Question

Express the prime factorization of 84 with exponents. Responses 3 ⋅4⋅7 3 ⋅4⋅7 - no response given 22 ⋅ 3 ⋅ 7 22 ⋅ 3 ⋅ 7 - no response given 2 ⋅ 32 ⋅7 2 ⋅ 32 ⋅7 - no response given 2 ⋅ 3 ⋅ 7

Answer 1

To determine the prime factorization of 84 with exponents, we'll first find its prime factors.

1. Start by dividing 84 by 2 (the smallest prime number):
\[
84 \div 2 = 42
\]
2. Divide 42 by 2 again:
\[
42 \div 2 = 21
\]
3. Now, divide 21 by 3 (the next smallest prime number):
\[
21 \div 3 = 7
\]
4. Finally, 7 is a prime number itself.

Now we put together the prime factors:
- The number 2 appears twice (from the divisions by 2).
- The number 3 appears once (from the division by 3).
- The number 7 appears once (as a prime number).

Thus, the prime factorization of 84 is:
\[
2^2 \cdot 3^1 \cdot 7^1
\]

In exponent form, this can be simplified as:
\[
2^2 \cdot 3 \cdot 7
\]

So the correct expression for the prime factorization of 84 is \(2^2 \cdot 3 \cdot 7\).