91Ó°ÊÓ

When dealing with adding or subtracting rational expressions, finding a common denominator is key. The least common multiple (LCM) is the smallest number that can be divided evenly by all given denominators.
For instance, in the exercise involving the expressions \(\frac{9t}{5}\) and \(\frac{3}{4}\), we need to find the LCM of 5 and 4. Let's begin by listing their multiples:

• Multiples of 5: 5, 10, 15, 20, 25, ...
• Multiples of 4: 4, 8, 12, 16, 20, ...

By comparing these lists, we see that the number 20 appears first in both sequences. Thus, the LCM is 20. The LCM helps us create a common base for the fractions, allowing us to combine them easily.

To add or subtract rational expressions, you must first rewrite them with a common denominator, using equivalent fractions. An equivalent fraction is a different way of expressing the same value or proportion by using a new numerator and denominator.
After finding the LCM, we adjust both fractions to have this common denominator. For the expression \(\frac{9t}{5}\), multiply both numerator and denominator by 4 to get \(\frac{36t}{20}\). Similarly, for \(\frac{3}{4}\), multiply both by 5, resulting in \(\frac{15}{20}\).
This step ensures that the fractions are expressed in terms of the same basic unit (the denominator), making it possible to directly add or subtract their numerators.

Simplifying expressions means reducing them to their simplest form. Once rational expressions have the same denominator, you can add their numerators straightaway. Our example becomes \(\frac{36t}{20} + \frac{15}{20}\).
Adding the numerators together, we have:

• Numerator: \(36t + 15\)
• Denominator: 20

Thus, the expression simplifies to \(\frac{36t + 15}{20}\).
Always look for further opportunities to simplify, such as factoring the numerator. This ensures the solution is in its most reduced form, allowing easier interpretation and use in subsequent calculations.