To find an LCM it helps to factor the expressions. Since we're looking for the LCM of the denominators we will factor the denominators
is a difference of squares so it will factor according to that pattern:
The other denominator has a GCF:
If you have trouble seeing what the LCM is from these factors, it can be helpful to write them in a certain way. (It can also help to rewrite number factors in prime factors. So I will rewrite the 8 as 2*2*2.):
x^2-64 = (x+8) * (x-8)
8x-64 = (x-8) * 2 * 2 * 2
Note how each column represents a different factor. This is why there is a blank below the (x+8). There is no (x+8) factor in 8x-64. And this is why the (x-8)'s are lined up. And this is why the 2's all all out at the end. There are no factors of 2 in . Once the factors are written like this, then the LCM will be the product of each column's factor:
x^2-64 = (x+8) * (x-8)
8x-64 = (x-8) * 2 * 2 * 2
LCM = (x+8) * (x-8) * 2 * 2 * 2
LCM's are most useful in factored form , this may be an acceptable answer. If not, then you just multiply it out. The easy way to multiply is to multiply the 2's to get 8, then multiply the (x+8)(x-8) to get and then multiply these to get:
LCM =
P.S. Another advantage of writing the factors as I have done, is that you can see what you need to multiply each denominator by to get it to change to the LCM. Comparing the and LCM lines above we can see that the is missing the 3 factors of 2 (or 8). So we would multiply the numerator and denominator of the first fraction by 8 to turn the denominator into the LCM. And comparing the 8x-64 and LCM lines above we can see that the only missing factor is (x+8). So we would multiply the numerator and denominator of the second fraction by (x+8) to change its denominator into the LCM.