Question

Determine the exponent of the prime factor 3 when expressing the prime factorization of 54. Responses 3 3 - no response given 0 0 - no response given 1 1 - no response given 2

Answer 1

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

We can start by dividing 54 by the smallest prime number, which is 2:

\[
54 \div 2 = 27
\]

So, we have:

\[
54 = 2 \times 27
\]

Next, we factor 27. Since 27 is an odd number, we try the next smallest prime number, which is 3:

\[
27 \div 3 = 9
\]

Thus, we can express 27 as:

\[
27 = 3 \times 9
\]

Now, we factor 9:

\[
9 \div 3 = 3
\]

Thus, we have:

\[
9 = 3 \times 3
\]

Putting all of this together, we can write the prime factorization of 54 as:

\[
54 = 2 \times 3 \times 3 \times 3
\]

This can be expressed using exponents as:

\[
54 = 2^1 \times 3^3
\]

From this, we see that the exponent of the prime factor 3 in the prime factorization of 54 is 3.

Thus, the answer is:

**3**