Physics - Unit & Dimension Question with Solution | TestHub

$E=G^p{h}^q{c}^r$ $$\begin{array}{rl}& \left[{\mathrm{M}}^1{\mathrm{L}}^2{\mathrm{T}}^{-2}\right]={\left[{\mathrm{M}}^{-1}{\mathrm{L}}^3{\mathrm{T}}^{-2}\right]}^p{\left[{\mathrm{ML}}^2{\mathrm{T}}^{-1}\right]}^q{\left[{\mathrm{LT}}^{-1}\right]}^r\\ & ={\mathrm{M}}^{-p+q}{\mathrm{L}}^{3p+2q+r}{\mathrm{T}}^{-2p-q-r}\end{array}$$ Applying principle of homogeneity of dimensions, we get $$\begin{array}{rl}& -p+q=1\\ & 3p+2q+r=2\\ & -2p-q-r=-2\end{array}$$ Add (iii) and (iv), $p+q=0$ Add (ii) and (v), we get $q=\frac{1}{2}$ From (ii), we get $p=q-1=\frac{1}{2}-1=-\frac{1}{2}$. Put in (iii), we get $-\frac{3}{2}+1+r=2,r=5/2$