Question

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

Answer 1

What rational number could be used as an exponent to rewrite 23−−√4
?

Answer 2

could you not do the box thing and put it how its soupose to be

Answer 3

Just the answer normaly

Answer 4

so the answer is 1/4?

Answer 5

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

1. 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 \( 377 \).

2. 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).

3. 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:
\[
\boxed{25}
\]

Answer 6

To rewrite \( \sqrt[4]{23} \) using a rational number as an exponent, we start with the understanding of roots in terms of exponents. The fourth root of a number can be expressed as that number raised to the power of \( \frac{1}{4} \).

Thus, we can express \( \sqrt[4]{23} \) as:

\[
\sqrt[4]{23} = 23^{\frac{1}{4}}
\]

Here, \( \frac{1}{4} \) is the rational number that is used as the exponent.

So, the rational number that could be used as an exponent to rewrite \( \sqrt[4]{23} \) is:

\[
\boxed{\frac{1}{4}}
\]

Answer 7

Certainly! The rational number that could be used as an exponent to rewrite \( \sqrt[4]{23} \) is:

\(\frac{1}{4}\).

So, we can express it as:

\[
\sqrt[4]{23} = 23^{\frac{1}{4}}
\]

Answer 8

The rational number is \( \frac{1}{4} \).

Answer 9

Yes, the answer is \( \frac{1}{4} \).