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The greatest common divisor (GCD) is an essential concept in simplifying fractions. The GCD of two numbers is the largest number that divides both of them without leaving a remainder. For instance, to simplify \(\frac{12}{28}\), we need to find the GCD of 12 and 28.

The prime factorization of 12 is 2 x 2 x 3, and of 28 is 2 x 2 x 7. The common factors here are two 2s. Hence, the GCD of 12 and 28 is 4.

Once you find the GCD, you divide both the numerator and denominator by this number. For \(\frac{12}{28}\), you divide both 12 and 28 by 4 to get \(\frac{3}{7}\).

Remember, finding the GCD simplifies the fraction to its lowest terms in just one step, making it easier to grasp and apply.

Rational expressions are fractions that have polynomials as their numerators and/or denominators. Simplifying these expressions follows the same principles as simplifying regular fractions.

Take \(\frac{12}{28}\) for example, which is a simple rational expression. The process involves finding the GCD of the numerator and the denominator, and then dividing both by this number. This reduction helps in solving mathematical problems more efficiently.

Understanding rational expressions is critical for algebra and calculus, where you often need to simplify complex fractions to solve equations or integrals.

A fraction is in its lowest terms when the numerator and denominator have no common divisors other than 1. To reduce a fraction to its lowest terms, you divide both the numerator and denominator by their GCD.

For instance, \(\frac{12}{28}\) can be simplified by dividing both 12 and 28 by their GCD, which is 4. This results in \(\frac{3}{7}\), a fraction that cannot be simplified further.

When a fraction is in its lowest terms, it is easier to use in calculations and comparisons. It also provides a clearer understanding of the relationship between the numerator and the denominator, making mathematical problems simpler and more intuitive to solve.