if dimensions of length are expressed as g^x c^y h^z where g,c and h are the universal gravitational costant, speed of light and planck`s constant respectively. what is the value of x,y and z?

Dimensions of Length = M⁰L¹T⁰

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                     Gravitational constant = M^-1L³T^-2

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                     Speed of light = M⁰L¹T^-1

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                      Plancks constant = M¹L²T^-1    

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Putting them in the equations:-

M0L1T0 = (M^-1L³T^-2)^X * (M⁰L¹T^-1)^y * (M¹L²T^-1)^Z

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            = M^-x * L^3x * T^-2x * M⁰ * L^y * T^-y * M^z * L^2z * T^-z

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            = M^-x * M⁰ * M^z * L^y *  L^3x * L^2z * T^-2x * T^-y * T^-z

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            = M^z-x * L^{3x+2y + 2z} * T ^-(2x+y+z)

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Therefore equating the respective squares.

1) M⁰ = M^z-x

Hence z-x = 0 ; X=Z........... 1

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2) L¹ = L^{3x+2y + 2z}

Hence 3x+2y+2z = 1

FROM 1 :- 3z + 2z + 2y = 1

                5z + 2y = 1

                y = (1 - 5z) / 2 .................. 2

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3) T⁰ = T ^-(2x+y+z) 

Hence 2x + y + z = 0

Replacing values of X and Y from 1 and 2 :-

2z + (1-5z)/2  + z = 0

3z + (1-5z)/2 = 0

(6z + 1 - 5z) / 2 = 0

z + 1 = 0

Z = -1

Therefore X = -1

And Y = [1-5(-1)] /2 = 3