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A rational expression is a fraction where the numerator and the denominator are both polynomials. Rational expressions are similar to rational numbers, but they include variables. An important aspect of working with rational expressions is simplifying them whenever possible. You can simplify a rational expression by factoring both the numerator and the denominator, and then canceling out the common factors. This makes it easier to work with and understand the expression. For example, in the expression \(\frac{\sqrt{15}-3}{6}\), we see that the denominator (6) is already in its simplest form. The goal is now to look at the rational expression and ensure it is as simplified as possible, which leads us into the next core concept.

Simplifying radicals involves expressing a radical expression in its simplest form. For instance, radicals like \(\sqrt{15}\) cannot be simplified further because 15 is not a perfect square and has no other perfect square factors. Simplifying radicals is essential when dealing with expressions under square root signs to make calculations easier.In our example, the radical expression is \(\sqrt{15}\), and it remains unchanged. However, simplifying radicals helps ensure that every part of our expression is in the simplest form possible before proceeding with other operations, such as addition, subtraction, or division.

Working with fractions involves a variety of operations including addition, subtraction, multiplication, and division. In the context of the given exercise, we focus on simplifying the fraction. When rationalizing the expression \(\frac{\sqrt{15}-3}{6}\), we rewrite it by dividing each term in the numerator by the denominator. This gives us two separate fractions: \(\frac{\sqrt{15}}{6}\) and \(\frac{3}{6}\). The fraction \(\frac{3}{6}\) simplifies to \(\frac{1}{2}\). Therefore, the simplified expression becomes \(\frac{\sqrt{15}}{6} - \frac{1}{2}\).Understanding how to properly perform operations on fractions is key to solving these types of problems accurately. Remember to always simplify your final answer.