To express \(100 \frac{2}{3}\) as a product of identical factors, we first convert the mixed number to an improper fraction.
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Convert \(100 \frac{2}{3}\): \[ 100 \frac{2}{3} = 100 + \frac{2}{3} = \frac{300}{3} + \frac{2}{3} = \frac{302}{3} \]
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Now, we want to find a way to express \(\frac{302}{3}\) as a product of identical factors. We can factor out the common terms.
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Notice that \(\frac{302}{3}\) can be expressed similarly as: \[ = \frac{1}{3} \times 302 \]
This doesn't lead to identical factors yet, so let's think of \(100 \frac{2}{3}\) as a product involving 3:
- We know that \(100 \approx (10^2)\) so, we can play around with that:
Since we want to write it in the form of perfect squares or cubes, let's look at: \[ 302 = 100 \cdot 3 + 2 \rightarrow 302 = (100 + \frac{2}{3}) = 100.6666667 \text { in decimal} \]
While \(302\) equals to \((3 \cdot 100 + 2)\), this is not easily forming product identically by just pure multiplication/division.
Thus the simplest related breakdown to product of identical factors is not possible using classic algebraic means but simply recognizing the identity can guide us as closely to: \[ \frac{302}{3} = 100 \frac{2}{3} = \frac{1}{3} \cdot 302 \]
For a clear product analysis: If expressed completely, Keep it as: \[ \frac{302}{3} \quad \text{(which is } 100 \frac{2}{3} \text{ in simplest fractional terms) } \]
For a step-by-step approach, the factors could yield deeper algebra but finding a standard compact factorization yields maximum readability here. Thus seeing \(100 \cdots/3\) leads fundamentally as \(100.66667\) rather than forcing products towards square/cube.
Should there be focused values for identical checking, choosing something else may still facilitate clearer fractions when going outward towards roots or higher integers naturally will help—otherwise, any square root \(x^2\) or cubic doesn't neatly factorialize there present.
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