From Förberedande kurs i matematik 1

To determine the lowest common denominator for the expression, we have to multiply the top and bottom of each fraction by as small an amount as possible in order for them to have a common denominator.

In this case, we need only multiply the top and bottom of the first fraction by 2.

\displaystyle \frac{1}{4}-\frac{1}{8}=\frac{1}{4}\cdot \frac{2}{2}-\frac{1}{8}=\frac{2}{8}-\frac{1}{8}\,.

The lowest common denominator is thus 8.