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 Complex Fractions

Before introducing the definition of complex fraction, let's remember:

  • A simple fraction is a fraction containing no other fractions as part of its numerator or its denominator.
  • A simple fraction can be simplified. What we intend to do when simplifying a simple fraction is to obtain an equivalent fraction that looks simpler (i.e. it has less terms in its numerator and denominator).

Now that we have this clear, we can learn that

  • complex fractions are fractions whose numerator and/or denominator themselves are expressions containing other fractions.
  • A complex fraction can also be simplfied. What we mean by simplifying a complex fraction is to rearrange it into an equivalent simple fraction which is in simplest form.

You should already know how to simplify a simple fraction. But, how do we simplify a complex fraction?

When the numerator and denominator of a complex fraction are just single simple fractions themselves, simplifying it turns out to be very easy. We just use the wellknown “invert and multiply” rule: multiply the fraction in the numerator by the reciprocal of the fraction in the denominator:

We have turned the initial complex fraction on the left into a single simple fraction on the right. After you have done this, all that is left is to check whether the simple fraction on the right can be simplified any further.

Be careful, because this step is justified only if the numerator and denominator of the original complex fraction are both single simple fractions.

Example 1:



The strategy illustrated in the previous examples is easily applied to this complex fraction. First, the numerator can be simplified to give

The denominator can be simplified to give

Substituting these simplified expressions back into the original complex fraction then gives

after cancelling common factors in the resulting numerator and denominator. Since no further factorization is possible in either the numerator or the denominator of this fraction, and since no common factors still exist that can be cancelled between the numerator and denominator of this last expression, it must be the required most simplified form of the original complex fraction.


Example 2:



You can use this example as a practice problem. Try carrying out the required simplification first before you look at the outline of the solution that follows. This example has essentially the same structure as that of Example 1 above. So, first simplify the numerator

Then, simplify the denominator:


The numerator and denominator of this last expression are both fully factored and there are no common factors in them which can be cancelled. So this last fraction must be the required simplified form of the original complex fraction.



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