To perform a prime factorization of the number \(198\), we can break it down into its prime factors.
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Start by dividing \(198\) by the smallest prime number, which is \(2\): \[ 198 \div 2 = 99 \] Thus, \(198 = 2 \times 99\).
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Next, we factor \(99\). The smallest prime that divides \(99\) is \(3\): \[ 99 \div 3 = 33 \] Therefore, \(99 = 3 \times 33\) and we have \(198 = 2 \times 3 \times 33\).
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Now, we continue factoring \(33\). The smallest prime dividing \(33\) is \(3\) again: \[ 33 \div 3 = 11 \] So \(33 = 3 \times 11\) and we have \(198 = 2 \times 3 \times 3 \times 11\), or \(198 = 2 \times 3^2 \times 11\).
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Now, \(11\) is a prime number and cannot be factored further.
Therefore, the prime factorization of \(198\) is: \[ 198 = 2^1 \times 3^2 \times 11^1 \]
In summary, the prime factors of \(198\) are \(2\), \(3\), and \(11\) in exponential form \(2^1, 3^2, 11^1\).