Physics - Unit & Dimension Question with Solution | TestHub

As, gravitational constant,

$\left[G\right]=\left[{\mathrm{M}}^{-1} {\mathrm{L}}^3 {\mathrm{T}}^{-2}\right]$

Planck's constant, $\left[h\right]=\left[{\mathrm{ML}}^2 {\mathrm{T}}^{-1}\right]$

Hence, $\left[G\right]\left[h\right]=\left[{\mathrm{M}}^{-1}{\mathrm{L}}^3{\mathrm{T}}^{-2}\right]\left[{\mathrm{ML}}^2{\mathrm{T}}^{-1}\right]$

$=\left[{\mathrm{M}}^0{\mathrm{L}}^5 {\mathrm{T}}^{-3}\right]$

Velocity of light, $\left[c\right]=\left[{\mathrm{LT}}^{-1}\right]$

Now, ${\left[\frac{Gh}{c^3}\right]}^{1/2}=\frac{{\left[L^5{T}^{-3}\right]}^{1/2}}{{\left[L^3{T}^{-3}\right]}^{1/2}}$$={\left[{\mathrm{L}}^2\right]}^{1/2}=\left[\mathrm{L}\right]$

Hence $\left[\mathrm{L}\right]=$ length