Answer:
(A), (B) and (E)
Step-by-step explanation:
The given expression is:
![(\frac{750}{512})^\frac{1}{3}](https://tex.z-dn.net/?f=%28%5Cfrac%7B750%7D%7B512%7D%29%5E%5Cfrac%7B1%7D%7B3%7D)
We know that ![(\frac{x}{y})^\frac{1}{n}=\sqrt[n]{\frac{x}{y}}=\frac{\sqrt[n]{x}}{\sqrt[n]{y}}](https://tex.z-dn.net/?f=%28%5Cfrac%7Bx%7D%7By%7D%29%5E%5Cfrac%7B1%7D%7Bn%7D%3D%5Csqrt%5Bn%5D%7B%5Cfrac%7Bx%7D%7By%7D%7D%3D%5Cfrac%7B%5Csqrt%5Bn%5D%7Bx%7D%7D%7B%5Csqrt%5Bn%5D%7By%7D%7D)
(A) The given option is:
![\frac{\sqrt[3]{750}}{\sqrt[3]{512}}](https://tex.z-dn.net/?f=%5Cfrac%7B%5Csqrt%5B3%5D%7B750%7D%7D%7B%5Csqrt%5B3%5D%7B512%7D%7D)
Now, the given expression can be solved as:
![(\frac{750}{512})^\frac{1}{3}=\frac{\sqrt[3]{750}}{\sqrt[3]{512}}](https://tex.z-dn.net/?f=%28%5Cfrac%7B750%7D%7B512%7D%29%5E%5Cfrac%7B1%7D%7B3%7D%3D%5Cfrac%7B%5Csqrt%5B3%5D%7B750%7D%7D%7B%5Csqrt%5B3%5D%7B512%7D%7D)
which is equivalent to the given option, thus this option is correct.
(B) The given option is:
![\sqrt[3]{\frac{750}{512}}](https://tex.z-dn.net/?f=%5Csqrt%5B3%5D%7B%5Cfrac%7B750%7D%7B512%7D%7D)
Now, the given expression can be solved as:
![(\frac{750}{512})^\frac{1}{3}=\sqrt[3]{\frac{750}{512}}](https://tex.z-dn.net/?f=%28%5Cfrac%7B750%7D%7B512%7D%29%5E%5Cfrac%7B1%7D%7B3%7D%3D%5Csqrt%5B3%5D%7B%5Cfrac%7B750%7D%7B512%7D%7D)
which is equivalent to the given option, thus this option is correct.
(C) The given option is:
![\frac{5}{8}](https://tex.z-dn.net/?f=%5Cfrac%7B5%7D%7B8%7D)
Now, the given expression can be solved as:
![(\frac{750}{512})^\frac{1}{3}=(\frac{375}{256})^\frac{1}{3}](https://tex.z-dn.net/?f=%28%5Cfrac%7B750%7D%7B512%7D%29%5E%5Cfrac%7B1%7D%7B3%7D%3D%28%5Cfrac%7B375%7D%7B256%7D%29%5E%5Cfrac%7B1%7D%7B3%7D)
which is not equivalent to the given option, thus this option is incorrect.
(D) The given option is:
![\frac{750}{\sqrt[3]{512}}](https://tex.z-dn.net/?f=%5Cfrac%7B750%7D%7B%5Csqrt%5B3%5D%7B512%7D%7D)
Now, the given expression can be solved as:
![(\frac{750}{512})^\frac{1}{3}=\frac{\sqrt[3]{750}}{\sqrt[3]{512}}](https://tex.z-dn.net/?f=%28%5Cfrac%7B750%7D%7B512%7D%29%5E%5Cfrac%7B1%7D%7B3%7D%3D%5Cfrac%7B%5Csqrt%5B3%5D%7B750%7D%7D%7B%5Csqrt%5B3%5D%7B512%7D%7D)
which is not equivalent to the given option, thus this option is incorrect.
(E) The given option is:
![\frac{5}{8}\sqrt[3]{6}](https://tex.z-dn.net/?f=%5Cfrac%7B5%7D%7B8%7D%5Csqrt%5B3%5D%7B6%7D)
Now, the given expression can be solved as:
![(\frac{750}{512})^\frac{1}{3}=\frac{5}{8}\sqrt[3]{6}](https://tex.z-dn.net/?f=%28%5Cfrac%7B750%7D%7B512%7D%29%5E%5Cfrac%7B1%7D%7B3%7D%3D%5Cfrac%7B5%7D%7B8%7D%5Csqrt%5B3%5D%7B6%7D)
which is equivalent to the given option, thus this option is correct.
(F) The given option is:
\frac{\sqrt[3]{750}}{512}
Now, the given expression can be solved as:
![(\frac{750}{512})^\frac{1}{3}=\frac{\sqrt[3]{750}}{\sqrt[3]{512}}](https://tex.z-dn.net/?f=%28%5Cfrac%7B750%7D%7B512%7D%29%5E%5Cfrac%7B1%7D%7B3%7D%3D%5Cfrac%7B%5Csqrt%5B3%5D%7B750%7D%7D%7B%5Csqrt%5B3%5D%7B512%7D%7D)
which is not equivalent to the given option, thus this option is incorrect.