A number is written with the following factorization: 2 to the power of 2 times 3 times 5 to the power of 4 times 8 times 11 to the power of 2. Is this factorization a prime factorization? Explain why or why not. If it is not correct, give the correct prime factorization of the number.
8 is not prime. 8 = 2^3
so, getting rid of all your words,
2^2 3 5^4 8 11^2
= 2^2 * 3 * 5^4 * 2^3 * 11^2
now collect like bases.
To determine if a factorization is a prime factorization, we need to check if all the factors are prime numbers and that they are listed in ascending order.
Let's examine the given factorization: 2^2 * 3 * 5^4 * 8 * 11^2.
First, we notice that 8 is not a prime number; it can be further factored into 2^3. So, we can rewrite the factorization as:
2^2 * 3 * 5^4 * (2^3) * 11^2.
Now, we have all prime factors listed, but they are not in ascending order. To achieve an ascending order, let's reorder the factors:
2^2 * (2^3) * 3 * 5^4 * 11^2.
Simplifying the expression further, we can combine the common bases:
2^(2 + 3) * 3^1 * 5^4 * 11^2.
This simplifies to:
2^5 * 3 * 5^4 * 11^2.
Now, we have the prime factorization of the number, which is:
2^5 * 3 * 5^4 * 11^2.
So, the given factorization is not a correct prime factorization, but the correct prime factorization of the number is 2^5 * 3 * 5^4 * 11^2.