Why is it necessary to use a common denominator to add two rational expressions, but not to find the product .

Answer 1

You CAN find a common denominator when finding the product, but it is not useful to do so. In fact, it complicates things.

Suppose you want to multiply 2/3 times 5/6

You get 10/18

To simplify this, you divide top and bottom by the greatest factor common to both, namely 2, and you get 5/9

But....

Suppose you decided to convert the first fraction to 6ths first.

You get 4/6 times 5/6 = 20/36. Now you find the greatest factor common to both top and bottom. It's no longer good enough to divide by 2. You have to divide by 4 to get 5/9.


In general, when you multiply and simplify (or if you cancel before you multiply) you are trying to ELIMINATE common factors in numerator and denominator. Finding a common denominator first, however, INSERTS an extra factor which then has to be divided out at the end anyway. ( in the example, 4 contains the original common factor of 2 times the extra 2 we intoduced to get the common denominator.)

So it is counter-productive.

Answer 2

What is 1/3 of a pie plus 1/4 of a pie??
It is pretty hard to solve that without changing to a common denominator

What is 1/3 a pie times 1/4 of a pie. Well, in reality you are taking 1/3 of a pie and dividing that up into 4 pieces. So now that we have that 1/3 of a pie broken into 4 pieces, we only want one piece (given by the numerator). Thus that piece is of size 1/12. Skip that whole process by simply multiplying 1/4 by 1/3...yielding 1/12

I hope that helps!