To rewrite \( 125 \frac{2}{3} \) as a product of identical factors, we first convert the mixed number into an improper fraction:
\[ 125 \frac{2}{3} = \frac{125 \cdot 3 + 2}{3} = \frac{375 + 2}{3} = \frac{377}{3} \]
Next, we need to express \( \frac{377}{3} \) in a way that represents a product of identical factors. First, let's express \( 377 \) as a product of its prime factors. \( 377 \) can be broken down as follows:
\[ 377 = 13 \times 29 \]
This means we can express the improper fraction as follows:
\[ \frac{377}{3} = \frac{13 \times 29}{3} \]
However, this does not lead us to identical factors. Let's redo our calculations to find an expression in terms of identical factors.
We can convert \( 125 \) into a product form: \[ 125 = 5^3 \]
We can express \( \frac{2}{3} \) as: \[ \frac{2}{3} \text{ remains unchanged.} \]
Thus: \[ 125 \frac{2}{3} = 5^3 + \frac{2}{3} \]
However, since the goal requires simplifying the term, we revert to our product: \[ = 125 \cdot \frac{2}{3} = \frac{125 \cdot 2}{3} = \frac{250}{3} \]
The expression \( 125 \frac{2}{3} \) equals \( 125 \times \frac{1}{3} \).
To find identical factors, we can represent it as: \[ \frac{(5^3 \cdot 2)}{3} = \frac{10 \cdot 5^2}{3} \]
However, not distinctly identical.
Conclusively:
While there isn't a direct way to express \( 125 \frac{2}{3} \) as a clean product of identical factors, the best simplified expression of the mixed number in a format close to that request would have been:
But mathematically settled: \[ = (125) \text{ combined with } \text{ other whereas leading fractions do not yield identical factors collectively.} \]
Thus the final answer is: \[ 125 \frac{2}{3} = \frac{377}{3} \text{ or simplified as } 125 \text{ and computations make challenging identical factored terms.} \]
An alternate simplest form for clear numerical representation remains \( 125 \frac{2}{3} \) as is rather than forcing identical since variations yield none.
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