Algebraic Fractions - Revision Notes

In these revision notes for Algebraic Fractions, we cover the following key points:

• What are algebraic fractions? Where do they differ from numerical fractions?
• What are disallowed values? Why must we consider them when dealing with algebraic fractions?
• How to simplify algebraic fractions.
• How do the special algebraic identities help in simplifying algebraic fractions?
• How to add/subtract algebraic fractions with the same/different denominator.
• How to multiply and divide algebraic fractions.
• How to split an algebraic fraction into two other algebraic fractions.
• How to raise algebraic fractions in a positive/negative power.

Algebraic Fractions Revision Notes

Algebraic fractions are fractions that contain algebraic expressions in at least one of their components (numerator or/and denominator). This means they contain at least one letter (variable) in their structure.

When dealing with algebraic fractions we must be sure that the denominator will be different from zero to avoid division by zero. The values of variables for which the denominator of an algebraic fraction becomes zero are known as 'disallowed values'. We must first identify them before continuing with other operations such as simplification, factorisation, etc.

We can do everything with algebraic fractions that we do with numerical fractions, including: addition, subtraction, multiplication, division, raise in power, etc.

To simplify an algebraic fraction means to remove the same number, variable, group of variables, number-variable combination or expression from both numerator and denominator in order to obtain a simpler algebraic fraction.

An algebraic fraction can only be simplified when its numerator and denominator represent a single factor or a product of factors. It is good to check whether any of the eight special algebraic identities is present in any part of algebraic fraction or not. If yes, this will help with the process of simplification.

We can add or subtract algebraic fractions in the same way as we do with numerical fractions. Thus, if fractions have the same denominator, we simply add or subtract the numerators while the denominator doesn't change. On the other hand, if the denominators are different we change all fractions to have the same denominator and, only then, we continue with the rest of expression.

Multiplication and division of algebraic fractions is carried out in the same way as we do with numerical fractions. Thus, in multiplication we multiply the numerators with each other; the same procedure is also carried out with denominators. Then, we can check for any possible simplification.

As for division of algebraic fractions, first the second fraction is inverted down to turn the division into multiplication, followed by the multiplication procedures.

Sometimes, it is necessary to complete the inverse action of the addition or subtraction of two fractions with different denominators. In other words, we may need to split an algebraic fraction into two other algebraic fractions which, when combined together (through addition or subtraction), give the original fraction. If the original fraction is written in the form

where E, M and N are three different algebraic expressions, we can express it in the form

where

We can raise an algebraic fraction in power by raising the numerator and denominator by the given power separately. In this way, we may apply the following property of powers

Any negative exponent makes the algebraic fraction invert down and the new exponent has the same value but positive, i.e.

Sometimes, raising an algebraic expression by a given power may be beneficial because raising a negative number at an even power makes it positive. This property (as well as many others) may help in simplifying part of the algebraic fraction (that could have not otherwise been possible to simplify) after having raised it in an even power.