To rewrite the expression \( 125 \frac{2}{3} \) as a product of identical factors, we first convert the mixed number into an improper fraction.
- Convert \( 125 \frac{2}{3} \) to an improper fraction: \[ 125 \frac{2}{3} = 125 + \frac{2}{3} = \frac{125 \times 3}{3} + \frac{2}{3} = \frac{375 + 2}{3} = \frac{377}{3} \]
Next, we need to express \( 377 \) as a product of identical factors. First, we will find the prime factorization of \( 377 \).
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Prime factorization of \( 377 \):
- Check divisibility by small prime numbers:
- \( 377 \) is not even, so not divisible by \( 2 \).
- Sum of digits \( 3 + 7 + 7 = 17 \) is not divisible by \( 3 \), so \( 377 \) isn't divisible by \( 3 \).
- It doesn’t end in \( 0 \) or \( 5 \), so it’s not divisible by \( 5 \).
- Checking \( 7 \): \( 377 ÷ 7 ≈ 53.857 \) (not divisible).
- Checking \( 11 \): \( 377 ÷ 11 ≈ 34.273 \) (not divisible).
- Checking \( 13 \): \( 377 ÷ 13 ≈ 29 \) (not divisible).
- Checking \( 19 \): \( 377 \div 19 = 19.842 \) (not divisible).
- \( 377 = 13 \times 29 \) (both are primes).
- Check divisibility by small prime numbers:
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Thus, the improper fraction is: \[ \frac{377}{3} = \frac{13 \times 29}{3} \]
Now we rewrite \( 377 \) as a product of identical factors (though we realize it cannot be expressed in identical factors directly due to its prime factorization as \( 13 \times 29 \)). However, we can approach it as: \[ = 125 \frac{2}{3} = \frac{377}{3} \approx \text{(there isn't a simpler product as we cannot factor } 3 \text{ as a product of identical factors.)} \]
However, to represent \( 125 \frac{2}{3} \) symbolically, one can write it as: \[ 125^{2/3} = (125^{1/3})^2 \]
Therefore, the answer to the request of representing \( 125 \frac{2}{3} \) as a product of identical factors can be represented using cube roots: \[ = (5^3)^{2/3} = (5^{3*2/3}) = 5^2 = 25 \] Thus, \( 125^{2/3} = 25\)
Final answer: \[ \boxed{25} \]