To express the number 12523 as a product of identical factors, we need to start by factorizing it.
First, we can check if it can be expressed as a power of an integer:
- Estimate the cube root of 12523. The cube root of \( 12523 \) is approximately \( 22.6 \).
- Check if it can be represented as \( n^3 \) where \( n \) is an integer close to 22. The closest integers are 22 and 23.
Calculating:
- \( 22^3 = 10648 \) (too low)
- \( 23^3 = 12167 \) (still low)
- \( 24^3 = 13824 \) (too high)
Thus, 12523 does not seem to be a perfect cube.
Now, let’s check if we can factor it further or find any higher roots. Trying the prime factorization method would reveal if there are any simpler roots or expressions.
By doing appropriate factorizations, we see that \( 12523 = 23 \times 545 \).
Further breaking down \( 545 \):
- \( 545 = 5 \times 109 \), where 109 is a prime number.
Thus, the prime factorization of 12523 is:
\[ 12523 = 5 \times 23 \times 109. \]
Unfortunately, there are no identical factors we can extract from this expression since all factors are distinct primes.
The number cannot be expressed as a product of identical factors (like squares, cubes, etc.) in the simplest way because it isn't a perfect power. Therefore, we cannot further simplify it under the request of the expression as a product of identical factors.
Hence, 12523 can be kept as it is or shown as:
\[ 12523 = 5^1 \times 23^1 \times 109^1 \]
Without additional context or constraints, this would be the simplest and most accurate representation. Would you like to find a specific form like a square or cube?