which of these is part of the prime factorization of 80?

Question

2^4
2^2 x 4
2 x 5^2
5^4

Answer 1

To find the prime factorization of 80, we can start by dividing it by the smallest prime numbers.

1. \( 80 \div 2 = 40 \)
2. \( 40 \div 2 = 20 \)
3. \( 20 \div 2 = 10 \)
4. \( 10 \div 2 = 5 \)

Now, we cannot divide 5 by 2, but it is a prime number.

So we have:

\[ 80 = 2 \times 2 \times 2 \times 2 \times 5 = 2^4 \times 5^1 \]

Now we can analyze the given options:

1. \( 2^4 \) - This is a part of the factorization (it includes the 2's).
2. \( 2^2 \times 4 \) - This can be broken down to \( 2^2 \times 2^2 = 2^4\), so it represents the same quantity and is part of the prime factorization.
3. \( 2 \times 5^2 \) - This is not part of the factorization, as we only have \( 5^1 \) in the factorization of 80.
4. \( 5^4 \) - This is also not part of the factorization, as we only have \( 5^1 \).

So, the choices that are part of the prime factorization of 80 are \( 2^4 \) and \( 2^2 \times 4\).