The expression you provided is:

[ \frac{\sqrt[4]{(32x)^5}}{\sqrt[4]{y^5}} ]

Step-by-step solution:

1. Simplify the radicals: Both the numerator and the denominator have the same root (4th root). We can rewrite the expression as a single 4th root:

   [ \frac{\sqrt[4]{(32x)^5}}{\sqrt[4]{y^5}} = \sqrt[4]{\frac{(32x)^5}{y^5}} ]

2. Factor the powers: We can distribute the powers inside the root:

   [ \sqrt[4]{\frac{(32x)^5}{y^5}} = \sqrt[4]{\left(\frac{32x}{y}\right)^5} ]

3. Simplify the 4th root: When you have an expression like $\sqrt[4]{a^5}$, you can simplify it as $a^{5/4}$. Applying this to the entire expression:

   $\sqrt[4]{\left(\frac{32x}{y}\right)^5} = \left(\frac{32x}{y}\right)^{5/4}$

Thus, the simplified expression is:

$\left(\frac{32x}{y}\right)^{5/4}$

Final Answer:

$\left(\frac{32x}{y}\right)^{5/4}$

Do you want further details or have any questions?

Here are 5 relative questions to expand on the solution:

1. How would the answer change if the powers were different?
2. What is the significance of rational exponents in this simplification?
3. Can you explain how to handle different root powers?
4. How do you apply exponent rules to radicals?
5. What happens if the base of the expression includes negative numbers?

Tip: When dealing with radicals, always check if you can simplify them by combining the powers and the roots into a single term!