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.
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Start by dividing 54 by the smallest prime number, which is 2: \[ 54 \div 2 = 27 \] So, we have \(54 = 2 \times 27\).
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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 \(54 = 2 \times 3 \times 9\).
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We continue factoring 9: \[ 9 \div 3 = 3 \] So, \(9 = 3 \times 3\), or \(9 = 3^2\).
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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.