Multiplying and Dividing Fractions

Multiplying one algebraic fraction by another algebraic fraction, or dividing one algebraic fraction by another algebraic fraction, brings together three topics with which you are already familiar:

(i) the basic formalism:

and

(ii) the methods of simplifying fractions by cancellation of factors common to the numerator and denominator

and

(iii) using the laws of exponents when the factors being multiplied or divided are powers:

These laws of exponents imply, among other things, that we can move factors from the numerator to the denominator or the reverse, as long as we change the sign of the exponent at the same time. This property can make cancellations involving powers of a specific symbol very easy. For instance, we can do

or

or even

The rest of this document consists of a collection of examples illustrating how these three sets of properties can be used to carry out multiplication and division with algebraic fractions, ensuring that the final result is in simplest possible form.

Example 1: Simplify.

solution: Writing the two sets of brackets next to each other in this fashion indicates that the two fractions are to be multiplied. So

This result is technically correct, but many people view negative exponents as unattractive and somewhat conducive to misunderstanding or misinterpretation. Most practitioners would recommend writing the final answer here in the form of a proper fraction containing no negative exponents. Thus, the best way to state the final answer is