Question

Rewrite the expression 125 2/3 as a product of identical factors to find the product. Submit your answer in simplified form. Remember to simplify using the denominator of the exponent before applying the numerator on the exponent

Answer 1

No

Answer 2

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

1. Convert \( 125 \frac{2}{3} \) to an improper fraction:
\[
125 \frac{2}{3} = 125 + \frac{2}{3} = \frac{125 \cdot 3}{3} + \frac{2}{3} = \frac{375 + 2}{3} = \frac{377}{3}
\]

Next, we want to express \( \frac{377}{3} \) as a product of identical factors.

2. Since \( 377 = 125 + 2 \) doesn't factor nicely, let's look at \( 125 \).
\[
125 = 5^3
\]

Thus, \( 125 \frac{2}{3} = \frac{5^3}{3} + \frac{2}{3} = \frac{5^3 + 2}{3} \), but \( 5^3 + 2 \) does not lead to a simpler form.

3. Since we are interested in representing \( 125 \frac{2}{3} \) in exponents:
\[
125 \frac{2}{3} = 5^3 \cdot \left( \frac{2}{3} \right)
\]

Applying the exponent property, simplify \( 5^3 \cdot \frac{2}{3} \):
\[
= \frac{5^3 \cdot 2}{3}
\]

While we can't simplify this significantly further, we can express the fraction:
\[
= \frac{2 \times 125}{3} = \frac{250}{3}
\]

Thus, the final simplified form of the expression \( 125 \frac{2}{3} \) as a product of identical factors is:
\[
= \frac{250}{3}
\]

Therefore, the answer is:
\[
\frac{250}{3}
\]

Answer 3

I apologize for any confusion. Let's correctly rewrite \( 125 \frac{2}{3} \) as a product of identical factors and simplify correctly.

1. First, let's convert the mixed number \( 125 \frac{2}{3} \) into an improper fraction:
\[
125 \frac{2}{3} = 125 + \frac{2}{3} = \frac{125 \cdot 3}{3} + \frac{2}{3} = \frac{375 + 2}{3} = \frac{377}{3}
\]

2. Now, we can write \( \frac{377}{3} \) in terms of powers and see if there's a way to express it as a product of identical factors. However, \( 377 \) is not a perfect power, so let's break this down:

3. Instead, \( 125 \) can be factored as \( 5^3 \) because:
\[
125 = 5^3
\]
Thus, we can treat \( 125 \frac{2}{3} \) as:
\[
125 \frac{2}{3} = 125 \cdot \frac{2}{3} = 5^3 \cdot \frac{2}{3}
\]

4. To express this using exponents, recognize that \( \frac{2}{3} \) does not relate to the base \( 5 \). However, we want to express \( 5^3 \) as identical factors:
\[
5^3 = 5 \cdot 5 \cdot 5
\]

So, we get:
\[
125 \frac{2}{3} = \frac{2 \cdot (5 \cdot 5 \cdot 5)}{3}
\]

Since we're looking for the product of identical factors involving exponents:

5. To express \( 125 \frac{2}{3} \) in simplified form without fractions, we might treat it as:
\[
125 \frac{2}{3} = (5^{3} \cdot 2) \cdot \frac{1}{3}
\]

The simplified expression remains:
\[
\frac{250}{3} \text{ (This was our earlier result correctly presented, not an identical factor of 125 itself).}
\]

Alternatively, if looking just for products of identical factors relating back to mixed number and factoring:
\( 125 = 5^3 \)
But not leading to the same profound simplification as \( 125 \).

Thus, a clearer result obtaining identical factor solution:
- Expressed \( 125 \frac{2}{3} \) as:
- **Identical factors are \( 5, 5, 5 \).** Further factoring with \( 2/3 \).

If there’s any specific format or characteristics you're looking for that I misunderstood, please let me know!