Express the fotbwing numbers in question tod as prolucts of prime actorsin index Mofation using ore of the abone methods 12 (b) 40 (c) 3780 (d) 750

To express the given numbers as products of prime factors in index notation, we will factor each number into its prime components. ### (a) 12 1. Start with 12. 2. Divide by the smallest prime number, which is 2: \( 12 \div 2 = 6 \) 3. Divide 6 by 2: \( 6 \div 2 = 3 \) 4. 3 is a prime number. So, the prime factorization of 12 is: \[ 12 = 2^2 \times 3^1 \] ### (b) 40 1. Start with 40. 2. Divide by 2: \( 40 \div 2 = 20 \) 3. Divide 20 by 2: \( 20 \div 2 = 10 \) 4. Divide 10 by 2: \( 10 \div 2 = 5 \) 5. 5 is a prime number. So, the prime factorization of 40 is: \[ 40 = 2^3 \times 5^1 \] ### (c) 3780 1. Start with 3780. 2. Divide by 2: \( 3780 \div 2 = 1890 \) 3. Divide 1890 by 2: \( 1890 \div 2 = 945 \) 4. Divide 945 by 3: \( 945 \div 3 = 315 \) 5. Divide 315 by 3: \( 315 \div 3 = 105 \) 6. Divide 105 by 3: \( 105 \div 3 = 35 \) 7. Divide 35 by 5: \( 35 \div 5 = 7 \) 8. 7 is a prime number. So, the prime factorization of 3780 is: \[ 3780 = 2^2 \times 3^3 \times 5^1 \times 7^1 \] ### (d) 750 1. Start with 750. 2. Divide by 2: \( 750 \div 2 = 375 \) 3. Divide 375 by 3: \( 375 \div 3 = 125 \) 4. Divide 125 by 5: \( 125 \div 5 = 25 \) 5. Divide 25 by 5: \( 25 \div 5 = 5 \) 6. 5 is a prime number. So, the prime factorization of 750 is: \[ 750 = 2^1 \times 3^1 \times 5^3 \] ### Summary of Prime Factorizations - \( 12 = 2^2 \times 3^1 \) - \( 40 = 2^3 \times 5^1 \) - \( 3780 = 2^2 \times 3^3 \times 5^1 \times 7^1 \) - \( 750 = 2^1 \times 3^1 \times 5^3 \)