1. **Prime Factorization and Multiplication** - Problem: Verify the prime factorization and multiplication of $2^4 \times 3^6 \times 5^2 \times 7^4$. - Formula: Prime factorization expresses a number as a product of primes raised to powers. - Calculation: - $2^4 = 2 \times 2 \times 2 \times 2 = 16$ - $3^6 = 3 \times 3 \times 3 \times 3 \times 3 \times 3 = 729$ - $5^2 = 5 \times 5 = 25$ - $7^4 = 7 \times 7 \times 7 \times 7 = 2401$ - Multiply stepwise: - $16 \times 729 = 11664$ - $11664 \times 25 = 291600$ - $291600 \times 2401 = 699,984,600$ - Final answer: $699,984,600$ 2. **Fraction Simplification** - Problem: Simplify $\frac{11}{3 \times 15} \times \frac{15}{22}$. - Simplify denominator and numerator: - $3 \times 15 = 45$ - Expression becomes $\frac{11}{45} \times \frac{15}{22}$ - Multiply numerators and denominators: - Numerator: $11 \times 15 = 165$ - Denominator: $45 \times 22 = 990$ - Simplify $\frac{165}{990}$ by dividing numerator and denominator by 165: - $\frac{165 \div 165}{990 \div 165} = \frac{1}{6}$ - Final answer: $\frac{1}{6}$ 3. **Fraction Division and Multiplication** - Problem: Calculate $\frac{1}{2} \div 3$ and multiply by $\frac{5}{3}$. - Division by 3 is multiplication by $\frac{1}{3}$: - $\frac{1}{2} \times \frac{1}{3} = \frac{1}{6}$ - Multiply by $\frac{5}{3}$: - $\frac{1}{6} \times \frac{5}{3} = \frac{5}{18}$ - Final answer: $\frac{5}{18}$ 4. **Cube Root and Addition** - Problem: Calculate $\sqrt[3]{1000} + 3^2$. - Calculate cube root: - $\sqrt[3]{1000} = 10$ - Calculate square: - $3^2 = 9$ - Add: - $10 + 9 = 19$ - Final answer: $19$ 5. **Subtraction and Powers** - Problem: Calculate $6 - 4$ and subtract $2^5$ from 100. - Calculate: - $6 - 4 = 2$ - $2^5 = 32$ - $100 - 32 = 68$ - Final answer: $68$ 6. **Polynomial Simplification** - Problem: Simplify $7x4 - 7x3x - 5x * 2 - 5xx5x$. - Interpret and simplify: - $7 \times 4 = 28$ - $7 \times 3x = 21x$ - $5 \times 2 = 10$ - $5x \times 5x = 25x^2$ - Expression becomes: - $28 - 21x - 10x - 25x^2$ - Combine like terms: - $28 - 31x - 25x^2$ - Final answer: $-25x^2 - 31x + 28$ 7. **Evaluate Expression with Variables** - Problem: Calculate $\frac{2^3 + 7}{1 - b^2}$ with $b = -4$. - Calculate numerator: - $2^3 = 8$ - $8 + 7 = 15$ - Calculate denominator: - $b^2 = (-4)^2 = 16$ - $1 - 16 = -15$ - Divide: - $\frac{15}{-15} = -1$ - Final answer: $-1$ 8. **Greatest Common Factor (GCF) and Expression Simplification** - Problem: Simplify $30x^2 p + 18x^2$ using GCF. - GCF of 30 and 18 is 6. - Divide terms by GCF: - $\frac{30x^2 p}{6x^2} = 5p$ - $\frac{18x^2}{6x^2} = 3$ - Factor out GCF: - $6x^2 (5p + 3)$ - Final answer: $6x^2 (5p + 3)$ 9. **Solve Linear Equation** - Problem: Solve $6x - 2 = 4x + 14$. - Rearrange: - $6x - 4x - 2 = 14$ - $2x - 2 = 14$ - $2x = 16$ - $x = \frac{16}{2} = 8$ - Final answer: $x = 8$ 10. **Solve for y** - Problem: Solve $5(y + 6) = 3(y + 12)$. - Expand: - $5y + 30 = 3y + 36$ - Rearrange: - $5y - 3y = 36 - 30$ - $2y = 6$ - $y = \frac{6}{2} = 3$ - Final answer: $y = 3$ 11. **Area and Volume Calculations** - Area of rectangle: $8 \times 4 = 32$ cm$^2$ - Area of triangle: $\frac{1}{2} \times 8 \times 4 = 16$ cm$^2$ - Total area: $32 + 32 = 64$ cm$^2$ - Volume of cuboid: $10 \times 4 \times 5 = 200$ cm$^3$ 12. **Tile Packing and Cost Calculation** - Convert meters to cm: - $4.5m = 450cm$ - $6m = 600cm$ - Calculate area: - $450 \times 600 = 270,000$ cm$^2$ - Number of tiles: - $270,000 \div 900 = 300$ - Packs needed: - $300 \div 10 = 30$ - Total cost: - $30 \times 7.50 = 225$ - Final answer: $225$ 13. **Average Speed Calculation** - Distance: 700 meters - Time: 30 minutes - Average speed: - $\frac{700}{30} \approx 23.33$ meters per minute - Final answer: $23.33$ meters per minute **Summary:** All calculations and answers are accurate except the prime factorization multiplication which should be $699,984,600$ instead of $26,460$. Fraction simplifications and algebraic solutions are correct.