Multiplying and Dividing Rational Expressions

If you take a pizza and divide it into 8 (equal) slices, each slice represents 1/8 of the pizza.

. If you take one of these slices and divide it into 2 equal pieces, what part of the original pizza is each of these new slices? Cutting each of the original slices in two will give 16 slices. Therefore, the answer to the last question is that each smaller piece will be 1/16 of the pizza.
. You can also look at this problem as taking 1/2 of each slice (1/8 of the pizza). Hence,
.

In each case, the 1/8 represents how much pizza, and the ÷ 2 or * 1/2 represents the action performed upon the pizza. For this reason, a common denominator is not needed.

Multiply:

The basic rules for multiplication are that numerators are multiplied together and denominators are multiplied together. A consequence of this is that the magnitude of the numbers can get very large. To help you get the “more reasonable reduced” answer, factor the numerator and the denominator.

(Yes, you remember a quicker way to reduce this problem. But before you do that, be sure you understand that what you are about to do is really the same process as the one we just described—without the need to put all the factors into one fraction. The only operation in the numerator and in the denominator is multiplication, and the concept of “canceling” is really division of the numerator and denominator by the same number. You will find that you won't struggle with this in multiplication and division problems as you might when facing an addition or subtraction problem.)

Extending this practice to algebraic expressions, we find that the problem

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