4. \( (x y)^{-1} \)

To find the inverse of the product \( (x y)^{-1} \), remember that when we deal with products of numbers or variables, the inverse is applied to each factor. The rule is that the inverse of a product is the product of the inverses in reverse order. Thus, \( (x y)^{-1} = y^{-1} x^{-1} \). In other words, multiplying two things together and then taking the inverse is like flipping them around and taking their individual inverses! This principle is super useful in various fields like algebra, physics, and engineering, particularly when solving equations involving products. For instance, if you're dealing with fractions in calculus, knowing how to manipulate inverses can simplify expressions that otherwise seem cumbersome. It's like having a secret weapon in your mathematical toolkit—whipping out those inverses makes everything a little more manageable!