To find the prime factorization of 54 using the factor tree you've described, we start with:
- \(54\) branches into \(9\) and \(6\).
- \(9\) branches into \(3\) and \(3\), both of which are prime numbers.
- \(6\) branches into \(2\) and \(3\), where \(2\) and \(3\) are also prime numbers.
Putting it all together, the prime factors of \(54\) are \(2\), \(3\), \(3\), and \(3\). Therefore, we can express the prime factorization as:
\[ 54 = 2 \times 3^3 \]
Now, let's check the options provided:
- \(9 \times 6\) - This is correct, as it describes the multiplication leading back to 54.
- \(2 \times 3\) - This is not the complete factorization for 54, so this is incorrect (54 is 2 times 3 times a 3).
- \(2 \times 3 \times 3 \times 3\) - This is a correct (and expanded) representation of the prime factorization of 54.
- \(2^3 \times 3 \times 3\) - This is incorrect because \(2^3\) is actually \(8\), not \(2\).
- \(2 \times 3^3\) - This is a correct and concise representation of the prime factorization of 54.
Thus, the correct options are:
- \(9 \times 6\)
- \(2 \times 3 \times 3 \times 3\)
- \(2 \times 3^3\)
In summary, the prime factorization of 54 is accurately represented as \(2 \times 3^3\).
3 answers