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If the velocity of light $c$, universal gravitational constant $G$ and planck's constant $h$ are chosen as fundamental quantities. The dimensions of mass in the new system is
$\left[h^{\frac{1}{2}} c^{-\frac{1}{2}} G^1\right]$
$\left[ h ^1 c ^1 G ^{-1}\right]$
$\left[ h ^{-\frac{1}{2}} c ^{\frac{1}{2}} G ^{\frac{1}{2}}\right]$
$\left[h^{\frac{1}{2}} c^{\frac{1}{2}} G ^{-\frac{1}{2}}\right]$
Solution
Say dimensional formale of mass is $H ^{ x } C ^{ y } G ^z$
$M ^1=\left( ML ^2 T ^{-1}\right)^{ x }\left( LT ^{-1}\right)\left( M ^{-1} L ^3 T ^{-2}\right)^Z$
$M ^1 L ^0 T ^0= M ^{ x – z } L ^{2 x + y +3 z} T ^{- x – y -2 z}$
on comparing both side
$x-z=1$
$2 x+y+3 z=0$
$-x-y-2 z=0$
On solving above equations we get
$x=\frac{1}{2} \quad y=\frac{1}{2} \quad z=\frac{-1}{2}$
