Multiply Rational Expressions

How to Multiply Rational Expressions

Multiplication of rational expressions is similar to the multiplication of fractions of rational numbers. Let us consider an example of multiplying fractions, 3/4 x 7/2. Here the method to be followed is, first multiply the terms in the numerator and then multiply the terms in the denominator which gives 3x7/4x2 = 21/8.

Then any further simplification possible is carried out. In this example there was no possibility of cancelations as there were no common factors and hence the fraction got by multiplying the numbers in the numerator and the numbers in the denominators would be the final answer. Similar set of steps are used while multiplying rational expressions.

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A rational expression  consist of coefficients which are constants for a given term and hence first the coefficient parts are simplified then the product of the variable parts in the numerator is found, then the product of the variable part in the denominator is found and finally the products are written in the simplified form. In this process we use exponential laws and common factors for simplification.

Let us consider an example for a better understanding. Multiply Rational Expressions 4x2y2/3x and 3x/4y.  The product of the given expressions would be,  4x2y2/3x . 3x/4y Here first the coefficients are simplified and any cancelations possible are carried out. 4 and 3 get canceled and we get x2y2/x . x/y, this can be written as x.x.y.y/x. x/y; x and y are the common factors which can be canceled and we get x2y which is the final answer. Here we can use the exponential rule in simplification.

To multiply rational expressions involving polynomials the steps to be followed are:
First factor the terms in the numerator and the denominator
Reduce all the common factors
Further simplify the terms by either multiplying the numerators and denominators or leave the product in the fraction form

Example: Multiply (x2-9)/(x2+6x+9) and (3x+9)/(3x-9)
Solution: First factor the terms in the numerator and denominator
(x2- 9)= (x+3)(x-3) [special products]
(x2+6x+9)= (x+3)(x+3) [(a+b)2=a2+2ab+b2]
(3x+9)=3(x+3)                               [taking out common factor]
(3x-9) = 3(x-3)                               [taking out common factor]

The product of the expressions would be in the form, (x2-9)/(x2+6x+9) . (3x+9)/(3x-9)
Re-writing the given expressions as factors gives,
(x+3)(x-3)/(x+3)(x+3)  .  3(x+3)/(x-3)
Reducing the common factors gives,
(x+3)(x-3)/(x+3)(x+3)  .  3(x+3)/3(x-3)
Canceling the common terms gives 3/3 = 1, the final answer!

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Multiplying Rational Expressions Solver performs the multiplication and division of rational numbers when the expressions are entered in the given fields. This helps to check the answers.