Answer:
A. ![(\sqrt[3]{125})^9\ and\ (125)^{\frac{9}{3}}](https://tex.z-dn.net/?f=%28%5Csqrt%5B3%5D%7B125%7D%29%5E9%5C%20and%5C%20%28125%29%5E%7B%5Cfrac%7B9%7D%7B3%7D%7D)
D. 
Step-by-step explanation:
Equivalent expressions are those expressions that simplify to same form.
Now, let us check each of the given options.
Option A:
![(\sqrt[3]{125})^9\ and\ (125)^{\frac{9}{3}}](https://tex.z-dn.net/?f=%28%5Csqrt%5B3%5D%7B125%7D%29%5E9%5C%20and%5C%20%28125%29%5E%7B%5Cfrac%7B9%7D%7B3%7D%7D)
We know that,
![\sqrt[n]{x} =x^{\frac{1}{n}}](https://tex.z-dn.net/?f=%5Csqrt%5Bn%5D%7Bx%7D%20%3Dx%5E%7B%5Cfrac%7B1%7D%7Bn%7D%7D)
Therefore, ![\sqrt[3]{125} =(125)^{\frac{1}{3}}](https://tex.z-dn.net/?f=%5Csqrt%5B3%5D%7B125%7D%20%3D%28125%29%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D)
Thus the first expression becomes;

Now, using law of indices
, we get

Therefore,
are equivalent.
Option B:

Consider the second expression 
We know that,


Therefore,
. Hence, the expressions
are not equivalent.
Option C:

We know that,
![x^{\frac{1}{n}}=\sqrt[n]{x}](https://tex.z-dn.net/?f=x%5E%7B%5Cfrac%7B1%7D%7Bn%7D%7D%3D%5Csqrt%5Bn%5D%7Bx%7D)
Therefore, ![4^{\frac{1}{5}}=\sqrt[5]{4}](https://tex.z-dn.net/?f=4%5E%7B%5Cfrac%7B1%7D%7B5%7D%7D%3D%5Csqrt%5B5%5D%7B4%7D)
Now, ![\sqrt[5]{4}\ne (\sqrt 4)^5](https://tex.z-dn.net/?f=%5Csqrt%5B5%5D%7B4%7D%5Cne%20%28%5Csqrt%204%29%5E5)
Therefore, the expressions
are not equivalent.
Option D:

Using law of indices
, we get

Now, we know that, 
So, 
Therefore,
are equivalent.