To express the number 10032 as a product of identical factors, we first need to find its prime factorization.
- Divide 10032 by small prime numbers:
- 10032 is even, so divide by 2: \[ 10032 \div 2 = 5016 \]
- Continuing with 2: \[ 5016 \div 2 = 2508 \]
- Continuing with 2: \[ 2508 \div 2 = 1254 \]
- Continuing with 2: \[ 1254 \div 2 = 627 \]
- Now, 627 is not divisible by 2. Check 3: \[ 627 \div 3 = 209 \]
- 209 is not divisible by 3. Check 11 (the next smallest prime): \[ 209 \div 11 = 19 \]
- 19 is a prime number.
So the complete prime factorization of 10032 is: \[ 10032 = 2^4 \times 3^1 \times 11^1 \times 19^1 \]
To express this as a product of identical factors, we need to determine the largest identical factor. The factor with the highest exponent here is \( 2 \), so we can rewrite the expression as follows.
Since \( 2^4 = 16 \), we can find pairs: \[ 10032 = (2^2 \times 3 \times 11 \times 19) \times (2^2) = (4 \times 3 \times 11 \times 19) \times 4 \]
Calculating \( 4 \times 3 \times 11 \times 19 \):
- First, \( 3 \times 11 = 33 \)
- Then, \( 33 \times 19 = 627 \)
- Finally, \( 4 \times 627 = 2508 \)
Thus: \[ 10032 = 2508 \times 4 \]
In simplified form, the product of identical factors for 10032 could be expressed as: \[ (4 \times 3 \times 11 \times 19) \times 4 \]
This yields: \[ 10032 = (4 \times 627) \rightarrow (4^4) \text{ (since 4 is the simplest identical factor)} \]
Thus, \( 10032 \) can ultimately be simplified as: \[ = (2^2) \cdot (2^2 \cdot 3 \cdot 11 \cdot 19)^4 \]
Putting this together yields:
\[ 10032 = (2^2) \cdot (3 \times 11 \times 19)^4 \]
Thus the product of identical factors is: \[ (4 \cdot 627) \times 4 \text{ or simply \( 10032 \) expressed in its simplest factor form: } \] \[ 2^4 \cdot 3^1 \cdot 11^1 \cdot 19^1. \]