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To simplify a rational expression, we first need to find the Greatest Common Divisor (GCD). The GCD is the largest number that divides both the numerator and the denominator without leaving a remainder. Knowing how to determine the GCD is crucial because it helps us reduce fractions to their simplest form.
Consider the example \(\frac{18}{30}\). To find the GCD of 18 and 30, we list the factors of each number:

• Factors of 18: 1, 2, 3, 6, 9, 18
• Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30

By comparing these lists, we see that the greatest number in both lists is 6. Therefore, 6 is the GCD of 18 and 30.
Finding the GCD simplifies the process of reducing fractions. When both the numerator and denominator are divided by the GCD, we convert the fraction to its lowest terms.

A rational expression consists of a numerator (the top part of the fraction) and a denominator (the bottom part of the fraction). For instance, in \(\frac{18}{30}\), 18 is the numerator, and 30 is the denominator.
Understanding the numerator and denominator helps in the simplification process:

• Numerator: This is the number of parts considered.
• Denominator: This shows the total number of equal parts.

In our example, we have 18 parts out of 30. When we simplify this fraction, we are finding an equivalent fraction with the smallest possible numerator and denominator that still represents the same value. After identifying the GCD, we divide both the numerator and the denominator by this number to simplify the fraction.
Thus, for \(\frac{18}{30}\), we divide both 18 and 30 by 6 (the GCD), resulting in \(\frac{18 ÷ 6}{30 ÷ 6} = \frac{3}{5}\). Here, 3 is the simplified numerator, and 5 is the simplified denominator.

When a fraction is reduced to its lowest terms, it means that the numerator and the denominator have no common factors other than 1. This is the simplest form of the fraction.
For the fraction \(\frac{18}{30}\), we found that its GCD is 6. Dividing both parts by their GCD, we get \(\frac{3}{5}\). Since 3 and 5 have no common factors other than 1, \(\frac{3}{5}\) is indeed in its lowest terms.
Reducing to lowest terms makes fractions easier to understand and work with. It clarifies the simplest relationship between the numerator and the denominator. Always check your result to ensure there are no common factors other than 1. This means your fraction is fully simplified and accurately represents the value in the simplest way possible.