# Reduce a fraction to its lowest terms

Simplify or reduce a fraction is to convert it into an equivalent fraction whose terms are prime to each other. When this occurs, the fraction is irreducible and then it is in its plus simple expression or to your minimal expression. There are several ways to reduce fractions; One of them is to divide the numerator and denominator by the same factor in common, another is using prime factors.

## How to simplify or reduce a fraction

### General method

We divide the numerator and denominator by their common factors until they are prime to each other.

Example: Simplify the following fraction: (ax3) / (4x5Y).

We divide the numerator and denominator by x3.

(ax3) / (4x5y) = (ax3 ÷ x3) / (4x5y ÷ x3) = a / (4x2Y)

Like ay 4x2and they do not have any factor in common, this fraction is irreducible, that is to say, it is in its minimum expression.

#### If the terms of the fraction are polynomials

We factor polynomials as far as possible and remove factors common to the numerator and denominator.

Example: Simplify the following fraction: (x2 + and2) / (x4 - Y4).

Whereas x4 - Y4 = (x2 + and2) (x2 - Y2), we have:

(x2 + and2) / (x4 - Y4) = (x2 + and2) / [(x2 + and2) (x2 - Y2)] = 1 / (x2 - Y2)

Like 1 y (x2 - Y2) have no factor in common, this fraction is irreducible, that is, it is in its simplest form.

### Prime factorization method.

1. We find the prime factors of the numerator and denominator.
2. We write the prime factorization of each number.
3. We eliminate common factors.

Example: Simplify the following fraction: (48ab2) / (84a3 b3)

First we find the prime factorization of the numerator:

48ab2 = 24∙ 3 ∙ a ∙ b2

Then the denominator:

84a3 b3 = 22∙ 3 ∙ 7 ∙ a3∙ b3

We write the fraction with the prime factors in it:

(48ab2) / (84a3 b3) = (24∙ 3 ∙ a ∙ b2) / (22∙ 3 ∙ 7 ∙ a3∙ b3) =

We cancel the common factors:

= (22) / (7a2b) =

= 4 / (7a2b)

Like 4 and 7a2b have no factor in common, this fraction is irreducible, that is, it is in its simplest form.