To rewrite the expression \( 10032 \) as a product of identical factors, we first need to break it down using prime factorization.
Let's find the prime factorization of \( 10032 \):
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Since \( 10032 \) is even, divide by \( 2 \): \[ 10032 \div 2 = 5016 \]
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Continuing to divide by \( 2 \): \[ 5016 \div 2 = 2508 \] \[ 2508 \div 2 = 1254 \] \[ 1254 \div 2 = 627 \]
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Now \( 627 \) is not even, so let's try dividing by \( 3 \): \[ 627 \div 3 = 209 \]
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Next, check \( 209 \). It is not divisible by \( 3 \) (the sum of digits \( 2 + 0 + 9 = 11 \) does not divide by \( 3 \)). Check divisibility by \( 11 \): \[ 209 \div 11 = 19 \]
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Now \( 19 \) is prime.
So, the complete prime factorization of \( 10032 \) is: \[ 10032 = 2^4 \cdot 3^1 \cdot 11^1 \cdot 19^1 \]
Now, we can express \( 10032 \) as a product of identical factors:
\[ 10032 = (2^4) \cdot (3^1) \cdot (11^1) \cdot (19^1) \]
If we simplify it to use the smallest individual factors: \[ 10032 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 11 \cdot 19 \]
Hence, in simplified form, you can write the product as: \[ (2^4) \cdot 3 \cdot 11 \cdot 19 \]
You can also express all identical factors for the exponent: \[ 10032 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 11 \cdot 19 \]
If an exact formatted answer is needed: \[ 10032 = 2^4 \cdot 3^1 \cdot 11^1 \cdot 19^1 \]
That can also be written in simple terms: \[ 2^4 \cdot 3 \cdot 11 \cdot 19 \]
This representation shows \( 10032 \) as a product of its identical factors, each represented by its prime factorization.