Express the prime factorization of 1,200 as a product of prime numbers with exponents.(2 points)%0D%0A4----2 = 1,200
The prime factorization of 1,200 is:
1,200 = 2 × 2 × 2 × 2 × 3 × 5 × 5
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To express the prime factorization of 1,200, we need to find the prime numbers that multiply together to give 1,200.
We start by dividing 1,200 by the smallest prime number, which is 2:
1,200 ÷ 2 = 600
We continue dividing the quotient (600) by 2 repeatedly until we cannot divide any further:
600 ÷ 2 = 300
300 ÷ 2 = 150
150 ÷ 2 = 75
At this point, we cannot divide 75 by 2, so we move on to the next prime number, which is 3:
75 ÷ 3 = 25
We continue dividing by 3 until we cannot divide any further:
25 ÷ 3 = 8.33...
At this point, we move on to the next prime number, which is 5, since there is no smaller prime number that can divide 8.33... without leaving a decimal fraction. However, 5 does not divide 8.33..., so we move on to the next prime number, which is 7:
8.33... ÷ 7 = 1.19...
We can see that 7 does not divide evenly into 1.19..., so we have found all the prime factors of 1,200. We can express the prime factorization of 1,200 as:
1,200 = 2^4 * 3^1 * 5^0 * 7^1
Here, the exponents indicate how many times each prime factor appears. So, the prime factorization of 1,200 is 2 raised to the power of 4, multiplied by 3 raised to the power of 1, multiplied by 5 raised to the power of 0 (which means it does not appear in the factorization), multiplied by 7 raised to the power of 1.