In general, a fraction is the quotient of two algebraic expressions

a / b with b ≠ 0

In which the numerator represents the dividend and the denominator the divisor. Thus, 4/8 represents the quotient of a division in which the numerator 4 is the dividend and 8 the divisor. Fractions usually need to be **simplified**, that is to say, **transformed into a simpler equivalent fraction**. Example: 4/8 »1/2. In the simplification of fractions, the numerator and denominator are divided by the same factor in common.

**Definition**

**Simplify 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 **simplest expression** or to your **minimal expression**.

**How to simplify a fraction**

**1. When the terms of the fraction are monomials**

The numerator and denominator are divided by their common factors until they are prime to each other. Example: Simplify the following fraction: (2a) / (8a^{2}b).

We divide the numerator and denominator by 2a:

(2a ÷ 2a) / (8a^{2}b ÷ 2a) = (1 ∙ 1) / (4 ∙ a ∙ b) = 1 / 4ab

Since 1 and 4ab have no factor in common, this fraction is irreducible.

**2. When the terms of the fraction are polynomials**

Polynomials are factored as much as possible and factors common to the numerator and denominator are removed. Example: Simplify the following fraction: 3ab / (2a^{2}x + 2a^{3}).

- We factor the denominator:

3ab / (2a^{2}x + 2a^{3}) = 3ab / [2a^{2} (x + a)]

- We divide the numerator and denominator between a:

(3ab ÷ a) / (2a^{2}(x + a) ÷ a) = 3b / [2a (x + a^{2})]

Thus:

3ab / (2a^{2}x + 2a^{3}) = 3b / [2a (x + a^{2})]

As 3b and 2a (x + a^{2}) have no factor in common, this fraction is irreducible.

**Exercises**

- Simplify or reduce to its simplest expression:

a) (ax^{3}) / (4x^{5}Y)

b) (6x^{2}y^{3}) / 3x

c) (x^{2 }- Y^{2}) / (x^{2 }+ 2xy + y^{2})

d) (x^{2 }+ and^{2}) / (x^{4 }- Y^{4})

a) We must simplify a fraction whose terms are monomials, therefore, 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

The fraction is in simplest form.

b) The terms of this fraction are also monomials; We divide the numerator and denominator by 3x.

(6x^{2}y^{3}) / 3x = (6x^{2}y^{3 }÷ 3x) / (3x ÷ 3x) = 2xy^{3 }/ 1

Note that when simplifying all the factors of the denominator disappeared, it remains in this 1, which can be removed. The result is an integer expression:

(6x^{2}y^{3}) / 3x = 2xy^{3 }

The fraction is in simplest form.

c) We must simplify a fraction whose terms are polynomials. Considering that:

x^{2 }- Y^{2 }= (x + y) (x - y)

x^{2 }+ 2xy + y^{2 }= (x + y) (x + y)

Thus:

(x^{2 }- Y^{2}) / (x^{2 }+ 2xy + y^{2}) = [(x + y) (x - y)] / [(x + y) (x + y)] = (x - y) / (x + y)

The fraction is in simplest form.

d) Like the previous fraction, its terms are polynomials. Considering that:

x^{2 }- Y^{2 }= (x + y) (x - y)

We have to:

(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 })

The fraction is in simplest form.