Question

Rewrite the expression 125 2/3 as a product of identical factors to . Submit your answer in simplified form.

Answer 1

To rewrite the mixed number \( 125 \frac{2}{3} \) as a product of identical factors, we will first convert it into an improper fraction.

1. Convert the mixed number to an improper fraction:
\[
125 \frac{2}{3} = \frac{3 \times 125 + 2}{3} = \frac{375 + 2}{3} = \frac{377}{3}
\]

Now, we want to find the prime factorization of \( 377 \) to see if we can express it in terms of identical factors.

2. Factor \( 377 \):
- Check if \( 377 \) is divisible by any prime number:
- It is odd, so not divisible by \( 2 \).
- Sum of digits \( 3 + 7 + 7 = 17 \) (not divisible by \( 3 \)).
- It does not end with \( 0 \) or \( 5 \) (not divisible by \( 5 \)).
- Dividing \( 377 \) by \( 7 \),
\[
377 \div 7 \approx 53.857 \quad (\text{not divisible})
\]
- Dividing \( 377 \) by \( 11 \),
\[
377 \div 11 \approx 34.273 \quad (\text{not divisible})
\]
- Dividing \( 377 \) by \( 13 \),
\[
377 \div 13 = 29 \quad (\text{exactly divisible})
\]

Thus, the prime factorization of \( 377 \) is:
\[
377 = 13 \times 29
\]

3. Rewrite the improper fraction:
\[
\frac{377}{3} = \frac{13 \times 29}{3}
\]

Since \( 3 \), \( 13 \), and \( 29 \) are all prime, we focus on trying to express \( 125 \frac{2}{3} \) in a simpler form.

4. Now let’s consider \( 125 \frac{2}{3} \) altogether:

Realizing that \( 125 = 5^3 \) can be part of the factors.

So, the final expression \( 125 \frac{2}{3} \) can be seen as:
\[
125 \frac{2}{3} = \frac{125 \times 2}{3} = \frac{250}{3}
\]

However, in terms of simple depiction:

The simplified way to write this in identical factors using \( 5^3 \) as our starting point (and assuming 2 and 3 get carried along):

Thus expressing \( 125 \frac{2}{3} \) in terms of using identical factors we get:

Finally, we can express \( 125 \frac{2}{3} \) as a product of factors:
\[
\frac{250}{3} \quad \text{which factors down simply to base 5 with constant koefficients and gives us} \quad 250 \text{as } (5 \times 5 \times 5) \text{ and } 2, 3 \text{ as divisors.}
\]

Explicitly:

The answer can be summarized as:
\[
\frac{250}{3} = \frac{5^3 \cdot 2}{3}
\]

If you need purely the product of identical factors representation for practical applications, identify \( 5^3 \) (in general product forms), as constants and taking into consideration the fractions adjustments thereafter.

Thus, as simplified value:

In a box, we conclude:
\[
125\frac{2}{3} \rightarrow \frac{(5^3) \cdot (2)}{3}
\]
In identical groupings!