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

To express the prime factorization of 1,200, we first break it down into its prime factors.

1. Start by dividing by 2 (the smallest prime number):

   • 1200 ÷ 2 = 600
   • 600 ÷ 2 = 300
   • 300 ÷ 2 = 150
   • 150 ÷ 2 = 75

2. Now, divide by 3 (the next smallest prime number):

   • 75 ÷ 3 = 25

3. Finally, divide by 5:

   • 25 ÷ 5 = 5
   • 5 ÷ 5 = 1

So, the prime factorization of 1,200 can be collected as:

• \( 2^4 \) (since 2 was used 4 times)
• \( 3^1 \) (since 3 was used once)
• \( 5^2 \) (since 5 was also used 2 times)

Putting this all together, the prime factorization of 1,200 is: \[ 2^4 \cdot 3^1 \cdot 5^2 \]

Thus, the answer is: \[ 2^4 \cdot 3^1 \cdot 5^2 = 1200 \]