# If velocity of light c, Planck’s constant h and gravitational contant G are taken as fundamental quantities then express mass, length and time in terms of dimensions of these quantities.

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by akashPant

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by akashPant

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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*

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

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