** 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: (ax^{3}) / (4x^{5}Y).

We divide the numerator and denominator by x^{3}.

(ax^{3}) / (4x^{5}y) = (ax^{3 }÷ x^{3}) / (4x^{5}y ÷ x^{3}) = a / (4x^{2}Y)

Like ay 4x^{2}and 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: (x^{2 }+ and^{2}) / (x^{4 }- Y^{4}).

Whereas x^{4 }- Y^{4} = (x^{2 }+ and^{2}) (x^{2 }- Y^{2}), we have:

(x^{2 }+ and^{2}) / (x^{4 }- Y^{4}) = (x^{2 }+ and^{2}) / [(x^{2 }+ and^{2}) (x^{2 }- Y^{2})] = 1 / (x^{2 }- Y^{2})

Like 1 y (x^{2 }- Y^{2}) have no factor in common, this fraction is irreducible, that is, it is in its simplest form.

**Prime factorization method.**

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

**Example**: Simplify the following fraction: (48ab^{2}) / (84a^{3} b^{3})

First we find the prime factorization of the numerator:

48ab^{2} = 2^{4}∙ 3 ∙ a ∙ b^{2}

Then the denominator:

84a^{3} b^{3 }= 2^{2}∙ 3 ∙ 7 ∙ a^{3}∙ b^{3}

We write the fraction with the prime factors in it:

(48ab^{2}) / (84a^{3} b^{3}) = (2^{4}∙ 3 ∙ a ∙ b^{2}) / (2^{2}∙ 3 ∙ 7 ∙ a^{3}∙ b^{3}) =

We cancel the common factors:

= (2^{2}) / (7a^{2}b) =

= 4 / (7a^{2}b)

Like 4 and 7a^{2}b have no factor in common, this fraction is irreducible, that is, it is in its simplest form.