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
a) sqrt(36) Step 1: Find the prime factorization of 36. 36 = 2 × 18 18 = 2 × 9 9 = 3 × 3 So, the prime factorization of 36 is 2 × 2 × 3 × 3. Step 2: Group the prime factors into pairs and take one factor from each pair. sqrt(36) = sqrt(2 × 2 × 3 × 3) sqrt(36) = sqrt(2^2 × 3^2) sqrt(36) = 2 × 3 Step 3: Multiply the factors. sqrt(36) = 6 The square root of 36 is 6. b) sqrt(144) Step 1: Find the prime factorization of 144. 144 = 2 × 72 72 = 2 × 36 36 = 2 × 18 18 = 2 × 9 9 = 3 × 3 So, the prime factorization of 144 is 2 × 2 × 2 × 2 × 3 × 3. Step 2: Group the prime factors into pairs and take one factor from each pair. sqrt(144) = sqrt(2 × 2 × 2 × 2 × 3 × 3) sqrt(144) = sqrt(2^2 × 2^2 × 3^2) sqrt(144) = 2 × 2 × 3 Step 3: Multiply the factors. sqrt(144) = 12 The square root of 144 is 12. c) sqrt(324) Step 1: Find the prime factorization of 324. 324 = 2 × 162 162 = 2 × 81 81 = 3 × 27 27 = 3 × 9 9 = 3 × 3 So, the prime factorization of 324 is 2 × 2 × 3 × 3 × 3 × 3. Step 2: Group the prime factors into pairs and take one factor from each pair. sqrt(324) = sqrt(2 × 2 × 3 × 3 × 3 × 3) sqrt(324) = sqrt(2^2 × 3^2 × 3^2) sqrt(324) = 2 × 3 × 3 Step 3: Multiply the factors. sqrt(324) = 18 The square root of 324 is 18. d) sqrt(1296) Step 1: Find the prime factorization of 1296. 1296 = 2 × 648 648 = 2 × 324 324 = 2 × 162 162 = 2 × 81 81 = 3 × 27 27 = 3 × 9 9 = 3 × 3 So, the prime factorization of 1296 is 2 × 2 × 2 × 2 × 3 × 3 × 3 × 3. Step 2: Group the prime factors into pairs and take one factor from each pair. sqrt(1296) = sqrt(2 × 2 × 2 × 2 × 3 × 3 × 3 × 3) sqrt(1296) = sqrt(2^2 × 2^2 × 3^2 × 3^2) sqrt(1296) = 2 × 2 × 3 × 3 Step 3: Multiply the factors. sqrt(1296) = 36 The square root of 1296 is 36. Problem solving: Use any three examples to show that the square of any odd number is always an odd number. Example 1: Choose the odd number 1. 1^2 = 1 × 1 = 1 Since 1 is not divisible by 2, it is an odd number. Example 2: Choose the odd number 3. 3^2 = 3 × 3 = 9 Since 9 is not divisible by 2, it is an odd number. Example 3: Choose the odd number 5. 5^2 = 5 × 5 = 25 Since 25 is not divisible by 2, it is an odd number. These examples demonstrate that the square of an odd number is always an odd number.