Which is a prime factorization?
a. 5^2 x 7^2 x 18^3
b. 5^3 x 7^2 x 19^2
c. 6^2 x 7^3 x 18^2
d. 4^2 x 5^2 x 15^3
Prime factorization will decompose an integer number into powers of its prime factors.
Therefore there should not be any composite factors. Composite factors are numbers which can be further decomposed into prime factors, such as 6=2*3, or 18=2*3*3=2*3².
Look at the last column of all the answers and decide which ones are not prime factorizations.
The answer is B because all factors in 5^3 x 7^2 x 19^2 are prime. For example, 5,7,19, 3 and 2 cannot be decomposed. An example of decomposed is 9 where it can be divided by a number other than 1 and the number itself.
To find the prime factorization of a given number, you need to express it as the product of prime numbers. Let's analyze each option:
a. 5^2 x 7^2 x 18^3
To simplify this expression, we should break down 18 into its prime factors. 18 can be expressed as 2 x 3^2. Therefore, we have:
5^2 x 7^2 x 2^3 x 3^6
Now, we have the prime factorization of option a.
b. 5^3 x 7^2 x 19^2
Option b is already expressed as the product of prime numbers. Therefore, this is the prime factorization of option b.
c. 6^2 x 7^3 x 18^2
Similar to option a, let's break down 6 and 18 into their prime factors:
6 can be expressed as 2 x 3, so we have:
2^2 x 3^2 x 7^3 x 2^2 x 3^2
Simplifying this further, we get:
2^4 x 3^4 x 7^3
Now, we have the prime factorization of option c.
d. 4^2 x 5^2 x 15^3
Again, let's break down 15 into its prime factors:
15 can be expressed as 3 x 5, so we have:
2^4 x 5^2 x 3^6 x 3^2 x 5^3
Simplifying this further, we get:
2^4 x 3^8 x 5^5
Now, we have the prime factorization of option d.
Therefore, the correct answer is:
b. 5^3 x 7^2 x 19^2