Expert: Paul Klarreich - 11/18/2005

Question
x-2         3
-------,    ----
3x(x-2)     6x2 This is 6x squared not 6 times 2

These are fractions.  How do you find the LCM?

Thank You

Answer
Hi, David,

x-2         3
-------,    ----
3x(x-2)     6x^2

These are fractions. How do you find the LCM?

Easy.  All LCM-ing involves factoring.  Factor each denominator, then assemble enough factors to cover each one.

3x(x - 2) is factored.  You need one each of 3, x, (x - 2)
6x^2 = 3(2)x x.  You need a 3, a 2, and two x's.

So the LCM will have to have a 3, a 2, two x'2 and an (x - 2).

The LCM is  3(2)x^2(x - 2).

Of course, the example as you give it has a bit of a 'wrinkle'.  The two fractions can be reduced by canceling a factor of (x-2) in the first fraction, and a factor of 3 in the second.  So the first fraction becomes just  1/3x and the second becomes 1/2x^2;  that makes things simpler.

Going a bit further, if you are combining (adding or subtracting) fractions, then you will be 'building' the fractions to the LCM.  [Building fractions is the reverse of reducing fractions.]

When you do that, you will simply look at each denominator in factored form, and see what LCM factors are missing, then multiply top and bottom by those factors.