\( \frac{\sqrt{p^{2}-q^{2}} \times(p+q)^{\frac{5}{2}}}{(p-q)^{\frac{1}{2}}} \) if \( p \neq q \)

To help understand this expression better, let's explore its real-world applications. This mathematical form appears frequently in physics and engineering, particularly in problems involving energy or forces where variables represent different physical quantities. The manipulation of such expressions can lead to solutions that describe systems in equilibrium or calculate moment of inertia. Now, when tackling such expressions, be mindful of common pitfalls. One common mistake is to misinterpret the square root of a difference, \( \sqrt{p^2 - q^2} \), especially if either \( p \) or \( q \) is complex or negative. Always ensure that the values being used maintain the conditions necessary to keep the expression defined and valid!