b. \( \sqrt[4]{\frac{x^{4}}{y^{8}}} \)

Did you know that simplifying roots can make complex expressions feel like a breeze? When you tackle \( \sqrt[4]{\frac{x^{4}}{y^{8}}} \), you're basically looking for the fourth root of the numerator and the denominator separately. In this case, it simplifies beautifully to \( \frac{x}{y^2} \), showcasing how roots can elegantly condense information! Speaking of real-world applications, this process isn't just academic! Simplifying expressions like these is fundamental in fields such as engineering and physics, where precise calculations are crucial. For instance, when dealing with formulas for resistance or energy loss, applying root simplifications can help derive manageable equations for real problems, making complex scenarios much more understandable and solvable!