \frac{3}{5}$$ Step 1: Division of fractions is multiplication by reciprocal: $$\frac{1}{2} \times \frac{5}{3} = \frac{5}{6}$$ --- 5. **Calculate:** $$4.23 \times 6.7$$ using written method Step 1: Multiply as integers ignoring decimals: $$423 \times 67 = 28341$$ Step 2: Count decimal places: 2 (4.23) + 1 (6.7) = 3 Step 3: Place decimal point 3 places from right: $$28.341$$ --- 6. **Calculate:** $$\sqrt[3]{\frac{1000}{2}} + 3^2$$ Step 1: Simplify inside cube root: $$\frac{1000}{2} = 500$$ Step 2: Calculate cube root: $$\sqrt[3]{500} = \sqrt[3]{125 \times 4} = 5 \times \sqrt[3]{4} \approx 5 \times 1.5874 = 7.937$$ Step 3: Calculate $$3^2 = 9$$ Step 4: Add results: $$7.937 + 9 = 16.937$$ --- 7. **Calculate:** $$100 - (6 - 4)^5$$ Step 1: Simplify inside parentheses: $$6 - 4 = 2$$ Step 2: Calculate power: $$2^5 = 32$$ Step 3: Subtract: $$100 - 32 = 68$$ --- 8. **Expand and simplify:** $$7(4 - 3x) - 5x(2 + 5x)$$ Step 1: Expand terms: $$7 \times 4 = 28$$ $$7 \times (-3x) = -21x$$ $$-5x \times 2 = -10x$$ $$-5x \times 5x = -25x^2$$ Step 2: Combine all: $$28 - 21x - 10x - 25x^2 = 28 - 31x - 25x^2$$ --- 9. **Calculate:** $$\frac{a^3 + 7}{1 - b^2}$$ when $$a=2$$ and $$b=-4$$ Step 1: Calculate numerator: $$a^3 + 7 = 2^3 + 7 = 8 + 7 = 15$$ Step 2: Calculate denominator: $$1 - b^2 = 1 - (-4)^2 = 1 - 16 = -15$$ Step 3: Calculate value: $$\frac{15}{-15} = -1$$ --- 10. **Factorise fully:** $$30x^2p + 18x^2$$ Step 1: Find common factors: Common factor is $$6x^2$$ Step 2: Factor out: $$6x^2(5p + 3)$$ --- 11. **Solve for x:** $$6x - 2 = 4x + 14$$ Step 1: Subtract $$4x$$ from both sides: $$6x - 4x - 2 = 14$$ $$2x - 2 = 14$$ Step 2: Add 2 to both sides: $$2x = 16$$ Step 3: Divide both sides by 2: $$x = 8$$ --- 12. **Solve for y:** $$5(y + 6) = 3(y + 12)$$ Step 1: Expand both sides: $$5y + 30 = 3y + 36$$ Step 2: Subtract $$3y$$ from both sides: $$2y + 30 = 36$$ Step 3: Subtract 30 from both sides: $$2y = 6$$ Step 4: Divide both sides by 2: $$y = 3$$ --- 13. **Cost to tile bathroom wall:** Step 1: Calculate wall area: $$4.5 \times 6 = 27 \text{ m}^2 = 270000 \text{ cm}^2$$ Step 2: Tile area per tile: $$900 \text{ cm}^2$$ Step 3: Number of tiles needed: $$\frac{270000}{900} = 300$$ Step 4: Number of packs (10 tiles per pack): $$\frac{300}{10} = 30$$ Step 5: Cost per pack: 7.50 Step 6: Total cost: $$30 \times 7.50 = 225$$ --- 14. **Area of shape (rectangle + 2 isosceles triangles):** Step 1: Rectangle area: $$8 \times 4 = 32 \text{ cm}^2$$ Step 2: Triangle base = 4 cm (height of rectangle), height unknown but assume equal to 4 cm (isosceles with base 4 and height 4 for calculation) Triangle area: $$\frac{1}{2} \times 4 \times 4 = 8 \text{ cm}^2$$ Step 3: Two triangles area: $$2 \times 8 = 16 \text{ cm}^2$$ Step 4: Total area: $$32 + 16 = 48 \text{ cm}^2$$ --- 15. **Volume of cuboid given face areas:** Step 1: Let dimensions be $$l, w, h$$. Given: $$lw = 12$$ $$lh = 20$$ $$wh = 15$$ Step 2: Multiply all: $$(lw)(lh)(wh) = 12 \times 20 \times 15 = 3600$$ Step 3: Left side equals: $$(lwh)^2 = 3600$$ Step 4: Take square root: $$lwh = \sqrt{3600} = 60$$ Step 5: Volume = $$lwh = 60$$ --- 16. **Distance from home to school:** From graph, distance at 10 minutes is 400 m, at 20 minutes still 400 m, at 30 minutes 700 m. Distance from home to school = 700 m --- 17. **Describe motion from 10 to 20 minutes:** Distance remains constant at 400 m, so Iman is stationary or resting. --- 18. **Time to walk to school:** From 0 to 30 minutes, total distance 700 m. Time taken = 30 minutes --- 19. **Average speed:** $$\text{Speed} = \frac{\text{Distance}}{\text{Time}} = \frac{700}{30} \approx 23.33 \text{ m/min}$$ --- 20. **Harbour water depth graph:** a) Maximum depth is approximately 8 m b) Minimum depth is approximately 4 m c) Time between successive high tides (from 3 to 15 hours) = 12 hours --- **Total questions answered: 20**