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planck's constant speed of light in vacuum and Newton's gravitation constant are 3 fundamental constant which of the following combination of three has the dimension of length

Units of Planck's constant = J s = N m  s  = kg m s-2 m s = kg m2 s-1

Dimensions of Planck's constant = M L2 T-1

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Units of gravitational constant  = N m2 kg-2  = kg m s-2  m2 kg-2 = kg-1 m3 s-2

Dimension of gravitational constant = M-1 L3 T-2

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Units of speed = m/s-1

Dimension of speed = L T-1

If the combination of planck's constant , gravitational constant and speed of light gives length , then

[ M L2 T-1 ]a  [ M-1 L3 T-2 ]b  [ L T-1  ]c  = L

By equating the exponents on both sides, we get

a - b = 0   or  a = b

2 a + 3b + c = 1  or  5a + c = 1  ....................(1)

-a -2b -c = 0 or  3a+c = 0 .....................(2)

by solving eqn.(1) and  (2) , we get

a = 1/2  and  c = -3/2

Hence b = 1/2

hence we get

Hence the combination  ( h1/2 G1/2 c-3/2 ) gives unit of length

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