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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 â€œdivide outâ€ 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

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.

Caution

Although it is true that

we cannot divide out the 2â€™s in this expression because the 2â€™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

Reducing

Reduce each rational expression to its lowest terms.

Solution

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

 Factor. Divide out the common factors.

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