Question

Determine the exponent of the
prime factor 3 when
expressing the prime
factorization of 54.

Answer 1

To determine the exponent of the prime factor 3 in the prime factorization of 54, we first need to find the prime factorization of 54.

1. Start by dividing 54 by the smallest prime number, which is 2:
\[
54 \div 2 = 27
\]
So, we have \(54 = 2 \times 27\).

2. Next, we factor 27. The smallest prime number that divides 27 is 3:
\[
27 \div 3 = 9
\]
Thus, we have \(27 = 3 \times 9\), leading to \(54 = 2 \times 3 \times 9\).

3. We continue factoring 9:
\[
9 \div 3 = 3
\]
So, \(9 = 3 \times 3\), or \(9 = 3^2\).

4. Combining all the factors together, we have:
\[
54 = 2 \times 3 \times 3 \times 3 = 2 \times 3^3.
\]

Thus, the prime factorization of 54 is:
\[
54 = 2^1 \times 3^3.
\]

The exponent of the prime factor 3 in this factorization is **3**.