# Algebraic fractions

Algebraic fractions are those that can be represented as the quotient of two polynomials, that is, as the division between two algebraic expressions that contain numbers and letters. It should be noted that both the numerator and the denominator of an algebraic fraction can contain additions, subtractions, multiplications or even powers.

Another point to keep in mind is that the result of an algebraic fraction must exist, so the denominator must be non-zero.

That is, the following condition is met, where A (x) and B (x) are the polynomials that form the algebraic fraction:

Some examples of algebraic fractions can be the following:

## Equivalent algebraic fractions

Two algebraic fractions are equivalent when the following is true:

This means that the result of both fractions is the same, and furthermore, the product of multiplying the numerator of the first fraction by the denominator of the second is equal to the product of the denominator of the first fraction by the numerator of the second.

We must take into account that to construct a fraction equivalent to the one we already have, we can multiply both the numerator and the denominator by the same number or by the same algebraic expression. For example, if we have the following fractions:

We verify that both fractions are equivalent and the following can also be noted:

That is, as we mentioned previously, when we multiply both the numerator and the denominator by the same algebraic expression, we obtain an equivalent algebraic fraction.

## Types of algebraic fractions

Fractions can be classified into:

• Simple: They are the ones we have observed throughout the article, where neither the numerator nor the denominator contain another fraction.
• Complex: The numerator and / or the denominator contain another fraction. An example can be the following:

Another way to classify algebraic fractions is as follows:

• Rational: When the variable is raised to a power that is not a fraction (like the examples we have seen throughout the article).
• Irrational: When the variable is raised to a power that is a fraction, as is the following case:

In the example, we could rationalize the fraction by replacing the variable with another that allows us not to have fractions as powers. So if x 1/2 = y and we replace in the equation we will have the following:

The idea is to find the least common multiple of the indices of the roots, which, in this case, is 1/2 (1 * 1/2). So if we have the following irrational equation:

We must first find the least common multiple of the indices of the roots, which would be: 2 * 5 = 10. So, we will have a variable y = x 1/10 . If we replace in the fraction, we will now have a rational fraction: