# Simplifying fractions 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) / (8a2b).

We divide the numerator and denominator by 2a:

(2a ÷ 2a) / (8a2b ÷ 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 / (2a2x + 2a3).

• We factor the denominator:

3ab / (2a2x + 2a3) = 3ab / [2a2 (x + a)]

• We divide the numerator and denominator between a:

(3ab ÷ a) / (2a2(x + a) ÷ a) = 3b / [2a (x + a2)]

Thus:

3ab / (2a2x + 2a3) = 3b / [2a (x + a2)]

As 3b and 2a (x + a2) have no factor in common, this fraction is irreducible.

## Exercises

1. Simplify or reduce to its simplest expression:

a) (ax3) / (4x5Y)

b) (6x2y3) / 3x

c) (x2 - Y2) / (x2 + 2xy + y2)

d) (x2 + and2) / (x4 - Y4)

a) We must simplify a fraction whose terms are monomials, therefore, we divide the numerator and denominator by x3.

(ax3) / (4x5y) = (ax3 ÷ x3) / (4x5y ÷ x3) = a / 4x2y

The fraction is in simplest form.

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

(6x2y3) / 3x = (6x2y3 ÷ 3x) / (3x ÷ 3x) = 2xy3 / 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:

(6x2y3) / 3x = 2xy3

The fraction is in simplest form.

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

x2 - Y2 = (x + y) (x - y)

x2 + 2xy + y2 = (x + y) (x + y)

Thus:

(x2 - Y2) / (x2 + 2xy + y2) = [(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:

x2 - Y2 = (x + y) (x - y)

We have to:

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

The fraction is in simplest form.