Express the prime factorization of 1,200 as a product of prime numbers with exponents.(2 points)

Question

___^4 x ___ X ____^2=1,200

Answer 1

To find the prime factorization of 1,200, we start with dividing it by the smallest prime numbers.

1. Divide by 2:

   • \(1200 \div 2 = 600\)
   • \(600 \div 2 = 300\)
   • \(300 \div 2 = 150\)
   • \(150 \div 2 = 75\) (At this point 75 isn't divisible by 2, we switch to the next prime number)

2. Divide by 3:

   • \(75 \div 3 = 25\) (25 isn't divisible by 3, we switch to the next prime number)

3. Divide by 5:

   • \(25 \div 5 = 5\)
   • \(5 \div 5 = 1\)

Putting it all together, we have:

• From 2, we divided 4 times: \(2^4\)
• From 3, we used it once: \(3^1\)
• From 5, we divided 2 times: \(5^2\)

Thus, the prime factorization of 1,200 is: \[ 1,200 = 2^4 \times 3^1 \times 5^2 \]

So, in the given format:

• \(2^4 \times 3^1 \times 5^2 = 1,200\)

Therefore, filling in the blanks: 2^4 x 3 x 5^2 = 1,200.