A number is written with the following factorization:2² x 3 x 5⁴ x 8 x 11².Is this factorization a prime factorization? Explain why or why not. If it is not correct, give the correct prime factorization of the number.
It is not, since 8 is not prime
8 = 2^3
so work that into the prime factors
To determine whether the factorization provided is a prime factorization, we need to check if all the factors are prime numbers and if they are raised to their respective exponents.
Let's analyze the given factorization:
2² x 3 x 5⁴ x 8 x 11²
Starting with 8, we know that it can be factored as 2³, so we can rewrite the factorization as:
2² x 3 x 5⁴ x 2³ x 11²
Now, let's analyze the factor 2. Since we have a 2² term and a 2³ term, we can simplify it to a single term of 2⁵:
2⁵ x 3 x 5⁴ x 11²
Now, the factorization looks like:
2⁵ x 3 x 5⁴ x 11²
Now, let's check if the remaining factors (3, 5, and 11) are prime numbers, which means they cannot be factored further.
3, 5, and 11 are prime numbers, as they do not have any other factors besides 1 and themselves.
Therefore, the factorization 2⁵ x 3 x 5⁴ x 11² is a prime factorization since all the factors are prime numbers and they are raised to their respective exponents.