Question

Express the prime factorization of 1200 as a product of prime numbers with exponents. (2 points)

Answer 1

To find the prime factorization of 1200, we start by dividing the number by the smallest prime numbers and continuing this process with the quotient until we reach 1.

1. Start with 1200 and divide by 2 (the smallest prime):
\[
1200 \div 2 = 600
\]
\[
600 \div 2 = 300
\]
\[
300 \div 2 = 150
\]
\[
150 \div 2 = 75
\]

We have divided by 2 a total of 4 times.

2. Next, we divide 75 by the next smallest prime number, which is 3:
\[
75 \div 3 = 25
\]

3. Finally, we divide 25 by 5:
\[
25 \div 5 = 5
\]
\[
5 \div 5 = 1
\]

We have divided by 5 a total of 2 times.

Now, combining all the factors, we have:
- 2 used 4 times
- 3 used 1 time
- 5 used 2 times

Thus, we can express the prime factorization of 1200 as:
\[
1200 = 2^4 \times 3^1 \times 5^2
\]
This is the prime factorization of 1200 in exponential form.