Given the relation, $m=\frac{m_{0}}{\left(1-v^{2}\right)^{\frac{1}{2}}}$

Dimension of $m= M ^{1} L ^{0} T ^{0}$

Dimension of $m_{0}= M ^{1} L ^{0} T ^{0}$

Dimension of $v= M ^{0} L ^{1} T ^{-1}$

Dimension of $v^{2}= M ^{0} L ^{2} T ^{-2}$

Dimension of $c= M ^{0} L ^{1} T ^{-1}$

The given formula will be dimensionally correct only when the dimension of L.H.S is the same as that of R.H.S.

This is only possible when the factor, $\left(1-v^{2}\right)^{1 / 2}$ is dimensionless i.e., $\left(1-v^{2}\right)$ is dimensionless. This is only possible if $v^{2}$ is divided by $c^{2} .$

Hence, the correct relation is

$m=\frac{m_{0}}{\left(1-\frac{v^{2}}{c^{2}}\right)^{\frac{1}{2}}}$