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Simplifying Fractions 3
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Slope of a Line
Percent Introduced
Reducing Rational Expressions to Lowest Terms
The Hyperbola
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Multiplication by 75
Solving Quadratic Equations Using the Quadratic Formula
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Basic Algebraic Operations and Simplification
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The Standard Form of a Quadratic Equation
The Discriminant
Dividing Monomials Using the Quotient Rule
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Steps for Solving Linear Equations
Quadratic Expressions Complete Squares
Fractions 1
Properties of Negative Exponents
Factoring Perfect Square Trinomials
Solving Quadratic Equations Using the Square Root Property
Dividing Rational Expressions
Quadratic Equations with Imaginary Solutions
Factoring Trinomials Using Patterns
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Reducing Rational Expressions to Lowest Terms

Each rational number can be written in infinitely many equivalent forms. For example,

Each equivalent form of is obtained from by multiplying both numerator and denominator by the same nonzero number. For example,

Note that we are actually multiplying by equivalent forms of 1, the multiplicative identity. If we start with and convert it into , we are simplifying by reducing to its lowest terms.We can reduce as follows:

A rational number is expressed in its lowest terms when the numerator and denominator have no common factors other than 1. In reducing , we divide the numerator and denominator by the common factor 2, or “divide out” the common factor 2. We can multiply or divide both numerator and denominator of a rational number by the same nonzero number without changing the value of the rational number. This fact is called the basic principle of rational numbers.


Basic Principle of Rational Numbers

If is a rational number and c is a nonzero real number, then

Helpful hint

Most students learn to convert into by dividing 3 into 6 to get 2 and then multiply 2 by 2 to get 4. In algebra it is better to do this conversion by multiplying the numerator and denominator of by 2 as shown here.



Although it is true that

we cannot divide out the 2’s in this expression because the 2’s are not factors. We can divide out only common factors when reducing fractions.

Just as a rational number has infinitely many equivalent forms, a rational expression also has infinitely many equivalent forms. To reduce rational expressions to its lowest terms, we follow exactly the same procedure as we do for rational numbers: Factor the numerator and denominator completely, then divide out all common factors.


Example 1


Reduce each rational expression to its lowest terms.


a) Factor 18 as 2 · 32 and 42 as 2 · 3 · 7:

  Divide out the common factors.


b) Because this expression is already factored, we use the quotient rule for exponents to reduce:


Helpful hint

A negative sign in a fraction can be placed in three locations:

The same goes for rational expressions:


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