To determine which expression is equivalent to [tex]\(\left(\frac{125^2}{125^{\frac{4}{3}}}\right)\)[/tex], we need to simplify the given expression step by step.
Let's start with the expression:
[tex]\[
\left(\frac{125^2}{125^{\frac{4}{3}}}\right)
\][/tex]
Using the properties of exponents, specifically the rule [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex], we can combine the exponents in the numerator and the denominator as follows:
[tex]\[
\frac{125^2}{125^{\frac{4}{3}}} = 125^{2 - \frac{4}{3}}
\][/tex]
Next, we need to simplify the exponent [tex]\(2 - \frac{4}{3}\)[/tex]. To do this, we find a common denominator for the fractions. The common denominator between 2 (which can be written as [tex]\(\frac{6}{3}\)[/tex] for consistency) and [tex]\(\frac{4}{3}\)[/tex] is 3:
[tex]\[
2 - \frac{4}{3} = \frac{6}{3} - \frac{4}{3} = \frac{6-4}{3} = \frac{2}{3}
\][/tex]
So, we have:
[tex]\[
125^{2 - \frac{4}{3}} = 125^{\frac{2}{3}}
\][/tex]
At this point, we need to evaluate [tex]\(125^{\frac{2}{3}}\)[/tex]. The exponent [tex]\(\frac{2}{3}\)[/tex] means taking the cube root of 125 and then squaring the result.
First, the cube root of 125 is:
[tex]\[
\sqrt[3]{125} = 5
\][/tex]
Then, we square the result of the cube root:
[tex]\[
5^2 = 25
\][/tex]
Hence, we find that:
[tex]\[
125^{\frac{2}{3}} = 25
\][/tex]
Therefore, the expression [tex]\(\left(\frac{125^2}{125^{\frac{4}{3}}}\right)\)[/tex] simplifies to 25.
Thus, the correct answer is:
[tex]\[
\boxed{25}
\][/tex]
