Given relation: m = \frac{m_0}{(1 - v^2)^{\frac{1}{2}}}
Missing constant: Speed of light = c
Step 1: Writing the dimensional formulae for given physical quantities
Moving mass [math] m = [M^1 L^0 T^0] [/math]
Rest mass [math] m = [M^1 L^0 T^0] [/math]
Speed [math]v=[M^0L^1T^{-1}]\ \Rightarrow\ v^2=[M^0L^2T^{-2}][/math]
Speed of light [math] c = [M^0 L^1 T^{-1}] [/math]
Step 2: Setting up a correct relation
According to the principle of homogeneity, the given formula will be dimensionally correct only if the dimensions of L.H.S and R.H.S are the same. This is only possible when the factor (1 - v^2)^{\frac{1}{2}} has no dimension.
Just dividing v^2 \text{ by } c^2 makes this possible. Therefore, the correct relationship is m = \frac{m_0}{(1 - \frac{v^2}{c^2})^{\frac{1}{2}}}