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When dealing with fractions, we often need to find a common denominator to combine them. This is especially true for rational expressions, which are fractions involving variables.
A common denominator is a shared multiple of the denominators of the fractions. It allows us to rewrite the fractions so they have the same bottom part (denominator), making it easier to add or subtract them. For instance, in the expression \(\frac{2}{3 x}+\frac{1}{5 x}\), the denominators are \(\frac{3x}{3x}\) and \(\frac{5x}{5x}\). The common multiple of these is \(\frac{15x}{15x}\). This transformation helps streamline the process of combining the fractions.
Let's break it down:
* Rewrite each fraction with the common denominator: \(\frac{2}{3 x} = \frac{2 \times 5}{3 x \times 5} = \frac{10}{15 x}\) and \(\frac{1}{5 x} = \frac{1 \times 3}{5 x \times 3} = \frac{3}{15 x}\).

Arithmetic with fractions involves basic operations like addition, subtraction, multiplication, and division. Each operation has specific rules. For addition and subtraction, as seen in the problem \(\frac{2}{3 x}+\frac{1}{5 x}\), having a common denominator is crucial.
Once fractions have the same denominator, you can combine them by simply adding or subtracting their numerators while keeping the denominator constant. For our case, the fractions become \(\frac{10}{15 x} + \frac{3}{15 x}\). Now, add the numerators: \[ 10 + 3 = 13 \]. This gives \(\frac{13}{15 x}\).
This is a basic yet essential skill for handling more complex rational expressions and equations. Understanding and practicing these operations can significantly ease working with fractions.

Simplifying rational expressions involves reducing them to their simplest form. This often means factoring polynomials and canceling out common factors in the numerator and the denominator.
For the given problem, \(\frac{2}{3 x}+\frac{1}{5 x}\), we did not need factoring, but recognizing that both fractions shared the variable 'x' in the denominator was key.
By finding a common denominator, rewriting the fractions, and combining them, we have effectively simplified the expression to \(\frac{13}{15 x}\). This is the simplest form for this case as there are no further common factors or terms to reduce.
Mastery of simplifying rational expressions is foundational for algebra and calculus, underpinning many mathematical techniques and problem-solving methods.