A **algebraic expression** it is a combination of variables, numbers and signs of operations (addition, subtraction, multiplication, division, empowerment, settlement, logarithms). In a **algebraic fraction***, *the numerator and denominator are algebraic expressions. It has the same properties as a numerical fraction.

**Definition**

An algebraic fraction is a fraction whose numerator and denominator are algebraic expressions. For example: 2x / (2 + √x) is a fraction whose numerator is a monomial and the denominator a binomial.

**Simplification**

For **simplify or reduce a** **algebraic fraction** we must 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 an algebraic fraction?**

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

**Example**: 4x / (16x^{2}Y)

We divide the numerator and denominator by 4x:

(4x ÷ 4x) / (16x^{2}y ÷ 4x) = (1 ∙ 1) / (4 ∙ x ∙ y) = 1 / 4xy

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

**Add and subtract**

**If the fractions have the same denominator**

- We simplify the given fractions if possible.
- We add or subtract the numerators of the fractions and the common denominator remains.
- We simplify the fraction that results, if possible.

**Example**: Add the following fractions: (√x + 1) / 9; (2 + x) / 9

Both fractions have the same denominator, therefore, we add the numerators and keep the common denominator:

[(√x + 1) / 9] + [(2 + x) / 9] = (√x + x + 3) / 9

### If the fractions have a different denominator

- We simplify the given fractions if possible.
- We reduce the fractions given to the lowest common denominator.
- We add or subtract the numerators of the resulting fractions and keep the denominator obtained in step 2.
- We simplify the fraction that results, if possible.

**Example**: subtract the following fractions: (x + 12) / 4; 3 / (x + 1)

Both fractions have different denominators, we must reduce them to the lowest common denominator. To do this, we find the lcm of the denominators: lcm = 4 (x + 1). We divide the lcm by the denominators of the fractions:

4 (x + 1) ÷ 4 = (x + 1) and 4 (x + 1) ÷ (x + 1) = 4

The quotients obtained are multiplied by the respective numerators, that is:

[(x + 12) / 4] - [3 / (x + 1)] = [(x + 12) (x + 1)] / [4 (x + 1)] - [(3 · 4) / 4 (x + 1)] =

= [(x^{2 }+ 13x + 12) / 4 (x + 1)] - [12/4 (x + 1)] =

We subtract the numerators

[(x^{2 }+ 13x + 12) / 4 (x + 1)] - [12/4 (x + 1)] = (x^{2 }+ 13x) / 4 (x + 1)

**Multiplication and division**

**Rules for multiplying algebraic fractions**

- We factorize (as much as possible) the terms of the fractions we want to multiply.
- We simplify, removing the common factors in the numerators and denominators.
- We multiply the numerators together and the result is written as the numerator of the resulting fraction.
- We multiply the denominators together and the result is written as the denominator of the resulting fraction.
- We simplify the resulting fraction, if possible.

**Example**: multiply the following fractions: 25x / (5x + 5); 16x / (4x^{2})

[25x / (5x + 5)] × [16x / (4x^{2})] = {5^{2}x / [5 (x + 1)]} × [4^{2}x / (4x^{2})] = [5x / (x + 1)] × (4 / x) =

= [(5x) (4)] / [(x + 1) (x)] = 20x / [x (x + 1)]

We simplify the fraction:

20x / [x (x + 1)] = 20 / (x + 1)

**Rules for dividing algebraic fractions**

- We factorize (as much as possible) the terms of the fractions we want to divide.
- We simplify, removing the common factors in the numerators and denominators.
- We multiply the first fraction by the
**reciprocal**of the second. - We simplify the resulting fraction, if possible.

** Example**: divide the following fractions: 7x / (√49x + 7); 3 / (√x + 1)

The reciprocal of the second fraction is:

(√x + 1) / 3

So:

[7x / (√49x + 7)] ÷ [3 / (√x + 1)] = [7x / (√49x + 7)] × [(√x + 1) / 3] =

= [(7 ∙ x) / (√7^{2}x + 1)] × [(√x + 1) / 3] = [(7 ∙ x) / 7 (√x + 1)] × [(√x + 1) / 3] =

= [x / (√x + 1)] × [(√x + 1) / 3] =

= [x (√x + 1)] / [3 (√x + 1)] =

= x / 3