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Simplifying fractions means reducing them to their simplest form. If possible, we aim to make both the numerator and the denominator as small as possible. For example, take \( \frac{ 8 }{ 12 } \). This fraction can be simplified by finding the greatest common divisor (GCD) of both the numerator and the denominator, which is 4 in this case.

Hence: \( \frac{ 8 }{ 12 } = \frac{ 8 ÷ 4 }{ 12 ÷ 4 } = \frac{ 2 }{ 3 } \)

Simplifying fractions is important because it makes further calculations easier. In the provided exercise, we simplified \( \frac{b}{b-a} \) to \( -\frac{b}{a-b} \). This step is crucial as it sets up the fraction for further operations by making both denominators the same.

To add or subtract fractions, they must have the same denominator. A common denominator can be thought of as a shared base that allows for the fractions to be combined.

In mathematical terms, if you have two fractions: \( \frac{ a }{ c } \) and \( \frac{ b }{ d } \), you need a common denominator to perform operations. The least common denominator (LCD) is often the easiest choice, though any common denominator will work.

In the given exercise, the common denominator for both fractions was \( a-b \). This step was fairly straightforward because we simplified the second fraction in the first step. Thus, both fractions transformed into \( \frac{a}{a-b} \) and \( -\frac{b}{a-b} \). Now, they can be combined into one fraction: \( \frac{a - b}{a - b} \).

Algebraic expressions consist of variables, constants, and operations (like addition, subtraction, multiplication, and division). They may look complicated at first, but understanding the basic operations can make working with them easier.

In the provided problem, the expressions \( \frac{a}{a-b} \) and \( \frac{b}{b-a} \) are examples of algebraic fractions. Simplification and finding common denominators are essential skills when dealing with these expressions.

Once both fractions have the same denominator as shown, you can combine their numerators. Then you can manipulate the resulting expression more easily. In our solution, after subtraction, the result became \( \frac{a - b}{a - b} \), which simplifies to 1. This is a vital concept as it shows how algebraic fractions can be simplified to the most basic form, often making more complex problems easier to solve.