c = L T⁻¹
h = Joules second = energy / frequency = M L² T⁻¹
G = Newtons meter² / kg² = M⁻¹ L³ T⁻²
M = c^k h^m G^n = M^(m-n) L^(k+2m+3n) T^(-k-m -2n)
=> k+2m+3n = 0, m - n = 1, and - k - m - 2 n = 0
=> 2m +3n = m + 2n, m = -n and so n = -1/2, m = 1/2 and k = 1/2
M = √c √h / √G
L = c^p h^q G^r = M^(q - r) L^(p +2q + 3 r) T^(-p - q - 2 r)
=> q - r = 0 , p +2q + 3 r = 1 and -p - q - 2 r = 0
=> p+ 5r =1 , p = - 3r => r = 1/2, p = -3/2 , q = 1/2
L = √h √G / √c³
T = c^x h^y G^z = M^(y-z) L^(x+2y+3z) T^(-x-y-2z)
=> x+2y+3z = 0 , y-z = 0 and -x-y-2z = 1
=> x = - 5 z z=1/2 x = -5/2 y = 1/2
T = √h √G / √c⁵
