To express \(10032\) as a product of identical factors, we first need to find its prime factorization.
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Start dividing \(10032\) by the smallest prime number, \(2\):
- \(10032 \div 2 = 5016\)
- \(5016 \div 2 = 2508\)
- \(2508 \div 2 = 1254\)
- \(1254 \div 2 = 627\) (no longer divisible by \(2\))
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Now, proceed with the next prime number, \(3\):
- \(627 \div 3 = 209\) (no longer divisible by \(3\))
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Next, check \(209\) for divisibility by \(11\) (next prime):
- \(209 \div 11 = 19\) (both \(11\) and \(19\) are primes)
Now we have the complete prime factorization: \[10032 = 2^4 \times 3^1 \times 11^1 \times 19^1\]
To express \(10032\) as a product of identical factors, we note we have \(4\) factors of \(2\). The combinations will be from \(2^4\), giving us \(16\):
Thus, we can express \(10032\) as: \[10032 = (2^4 \times 3 \times 11 \times 19)\]
To simplify and express it as a product of the same factor: \[10032 = (48)^2 \times 19\]
Recognizing similar prime factors and rearranging, we arrive at using \(64\), since \(2^6=64\).
We might choose to return to the base prime factor, hence:
Final answer: \[10032 = 64 \times 156\]
However, noting we want identical factors: We can break that down, thus finally concluding that: \[10032 = 32^2 \times 3^1 \times 11^1\] And recognizing \(32\): Thus, converting those into products, noting that \(4 \times 4 = 16 \), thus keeping wanted factors.
In simplest, finished final notation: 10032 = 32² * 3 * 11 * 19, thus output final requested to submit: 10032 = 2^4 * 3 * 11 * 19 or more simplified *10032 = (32^2)* (Please write in the form required output should be noted as indicated).
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