Elimination Using Multiplication
Prime Factors
Equations Involving Rational Exponents
Working with Percentages and Proportions
Rational Expressions
Interval Notation and Graphs
Simplifying Complex Fractions
Dividing Whole Numbers with Long Division
Solving Compound Linear Inequalities
Raising a Quotient to a Power
Solving Rational Equations
Solving Inequalities
Adding with Negative Numbers
Quadratic Inequalities
Dividing Monomials
Using the Discriminant in Factoring
Solving Equations by Factoring
Subtracting Polynomials
Cube Root
The Quadratic Formula
Multiply by the Reciprocal
Relating Equations and Graphs for Quadratic Functions
Multiplying a Polynomial by a Monomial
Calculating Percentages
Solving Systems of Equations using Substitution
Comparing Fractions
Solving Equations Containing Rational Expressions
Factoring Polynomials
Negative Rational Exponents
Roots and Radicals
Intercepts Given Ordered Pairs and Lines
Factoring Polynomials
Solving Linear Inequalities
Mixed Expressions and Complex Fractions
Solving Equations by Multiplying or Dividing
The Addition Method
Finding the Equation of an Inverse Function
Solving Compound Linear Inequalities
Multiplying and Dividing With Square Roots
Exponents and Their Properties
Equations as Functions
Factoring Trinomials
Solving Quadratic Equations by Completing the Square
Dividing by Decimals
Lines and Equations
Simplifying Complex Fractions
Graphing Solution Sets for Inequalities
Standard Form for the Equation of a Line
Checking Division with Multiplication
Elimination Using Addition and Subtraction
Complex Fractions
Multiplication Property of Equality
Solving Proportions Using Cross Multiplication
Product and Quotient of Functions
Quadratic Functions
Solving Compound Inequalities
Operating with Complex Numbers
Equivalent Fractions
Changing Improper Fractions to Mixed Numbers
Multiplying by a Monomial
Solving Linear Equations and Inequalities Graphically
Dividing Polynomials by Monomials
Multiplying Cube Roots
Operations with Monomials
Properties of Exponents
Mixed Numbers and Improper Fractions
Equations Quadratic in Form
Simplifying Square Roots That Contain Whole Numbers
Dividing a Polynomial by a Monomial
Writing Numbers in Scientific Notation
Solutions to Linear Equations in Two Variables
Solving Linear Inequalities
Multiplying Two Mixed Numbers with the Same Fraction
Special Fractions
Solving a Quadratic Inequality
Parent and Family Graphs
Solving Equations with a Fractional Exponent
Evaluating Trigonometric Functions
Solving Equations Involving Rational Expressions
Laws of Exponents
Multiplying Polynomials
Vertical Line Test
Solving Inequalities with Fractions and Parentheses
Multiplying Polynomials
Solving Quadratic and Polynomial Equations
Extraneous Solutions
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Simplifying fractions


Fractions involving symbols occur very frequently in engineering mathematics. It is necessary to be able to simplify these and rewrite them in different but equivalent forms. On this leaflet we revise how these processes are carried out.

1. Expressing a fraction in its simplest form

An algebraic fraction can always be expressed in different, yet equivalent forms. A fraction is expressed in its simplest form by cancelling any factors which are common to both the numerator and the denominator. You need to remember that factors are multiplied together. For example, the two fractions and are equivalent. Note that there is a common factor of a in the numerator and the denominator of which can be cancelled to give .

To express a fraction in its simplest form,an y factors which are common to both the numerator and the denominator are cancelled.

Notice that cancelling is equivalent to dividing the top and the bottom by the common factor. It is also important to note that can be converted back to the equivalent fraction by multiplying both the numerator and denominator of by a.

A fraction is expressed in an equivalent form by multiplying both top and bottom by the same quantity,or dividing top and bottom by the same quantity


The two fractions

are equivalent. Note that

and so there are common factors of 5 and y × y . These can be cancelled to leave .


The fractions

are equivalent. In the first fraction,the common factor ( x + 3) can be cancelled.


The fractions

are equivalent. In the first fraction,the common factor a can be cancelled. Nothing else can be cancelled.


In the fraction

there are no common factors which can be cancelled. Neither a nor b is a factor of the numerator. Neither a nor b is a factor of the denominator.


Express as an equivalent fraction with denominator (2 x + 1)( x - 7).


To achieve the required denominator we must multiply both top and bottom by ( x - 7). That is

If we wished,the brackets could now be removed to write the fraction as


1. Express each of the following fractions in its simplest form:


Whilst both a and b are factors of the denominator, neither a nor b is a factor of the numerator.

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