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Question 2

Expression for time in terms of $$G$$ (universal gravitational constant), $$h$$ (Planck constant) and $$c$$ (speed of light) is proportional to:

$$[t] = \text{M}^0\text{L}^0\text{T}^1$$

$$[G] = \text{M}^{-1}\text{L}^3\text{T}^{-2}$$

$$[h] = \text{M}^1\text{L}^2\text{T}^{-1}$$

$$[c] = \text{M}^0\text{L}^1\text{T}^{-1}$$

$$\text{M}^0\text{L}^0\text{T}^1 = (\text{M}^{-1}\text{L}^3\text{T}^{-2})^x (\text{M}^1\text{L}^2\text{T}^{-1})^y (\text{M}^0\text{L}^1\text{T}^{-1})^z$$

$$\text{M}^0\text{L}^0\text{T}^1 = \text{M}^{-x+y}\text{L}^{3x+2y+z}\text{T}^{-2x-y-z}$$

$$\implies x = \frac{1}{2}, \, y = \frac{1}{2}, \, z = -\frac{5}{2}$$

$$t \propto G^{1/2} h^{1/2} c^{-5/2} = \sqrt{\frac{Gh}{c^5}}$$

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