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When faced with the challenge of rationalizing a denominator, particularly those containing square roots, we utilize the concept of a conjugate. The conjugate of a denominator is a critical tool in mathematics for simplifying expressions that contain radicals. The term 'conjugate' essentially refers to a binomial formed by changing the sign between two terms. This technique might not be immediately apparent in our exercise because we are dealing with a single square root, but let's explore a more general scenario.

Imagine we have a denominator of the form \(a + \sqrt{b}\). The conjugate in this case would be \(a - \sqrt{b}\). Multiplying the original expression by this conjugate would eliminate the radical in the denominator. Even though our given problem does not have an explicit binomial, we treat \(\sqrt{xy}\) as if it were \(0 + \sqrt{xy}\), thus its conjugate is simply \(\sqrt{xy}\) once again.

By multiplying both the numerator and the denominator by this 'conjugate', the square root is effectively removed from the denominator, achieving the goal of rationalization. In our exercise, this results in the numerator containing the square root, while the denominator becomes a 'regular' number, without any radical.

After the multiplication with the conjugate, simplifying rational expressions becomes the focus. Simplification is an essential skill in algebra that makes expressions easier to work with. Once we have removed the radicals from the denominator, we can look at the expression's numerators and denominators separately to break them down into their simplest forms.

In the given exercise, after multiplying by the conjugate, the denominator is simplified to \(xy\), the product of the two variables under the radical. The numerator consisting of \(3\sqrt{xy}\) remains as is because it has been simplified as far as it can be with the given information. However, students should remain vigilant in case there are common factors in the numerator and denominator as these should be divided out to fully simplify the expression. Remember, the goal is to express the solution with no radicals in the denominator and the terms in their simplest possible form.

When the task is to rationalize the denominator, we often have to multiply fractions. The general rule for multiplying fractions is straightforward: we multiply the numerators together and the denominators together. This step-by-step approach breaks down when multiplication introduces a radical into the mix, requiring an additional step of rationalization.

For our exercise, the multiplication is simple, with the numerator \(3\) and the conjugate \(\sqrt{xy}\) being multiplied to yield \(3\sqrt{xy}\). Similarly, the denominator \(\sqrt{xy}\) is multiplied by itself, giving \(xy\). Multiplication of fractions, therefore, serves as the foundation for the process of rationalization. By understanding how to multiply fractions, especially when one or more radicals are involved, students can successfully maneuver through the process of rationalizing denominators and simplifying algebraic expressions.