This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Okay, let's tackle the 'a' parts of those problems. Problem 9a: Step 1: The problem asks to find the Highest Common Factor (HCF) of 18a^2y and 27x^2y^2. Step 2: Find the prime factorization of the numerical coefficients: 18 = 2 × 3^2 27 = 3^3 Step 3: Identify the common prime factors and their lowest powers: The common prime factor is 3. The lowest power of 3 is 3^2. HCF of 18 and 27 is 3^2 = 9. Step 4: Identify the common variables and their lowest powers: For variable 'a', it appears only in 18a^2y. For variable 'x', it appears only in 27x^2y^2. For variable 'y', it appears in both terms. The lowest power of 'y' is y^1 (from 18a^2y). Step 5: Multiply the common numerical HCF by the common variables with their lowest powers. The HCF is 9y. Problem 10a: Step 1: The problem asks to find the sum of the prime factors of 180. Step 2: Find the prime factorization of 180. 180 = 18 × 10 18 = 2 × 9 = 2 × 3^2 10 = 2 × 5 Step 3: Combine the prime factors: 180 = (2 × 3^2) × (2 × 5) = 2^2 × 3^2 × 5^1. Step 4: The distinct prime factors are 2, 3, and 5. Step 5: Find the sum of these distinct prime factors: Sum = 2 + 3 + 5 = 10. The sum of the prime factors of 180 is 10. Problem 11a: Step 1: The problem states that the sum of 8 and a certain number is equal to the product of the number and 3. We need to find the number. Step 2: Let the certain number be x. Step 3: Translate the statement into an equation: "The sum of 8 and a certain number": 8 + x "The product of the number and 3": 3x "is equal to": 8 + x = 3x Step 4: Solve the equation for x. Subtract x from both sides: 8 = 3x - x 8 = 2x Step 5: Divide by 2: x = (8)/(2) x = 4 The number is 4. Problem 12a: Step 1: The problem asks to round off 0.01256 to three significant figures. Step 2: Identify the significant figures. Non-zero digits are always significant. Leading zeros (zeros before non-zero digits) are not significant. In 0.01256, the first significant figure is 1. The second significant figure is 2. The third significant figure is 5. Step 3: Look at the digit immediately after the third significant figure, which is 6. Step 4: Since 6 is 5 or greater, round up the third significant figure (5). 5 becomes 6. Step 5: Keep the leading zeros as placeholders for the decimal point. The number rounded to three significant figures is 0.0126. That's 2 down. 3 left today — send the next one.